ABELIAN VARIETIES
DAVID MUWORD
Witl Appendictt iy
C F Ramahujaji and Yum
!ta institute of tundahentai, beseaboh Bombay
OXFORD UNIVERSITY PRESS

CONTENTS
CHiFTlE
I AKALXTKJ
1 Complex Tori
2 Line bundles on a complex torus
3 Aigebraizability of tori
II Algibraio Thborv tia Varieties.
4 Definition of ftbelian varieties
5 Oohomology and base change
8 The theorem of the cube: I
7 Dividing varieties by finite groups
g The dual abelian variety obar 0
9 The case k = C
III. Alghbbaio Thboby^via Schbmss
10. The theorem of t lie cube: II
11 Basic theory of group schemes .
1! Quotients by finite group schemes
13, Tbe dual abelian variety in any characteristic .
14, Duality theory of finite commutative group floheme*
15, Applications to abelian varieties
IS. Cohomology of line bundles
IT. Very ample line bundles
FV HomfZ X) AND THB I ADIO BIPB*S»KTATIOH
IS
IB
SO
SI.
?S
23
24
Etale coverings
Struotnre of Hom(X X)
Riemann forms .
Positivity of the Bosati involution
Examples .
The group »(?)
The oase k = C
1
1
13
24
39
39
46
SB
Sff
74
B2
89
89
03
108
123
132
143
ISO
103
1ST
107
172
1B3
192
210
321
23fl

CONTENTS
Appendix I . Tie Theorem of Tate byC P Ra maim jam
Appendix II; Mordeli Weil Theorem by Yuri Mamn
Bibliography
Index
ANALTTIC THEORY
I Complex Tori We shall investigate in thU chapter a compact
oonneoted oomplax Lie group X of dimension g, i.e. a compact
oonneoted oomplax manifold of dimension g with a group structure
on the underlying Bet such that the mapa X X X-*-X, X •* X defined
by fayI—>*¦? an<* *'—>s Me holomorphio. Let 7 be the
tangent spaca to X at the identity point e e X. V is a complex
veotor apaoe Rooall that for eeery oowplex lie group X, with
tangent space V at«, for erary teF there is a unique holomorphto
boraonorpbism
suoh that cty, takes the unit tangent vector toCatOtoeeF (Cf
Hoohsohild, StnuXure 0/ Lie Qroupt, p. 79 and p. 196). MbreoTer
the function ^,A) in I and e is » holomorphio map C x V-+X The
expcnmtmi map exp: V ~+X is defined by exp(u) = #,A). Beoanse
of the uniqueness property characterizing ^, ^>H{I) — ^,(*<),
hence &(<) =-exp(lo). Therefore, if we identify as usual the tangent
space to V at 0 with 7 itself, the differential of exp at 0 is the
identity map of V onto 7. Returning to a compact connected Z
now, we first prove:
A) Xita commutative gronp
Proof. In fact, for x in X, define O, to be the conjugation map
X-+X, Ot (jO ¦=. xyx~». The differential (dCm), is an automorphism of
Funds:, y{dC^,\» a holomorphio map of Z into AutG)cH!nd{F).
Since End( F) ia a flnita-dimenuonal oomplex vector space and the
only holomorphio functions on a compact oonneoted complex
manifold are oonstants, we deduce that {dO,), is independent of
xeX, henoa (<K7,J, = (dCt)t =17. Now for any homomorphism
T:Xl-*Xt of complex Lie groups,

s abblian TAanmES
This follows from the uniquenses property characterizing the
homomorphismsii—> eipI<(to} from C to Xf It is easy to prove
from this that
Since (dCt\ =• \r, this shows that G,(expy) =expy, so exp(F) is
in the center of X. Sinoe d<exp) is the identity, it follows from the
implicit ftinotion theorem that exp defines a nomeotnorphism of a
neighborhood of 0 e F with a neighborhood of e in X. Since X is
oonneoted, this implies that exp(F) generates X as a group, and
it follows that X is commutative.
(!) Tht exponential map exp: V->¦ X t* a turjtciive homomor-
phttm of complex Lie group* with kernel a lattice' U en F, and induce*
an MomorpAwm 7jVZ X, i.e. Xisa complex tonu
Let *, jf 6 7. SinoeXis commutative, the mapC-tldenned by
fi—^{exp*«!).{exply)i8abokimorphiohomomorphism(and the image
of | ~ J by the tangent map is easily seen to be x + y. Now for any
*S T, the map (-+ eip(fa) is oharaoterized as the unique holomorphio
homomorphism, whose tangent map takes (~ I to x e F Henca
(expte).(exp ty)=expK*+y)alldPl>t'tl'nB' = 1. weflnd that exp
is a homomorphism. It is surjeotive sinoe on the one hand X ii
oonneoted while on the other hand exp( 7) contains a neighborhood
of* and henoe an open and closed subgroup of X. The kernel V is a rs
discrete subgroup of 7. since there is a neighborhood N of 0 in F i,
such that explj JT ^-X is injeotive. The induoed homomorpbiam
VjU-t-Xis hobmorphio by definition of structure of complex
manifold on 7(V, and is an algebraic isomorphism of groupe, The |
tangent map at the identity of this map is an isomorphism, and ':
henoe by the inverse function theorem, the inverse is holomorphio ¦
at e and henoe holomorphio everywhere on X (translations being
holomorphio isomorphisms on both VjU and X) Therefore X ia i
fsotnorpbia to V/U. Siase lattice* ere the only discrete subgroups i
of vector spaces with oompact quotient, V mnst he a lattice.
•n&at ipaHriiUitMlicmipciDentadbr
ANALYTIC THKOBY
I^om now on, we use additive notation for the group operation
in X- We will fix the notation ir: F-*Z for the exponential homo
for the rest of this chapter.
C) As an abstract grmtp, X i» divisible (i.e. nj — X formsZ,
fl ^ 0) and if for n e Z, n # 0, Xm is the subgroup of elements a»t»-
hfated by », XKs (Z/aZ)".
Pboop. By {2), we see that as a real Lie group X is iso
morphia to (R/Z) • = (8l)» where Sl is the circle group Henoe
we have C). '
D) We have canonicai ttomorphtsmt
a it i\ i group of alternating f-forms
i/(Zi)=| UX...XV—hZ
Froot. (F, w) is clearly the universal covering epace of X hence
17=. v' *@) is exactly jr^J.O). Sinoe for any good topologioal space X
H»(Z,Z) sHomfir.tX), Z),
the assertion is correct for r = 1. Then to prove it for all r it will
suffice to show that cup product induces an isomorphism
A'(iPtr, Z))
, Z) allr
(•}
Bat note that if (*) is correct for spaoan X! and X, with finitely
generated oohomotogies, then by the Kiinneth formula, {*) holds for
, x X, X)
. Z)]

4 ABEUAN VAMETIES
(Nota here that (*) far X,, X, implies J5f(Xi, Z) Is torsion-free,
hence the tor term in Kunneth disappears.) But our toniB X is
produot of 8l'e, for which (•) is trivially valid.
E) Computation of the groups J?*(X Q») when Of = tteaf of]
holomorphK p-formt on X
The oohomology groups B*(X, O*) are one of the most significant
invariants of any compact complex manifold X and their computa-
computation for a torus will take up the rest of this section
Let V = Tox be the tangent space to X at 0 (regarded as
complex vector space), and let T = Homc(K, C) be the oomplex oo-;
tangent space to X at 0. By translation with respeot to the group
lav on X, every oomplex p-oovootor * e A'T extends to a trains
lation invariant holomorphio p-form o>. on X. In fact, let Tt\X-* X
be the map TJ$) = x+y. Then define {»„), = T* J«). Moreover;
the map «i—*a>t defines a homomorphism of sheaves:
which is easily checked to be au isomorphism. In other words,
itaglobalit/ free sheaf of C^-modules. Since the only global section!
of f>z are oanetants, the global aecttons of 11* are exaatly the trant.1
lation-invariant y-forms u>.. In fact, because of the isomorphism
(*)weget
The mam result that we want is
Thbokim. If f —
\tomorphisms
for aU q hence
Onr proof of this (due to 0, P. Ramanujam and related to tha1
of Weil [W2]) depends on the well-known Dolbeftult resolution:
C) (hen then are naturd
ANALY11C THEOBY 6
trhore **¦' is the eheaf of C complex-valuod differential forms of
type-(p, q) on X, and d is the component of the exterior derivative d
mapping *•¦* to W*+1. For background on this, see Gnnning-
Rossi, Oh. tt. The'if'* are fine sheaves, henoothe above resolution
defines isomorphisms
H>(X 0 ) ~ ~ {3-oloscd @, ff)-forms on X)
' ' ~'~ ,g-])-formaonX)
Moreover, if ? => Vfi is the sheaf of 0" complex-valued funoliono
onX, then jnat as with holomorpbio forms, there is an isomorphism
taking S/i ® (t, to S/jtD,, where m, ta the tranBl&tion invariant
(p, (j)-form with value « 6 A'T ® AfT at 0. Note that these
translation-invariant, forms ?ua are all closed. In fact, since ai^t =
". A «j. i*1 i" sufficient to check this for a of degree A. 0) or @, 1).
Sinoe ir:7->- X ia a looal isomorphism, it is sufficient to ohetik that
&{**(<•>*)) = 0- But considering a itself (which ta in 3'$ T) as a
function on F, b*(ui.) = da. TherefoTO<I(ir*ui.) = <?* = 0
Now let A' «= © A'T be the exterior algebra on T Let a =
TCX V) Then via ^ we get an isomorphism
If we define a differential 3 on the set of spaces on the left by
S(/® k) <-> I/A a, then (because the «„ are closed), the complexes
^*) *« isomorphio. Therefore
Onraimisnawtoshowthattheinehuaont: A -»-a®cA deflnas
an isomorphism of oohomology, i.e. A* -+ B*(d ®c A*). We will
do this by Vanxwr aeries. Let ^ be the measure on X induced by
the Bactidean measure on V, and so normalised that the volume
of X is 1. We define a C-linear map ft: a -*¦ C by putting
= ffp- ^°r any vector spade If over C we denote by ftp the

fl ABELIAN VARIETIES
map ft®l»: a® W ^*-W: m particular, wo get ft map fiA u®cA
-»-A" whioh is A"-linear and such that n^i = IdA
I
= "¦
Since pA is A'-linear, it suffices to prove that )iACf)
Q. Choosing a basis «!,,..,<¦>„ of T, we can expand D/s
>(, The ooefltoients ftj are all of the form D{f), where
iJiflsomeinvariant veotor field onX Therefore the lemma follows
from the elementary fact that if/is &C -function on Aperiodic
with respect to the lattioe t/, and D is a trauslut ion-invariant veotor
field on 7, then
p
¦ 0
hit *= some Euobdoea volume element).
Let V* =Hom(l7, Z). If A e 17*, then A extends to an R lmoer
mop A: V -+R and we aan then form the funotion a;-+«i*uwon
7, This funotion is invariant under the action of U, hence it equals
»here e, is a CB-ftmction on X. Now define a C-linear
SjOir, wn
mapQi:a^CbyO,{/) = n.(^Jt/) = f e^,./.fi. More generally, for
any vector space W, define &: ftOc^-*- Jf by Qk (/® w) = f>(«-j/)^
The 0,(/) arc the Fourier coefficients of /• for every Jea®cw
we get tho expansion
The <J, are oompatible with C-linear maps W -* W' just as ft is
in partioular C»:a®c A'-*-A" is a AMinearmap
Vat the remainder of this proof, we choose a Henuitian norm |
I y on the oomplex vector space F. As UBual, this induces a norm |
on T, hence on the whole exterior algebra A*.
Moreover, define the mapping 0: V* -*¦ f as follows
R) c Hom,(F C) s [T © T]
*
projection
ANALYTIC THEORY 7
This makes U* into a lattioe in 71 henae by restriction we get a
norm || || on V* too
Lemma 2. A) The map f -*¦ {^,(/)}MW. «* on Momorpiwm
o_f q onto tite mi-tor spae.t. of at! maps Q: U* -± C de-crtawiQ at co
en [j A j|-", a« n, i.e.
B) For all ta
Proof. A) is standard Fourier analysis. To prove(B), note that
ff*(S«-») = 3(«"**") '- - 2iri«-*"u.Sjl = n*[- 2jr»e_, ®O(A}],
henoe 3«_) = — 2jrw j®6T(A). Therefore, by Lemma I for all
(- I)'
The following is well known.
Lemma 3. Let W fie a complex vector apace, D e Hnm^W, C)
Then D txtcnfa lo a map Dj : A'lT -tA'-'ff for all p ealhd
interior multiplication by 1) such that
A}
DJ{X A
{2} m
A XF)
w A
\f *D{Zt)X A A
-1 for all weA W
We are now all set to prove that»: A* -+0(ScA" 'fl ¦ homotopy
equivalence for 3-oohomology. Por every H e V*, \ t*0 define an
etenwnt A* e H.otoc{T,C) uei ig the Hermitian inner product <,>
on ?:

8
ABKUAN VARIETIES
Then 2w» A*(C(A)) = l,and ||A»|| < (Sw>—>. J A_|| '. For all u
we define &(a>) ea® A* by menus of its Fourier expansion as
folio wa :
0. (*(«)) = o
It is Miy to oheofe by Lemma 2 and some easy eHtimates that one
and only one Huoh Jb(tn) exists. Then ve B,3nert:
In faat, for any w e fl ® A', we can oheck that both sides have the
game Fourier coefficients. If A # 0,
-ftW
and Cttf^iB)) = 0. If A = 0, then 5,D8™) - 0 and Qt{dka) 0,
whaeQ0H=^i(f.A«)). This proves (•)
It follows immediately from (*) that p commutes with Z and that
• •^t ii homotopio to the identity. Thus t:A'-t-a®cA" induces
Uomorphisms in 9-cohomology as olaimed.
1 In the above isomorphism of B*{X ffx) with AT,
the enp prodaat pairing
H'ifj, eM) x JSf"(x, ez)—t- j^+"(z «»,)
oorrespends to the exterior product
This follows from general sheaf theory, since we resolved the sheaf
f>x of C-ftlgebrttf by a differential graded algebra (V"-*, 3), In such
a, case, oup prodoot ean be computed by multiplication in the
reaorsTDg sheaf (Godsmani §6.6).
1 The natural map induced by oep product
AHALYTOC THEOHY
2. The same method used in the proof of the theorem
enables one to compute the oohomology of the de Rham complex. Let
<C» = © Vr-t be the sheaf of C oomplex-valued n-forms. Then
d d
o >c > y > y >
is a fine resolution of the constant sheaf C, henoe as usual
f/»i T fi {d-closed n-fornu}
1 ' = rfRS" l)-forms}
Just as with (U, ;)-forms, we obtain the result: for all d-elosed
re-formfl in, there is a unique translation-invariant re-form w,
oGA"HomB(V, C}, such that
ui — it, = di), some (n 1) form i).
Therefore H"IX, C) k A"[HomB(K, C)], Once again, these isomor-
isomorphisms take oup product on the left hand side to exterior product
on the right. Also, since
this shows thai
H n,(F C)s T® T
T)
This is the famous Hodge decomposition.
3r A. ^loner took at what we iittve done so for roveals
that t tie Hi hint ion is n little oomnlicated. Consider the three
sheaves on X embedded in one another as follows :

10
ABELIAN VARIETIno
(Z and C being the iwnatant sheaves). Looking at their H1 a we '
have found three independent evaluations of these groups:
vta the isomorphism
7 Z)
C) =T©2
all spaces 7.
via the exect sequence
d
0 >-C t-V >
and r © T-
v a tbe exact sequence
0 > &z >?0D-
—>-0
and P
It IB only natural to assume that the vertical arrows connecting the
oohomology groups correspond, tinder these evaluations to the
canonical maps (a) Ham(U, Z)-* Hom([7, Z)®t C s Hom,(r, C),
and (b) projection of T$T onto f. Let us check that this does
occur.
B
PolBT 1 The map BH,X, C) -!-*¦ iT>(J, C,). This can be oom-
pnted hy comparing the two restilutions. LetC01: *1 = *l-*©»*1
-+ <PA be the projeotion. Cnl takes d-closod forma to (Mooed
forms hence we get a diagram:
d
0 *C >- ?° > Vi^toBa 1- 0
L
AKALYTIO THEORY
This gives us a oommutative diagram
projection
Thue ^ i» the expected map
2 3 m svrjtdwe
11
(X C)
Eonrr 2. The map J5f»(JC,Z) -%¦ fl^.C}. Let o 6 fl'(X, Z).
Hott does such an a determine a homomorphism a from jr,(J[)toZ?
If ^:S1-+XisaloopinXcorrespondingto an element [^] e w,(X),
then a([^]) is found by considering <t>*(a) e H\Sl Z) and using the
canonical isomorphism t: iT1!^1, Z) > Z
In particular, for all ueV, let ^,: fl1 -»¦ X be the loop
^(J) = w(to) (where S1 is parametrized by ( e R
ooneidsred mod Z)
Then a determines a e U* by the rule
we push o into the sheaf C: we get «(o) eif'fJT, C)
Acoording to one swond evaluation, there is a unique b e T © T
8uoh that if uit is the invariant 1-foim on X with value b at 0
then we get:

12
o>% I-
Pullmg baok to Sl we find:
>, Q
I
«(«(¦»¦
But now it ia an elementary matter to obeok that if ij is a 1-fom
on fl» (any auoh y is cloaed), if 8A7) is its image in tf»(S», C]
and if <{%)) is tbe image of 8[n) via the canonical isomorphism
)-^* C then
Therefore
3(») = *(«<<*))
So oisjuatthe restriotion o?the funotion ft on Fto P
Ifting oomptibility of our evaluations with oup products, we
heve even proven now that we have the following compatibilities
between the evaluations of the »"" oohotnology groups:
theory
of (inclusion U* c
13
projection onto j)=0 if=» faotor
2 Lute bundla oo a complex torus. We recall the well known
Tbsorim Tar all integers p > 0 Hr(C*,0) = @)
For a proof, cf, Gunning-Rossi, p. 28 and p. 184.
ConoLLABY. fP(C, 0*) = (l), ffltt jo>0 In particular aUholo
morphk liite bundles on Cs are trivial.
Pnotif Use the exact sequence
0 y Z > 0 > 6>* • >¦ 0
iHitl the fact tin U»(C* Z) =• @), alip > 0,
We wish to give a direct geometric description of every
(hoiomorphic) line bundle L on a oomplex toruej, By the corollary
the line bundle n*(L) on V. ia trivial. If we choose an iaomorphism
the canonical action of U on v*(L) (i.e. the action such that the
quotient of n*{L) by Pis justthe original bundle L) carries over by
means of x mto a linear action of V on the trivial bundle
covering the action of U on the base V by translations. Let us
j denote by H* tbe multiplicative group H'(V, <SJ) of nowhere
niHhing halomorpiiio functions on V. Siuoe the only holomorphio
iwtcmorphiains uf a line bundle fising the base are given by multi-
multiplication by non-vanishing hoiomorphic funotions, we aee that tbe
lion of U on C X V ie given by

it
ABELIAN VARIETIES
(a, *) i—*¦ 4Jfl, %) = (\[z) « * + ») all» e G (A)
where (^ e fl*. Writing down the condition that
thai
a a. 1 eoayde for pi
with coefficients in B* :
ANALYTIC TKKOBY
IS
Farther if the trivialfeation x « »"*red by multiplication by ,|
nowhere vanishing holomorphio function / on V, fc) is replawdj
by the oohomologoUB oooyole J:
^W •=«.(*)/(* + •)/(*)"'• 'I
Therefore we have defined a map from U><X •}) to H'lP. ^
But we can go in the other dweotion too. If we start with » 1-oooyol*
{*,) with coeffioientB in B*, then define a line bundle L on J as th.:
quotient of C X 7 by the action of V given by M >-*¦ feW-«
* +«). Therefore we have found on isomorphiBm
More generally, for any eheaf V on X, there is a natural ,,*f
A-iT'tP rftT.n'^H-^^HX^). ^6 definition and propertie»(|
Tare reoaUed in an appendix to thi» section. Since Bl{V. 0%) = A|
aU i > 1. the * defined in the appendix is also an ieomorph«D|
Let us oheck that the isomorphism just obtained and that of t^
appendix are the same. In 6wt( choose an open
by emoU enough connected open sets Vt- Then
(a)
tmon
(b) If «¦( ¦= restriction of » to J7( » l^i
morphtsm
(o) If^nFjft 0 then3«uefsuohthat
is a honwj
"I
The nap 4 °f the appendix by definition takes a group 1 oooycie
{,!_} to the Cooh 1-oooyole {/y},/M e r(V, n Vj. *|) defined by
But {/y} defines the line bundle L which is the union of trivial
line bundles C X Vt, modulo the patching
C x Pi C X rt
(a, I) I
l-Cx {V, n V,}
>- [n/v{x), x).
But #| is au isomorphism of C x IT, with C x Vj. so L can also
be described as the union of trivial line bundles C X W{, modulo
the patching
C x If, CxW,
Cx
-> C X
Now the disjoint union of C x Wt is just the line bundle C X V
pulled back to U Wt, and the set of above identifications in just
the equivalence relation on this pull-back bundle induoed by the
equivalence relation on C x V given by the action of the group V
Therefore, L is eiaotly C X V modulo V.
On any complex anatyt c space X the exact sequence
t According to ibM oonvantionB of tha appendix, w« thould take thautton of V
nn S« ns given by \v k) (t) " M* - u). u e fj, I « F. But then. If t, vMiArx the
MxaAilivn nboTP,ff[ f t—via* l-voaycle tar bhifl action, and aunvamljr, Thut,
¦¦mil swncialDd 1 Docyole la givnn by/ije r(Pi<1 Fj, ffj),
formula wfl hMT«abovft,

AKALTTIO THEORY
17
19 ABBLIAN VAWBTEB8
defines a oo boundary S : H*(X, 0J) ¦+ H*{X, Z) If aline bundle
L corresponds to a oohomology class A 6 H1(X, <PJ), then 6(>) is ]
called the first Ghent das,i of L. In our case, suppose L is defined j
as above by a 1-oooycle {es} with values in B*. We want to calculate jj
the first Cbern olass of the corresponding line bundle. First notic« |
that since if(F, Z) = @) for t > 0, it follows from the appendix'
that the maps $: B<(V, Z) = ^@, B"(V, Z)) -+ Bi{X, Z) ftr«
isomorphisms. If B is the ring of holomorphio functions on V we
have an exact sequence
Z E B*
(since V is simply oonneoted), bo that by the compatibility of i
with S (see Appendix) we get the diagram
Z)
Z)
Henoe identifying B*( V, Z) and H*(Xt Z) by the above homomor-j
phism, the Chern class ofL is simply !(ol {«„}). Write <
with/, holomorphic in V. Then by definition, &{tA{%}) mB*(V, Z) iij
given by the 2-oooyole F[uv u,) on V with coefficients in Z given by]
J(«i.».) =/„,(* + «i) -/.,+.,(*) + A,(*) e Z
Now ire use the following standard fact.
The map which associates to any map F;Vx O -*Ztiul
1 x U -+ Z (tejtMed tj, J^i,, »J = F{uv »,) ^- !¦(«„ i
mojM tne (Ttwtp o/ 2-cocydei ?*(P, Z) inio (Ae jpooe o/ i
itnajr moju [7 x V -* Z, and irtdvxxx an i
Pboof. Krat we oheok that if feZ*(U Z) S = AFia bilinear
We have
In this equation, instead of «,, u, and u,, substitute ut, u± and «,
(respectively «!, u, and u,) and oall the equation so obtained
(ii) (resp. (iii)). Then (i) + (ii) ~ (iii) gives us that
Since fi(u, u) = 0 and ?(«, u) = - JJ(o, «), it follows that S is
alternating bilinear. Now suppose F = h(i is a eo boundary Then
<?(«, + «,) +
Henoe A induoea a homomorphism B^U Z) ~t- Hom(A»l/ Z) s
A'Hom{P,Z).
Now, ainoe we have an isomorphism ^ of S*[O, Z) onto fl*(Z, Z)
where X is a torus, taking cup products to cup products {see
Appendix), and we know that H*(X, Z) is the exterior algebra on
H'lX, Z), it follows that H*(U, Z) is also the exterior algebra on
H*(I7, Z) = Hom(*7, Z). Thus, to prove that A is an isomorphism,
. it audioes to prove the last statement of the lemma. But now, if
f (reap. ])) is given by the horaomorphUm/(resp.(r)of GintoZ,f uij
is given by the 2-oooycle (see Appendix) c{s, f) = /(*).(((). bo that
A(i u ^) is given by the map: A(? u ijK*, (} = /(«)j{() -/(()((*) =
tfA »)(*,()•
Rimabk We have thus an isomorphism H'{X, Z) < ¦ ¦
A
B*{0, Z) —*- A'Hom([7, Z). This coincides with the isomor
phifim ff'(.X, Z)->-A'Som(G, Z) defined in §1, using cup product in
Z) and the isomorphism H\X,Z) -^-Hom(ET.Z). Infect
$ commutes with cup products and A has the property that
it maps cup product into exterior product, by the lemma and

18
ABEL1AK VARIETIES
•j,; B\V, Z) = Hom({7, Z) ¦+ H'{X, Z) is easily checked to coincide
with the inverse of the isomorphism of jl using the naturality of;
$ Thus, in future, we can unambiguously identify B*(X, Z) with^
A'Hom(t/, Z).
Returning to the line bundle l> arising from an {\} e Z*{V B*)
we state formally our conclusions as a
PitorosmoH. The Chern does of the line bundle corrttpondtng
to^e&iU.B*) it the alternating 2-form onV vnth valves in Z|
given by
S(«i. «.)=/«.(*+ «!>+/.,(*) /.<* + «*) /„,(*) (* arbitrary in V)
n
toftere
Oobollaey. // we exUnd B R-linearly to a map V x V -+ R.
(w. iy)=*E(z,y)forx,y?V,
Pboof. In foot, since E represents an element of E'(X, Z]|
in the image of H'(X, €1%) -+ B»(X, Z), its image by B\X, Z)
Et{X, (Pj) must be lero (and conversely). Now, this last map|
feotoncas as S'{X, Z) -1> fl'lX, C) -i> W'(Z, (?,). If
put How>tF,Q= Homef^C)© Ebiiic^utF, C) =T© f,
have established isomorphisma fl»(J, C) = A*(r® r)
(T®T)ffi (A1?), and H'fi, Gx) b A*fi and j" goes over into
projeotionA'frffiTJ^-A'r. Further,»W ia nothing but the :
lineal extension of JI (cf. Remark 3, §1), which again we denote by E?i
Write B-=B1+Es+Et, where BveA'r,B, 6 Asf; and #,3'i?|
The reality of ? implies that Et = jB,, so that j(B) =0 if and on
if Jf = S,, and this Holds if and only if E{x, y) = E(ix, iy).
Our nest aim is to give as explicitly as possible all tine bund
on the complex torn* X, or equivalently, to find the simplest kini
of representing coeycles {\} for all oohomology classes in B^U, H'^-
This in in turn equivalent to finding a system of functions {fjmj
holomorphto in V and satisfying (•)¦ '
ANALYTIC THEOBT 18
Thus we atsvme given to us an alternating form S; UxU -+Z
with B(ix, iy) — E{x, y) and we seek to find {/,} satisfying (*) and
(**}, Let us look for solutions fu which are linear in z {not neces-
necessarily vanishing at 0),
We use the following elementary result
Lemma Let V be a complex vector space There m a 1-1 eorrei
wmdenct between the Hermilian forms B on V and the real skew
tymmetric form* E on V satisfying the identity S(ix, iy) = B{x,y)
tri)«* w given by
E[x,y) =Ira H(x,y)
The pruof is left to the reader.
Let B. correspond to the given E then one checks immediately
that the functions
satisfy (**) for any oonatants /?„, and the reader can also check if
he likes that those are the only linear solutions of {**), holomorpbio
m z modulo ooboundaries. Substituting in (*), we get a further
condition *
for all Uj u,et/, Writing tj9, = y, + JH(»,»), thu reduoes to
7^ + VH~ V.1+«, +1 t*t%.»*) eiZ
Now it is still permissible to modify ifm by the ooboundary of a
CBneai form L on V, or what is the same, we may replace yM by
7, — ?(tt) with L:V-*C being C-linear, The above equation shows
that R*7. is additive in U, and hence extends to an R-linear map
X: V -+B, and there is a unique C-linear form L on Y with ReL = A
(viz., the form defined by ?(p) = A(ti) — tA(tv)). Modifying ybythii
i, we may assume that y is pure imaginary. Writing i(u)= e*rrn
we see that a haa to satisfy the conditions

A-BELIAK VARIETIES
W« oan oheok that given S, there always exists euoh an a
eqaivsjently, that there alwaya exists a map 8: V -+B such that
8(u,+«,)-S(«J-S(«t) si *(«„«,) (mod 1) foralU^u.6 1
Thifl is left ss an exercise to the reader
We have thus proved the
Lbniu Let B be a KermiUan form on V sttch that t/ fi = Im j
E(UxU) cZ ?ela: U-* C* =. {ie C* | |z| - 1} bt a map
»(«, +!»,)= «fa«»l^' O^WjJdllt,), Ui§U
Such maps sl exist for any given H at above. If we put
Diitinitiok. L(H,«) m tile guotunt o/ Cx F /or (A* aetum o/
Note that the map {B, a) >—> {«.} satisflea the condition that i
{t?>} oorreaponds to (H,, a,), {«i".<i11} oorreaponda to (
Therefore we have an isomorphism of line bandies
The main theorem of this section is
Thbosxm at AppiixHimsiBT. Any line bundle L on
mplex tontt X it ttomorphic to an L{II, a.) for a
(B n) tattsfyxT^tAt conditions of tite above Itmma Wehavti
ANALYTIC THKOBY
21
^^ere Pta I« '*« group of line bundles on X Pio°X lA« subgroup of
QfOst kHkK an iopalagically trivial and At Jast oerttotrf map w given, by
g , >. TinH {with the wool identification of H*(X, Z) with alternating
Pboov. We have already shown that ati alternating integral
2-fono S on U, oon aide red as an element in H*(,7, Z), maps into 0
in H'(X< ®z) ^ »nd only if E(ix, iy) = Bfr, y) when E ia extended
R-linearly to V X F; that ia, if and only ifitielrnHforH Hermitian.
Thus v is an iaomorphism. By definitions and the above lemma
stating ezistenoe of a far given ff,the first raw ia exact. Since the
typological triviality of a, line bundle L is equivalent to the vanishing
of its Chern class, and since v is an iaomorphiam, the second raw
is also exact.
To prove the theorem, it suffioea to show that \ is an isomorphism.
If ieHora(t/,Cf) with A(«>= 1, we can find g eH* = B"{V, €\)
with
If ^ is a oompoet set in ? with X + U = 7, it follows that for any
ieF, l?B}| < Sups|g(i)), aince [a| =1, Hence g «an only be a
constant, go » = 1, whioh shows that i is injeotive. Consider the
uimmuUtive diagram
fl (tr C} ^ ff'(l/ H) »• Ker[H'{[; B*) —*¦ 8*{V Z}]
¦I '!
w
C) ¦
¦ B*{X,Z)]-
where the vertical maps are isomorphisms and the maps denoted hy
1*1' are surjeotive. But we proved in §1 that H'(Z, C)-*H'(X, <HX)
u eurjeotive. It follows therefore that every line bundle LePic*{X)
is presentable in the form C X V modulo an action of D of form
,(A, z) =a (A.«(w), 2 + w), where «: tf-+C* is a homomorphism

22
ABELIAN VARIETIES
ANALYTIC THEORY
But as we saw on p. 20, by an automorphism of C x V,
oon always normalize suoh aotions ao that Image(a) c C*. Then
fore A is Burjective.
APPENDIX TO |2
We want to study eohomology of sheayeB in the situatio
T = XjQ, where O is a discrete group, acting freely and die
tinuously on a good topolagical space X (i.e. yieX, x has|
neighborhood U, such that f/±r><r(t7.) = 0, ali'o eO,<t
Let ir: X -*¦ Y be the projection.
First recall the definitions of the cohomology of abetraot |
Let if be a G-modulo and let
C{G if) = {group of funotions/ G* *¦ M)
S & >C?1 the map
if) = Ker(S); ^{0, Jf) Im(S)
if) -&(<},ld)IB>{Q,M)
= dorived functors of if (—»JH°(CJ if), where
JW)={meif|(r(ra) = m aUtreG} (also written if°).
Given a G-Iineor pairing if X Jf
- P of G-moduleiB
We want the result:
V flheft^es jF on Y, there is a natural map
It has the properties:
B an exact aeqaenoe of sheaves on Y, and
je exact, then we get a hoinomorphism from the Bohomology
Bequen«> of #'(<?..) to that of fl»(r,.).
(b) The natural maps 4 are oompatible with oup product
(c) H
H*(X *•#")= @) i> 1
then
ffl'ierii „¦#¦)) ^(r^)
is an laomorphiam
To define ^, ohoose a, oovenng { V^ of Y euch that for eaoli t
{I)' v-'iV,}=\i oiV,) V{ cl open such I hat res w fr, % p',
.«?
B) V*'J.l''1Bre exists at most one cr e 0 Bach that U(naUf ^0
call it cry if it exists.
Define a map from group 00-ohainB to Ceoh co-chains
It is easy to oheok that S^,, = 4t+is> hence the $p induce a map
$:H>(Q, V(X, w*^)) -*&&,&). Properties (a) and (b) follow
immediately by computation. To prove @), we use induction on
p: for p = 0, it is obvious. In general, embed Jf in an injeetive
V eheaf&' and let.?- =&'!&. Then we find

ABELIAN VARXETEB8
ANALYTIC THEORY
hence
It suffices to prove that V(tt* &') is an injective G-module
that J?'(X, ir*#"') = @), » > I, beoause then it follows th^
if(X, it*y) = @), i > 1, hence <f\,$t are isomorphisms by tlf
induotion hypothesis, hence <f/3 is on isomorphism. We need
LamtA If y is an injeclive <Pr-sheaf, then it*y U a fia
e>z-ttieaf and T{ir*y) an injective Q-modvAi.
For all G-modulea M, let M be the constant sheaf on ',
with value M. There is an obvious action of 0 on M oompatib
with its action on X. Then 0 acts also on t!^{M), so we can fot|
vt{M)a, It is easy to check that
Hom0(Af, r(ir*^)) = Homs (ir»(Jf)8 &)
So ]fif!C if,, then v^M^cn^MjP, hence
is surjeetive, hence Home(Mt, V(it*&)) -* HomgfJlfj, F(it* i
aurjeotive. This shows that r(**#") is injeetive. Secondly, 3* iiil
jective implies & flasque, and since n- is a local horaeomorphian
then rt*y is flasqaetoo. (Cf, Grothendieck, Sur ipalqttts
d'algebre. homotogxgw, Tohoku Math. J. A9ST), esp. Cij. V, p. 19S.)|
3 Algcbraoabiiity of ton. We have seen that any line bund
L on the complex torus X = VjU is isomorphio to a unique
bundle of the form L(B, a) where if: V x V -*¦ C is hermitiari wit
S — Im H integral on U X U, and a is a map V -+ C*. satisf
«(«i +ua) =ef"*"''"")a(u1),«(«i). L(H, a) is the quotient of Cx VI
the action of U given by
25
We now investigate the Motions of i(H, a). These sections are
in a natural one-one correspondence with sections 8 of the trivial
bundle CxF over P (i.e. holomorphic fiinotions S on F) which me
invariant under the above action of O, that is, which satisfy the
functional equation
*[« + u) = «.<*) 8{z) = «(tt). c»h«)+t.JB» « ((,), teV.ueV
guoh » function is culled a theta-functton for the hermitian form H
mid the maltiplicatar a
First oonsider the case when H ia degenerate Since B = Im Jf
Bud 5(«, y) = S(ix, y) + »J5(*, y), we have
It follows from the first expression for N that N is a complex
subapaoe of 7. Andsinoefi is integral on Ux U, it follows from the
second expression for N that N n V is a lattice in if. If fl is an
associated theta-fiinotion, we must have
Thus, ifZUacompaet subset ofj^ with 2^—K+{Sfr\ V) we must
have
for all z' e W. Therefore, by the maximum principle for holomorphio
functions, S{zg +z') = fl(^) for z' eN and fl ia constant on coaets
inodN. It follows from the earlier equality that if fl^O,thena(u)=l
for veNr) V. Thus, if ij: P-» VjN ia the natural map, we see
that any theta-function for (H, a) is of the form S>ij, where 8 is a
theta-function on VjN for the lattice tj([/), the hermitio-n form H
induced by H, and the muitiplioator 5 obtained from a by passage
to quotient from V to UjNn V. Now B is non-degenerate on
V=VjN. Thus the study of the theta-funotions for {H, a) is reduced
to the study of theta-functions for (if, 5) on the quotient V=VjN

26 ABEIJAK VARIETIES
and we may restrict onrselves to tbe case when H is nori-degenerat|
In particular, we see that if 8 is degenerate with null space N, if ]
vanishes at zs V, it vanishes on the ooset z+N, eo that any section
a of L{B, «) whioh vanishes at an x e X = K/t/ also vanishes on th^
ooset *+Xf where X' is the subtorus N[U r< & c X. In particular
ire see that if the sections of L{H, a.) deiine a morphiam of X
projeotive space at all, this morphism has to factor through
quotient torus XIX', X' = SjV n N. Thus L{H, «) cannot
ample if if is degenerate.
Next, suppose there is a complex subspaoe If c V of positivi
dimension such that B(u>,«) < 0 for w e W, to # 0. Let JT be
compact subset of V with F = U + IT. Let ^e V and we IF, a;
write w = d+n, deif, «e If. We have
I 91a, + ») I = 18{^ + d + «) ]
and since
ANALTTIO THEORY
27
Of the terms on the right, for fixed %, the first is a real negative]
definite qaadrattc form in at, the second linear in if and the third
is bounded {since d stays in a oompactset K), so that the expression
tends to — co aata-t-oo in W, and applying the maximum principled
to fifz,, + up) as a mnction of w, we conclude that ${% + 10) = 0?
hence 6 mO, Thus L(H, a) has no non-zero seotions in thia ease,"
Therefore, if B is not positive definite, L(H, a) oannot be ample.
From now on, we wort under the assumption that B is positi-
definite {and E = Im B integral on Ux V as always) We shall'
prove the following
Pbofobition. Ifhen B m positive dtfintii and E~lm B U
wprewed as a matrix wing a baste of U over Z we have
dun fl%T Lift «)) = dim [«pace of theta functions tmtt rwpeet (o
(B, a)]
The idea of the proof is aa follows. Since in ej?)>? occurs
n the exponential linearly, one might hope that by multiplying
8 by «**' where Q is a suitable quadratic funotion one wiil be able
to obtain periodicity for the new function with respect to a big
aublattice V' of U. We can then expand this periodic function as a
Fourier aeries, and the behavior of $ with respect to lattice points
not in V oan be expressed in terms of the Fourier coefficients.
This enables one to compute the number of linearly independent
solutions.
Let then ej,z) = a(«).e-a<1'1*+!•«"."> as usual, and let $ he a
holomorphio function on V satisfying 0{s-(-«) = t^{zN(z). If
B: V X V -*¦ C is any complex symmetric bilinear form, and if
wb put 9*{z) = e^1'*1'1* 8{z], 6*{z) satisfies the modified equation
for ali u 6 V. Now, we can choose a aublattice U' of U of rank g
{= dim V) auth that A) M(U' X W) = 0, and B) if W - R.G',
If n {/ = V. Then IK n iW is a complex subspace of V on which ?
uudhenceff is identically 0. Sinoeifia non-degenerate, W n iW~{0),
andBoV= W@ iW = C®zU'zC<g>iW. Since E(Wx W) = G,S
has a real symmetric restriction to W, and by the R.bove, there is a
unique symmetric complex bilinear B on V such that B | W x ff" =
ffj yr X W. By C-linearity in the first variable, B(z,w) - Btz,v>)
^E V. Since E\ U'X W = 0,a.\U'; V ->¦ Cf ia a homo-
, and we oan find a (Minear form X on V with A real on IT
Mid r(u) = e!-*Wll) for u 6 (/'. The funottonal equation for 8* shows
then that e-*wUm.8*it) is periodio with respect to the lattice V.
Let us write U' — Homz(I/', Z) c Homc{F, C), and expanding
in a Fourier aeries, we obtain the expression
Now, for any « e t/ and «' g !7', (iT - ?) («',«) =if(u,«') - B(u, a')
= - 2i Im fi(«, u') = 2» JS(«',«) and if « e U' is denned by «(u') =
',«) and extended C-linearlyto V we deduce that (ff- 5)(s,w)

ARBLJAN VAMETCEfl
ANALYTIC THEORY
29
= 2iu(z). Substituting the Fourier senas A) in the function
equation we get for any v eU,
»e *•"'
and comparing coefficients,
Thus, if if is the image of 17 under the homomorphism U -.
given by »i—>», we see that the et are uniquely determined qd?
they are specified for % running through a system of representativi
0 so «j — u, e U' and one oheoks that the relations B) obtained wii
u, mid u, for u are the same.) We shall check conversely tit
given any system {«,}„!?¦ of constants satisfying B), there existal
corresponding function, i.e. the aeries A) is the Fourier series of*
hobmorphio function. It suffices to check the uniform absolute (
vergenoe of (f) on compact subsets of V. Blruig a ^q g U' it su
to prove this for the solution cf such that et = Oif^ — %$$ if (
«,t = 1. Writing x=Z» + uf°rX 6X»+-^' when * lies in a oomp
set K c V, the series A) is majorized in absolute value by
hence by
oonat
const
e!"*"tJ)<
where ||-u|| denotes a suitable norm onJf, andjl a positive oonst«[|
determined by ^,, K, % and H. Since the sum V= Wffi iW ia din
we can find R-Iinear maps/, ifi: V -*-W such that z =^{z) +vji
Sinceu s real on W*ndB{W X W)= 0, weget
further, ili(tt) = O <=* «=^(t() ¦*»» itef-e^ tt=O, so that Im «(w)
is & negative definite quadratic form on if Thus, the above aeries
converges very rapidly.
We deduce that the dimension of the space of theta functions
for (H, a) is the index V'jM
Thus we have only to show that if U is a. free abeltan group of
order 2g, E a skew symmetric bilinear form on U into Z non-dege-
non-degenerate over Q, V a direct sumuiand of U of rank g on whioli
^ s« 0 and n the order of the ookernel of the map V -*¦ Komz(U\ Z)
deflned by «i—«> B(i u), tlien | dot S [ = b*. Tliis follows from the
diagram
0 0
with exaot ro;vs and columns, the definitions of a., ft, and y being
via E, using the fact that a and y are transposes of each other up to
sign, hence have ookernelaof same orders, and the fact that/) has a
cokernel of order [det J? |.
Weoan now prove the main tKooremof thisseotion.
Tubobbm or LKisoHBTa. Let X be a cotnfUx tonts VjV, H a
hervUtian form on V weh (hat S = Im H i» integrai en U xU.tta
map U -+¦ Cf, wM «(«i +«|) = «(ul) (ilit^e*'^^11** ond L =L(H, a)

30 ABELTAK VARIETIES
(A* awoeuUed Itns bundle on X Then (he following statements an (
ANALYTIC THEORY
31
A) Owen any complex eubtona T of X, then *s an integer N > 0,
a section a of h*" and lu>o points xt,zteX,x1 -x^eY such that <
B) The hermitian form H %» positive definite
C) The space of holomarphtc sections of IP* give an imbedding of
Xasa closed complex svbmnatfold tn a projecttve space for each n>i, |
Proof. We have already shown earlier that A) =. B), and:
C) » A) is clear It remains to assume B), and deduce C).
We use the expressions "theta-funotiona for (H, a)" and "Bechon
of L(H,0l)" interchangeably,' making the identifications indicated
earlier. We shall prove C) for n — 3 (the oases n > 3 being proved^
quite similarly).
Krstly, if Sis a seotion of L(H, a) and a, b e 7, then fi(z — a),-
8(z — b).8(i +o +6) is a section of LCfl, aa). In fact, on making |
the substitution » + « for z in this funotion, it acquires a factor
r(*-a,w) + iT/f(*-*,«)-!-flJ
vhioh proves the assertion. Henoe, if % e 7, there is a section |
^ of hx not vanishing at z,- In fact, one has only to take a non-zero
section 8 of L(B, a), which exists by the Proposition, and then to \
ohoose a, b g 7 such that 8(zt -a)jk(}, tf(z, - 6) & 0 and fl(z<, + a + b)
^0, and put <(> to be the product ff(z — a),8{t — b).8{z +a +b).j
Thus, if fl0,..,, 8t is a basis of the sections of L\ we got a well
defined holomorphic map
6 X vP*
given in terms of homogeneous coordinates, by
W o(i( U
Next we prove that 9 is on injective map. If not, there exist
*!, it, 6 F *y — z, fi P ant) a non-isno constant y e C* such that foi
sJltheta-functions^ for C/f, il), we have $(z,) = y^(i,). In parti-
, for any a, b e 7 and any theta-funcfcion 0 for (H, a) we have
ffe new consider both sides aa functions of a (fixing b), and take
logarithmic derivatives, bo as to eliminate y. Writing m for the
(meromorphio) d'fferential-- we obta'n the relation
- u[s, - a) + »(*! + o + *)¦=- w(«t a) + ai(*i + a + 6), a b e V
which means that the differential u>(zt + t) aifa + z) is translation
invariant in «, hence of the form dl(t), where 1 is a C-linear form
on V. But then, this is also the differential of log °^ + .hJ so that
8{ + )
ire obtain an identity
= J,.**>.((«4-*.)
foe some ^ eC* Writing <r = z, - *,, this may also bo written as
with a fixed A e C*. Making the substitution z i—> z + « (u e V)
using the functional equation for 0 and comparing thomultipliootors
cd both sides, we get that
or iff (cr, «) - J(«) e 2«fZ,« e P.
This implies that fflff(ff, «) _ i(u) = fffi{u, „) _ {(„) + *(H(a, «) -
?f(u, o)) = w/f(u, t7> - l{u) + 2mE(o, u) takes only pure imaginary
values for ail « e V, henoe the same holds for irff (w, tr) - I(u), »nd this
being complex linear in «, we must have irff(ti, cr) = i(u) for all
• eV. But then it follows that Zm.S(«,«) e 2wiZ for « e P, henoe
»6P1 = {«enJ(*,»NZ, V«6t7}, which is a latUce in 7
oontaining [7 as a sublattice of finite index. Since it $ U by
assumption, [T + Ztr = P, and the equation
shows that 6 is actually a 0-funotion for the lattice V + a Z, the
hermitian form H and a suitablemultiplioator it'oaU -\-Za extend
Ik,.

3B
ABEUAS VABIBTIBS
ANALYTIC THEORY
ing i. Now, this must hold for any section S of L(H, a), and to'
dimension of the apaoe of such OV is V(detpjE). the root of tbj
determinant of S for the lattice V. On the other hand, if U'
V + Z a 3 ff, the dimension of the space of theta-functiona for tfc
lattice P' and JJ and any multiplicator a' is yr(det[r.ff). tt'
Toot of the determinant of E on the lattice V. But since
have evidently
and since there are only finitely many possible h b extending j
tt follows that almost all theta-fonotiona for H, a and U ore
theta -functions for H, a', and V for any a.'. This is a oontradi
J; X ^-P* is injeotive
To complete the proof of tha theorem we have only to
that 6 mduoea an infective map of tangent spaces at all points
f 3
X If not, there is a &. e V and a tangent vector ? «: -- at ^
i 3*
not all a, — 0 mapped into the 0 veotor at 8{tt(^,}) in
[B then an a, 6 C such that for all j> <sV{X, LCH,
ie
, Take
fcr all ? as above where D = ^ «,_
$(x) = fl(* - a)B(z ~ b)9{z + a + 6) as before, with a, b e V
B e T(L(H, a)). If we put f(z) = D(log r?>(*}, we obtain
for allo, ieF One conclndes easily that Jin a linear (not
sarily homogeneous) function of z. Integrating the equation f{i)
IHlog 0)(z), we obtain that there Uan«eF, a^O oaoh that
all A eC, we have
for same constant c One concludes as in the earlier step (by
anting down the transformation formulae for both sides, for tbe
substitution g i—t- z + u, u e U) that for all X e C, Aa belongs to the
lattice 17-i = {* e f I iS («, ji) e Z, V « 6 IT|, This iB a contradiction
yfa next recall some definitions.
Let X be an algebraic variety over C. There is a canonically
assooiated analytic space structure on the underlying set of X. We
denote this analytic spaoe by XbDtl and its structure sheaf by <PZtM.
Often, we will not be bo explicit, and talk of holomorphio functions
on X, holomorphic maps from or into X, etc. Also, we shall say
that an analytic space X is algebraio or algebraisable if there is an
algebraic variety 7 fluoh that 7M = X. Note further that an
algebraio variety X is complete if and only if J^j is compact
(This is an easy consequence of Chow's lemma.)
We now recall the
Thsobsm ob Chow Let X be a complete algtbrak variety and T a
doatd analytic subset of X^. Then 7 « ZarUld closed in X
Chow proved the theorem for X~P*, but the above version
follows immediately from this and Chow's lemma
An easy oonsequenoe is that if X and 7 are complete algebnue
varieties and /: Iw -+ 7^ is a holomorphio map, then / considered
as » map from X to 7 is an algebraio morphiam. To prove this, let
r in X,,,, x 7M = (X x 7),^ be the graph of/; it is a closed onalytio
subset, hence a closed algebraic subset of X x 7. For every {x,f{x))
e V, the projection I* ¦+ X induces a local homomorphism of the
algebraio local rings eV -* ^/w>.v ^ ffi = e*z> *. = *(.jw».r-
Then I claim that fft-t- ffi, is an isomorphism. Firat, use the fact
that the projection .V -*-X is proper and bijeotive: by Zariski's
Ham Theorem, this means that &t is a finite ^-module. Let St«
(x,ni)),rjtf and @j» V
phism, we get a diagram;
Since r -*¦ X is an analytic isomor-

isomorABEL1AS VARIETIES
ANALYTIC THKOBT
In particular, 6^ ¦+ ff, is injeotive If m( = maximal ideal in
then dividing by j»J we get
henoe ml=tn1(Pt + >r^. Therefore the ffa-module 7aljmi0t beootnn
@) after ®* ®tS*H'- henoe .by Nakayama's lemma, mi = i
Therefore the $,-modulfe ^tl^i becomes @} after ®9[ $
hence by Nakayama's lemma, (P, =<V This shows that r -+ X x
an algebraio isomorphism, hence that/ is an algebraic inorphiam
In particular, we see that a oompaot, complex space has at
one algebraic structure.* One further foot that we will need is
if JCis a complete algebraio variety, every raeroraorphio function
t For noa-txunpaot complex «p«oa«, thii it quite fain. For instance. Sen* [81j
p. 108,hMgiv«ii«ioMlowlBg mamplBifof ewi7 l-dliwmaional obeli* • a
X ovai C, then )• m. nniqm klgsbraSo group G whioh ia * non-trivial
(u c%. gronp):
Iti««»ilroh«ik»dthatlf ffoi.U.(»bjebwiD)j>troiitiiraahi»f. than P((Pff)=<
But taking tii« anlTsnal covering of Q. do* nbtcki that analytically, ¦
a - vjv.
F-a t.dimonaional oomploi veotw apam, U •=¦ |
• C-l»Kiof V. IfCdrootMUMldinumeioiuilaBlne algabiaia group, given 1
C - {0}trader multiplioitioo, it follow* eatllj tb*t Sand Gn X G»art two i!
s of tbs lama ualftio gnnipl
on Jm 's a rational function on X, i.e. in the function field C(X);
this flan be proved similarly by considering the "graph " off, and
applying Chow'a theorem.
(Jetting baok to complex ton, we 1 ave
Y. Let X= VjU be a g-dimensxmal complex tuna The
fdlowing are tquivalerti
A) X in ihi compUx space aMooiated to a prtyedim alfebratt.
variety,
B) X is ths complex space attociattd to any algebraic variety
C) there ecu* g algebraically independent meromorphic- fvnctwm
on X
(*) there it a positive, definite hermtttan form H on V sutft thai
Im [H) is iitttgral on U X V
Phoof. (I) =*¦ B) a. C) are obvious, D)*{1> has been proved
m the theorem. It remains to prove C)=> D). Let ft /f
be the independent maromorphio functions. Let Df he the polar
divisor of ft, D = SD(, and L the line bundle associated to D
Then L admits j + 1 sections aB <r( suoh that whenever /( id
regular, o-( =/, o0. By the theorem of Appell-Hurabert, L = L(H, a)
foe some hermitian form H on V with lmH(V x V) c Z, some a.
Since Lhas sections at all, by the discussion preceding the theorem,
we know that B is positive ftemi-definite. Let Fo be its degenerate
aubspaoe, and XB = VajVanV the corresponding subtorus of X.
Then the quotient torus Z/Je equals (K/F0)/image([/), and B is
induoed by a positive definite H on Vj Vai suoh that Iva.(H) is integral
on the lattioe VjV n Va. Therefore, by the theorem, 2jXB is a pro
fective algebraio variety. But also, by the discussion earner, we
know that if a in any section of L, the sreroes of o are unions of
wsets of Xa. Applying this to the sections ? «,o-,, it follows that
4-1
all the analytic sets /, = oonstant are unions of oosets of TB, henoe
eaoh /, is induoed by a meromorphic function Jj on Z/Jfa Let
C(J/J0) be the function field of J/Jo; then

ANALYTIC THEORY
37
ABELIAS VARIETIES
0 = tr. defo C(/,,...,/,) < tr. degc C(X/X0) =.
Therefore dim Xo = 0, and B is non-degenerate
As an erampte, notice that we get immediately the algebraizabilitj j
of 1-dimensional tori: in fact, if X = C/{»+mui/n,meK}, with!
Im u» > 0 then tet
One checks immediately that lmB = E has values #A,1) =|:
E{a>, en) = 0 fi(u>, 1) = — B(lp «) = !> hence H aatisfies the|
hypotheses of the theorem. Moreover, several projecting
embeddings of X are very well-known in the classical theorjg
{of Hurwitz-Oourant)' for example, if I
p(z) = —, + > ,, _ „ „ ¦ ,.i - ln ,. _,u
WMt to show that for almost all lattices V c 7 qo element of
.lrom([f Z}) has image entirely in the middle factor TO T
ttoteK ItwiUsuffice to show that Al(Hom{[7,2))->A|3' is
Bat Hom(P,Z) projects into a lattice in T, and for
Stable choice of the lattice [/in F,Im(Hom(P,Z)Ms«> aibitrary
I ttioe in T. So the oonoluflions follow from
lattice* VcV.itemap
|
is the Weioratrass p-function, then p ia a meiomorphio function,a
periodic with respect to 1, ta, with double poles at the point*?
n -f- mm. The map; J
ii—y A,4>(z)> P (J)) (projcetive coon] natei):-
C ^ P' ?
induoes an ieomorptiiam of X with a plane cnbic ourve of the form ^
Jo X| = 4^5 + oXpX, + bXl (for suitable a, 4 depending on <•>). i
Ou the other hand, in dimensions > 2, it is easy to boo that almost
all tori ace non-algebraic. In fact, we can check that on almost all"
tori X, Ko(X) = Pic»(X) or equivalently (by the theorem of i
Appell-Humbert) that there is no skew-symmetrio E = V x V -> S ';.
which is (a) integral on UxU and (h) RatiB&zn E{ix,iy) = E(x,y),%
Let X = VjV, and put T Homc(V, C), 5"' ¦= Hom^^F, Cj-;
na before. Consider the map "i
z,,... ,2, are introduced in 7, and V is described by
. ^1, K.-< 23, then almost all can be inter-
Lted to mean all ? X 2<f-tuplea K) not lying on a countable
union of ((rt^fl) - IJ-dimensional ansilytic subaete.
proof of this lemma to the reader.
We leave the

II
ALGEBRAIC THEORY VIA VARIETIES
4. Definition of abdian varieties. We now turn to the study of
abelian varieties over an arbitrary algebraically closed field k.
An abdian variety Xis a mmpkle algebraic variety1
peer k witt a ffrovp law m: X x X -» X *ucft (loi m and the inverse
maip ore both morphtMna of varieties,
Hole that if k = C, then the underlying oomplex analytic space
of an iibelian variety is a compact complex analytio group, henoe by
the results of jl, it is a oomples torus. When i ^ C, the first aim
of the theory of abeliaa varieties is to show tliat an abelian variety
bm properties analogous to those enjoyed by a complex torus. Of
ooune, when char i = 0, many of these results can bo proven by
reduction to the case k = C (Lefsohetz's principle), bat when
tshar k^Q, this is by no means possibla. We shall want to answer
the following basic questions
Qcraairiotr 1, Structure of X as an abstract group.
We will show that X is a commutative and divisible group. We
will also show that if nx denotes multiplication by n (n an integer
> 0) on X, the kernel X% of itx, or what is the same, the gronp of
elements xe X suah that nx =• 0, has the following structure :
, m > 0
where i oan take every value in the range 0 < » < g — dun X
Th« integer i will be called thep-rankoiX.
Qoaswoir 2. Calculate the oohomology group &(X tV)
(Q* being the sheaf of p-forms on X).
As in the classical oases we have canonical tsomorph ami
AE^tJ.n1)]®! A[B (X ax)]
in p»rtlc*l*r th.t it ii trnduslbla

AMELIAS VAMEITJC8
ALOEBHAIO THEORY VIA VARIETIES
and
dim
Q } =g
We also show that ir^X) {in the algebraic sense, i.e. the
Jsetive limit of finite groups of unramified Galois coverings} :j
isomorphio to IKZ,)* in ohar 0 and to II(Zj)*' x 1L in ohaH
More preoiaaly, we shall show that if Y >. X is any morphia
auoh that a finite group O aota freely on T in such a way th|
X becomes the quotient of Y for this action there is an inted
n > 0 and a commutative diagram !
¦*¦ X
Purther, Y carries a Btruoture of abelian variety such that / andf j
are homomorphiaros. M
3 The atruoture of Pie X #,
We will show that there is an exact sequence i
0 > Pio°X —>¦ Pic X —y NS{X) —> 0 f,
where Pic'X has a natural structure of an abelian variety, an|[
NS(X) is a finitely generated free abelian group, whose rank p|i
oall&d the base number of X f ¦'
We will also try to find analogues of the olaasioal description i^
N8{X) by Kiemftnn forma E. J
Belated questions ore: (a) for a pair of ahehan varieties X, If:
show that Hom{X, Y) is a free abehan group on a finite tmmbei of- {
generators, (here Horn means the set of mapa whioh are both w;
phiamB of varieties and homomorphiamjj of groups), and (b) give |»
matrioial representation of this group of homomorpbisms (inthi-
oase we have the rapreaentation induoed in Hom(i/,(X)
sin)-
4 Oharacmenze ample line bandies. More generally,
mp the cohomology group* of arbitrary line bundles, and in
mrticuiar the dimension of the space of sectione-this is the
n-Roch problem.
t of the
(i) We start with the observation that on abelian variety w
wen/where non-smgutar, In fact, there has to exist a non-singular
po nt xa e X, and far x e X, the translation morphism T(r^-iy
X ~* X, given by T^^-tji^) = i.*o'I.y, is an automorphism of J
wrrying%tox bo that z ia again a non-singular point of X.
(ii) At a group, X it eomrnutatwe. We give two proofs, one hers
and the second a littte later. The first proof generalizes the proof we
gave in the classical ease. We consider not only the adjoint repre-
representation of X in the tangent space at the identity e, or in the space
of differentials at e, but in each of the spaces {<P,r,,/?K J,,) where 0Xjt
in the local ring of X at t and Wij its maximal ideal, For ieI, let
C,:X-+Jrbe defined by Gs(y)=xyx-l,BO that G,(e)=e. ThetiO,
juduoea an automorphism GJ, of the vector space 0xJWx^, deduced
by passage to quotient from the automorphism CJ: <SZ
local ring. This induces a set theoretic map y: X k
xi—tG^t and if we put ou the latter group the natural structure
cf an algebraic variety (viz. that induoed from the inolusion
AnttfP^/ffil^cEndtu^/iDli,,), thelaat being a finite-dimensional
vector space over k), one checks easily that this U a morpbiam of
varieties. Sinoe the latter is an afuae variety and X is complete and
connected, y must be a constant mapl Since y(e) is the identity,
we see that CJ» ia the identity for all * e X and n > 0. But since
0 fflj,, = @) this means that C*: 6>Xi, -+ (PZ|, ia the identity, so that
0, redocefl to the identity in a neighbourhood of e in X. Since X is
iirednoible, Gt U the identity on X for every % that ia, X is
oommutatTve.
From now on, we write the group law in X additively Moreover
ire will use the following notations: for ieI, we denote by

—.
ABXLLUT VARIETIES
Tm X -+X the translation morphwm Tt{y) = x + y and
map xi—ntj: will be denoted by nx
(iii) IjT^Txfi u the tangent tpaee atOto X ft<, u tin dual
Tj q of differentials, then there ia a natural isomorphism
¦?>
_-is the sheaf of regular \-form»onX, One defines this mapp.^
ing as follows To eaoh 6 e A,, ooneider the 1-foTro ui,onl defined'*: j
by {o>f)t = S'SjCfp that is, the unique translation invariant 1-fonoi-i
on X whose value at Ois 8. It is aheoked easily that w, isaregulaiM
1-form on X, Thus, we have a natural homomorphism as above,: !
Since at any points, T^, induces an isomorphism of the space of;:!
differentials at a: onto the space Q,,, it follows that the above homo.]; j
morphisms of sheaves induces an isomorphism of fibers at x reduced; |
modulo the majtimal ideal ffiXi# at x. It follows by NakayanwV.j
lemma that it is an isomorphism of sheaves. 4'
Since X is complete and oonneoted and H*(X, Sx) = k, it follow '¦ \
from the above isomorphism that the everywhere regular forma onft.j
Xare precisely the invariant forms 'tl
(iv) For tvery n not divisible by the ehoractenstu p of k the endo-'f\
morphtsm nz is surjetthe. j
For the proof, we moke the following observation. The tangent:;
space TXxxt9tin to XxX at {0, 0) splits oanonically into »||
direct sum Tx © Tt, where T, (resp. Tt) is the isomorphie image of •: i
TXfi under the immersion X-f-XxX given by x>—t- (i,0);
it ¦¦'":.
(resp x i—* @, x)), Identifying T, with TXi<> = T by the* i;
isomorphisms, we note that the differential d{m); Tty T ~»T of ; |
the addition map m: X x X -vX, m(as, y) = x + y, is nothing*;;
but addition of components: rf(ntX*i. *») = *i +*»¦ In fact, it ii j
suffioient to oheok this (by linearity) on the two summands T of; j
T($ T and for these, it follows from the fact that the composite! i
XxX-
¦ X and X ¦
-XxX-
- .Tare the identity., ;
AMEBBAIC THEORY VIA VAB1BT1BS 43
It follows by induction on n that for any n > 0 (and hence also
for n < 0). (dnx)o ia multiplication by n. Thua, if ji Jf n, dnx is an
isomorphism. If nx were not surjeotive, by the dimension theorem
dim, «jf'(<))> 0, and we oan therefore find a non-aero *( e T
tangent to nJ^O) at 0. But then, we would have dnx(t)~Q
(since nj'W is mapped into the single point 0 by ft,) which is
ji contradiction.
The following lemma besides having other important applioa
tions, gives a aecond proof of the oommutativity of X.
Rigidity Lsmma. {Farm. I) Let X be a complete variety, Yand Z
d«jf varieties, ami f: X X Y ~+Z a morphism ntch that for some
yteY,f{X x.{pQ})i!>atingUpotnt%of Z Then there u a mvrphan
p. Y -+Z such that if pt:X X Y -*> Y is the projection / = gapt.
Proof. Choose any point x, e X, and define g : Y -* Z by
8(jf) = /(^e. #)¦ 8inee X X r is a variety, to show that / = jr.p,,
it is suffioiont to show that these morphisms coincide on some open
subset of JTx Y. Let U be an sffine open neighbourhood of ^ in Z,
F = Z-V,&tidO = pt{f-i(F));tben Q is closed in 7 since Jt is
complete and hence pa is a dosed map. Further y0 $ G since
j(X x {jr0}) = {z^. Therefore Y - G ¦= V is a non-empty open suhaet
of Y. For eaoh yeV, the complete variety Ix(j) gets mapped
by / into the affme variety V, and hence to a single point of V. But
this means that for any x eX, y e V,f(z,y) ~f(xt,y) = g»pa[x,y),
md this proves our assertion
Corollary 1. // X and Y are abelian vartettes and /: X -+Y
it any morphimn, /(*) = h{x) + a wftere itta Sotootho^Amw o/ X
into Y and a e F
Phoof. Replacing/ by/ — /@), we may assume/@) = 0 and we
hive to show under this assomption that / is a homomorphisra.
Consider the morpliism XxX >¦ Y defined by $[x, y) =f(x + y)
-/(y) -/(x). Then ^(X X {0}) = ^({0} X X) - 0, so that jt follows
by fhe above lemma that ij, a 0 on X X X, or what is the same
that / is a homomorphiam

44 ABELIAM VARIETIES
Note that in the proof of the above oorollary, despite the additii
notation used, no use of the oommutativity of X was made. Thrii!
we may use it to give a aeoond proof of the oommutativity o:
Cobolmbi 2 X it a commutative group
Ip fact, the morphism of X into itself mapping each el<
onto its inverse is a homomorphiam by Corollary 1. Hence, foj* •
x, yeX, {xy\~lc=x 1y~*'=*y~~1x~1, and X is oommutative,
Cobollaxy 3 Lei X be an abdian variety {with bate point i
Then on (fte category of complete varieties with bate point the fwu&\
8 i—> Hom{?, X) (where Horn denote* morphumt pretervmg iaj^j
point) it linear; that is, far 8, T in this category, rtemtfurolmop |j
Hom(S,X)x HowifT.X)—yHom(8x T X) f j
given by U,g)i—> ft, ft(»,t) =f{t) + g(t) it a bisection -*|
I
Proof. That this map is injective follows by fixing t and t it (
turn to be the base points »t and tt of 8 and T, respectively, in thj \
eqnation h(s, t)=f{t) + g{t). Next, take any ft e Horn C x T,X§\
and put /(*) = ft(«, (,), ^1) = &(**.«). *=(«. 0 = *(«. ') -M ~ #&
Then k[8 x {(„}) =*({*,] X T) = 0, and it follows from the rigiditjr
lemma that i = 0. |
APPENDIX TO J4 I
We want to prove the following result J
Thbobbm Jje(X6eoeompteewirt«(y eeXojoitrf onrf j
a morpAum *wA (Aa( m(a, e) = m(e, r) =
an abelian variety with group law m and identity e.
Paoor. We shall denote i»(x y) aimply by xy. Introduce
Xx X
AWEBRAIO 1HB0RY VIA VAIOETIEfi 45
Then ^ * («. «) = l(e- *}}¦ *° thftt by tlie dimension theorem
dim (Image W = <iimtIX-;n-8moe-I><-yiB complete, this implies
that A is surjeotive. In particular, given lei, there is an i' e X
with *'* = «. Thus,if r' = {{at,sf)eX x X|asy = e}, and pj(i - 1, 2)
ia the t* projection of X x X, pt(V) = X. Choose an irreducible
component T of V with ft{D = X. Note that dim T > dim X.
If yj =p( i r, j»^l(«) =l(e.e)}. B0 that as*1" V the dimension
theorem, dim(Imagej?j)=dimX.8inoe T ia oomplete tbia implies
that p\ is surjeotive
Let ^:TxX-*X be defined by #(*', sc). y) =x\xy). Then
^(F X {e)) ={e} bo by the rigidity lemma, ?((*', x), y) ¦=#<«,«), Jf)) -Jf,
that ia,
In particular, if (x\ x) e F, then x'{x.x') = x'. Choose an
(&', x) e T, and multiply the last equation on the left by x"
to obtain
x-{x'(xz'))=x-x
But x'tf=e, and by A), jj"(f(«'))=jai'. Therefore f (* :c)er
then not only is x'x = e, but also xx1 = e.
Let X:F xXx X^X he the map x{(a',*),s,z) ^{as1. jr)z)
Sinoe x(F x {e} X {«}) = e, by the rigidity lemma,
Multiplying on the left by x' and using A), we get
Sujce x' ia arbitrary in X(pJ being aurjeotive), this shows that
multiplication is associative. Thus X ia a group with group law tit.
In particular, for any x0 e X, the translation on—* xax is an automor-
automorphism of X aa a variety, and we deduce that X is non-singular
This also snows that f is bijective. But also the tangent map of
(Sat {e,e) is an isomorphism since tji(x,e} = (*,«) and ^(e,nr) = (a:,a)
Thus ^ cannot be inseparable, so that by Zariski'e Main Theorem,
^iaaniaomorphism. The inverse of >ji is given by {x,y)>—^ (^"'.tf).
so this shows that y 1—v y~l is also a morphism, hence X is od
abeiian variety.

46
ABELIAM VABtETTBB
ALGEBRAIC THKOKY VIA VAB1ETIES
47
5 Cobomology and base change. At this point, we have to digress |j
to prove a theorem of Grothondieok on the behavior of oohomology |;j
groups of a family of vector bundles E? on a family of varietiei f j
Xf, parametrised by points ye T where Xr is assumed to be a |
flat family of varieties. An important consequence of thia result -I'
la the aemicontinnjty of the dimonaiona of the cohomology groups 1 [
We assume the following bivsic result.
Tkbobbm ///: X-+ Y is a proper morphim of locally noetherian q
preschemes and & a coherent theaf of <St-mwhiles on X. for all p > o %\
the direct image sheaves &/+(&) are coherent sheaves of &r-tnodulet, if i
Wereoallthe following definition. If /: X-t-Y is a morphism (iff)
preschemes and W a quasLcoherent sheaf on X, &• is said to be flat ''}..
over Y or f-fiat if for eaoh x e X, ^"^foc its natural structure of }'
ff^^j-module) is 0 rj(I)-flat. It is easily shown that this condition :;:
is equivalent to requiring that for U cX, ? c Y with [tend V afflno ¦.?
open and/(U) c Vtf(U) in flat <5T(V) -module. i
¦.¦¦
Tde main result of thia section is the following i
Let f: X ¦* Y be a proper morphiam of noeHwrian
scheme* with Y — Sped A affine, and & a coherent theaf on X, fiat
over Y. There is a finite complex K': O-t-JC ->¦ K1 -+... -+K*-*$
of finitely generated projecting A-modvies and an isomorphism of
functors
xY 8peo B #" ®^ B) =r R'(K ®4 B) (y> 0)
on the eattgory of A algebras B
Proof Choose a finite affine oovenng % = {V,}^ of X by affine
open aubseta. Then the Ceoh oomplex C" = C(a, &) = © C"(S(, y)
of alternating Oech ooohains on H with coefficients in ^ is a flnita
complex of jl-fiat modules, whose oohomotogies are iaomorphio to
the oohomology groups H*{X, &)
Moreover, for all A -algebras B, {f( xy SpeoB} is an affitie cover
ng of Z XrSpecE), and CA[, $r)®ABis the module of Cech
j-cochains of & <g)A 3 for thia covering. Therefore
HF(X xr Spec Bt^%A B) ~ HT{C®A B)
for all B, and, in fact, funntorially in B
We need the following busk; lemma
Lemma 1 Let C be a complex of A-mod&les (A any noethenan
ring) saoh thai the H*(C) are finitely generated A-modules and such
thatC* i= @) only if 0 < p < n. Then there exists a /complex K" of
finitely generated A-modvles eiwh that KT ¦? @) only if 0 < p < n
attd Kr t> free if 1 < p < » and a homomorphism of complexes
^: K'-t-C" such that $ induces isomorphisms H^K") ~^»- f^lC), all\
Moreover if the & are A-fiai, thai K" will be A-fiat too
We define by descending induction on m diagrams
gm gm+l
Put K' = 0 for p > n. Suppose we have defined {K*
f > m + I such tbet the following conditions hold:
3*} for
(iii) The ^" induces isomorphisms in coiiomology fl«{iT)-i>
H'(C) for q>m + 2, and a surjeotion ker 3"+1 -> ff»+'(CT)
(iv) The if' are A -free and finitely generated, (ji > t» + 1)
We then oonstruot JC" 8™ and ^, so as to satisfy {i)-(iii) vrith m h 1
replaced by m.
Suppose first that m >0. Let 5""+l be the kernel of the homo
morphiam kerS"+>-i-ff^+'ICT). Since tf"*1 is finitely generated

48 ABEUAK VARIETIES
over A (A being noetherian), we can find a finitely generated
module Km and asurjeotion d'\ K1" ^i"+'. Purther, since fl*i
ia a finitely generated j4-module, we can find a aurjeotion
A
—> B^iO") with K*"finitely generated and free. Let |i:Jt*"
be any lift of A, and &K"*-+C" the composite of n with the|[
inolusion Z»(tT)^C". We then put K" ^K'-QK"", and define|
S": K* -*iP"+1 by putting it equal to *ero on it*" and equal to g~i
on it1". Since 4m+i«dr{K'm) c 30", we can find &: X1" -+<?¦ euoh";
that 3D&i = &,+i°31'. We then define^,,: K™ -+O* as being equal'$
to ^, on K'm and fc on ^*". The conditions (i)-{iii) are evidently |
fulfilled with m insteed of m ¦+¦ 1, ;|
Suppose then that m = — 1, that ia, that {X*, ^p, 8^ have beeo.J
defined for p > 0 satisfying (i)-{iii). We then replace ?° by*
^°/ker 9° n Ker ^», and we take ^»: K*-fC* and 3»: IP1 -*. K1 to:|
be the induced mappings.- Putting K' — 0 for p < 0, we get •?
complex >;
X"-*-...-kZ"-*0 J
Al/JEBHAIC THEOBY VIA VARIETIES
and a homomorphism ^: ^* -»¦ C which by construction induoei
isomorphisms in oohomology. We have only to check that it0 is;
4-flat when all the C are 4-flat. Consider the 'mapping cylinder1*
complex L defined by IS = KT®C*-' foryeZ, and d: I? -* &*l'i;
defined by d(x, 0) = (fe. 4(x}), 3@, y) = @, - dy). If O" is th«|
opraplei obtained from C by shifting degrees by one (and making:?
» sign change in 3), C'» = C*, we have an eiaot sequence,of-
complexes 0 -+ C -+ IS -* K' -*¦ 0, and hence an eiaot cohomologj
sequence ¦;
and one sees from the definition that the oohomology maps
#»(?')-y.ff'+'tC")afl»(CT) are the ones induced by +:XM
Since these are all isomorphisms, Br[U) = {0) for all p e Z. But
thetl
-»¦ 0 is exact and the
modules ?' are flat for t > 1, henoe K" is jl-flat.
Applying the lemma to our oa*e we have a oomplei K and a
t-C such that
Note that K" is ji-projeotive, since it is .i-flat and finitely generated
aver a noetherian A. It remaina to oheok that for all ^-algebras
B, H'{K'®JlS)^-Hr{C^>1B) is an isomorphism too Thiaisa
consequence of
Lbhka 2 LetC,K be any finite complexes of fiat A module*
and l*t C •* K' be a homomorphism of complexes inducing iaomar
flMm3 w{C) ~^y H*(Xr) for all p. Then, for every A-algebra B
the maps HT{C ®j B) ~> Hr(K <&A B) are womorpiwnw
Pnoor. Construct the 'mapping cylinder' // exactly as in the
proof of Lemma 1. Aa before, we see that X* is an exact finite
complex of flat vl-modules. Then it is easy to aee that ail the
modules Z* = Ker(L» h L»+I) are flat too henoe
0 y & >?» >¦ Z*+x >-0
is a short exact sequence of flat A modules Therefore
0 > Z»®A B >- L»®j B —*¦ Z>+1@>4 B —> 0
is exact, from which it follows that L <gj B is exact But now
L' 0j B is the mapping cylinder of the map if* @j B -+ C ®A B
So uaing the oohomology sequence in reverse, it follows that
B*[1C ®x -S) -* B?tfT ®A B) are ieomorphisms.
!Fbr any morphism f:X-+Y and y e T, we denote by X? the fiber
avery of/ (i.e., the fiber produot X XrSpeo k{g), oonaidered an
a scheme over i(t/)), and forjF quasi-coherent on X, we denote by
i(y) on Xr
We have then the following important corollary

60
VARIETIES
ALGEBRAIC THEORY VIA VARIETIES
51
Corollary. JMX Y f and & beat tn the theorem {except that T$
need not be affine). Then we have; |
(a) For each p > 0, the function Y -+E defined by i
y i—+ dinlif,) H"{Xt, &t) U upper semiamtinuou) m Y #
(b) The function Y-^Z defined by 4
f-Q
is hcaUy constant on Y
Broof. The problem being local on F, we may assume Y affina, j ¦
Let JT" bo a complex as in the proposition; by further localization,v?
¦we may assume if" to be&fiee complex. Denote by <F: JC -»-iP+!.?
the Dohoundary of K We then have j
,) = d'mw [ker |
The first term being constant on Y,(h) followBon taking alternating ;
sum of (*) over all p. We assert that for any p > 0, the function ¦
/^ (y) — dim^ [Im^gifcly))] is lower semi-continuous on Y. In:<|
fact, if f is any integer > 0, and d^'.A-'K* -+ ATKP+1 is the m»p J
induced by dP, '?
end this sot is dosed since d* is a homumorphism of free finitely •'
generated modules, and hence ia defloribed by a matrix in A, and If
the above set is the set of common leros of all entries of the matrix. |
This prove* (a). >}
HoreoTer, the theorem gives the following criterion for putting S
together the oohomotogy groups of f along the fibres of / into »}•,
veotor bundle on Y %
Corollabt 3 Let X Y f and & be as above A#mme Y t> if
reduced and connected Then for all p the follovnng are
(i) y i—> ''™«ii> ^P(-X,i &,) w a amstaitt fvnctioi
(u) Brj^) i* a locally /ree tkcaf So Y and for nil y t Y the
natural map
it on tt&morphwn
If thttf cotidxlxons are fulfilled me have further that
is on isomorphism far all y e Y
Proof. Again assume Y affine, K' aa :n the proposition.
(ii)=- @ is obvious. To prove (i) => (ii), we neeci \m> lemmas.
Lkhma 1. If Y is reduced and & a coherent sheaf on Y such that
dim^J.^®ffp*(j0] ~r all y^Y then & is locally free of rank
ranY
Pnoor. For any yeY, let u, <,T e&~f lift generators of
¦^,®fc(S). Since »,,.„, tr, are extendable to sections in a neighbor-
neighborhood of y, we have a homomorpbism (r: t>-f\r -j>#"|f defined in a
neighborhood V of y. Then a is surjeetive on the stalks at y, by
Nakayama's lemma, so coker(ir) is sera at y and hanee in a neighbor-
neighborhood of y. Thus, wo may assume a to bt surjective. Then by
assumption, for every y' e V, the map
is an isomorphism. Thus, if D is the kernel of tr we have
D,. c $!„ <fy for each y' e V. Since Y is reduced, this means that
0 = @}. Thus it ia an isomorphism
We apply this in the following
Lehma 2. Let Y be a reduced noeStenan ajjine scheme and let

ABBLIAN VARIETIES
be a fomomorphism Of eoh&rent locally fret SYshtamt. If
Aim^Imj^® i(y))] w locally constant, then there are splittings
rath (bat, A
piwm i.e.
— @)
d sove o,
Di and ,
0 0
¦ 0t w on womor-
*
Pboo», By Lemma I, D/^(^) ia looaily free. If Y = Spec (i)p|
if = r(F, #¦), tf ^V{Y, O), then this means that NftiH) ia 4-pro4
jeotive. Therefore Jf splits into the direot sum of ^{if) and a secawjlt.
sabmodule isomorphio to NjtyM). Or, in sheaves, 0:0,8H,, whenl
D, = Im(f). Moreover, this shows that $Jf} is J-pTOJective, too.S
SO if tptits into the direct mm of Ker{^) and a second submodnlit^
isomorphioto^(if). Or, in sheaves,^ =#r,©#"t> where ^{#"j)^|
I
Now assume (i) holds. let ? be the oomplex given by the*
theorem. As in the proof of Corollary t, dim{Tm((?'~> 0 hit/))] anil
dim[Im(ii* ® i(ff))] ore locally oonstant. By Lemma 2, app!i«)#
first to (^:?" -*K*+\ and deoond to d^_,: K*~l !
we get splittings into protective modnles:
II
K, ,
where V.-^-i), Vi^-i"* *»' is " {|
i^ffl H, =Ker(d,), and &,•.&',-+Br+l is an inomorphiam. B?
followe immediately that
and ?P-'(r ®A B)
This proves (ii).
ALGEBRAIC THEORY VIA VARIETIES
3. Let X, Y, f and & be as above (mtlike Corollary
2 T w^ *°' ^e reduced). Assume jar some p that HT{XI/, ^t) — @)
all y e Y. Then the natural map
,'j o» MomorpSwm for ally
pBOOV. Again assume Y =Spna[A) K us in the theorem
for My 6 Y we know that
d" <P
K* >®%) »- JSf» <g> Jfc(jr> »- JC* > ® %)
is sxaot. Split tiie vector space KT @ l(y) into iFt 9 W,, where
Jpj = Image of JT*® fcftri, and Wt is mapped injectively to
K**l® 4(s}- To prove the wrollary at y, we can replace A by any
localisation As, (J e A, f(y) =? 0). If we do tltis for a suitable/, we
may assume that K* itaelf splits into a direct sum of free modules
Wx®Wt such that (a) i?,= IP(®%}, and (b) td c Im{iP-»)
To do this, just lift a basis of W1 to any elements in the image of
i?, and lift a baeia of Wt arbitrarily. But then since W,® i(jf)
-*K**l®l:(y) is injective, it follows that Wt -*-.JP+1 is also
injeotive if A is replaced again by a suitable localization At. But
then Im(?p-») n Wl = @), hence Wl — ImfiC). Since ff, is a
prujeotive module the surjeution ^T* -»¦ IF, -* 0 splits, and
)© T^!- It follows that we have eiaot sequences
K*
Therefore J3*-'(j;,^) e ^"'(Z, ^)® %) as required
Cobollaby 4, Le( J, 7, tmdf be aa above. If B*/»(^) = @)
0, torn H^X,,?,) = (Q) for all y e Y, and for k> ka.
Use Corollary 3 and decreasing mduotion on fe(.
6 LttX Y f and^bt. at above Then if Bis a fiat

ABBLIAH VARIETIES
tP(X xTSpeo B,&®AB) a
JS.
Pboof. This follows immediately from the fact that for B&tf
over A, and any complex JC, :|
Cobollaby 6, (Seesaw Theorem—provisional form). ag
complete variety, T any variety and L a line bundle on X X T ThtH%
|
the set
|
31,
y(t(istnvialon X X {(}}
« closed in T and if on X X Tltpt: X x Tl-+T1i»
then Lixx^zplM for some lint bundle M on Tt
Pboof. We first make the remark that a line bundle M on ii
complete variety X is trivial if and only if dim H"[X, M) > 0 «iii™
dim fl%T, If"') > 0 where M denotes the sheaf of seatkms o$
M. In faot, the necessity of these conditions is clear. 9nppoa«f;
conversely that they bold. The first implies the existence of t|
& '¦!¦
non-zero homomorphiim ffI h M, and the eeoond implies *f
non-zero homomorphiBm t>x-*-M~l, henoeon dualizing anon-zero;
I
homomorphism M
(Px. Hence t(o(l)) is a non-aero
of ffijf, and sinoe X is complete and connected, -r(a{l)) is a non-zeni|,
Boalar. This implies that rn is an iBomorphism, hence tf and t.|
are iaomorphtBnu.
It follows that 3\ is the Bet of points (of Tench that dim B\X X {i
L\Xx{i})>0and dim H<>{X x {t), L~11X X{*}) > 0, and it folio-
from Corollary 1 that I", ie closed. Replacing T by Tt (so T is no^
merely a reduced scheme of finite type over h) and L by its rest
to X x Tx, we may assume that L | X X {(} ib trivial for eaoh ( e 31,
Henoe dimH*{X X {!}, L\X X\ {(}) = I for all * 6 T, bo that hy|
Oorollary 2, p,* (?) = M is an invertible sheaf on T and
—
ALGEBRAIC THEORU VIA VARIETIEB SO
. ftn isoniorpbism. It clearly follows from the triviality of
j, IX X {') that the natural map p\{M) -*¦ L is an isomorphism.
Since H is the sheaf of sections of M, then p%M = L.
0 The tbconm of the cube: 1
ThkobKH. i*f X, 7 Ae eom'pUU variettea Z any variety and x., y,
osJ /„ **"* potnfo o« X, Y, and Z, respectively. If Lie any line bundU
o* X X 7 X Z whose restrictions to each of {xa} xYxZ, .
L m trivial.
B*maek. Let T be a oontravariant functor on the category of
complete varieties into the category Ab of abelian groups. Lei
.i^«i)eail)rByal*mof complete varieties, zf a base point of
X(, »nd let jtjI I0!i...xJ,-tIlx...xi,K...Xi,(I| indicating
the omission of the i-th factor X,) be the projection map, and
b,: J,X...x^(X...xXB->-X0 X...XX. the 'inclusion' defined by
Consider the bomomorphisms
4: Y[ T(X<, X X X, X
XX.)
defined by
One then proves by an easy induction on n that we have a natural
splitting r(Xox...xXJ = ImaffiKer 0. The functor T its said to
btolorder n (linear it n^l, quadratic if n=2, etc.) if a issurjeotive,
or equivalently ft is injeotive. (Note that the definition of a ii
independent of base points.)
Thus, the above theorem (when Z is also assumed complete)
ffl»j> be paraphrased an saying that the functor Pic J is a quadratic
funotot on the category of ooraplete varieties

ABELIAK VARIETIES
ALJEBBAIC THEORY VIA VARIETIES
57
Now if 2 A < » < 3) are ©ontravanant functors on compjs
f
varieties into Ab and Tt
y Tt and Tt
/ 9
g
Ti are nBh
transformationa auch that Tx —^->- T% —1-y T^ is an exact i
ence, and if 21, and T, are of order n, ho ia Ta, aa follows fron\|
esaotness of
0=Ker18Ji(X1),...,Xn)-».Ker^,i(Xu -XJ-*¦ Ker ^Jt(X»,
Thus we get a proof of the theorem of the cube when the baas f
is C by observing that we have an exact sequence
IP(X, 0)-+J5P(X, ®*)^>H\X, Z)
fuj ctorial in X and Hl(X, ff) is linear (hence quadratic)
Ha{X, Z) is quadratic in X, by Runneth forniuks.
Pboof of tub Theobbm {following Weil and Murre). Byl
'Seeaaw theorem', it is sufficient to prove that for every {x,i
the restriction of L to {x} x Y x {z} ifl trivial, since it is
given that L restricted to X x {ya} x Z is trivial. Tho foil
enables us to reduce tho proof of the theorem to the case when,
a complete non-singular curve.
Lbmha. Let X be any variety and ^ijt
an irreducible curve C on X containing xu atid x
Then
Pboof. We assume that dim X > 1. By the lemma of Chow,|
may assume X projectile. Moreover, by induction on dim X, i|
sufficient to find a sub variety Y of eodimension one b X oonteit|
Xq and zv We can find an X' > X birational, with X'
and dlaif-'fa) > 1. (In fant, if A ia any meromorphio fui
on X with indeterminaciea at xa and «,, X' can be taken to
closure of the graph of A in X X P1). IfX' cP" ia an imi
thereisahyperpldnotf of P*suoh that H n X' = Y' is hrredi
by a theorem of Bertini, and lTn/~l(^) j= 0 since dim
> 1. Then Y = /(Y') is irreducible in X and contains % and
and the lemma ia proved
Resuming the proof of Liie theorem, wo can find for any ,t e 1 an
irreducible complete curve Ct in X joining xotax. Let it: C -*¦ Ct
\a the normalization of 0, and*r':Gx YxZ-+Xx 7 X Z the induced
Tho hypotheses of the theorem are clearly fulfilled for the
bundle »'*(?) onC X Y A Z (with X replaced by C and xt by any
uojnt of Cying over ^5), and it is sufficient to prove the triviality
of this bundle, aince it woulii then follow that h restricted to
filx Yxfe)'" *r'v'a' ^or anY ^ e X and 2 e Z
Tim*, we assiimo X to bo a complete non-singular curve, and it is
even sufficient to show the existence of a non-void open subset
Z'of Z »\ioli that L rostrioted to XxYxZ' ia trivial, since we
TOulrl then have proved the triviality uf L |Xx ^xfz} for 2 E Z'
and it tvouhl foilow by continuity that this holds for all z e Z
Ltst Ql be the aheaf of regular 1-forms on X and let g~^
Jim Ma(X, O1) be the getiuu of X. We can clearly find g points
j'J,,..,ZI(onX3ucht:liatifZ)=i;Pi, dim H"(X, ?11®0X(~D))=Q.
Denoting by J), the first project inn X X Y x 2 -> X, let L' ho the
lioe bundle L' = L ® p*(Lx{D)) (whoi-e LX{D) is tho Ime bundle
aasfldaterl to ^X(I>}) on X X Y X Z, and for any (y, t) e Y X Z, let
?'(,,¦>^etneraa'riot'otl°f L' taX x fa) X {2}. Since L'^^ =LX{D)
ve iiovo dimH>(X, L1^) = dim H"(X, IV ® <9X(- D)) = 0 by
Kjanmnn-Booh so that the cloeed set F = [(y,zN Y X Z
JimH'{X, ?'(„_,>} > l}of Y x Z does not encounter Yx {zt}. But Y
being complete, we can find Z' open in Z and containing z0 suoh
thsjt YxZ'nF = C, ho that by restricting ourselves to Z', we
m«y assume Hl{X, i'(ftl)) — 0 for all (ji 2) 6 Yx Z But this means
lint for all (j;,3)e
IaTiewofCorollary2ioth6propoaitionifji1j :X X Yx Z^t-Y xZ
istha projection, ja,3^(L') is an invertible ahctaf on Y x Z of rank
one and for any fe.i), the natural miip j)ta4, (?')(g t(jf,z) -+
fl'iX,!/^) is an isomorphism. Let U be any open subset of Y x Z
on which jJ,,*(L') is trivial, and <tv el\U,Vu*(M)} = '

58
ABELIAK VABIFTIES
generating section. Let j)a be the divisor of zeros of av in p-^
Since for U,U' open in Y X Z, <rv and cr^. differon Vn W |
nowhere vanishing function, we have Dv n p'^{U n W) =
P~u{U n E7'), bo that we have an effeotive divisor B on X x 7 j|
such that D\p~aW) = IV For each (jw) e 7 X 2, the restri
of D to Jf x{y} x{*} is the divisor of zeros of a non-aero section
fl°(i'w,rt)- I" particular, 5 restricted to Ixjj) X {z,} and Xx(|
x {z} for any y e Y, z 6 Z must coincide with Z) = S Pj. Henos^j
PeX, P=^ Pt{i = t,...,g), the restriction of 5 to {P} x 7xZ
a Bopport 3 not meeting (fjxTx {io} or {P} x {sf0} x Z.
projection T ot 8 on Z is therefore a proper closed subset of
and since S is pure of oodirueneion one in {P} xY xZ,8 must;
of the form U {P)xYx Tt, Tt closed *nd of oodiuienBion 1 i^
t-l !
But BinceiSn {P} xfeo) X2=0, it follows that S~ 0, that iM
support of jD does not meet (P)x7xZ for P e X, P ?<]
t
HenoeZ>mu8t be of the form EndWJxrxZ), and restrio
toXx{y0}x{2o}, we aee that J3=S({J*,}x FxZ) Hence for
(y,z) e 7x Z, L'^ is the line bundle LX{D) and therefore|
restricted to Zxj})x[«] is trivial
Cokollaby 1 With X, T and Z as in tht proposition,
line bundle on, XxYxZis isomorphic to p*%(L) ® p*
where ptj w the projection ofX xYxZ onto iht product oft
andja factors, and L,M,P are line bundles on XxY
YxZ nspeclively.
Proof, This is a eonsequenoe of the remark preceding the pr|
of the theorem
%
Cobollaby 2 Let X be any variety, Y an abeltan variety, m
fff,h:X->Y7norp>»sm3.ThenforaULe1?io{Y) we&ree
ALOBBBAia THEORY VIA VARIETIES
h*L '.
the projection onto the i*
Yx Y -* Y and m =
where 0, $i,?i>n: Y X F ->¦ T are the 0 map, the projections, and
addition. Therefore q*M h trivial. By symmetry, M is trivial on
yx@)xr and Yx Fx(O) too. By the theorem,3f must be trivial
on y X 7 x r. Palling back if by the map \f,g,h): X-*-YxYxY,
the result follows.
Ooboliabt 3. I/X *s an abelian variety and tt 6 Z ihtn Jot aU
tint bundle) L
\X)*L
Proof. By Corollary 2with/ = (n
it follows that the "second difference",
—1^ and A-(-
1X)*L
hsnce for some line bundles -if, Mt, we must have
»<¦¦-»
Putting » = 0 showB that Mt ifl trivial and putting ft—1 showe
¦ * T
Jlj _ lit
Oobollaxy 4 (Thoorem of the square ) For all Ime bundles L
T*^, L® L ^ T*L® T\L

00 ABELIAN VARIETIES
Therefore tf ^(a;) = isom class of T*L® L ' t» P e(X)
a homomorphism from KtoYic(X),
Pboop. Apply Corollary 2 with X = Y,f and g constant mapi
with images x, y respectively, and A = identity. j
In terms of divisors Corollary 4 asserts that for any divisor fli
onX, andx,y eX, jj
ALOEBBAIC THEORY VIA VARIETEM
01
2»+1,JD
-f
(where » means linear equivalence),
In the rest of this book, we will always keep the notation ^ fbt
this very important map. Note that _
(*) ¦&,,«?, = 4>&1 + $l, ( + standing for the group law induced
J
K(L) = Ker(^) - {xeX | T*L = L).
Profohitiom K{L) i» a Zaristi-ctosed subgroup of X.
Pboof. Apply the Seesaw Theorem to the line bundles
m*L®p*L-la» lxl(m: XxX -*X being addition). It follows;;
that th&aet of igXsuch that«»*Z®yJi/~I is trivial on {x}xXl
is Zariski closed. But m*L^tp*L-1 |h xZ = T*L® Lr\ so thi«|
set is K(L).
Appltoation i. Let D be an effective divisor on an abelian-5
variety X and L = L(D) the associated line bundle. The following i
conditions are equivalent.
(i) The subgroup H —{xeX{T*{D) — ty of Z ia unite (equality 1
of divisors, not divisor classes).
(ii) K(L) is finite.
(iii) The linear system \2D\ has no base points and defines •
a finite morphiam X -+P*.
{ v) La ample on X
Proof. The implication (iii) a-(iv) is a general faot {BOA Oh. Ill
B,fl.l)or D.4-2)), We show next that (iv)=> (it). If K(L) is not
flnite, let Y be the connected oomponent of 0 of X{L), so that Y is
an abelian vaiie-y of positive dimension, and the restriction Lf of
ito Tis ample on Y. Further, ^(Z,,.):^,. for ally er.Hence.by
the Seesaw theorem ifm: Yx Y-+ Y is the addition andp,: YxY
+ r the projections, the line bundle m*{Lr) ®pt1ALjl)®ptfLLjl)
ja trivial on YxY Pulling back by the morpliism Y -+Y x Y,y
>-* is, ~y) P™ i8 W«»t ir@( - lF)*(ir) is trivial on Y, But Lr
is ample, and eo is ( — lj-l'fjjy) sinoe ~ lr is an automorphism of
f, so that Ly® ( — irJ'JLr) is again ample. This ia a contradiction
line* dim Y > 0, whioh proves that (iv) => (ii). The implication
(ii) =» (i) is trivial, since K{L)i H
Wa now show that (i) a. (iii). The linear system | ZD contams
the divisors THD) + Ttt{D), by Corollary 4, For any ueX, we
oan find an » 6 J such that Urtajlt Supp D, and this means that u^
Supp (T*W[D) + Ttt{D)). Thus, the linear system 12D | has no base
points, and defines a morphism j,: X -^P* If </, is not a finite
morphism, we oan find an irredaoibie cnrveCsueh that ^(G) =one
point. It follows that for all ? e 12D |, either B contains C or is
disjoint from C. In particular, for almost all z 6 X, G and T*{D) +
T*,{D) are disjoint. Now note the general fact
Limu tfCita eurvt onXandBUan irreducible divisor on X
mchthatCnB=0 then B in invariant under translation by 3^ xt
Pboot. If L — HE), then L w trivial on C since C and B are
disjoint. Therefore, f%L, reatrioted to C, has degree 0 for all
xeX. Bnt then 5^@) and B oan never intersect in a non-empty
finite set of points, since this would imply that T*{L) \0 had positive
degree; i.e. for all s, either Tm(G) and B are disjoint, or Tt(C) c B.
^teO.yeB. Then 3*^^@) and B meet at y. Therefore
c B, hence [f - k, -(-1,6 5. This proves the lemma.

62
ABBLIAK VARIETIES
If D = S«(D,, D, irreducible, then by the lemma, each i)( j,:
invariant under translation by all points xt — iE, x( e 0. This a)aM
tradicta (tj, hence we have proved that (i) => (iii). a:
This enables us to show trivially that an abelian variety X jjj|
projective. In fact, let U beany affine open subset oFX, D1 j)-J:
the components of X — V and Dtiie divisor D = S Bj- We williS
show that X* verifies (i) above. We may assume after a translatiojjf
that 0 e V, Then if = {xeX\ T?{D) =D} is a closed subgroup,!;
andfonefl, V is stable for T«. Since 0 6 U, it follows that R c V?:
and B being complete and U affine, B ia finite. IP:
Appuoatioi* 2. An abelian variety
for all n> 1, XB is finite.
is a divisible group,
Considering the homomorphiam nz: X-*X, it is olear tl
dim (ker n^) > 0 if and only if dim (Im(nz)) < dim J. Hence
prove »x nirjeotive, it suffices to check that Xn ~ ker(nz)
finite. But let ? be an ample line bundle on X Then
njis
alao amp]ij|
Since (— Ij) is an autoraorpliism of X,
and since }n(n + 1)> 0, i«(n - 1) |
ample. But then n*L cannot be trivial on any positive dimenBionil
snbvariety. Sinoe iji]IWIlJ) is trivial, kerfn^) must be finite,
0, we aee that n*L \a iJb||
3. We oan go even further and compute the onJi
of X,, when the aharaoteristia p of k does not divide n. We fiis
reoall some general facts. Xiet X and ? be complete varieties boti
of dimension n, and let /: X -*¦ F be a surjeotive morphism, Ttn
via /*, i(X) is a finite algebraic extension of ib( Y), and we dsfil
the degree, d (reap, aepomile ckgrM, tnjejwraiie degree) of / to
the degree [t(X)r A{7)] of thia extenaion (resp. [fc{
[fc{X): *{Ir)]*)- If/ 's separable, i,o. i(X) is separable over
then d is the oardinality of/"'(y) for almost all y e T.
inseparable the separable degree of k{X) over k(Y) instead io thi
ALOBBRAI0 THEORY VIA VARIETIES 63
oardinality of f~1iy) for almost all y. Moreover, a basic fact is
that if ^i> ¦¦¦ i Ai416 Cartier divisora on T, then we get the relation
between intersection numbers:
ffow suppose X and T are abelian varietiea. A homomoTphism
f: X ->¦ 7 is called an tK>p«iy if it is surjective, with finite kernel.
We have juat seen that nx: X-> X is an isogeny. Then every
isogony/ has a degree d, and since the cardinality of the kerne) of/,
#[ker/J, is the cardinality oij~l(y) for all y e T, we see that
# [ker/] = separable degree (/). (•)
Now take the case / = nz. Let D be an ample, symmetric divisor
(i.e. (— Ij)* D =D) on X; we have seen that ample D'b exist,
and then D + {— lz)*D ia both ample and aymmetrio. Then by
Corollary 3, n^D is linearly equivalent to ti'D Therefore, if
f = dim X,
degree
{D.... D)x =
hence degree (n,) =»¦
When is nt separable t Itpjfn, then by the result above, pjf degree
(»j) so t>x muat be separable. On the other hand, if p | n, then
ire saw in § ¦& that the differential d(n^) mapping Tj,, to Tf „ is 0.
Therefore, if a> is any invariant differential form on X, «J(ai) i*
(a) atill translation invariant, and (b) has value 0 in the cotangent
space to X at 0, henoe it is zero. Since the invariant differentials on
X generate the sheaf Qj over t>z, they generate the Jt(X)-moduIe
of ?(X)/it-differentills. Therefore we find that the induced map
nz on rational differentials Qt<z^ is 0 if j) | ». This implies that
the induced map on fc(X) mapB ?(X) into i(X)* and hence the
inseparable degree of pz in at least ]f.
It follows that if pjfn, § (Xn) = n". Thus XH is a finite abelian
group killed by n snoh that for all nt | m, X, contains exactly nt"

04 ABKLIAK VARIETIES
elements of order dividing m. It is elementary group theory that
the only suoh group is (Z/nZ)*. On the other hand, X, is annihilated
by p and is of order equal to the separable degree of pz, which
is jj'for aomeV with 0< t<g, so X, s (Z/pZ)'. Since X is divisible,
it follows by induction on m tbat for any m > 1,.
Summarizing, we have proved
Pbombition A) degree {nz) = n**
B) nz ttfxtrable -=*¦ p Jf n,
D) There is an integer i with 0< i< g such that for all m > 1
APPENDIX TO §6
We give an alternative proof of the foot tbat deg n^ = i»* for
«> 1, avoiding intersection theory
For an tnverlible sheaf i. on a complete variety X of dimension g,
and any coherent sheaf P on X, P^.(») =j((#'®?'1) is a polynomial
in ttof degree < g. We shall denote the coefficient of »* in this
polynomisJ by dL(&)jg\. so that d^t^") is an integer > 0. We call
^[9^1 the degree of L and denote it by degi The basio result ia
the following
A) Let X bt, a complete variety ut'tt on
' L. /or any wAflrenf liai/^mt X, M rank (#") &e tte dimen*ioii
twer the function field of X of the generic stalk of & or eqvivaUrtttp,
of the space of rational sections of '&'. Then we have
di(#-)=r»nk(J«-) degi
B) Let f: Y -* X be a swjective morpUsn of oompltte uoriettei of
the some, dimension g, and L an invertibU sheaf on X. Then
deg/*(L)=(deg/).(degZ)
Proof. A) It is a standard fact that we can find a nan-zero
coherent sheaf of ideals J and an eiaot sequenoe
ALGEBRAIC THEORY VIA VARIETIES
6fi
a torsion sheaf. Sinoe $~ has support of dimension
< dimX, x^SL") is a polynomial of degree < dim X, and the
»dditivity of gives us tbat
= dL {-/"•" ) = {rank*-). d^fS)
and using the exaot sequence 0 -±J -> G)z ¦+ ®i
that dk(J) = di(tf)=.degi.
0 we see
B) For each p > 0, we have a oanomeal womorphiam
iPp/,(/*(i")) = JP/.tff,}®^11. Takbg alternating aums in the
Leray spectral sequence H>(X, JP/*{/¦(?"}))• B*{Y,!*{??)) we
obtain that
%
Since there is a non-void open subset V of X such that
flt'(.U): f-^Uj-^V in a. Unite morphism, Jfft(ST) have supports
in a proper closed subset of X for p > 0. Further, JP/»(<Pr) is
dearly a coherent sheaf of rank = deg/. The assertion now
fallows on comparing coefficients of «' on both sides.
Now let X be an abelian variety, n a positive integer. Let L be an
ample symmetric line bundle onX; for any ample L, L®( — \X)*L
is both ample and symmetric, Then by Corollary 3, n?? ie iso-
morphie to ?"', Therefore, if g = dim X we get by the above
proposition that
deg nx. deg L = deg nj(i) = deg Lf
and amoe deg ? > 0 we get that deg nx — n**
degL
7. Dividing varieties by finite groups. Let /:X-* Y be a mor
phiam of algebroio varieties (over an algebraioaJIy closed field k)
I is said to be ttale if

06
ABBLIAlf VAJVEETIBS
(i) /is flat
(ii) for all jel.j"f(x) e T, if m, and m, are the maximal
ideals in <?„ and 8t, then /*(mr)Sp, =%
This is equivalent to assuming that / id a "formal isomorphism"
in the sense
(i)' for all xeX, u =/(*) ef, if ff,, S, are the completions
of 0m, Sfi then the natural map
h »,-•¦•.
ia an isomorphism
(Of. Mumford, /ntro Jiff, atom., p.353). When fc = C, it is also
equivalent to assuming that/is a local isomorphism of analytic
spaces. Oaf main result is
Let Xbt-cm algtbrme vartety, and Q a finite grovp 0/
y of X. Suppose that for any x a X, tit orbit Ot of x
w eontatned tn on ajpn< ape* «u6«d 0/ X. TAen (We m a potr
G w), «Ae« ri* a variety anJ itiX -±Y a morphitm, satisfying Qtt
follotoing conditions
(ii) if vt[$xf dtnatu iht mbeheaf of Q-invoriunls of
for the action of 0 on n-,!^) dedueni /mm (i). ife natitral
hamomorphitm <PT -nr+{Gx)° i* cm womorpftwm.
The fxur (T, it) m determined up to an womorphsm. by (hut
conditions. Tkt morphum -n %» finite ev.fj<xtive and ttparabU. Y it
If further G aefa freely on X {that it, if gx ^ x for any xeX and
anyg e0vnthg&t),vitem itaU worpAwm f
Pboo*. Sinoe the conditions (i) and (ii) determine the topology |
and structure sheaf of Y, the uniqueness assertion is trivial. Also, |
the problem of existence reduces to proving that if Y is the quotient ~J
XjG as a topologies 1 space, and U given the structure iheaf 8T => %
°, it is an algebraic variety. Suppose we knew the theorem
ALOBBRAIC rHBORY VIA VARIETIBB 07
(in Its entirety) to be valid when X is affine, For any xeX, let U
he an aflineopen subset of X containing Qx. Then V =fl (ft/'is an
affine open subset of X containing x and stable for Q. Thus X is
covered by 0-stable affine open sets V. Then each ir(U) is open in Y,
aadfr^'M^))"^' ao by the affine case of the theorem w{U), with
the restriction of <?r, is an affine variety, Bnt the open subsets
„{[/) cover Y, so the theorem would follow for X.
We may therefore assume X = &poc(A). Let A = fc[x, asp]
Then ff ftots on ^l by tbelaw ?(/)(«) ~f{g~'z), geQ,feA,xeX.
jjet v = order of 0. Fot/ g^ and 1 < k < v, denote by trrfj) the
elementary Bymmetrio function of degree k in {(F(/)}^a, and put
ff ^itroilajl^c^ptsothat B' is a finitely generated i-algebra, oon
turned in the algebra B = Aa of 0 inrariants of A But the a^ are
integral over S sinoe they satisfy the equation
if -OtixJX-1 + ... + (- l)'ff,(a,) - 0
«o Ji»» finite B1 -module. Sinoe B"cBcA andiS1 is noetherian, JB
is a finite JS' -module too and hence a finitely generated Jt-algebra
uidjiigafinite J3-module. If Y = Spec B, then Yis a variety and
we get a morphism it : X~* Y corresponding to the inclusion Be A,
and this morphism is finite and surjeotive. Next if R m the quotient
field of A, the action of O on A extends uniquely to an action on S.
hsnoea II ?D) e A", so that JS" is the quotient field of A" — JB
This proves that it ie a Galore extension of the quotient field
of S. In particular, jr is separable. Next, note that -rrt[0z)G
is 1 coherent sheaf on T since it is the intersection of kernels of
The natural homomorphism @r ~* '"*(^x)a induces an isomor
phiem of global sections, so it is an isomorphism. Next \ix,x' e X
have distinct orbits Gz ^ Ox' under Q, we can find an/ 6 A with

(S3
ABKI.TAK V^RIETILS
0 lor nil g e 0 ami If ^ —
/(gr) - 1 for nil u (. 0 /(91)
^ciJ siml ^M*)} -= 1, ^M*')) = 0, This shows that tt(x) #.
Tims "s ft set, r in tlui quotient of X by G. But it: X -+ y is a
ami !ien«3 n cloaed and continuous map, so F has the
topology too.
Only the last assort ion remains to be checked. Leta: e X, y =
and 3)! the maxinml iiietil of y in B. Considering A as a B-
dfiiiiLLtitype, lot BanAA be the completions of Band A respeiitiveli
for the 3J)-adi(s topology. Then, the natural homonvirphisrn B%
-* A U an isomorphism. Muititiver, ii is also the complol ion <$y
®?,r w't'1 reapcot In its maximal ideal. On the other luiiwi,
the only prinio tdeiils t.f A containing SRvl ar« the maximal idei
of tlio jjoitils ff(^}, ff a O, we have by the Chinese renmiinitor
a jiaturf.l isoniorphism
srliore ®Xfs is the completion of OXfI with reajiont to its inaximi!
ideal. The group 0 ants on A, and lienco on t-he two other rir^j
ocuutiing rubovy. On Ji(&BA, tlio action is given by
= fi®ft(«) inr h a G,br,B. a <s A. The faet that B ia the
G-itivaviiints in ^ otin be expressed by tlie exactness of
sequence of iJ-modulns
6®a M-..&® (*(*)-»).••¦) I
is exact, hance B is the subring of Q-invftriants in B ® aA sij|
On the other hand, the action of 1 e G induces an isomorphism!
SitiCii) Sis a flfit B module, it follows that
0 —*? —
ALGEBRAIC TffBOBY VIA VARIETIES 69
S^t- 0ZM> ttnd lf we identify II ffXi#I with n *x.t b7 m^ans of
these Isomorphisms, the notion of G on this ring may be described
w simply permuting the factors, i.&-M{Xf})f,{,={<ta~i,)}pQ, for any
1 g 0 and (fjjpo 6 II ^j .• Thus the invariants for this aotion can
[js iiloiitiflett with CJpJ, lor its diagonal immersion in II #*.*¦ We
thus deduce that the natural homo morphi sin Gy
, i.e. /is ^tale at x
BX:t is an iso
The condition that any G-orbit in X be contunet) in an
afftne open subset ia always verified when X is quasi-projactive.
Infant, if J[ is B locally closed aubsot of P* and if Xia its closure
inP* »"d xt(\ < i< »)iaaiiy finite set of points of X, we can always
find a hypersurfnco S in P* containing X — X but not any of the
jj. Th8nX~(Xni9) —X-(Xniff) isaffino and open in X and
tout tuna all the x--
When X and O are as above and (Y, ¦n) is the pair given by the
theorem, Y is called the gvoiUnt of X by G, mid is donotod by XfG,
Now let G act on X and let (J*,») be the quotient, und let &
be a coherent sheaf on Y. Since for any g efl, we have the eoinmitta
tive triangle
natural autoroorphiimj*: »*(#¦)-
over the action of j on X. Thus, 0 acts on **{&) in a manner
compatible with its action on X. By a coherent O-»htaf on X,
we ahull mean a coherent fj-module on which Q acts in n way
compatible with its action on X

70
ABELIAN VARIETIES
Proposition 2. Let 0 aet freely on X, and Y=XIO. TJlen
functor f\—t-ir'iP') is an equivalence between the category of c6,
Oy-modvkt and that of coherent Q-sheaves on X, whose tneer,«.|-
given by g ¦* w,^H. Locally fret sheaves correspond U> loeoiij
sheaves of the same rank.
Pboof. There are natural homomorphisms
*" ->¦ ^{w'&f, P a sheaf on Y
)¦ w*fa,(g)°> -+ 9, g a G-sheaf on X %
We will show that S and T are isomorphisms. We can again »btob|
X and r&reaffine. LetX = Spec A, Y = Spec B, where .8 = 4^
We must show that the natural maps $
S(Jtf): M ¦* (M®BA)°, M a ^-module, f
T(N): IN") ®BA-+N, N a G-ji-module |
are isomorphisms. But for all ,6-modules M the composition j
is the identity. Since A ie faithfully flat over B, 8(At) ® 1^ is «
isomorphism if and only if 8(M) is an isomorphism; therefore |t
will suffice to prove that all the T'a are isomorphisms. S
Now in the oase in which A is iaomorphic as a ring to Sx ... xS:
and in which 0 acts on Bx...xB by a simply transitive group of
permutations, it is quite obvious that T{N) is an isomorphiij
for every G-ji-module N. On the other hand, we oan reduce thj
proposition to this oase by taking completions. Letx ej, y =Ji0
and B = Sr>r. To show that T(N) U an isomorphism, it will snffiol; j
to show that
f
is an isomorphism for every yeT. But since B is flat over 3, tit);
module of O-invariants in N(&BB equals {N°)®BB (of. proof dj-
theorem) hence we get a diagram
ALOBBKAIC THEOfcY T1A VABLBTTES
71
T(N®BB)
henoe to
x JB
Since A®BB is isomorphio to fl
f(^®s^) 's Bn isomorphism
We want to study in more detail the oase when X is a oomplete
variety and 0 aots freely on X. We shall denote by G the group
Hom((?, i*) of fe*-valued characters of 0
Pboposition 3. In the above case, for all characters i G -*¦ k* let
iB={«6«*(<P,) Ig(a) = «(jr).a, allp e0}.
Then L^ is an invertibk sheaf on T, and the, multiplication tnirt[0z)
induces an isomorphism ,^.0 j^-+i^+^. TAe (wsijfnmen* on—>¦ L.
an isomorphism
P»oo». By Proposition 2, Ker (Pic Y ->¦ Pic Z] can be identified
as a set with actions of 0 on the trivial sheaf §x covering the action
ofQonX. Given any suoh action, the image of the unit section by
j e G is a nowhere vanishing section of 0x, and since X h» oompieta,
»non-ieroscalara"I({f}6i*;oleaily ecG-s-i* is a bomomorphism.
Conversely, given any suoh homomorpliism, we can define an action
a{Ooaffx ooveriug the action on the base by g(f) — oi^), (/oj).
Thus, as a set, we have a Injection G —*¦ Ker [Pic 7-J-Pio X] It
is easy to see that this is a group homomorphism
Given an action a of G on <9X corresponding to a character e, if we
denote the natural action of Gonw,((Pj) by (g,f)i—t-g{J) ==/aj-',
the action of Q on ir, (<PX) induced by thB action a is described by

ABBIJAN VARIETIES
72
Sinoe the corresponding invertible sheaf ?a is the set of invariant^
of ir$(<Pz) for this action, we get (hat
La^{aent(Ox)\g{a)~als)a}
Considered as subs heaves of it, D>x), we have evidently h^.L^cL -S.
On the other hand, since a nowhere zero section on an open imff:
of a line bundle on Y induces a nowhere zero section of the induced!
bundle on the inverse image of this open set, vie, aee that any genent-
ting section fe ?„(#) c r{tr~'{U), Sz) admits an inverse /-' jjg
v^{e>j)(V), which clearly proves that ?a ® L^-y LI+t ia 6urjeotive;J
Since both sides are invert]ble sheaves, tills is an isomorphism?
S. A) Suppose 0 is of order prime to the
teristio. Since the representations of 0 in all Ujo i:-vector
it,{@x)(Y) are completely reducible for every open V in ?, it ii;
easy to check that
whets the representation of 0 in all (lie vector apacea ^(l') eou-
tains no l-dimenaionalsubrcpresentation. If Ois aluo commutative
then we have simply ¦
mod lies
Since »»ir*^
proves alao v
Lobollaby If Q ha& order prime to the characterirtia, then for «jjT
coherent <Sr moduks^ & U a direct eummandof1«,(w+Jr). ;
Let us apply our results to abelian varietiea. The uuii
oonsequenoe is something like the fundamental theorem of
Galois theory. :
Tkbobkm i Lei X be an abeltan variety. Then there u a 14
correspondence between the two sets of objects:
73
Y1
+ Yt
ALGEBRAIC THEOBY VIA VARIETIES
tp) finite subgroups KnX
jh) separable Uogenus f:X -*¦ Y, uAere (we tsogente* /,:X
.v -r Y% art e<maid&red tquoX if there is an isomorphism A: Y
$h (JW /» = *°/ii fitcA « set up by K — ker (/),,
Phoof. Fnat start with a finite subgroup KcX. Then X acta
freely on X by tranalations, so we can form the quotient {XjK,f)
and / is an Male surjeotive finite morphism X -+ X(K. On the
other hand, XjK as a set U the quotient of the abstraot group X
by the subgroup K, hence it has a group etruoture. The group law
U in fact a roorphiBm. This follows by considering the diagram
fxf
XjK x X\K -
XjK
where m is the group law of X (» morphism) and n is the group
law of X/K {so far, juat a map). But it ia easy to check that
IjKxXjK ~ X xXjK xK, and since the morphism f°m:
X x X -s- XjK collapses the action of K x K, it faotora through
X x X)K x K i.e. « is also a morphism. Similarly, it oan be
uheoked that the inverse map on XfK is a morphium. Therefore
XjK is an algebraio group, finally, XjK is the image of ft
oompleto variety and therefore is complete. Thus XjK is an
abslian variety, and/;Z ¦+¦ XjK is » separable isogeny. Clearly the
kernel of/ is K.
Seoond, start with a separable isogeny f:X -*¦ Y. Let K be its
kernel, and as above form a new separable isogeny g: X ¦+ XjK
A moTphiam h in the diagram:

74
ALGEBRAIC THEORY VIA VARIETIES
exists, since / collapses the action of K, hence it factors through ¦
the quotient X\K. But h ia obviously bijeotive, and the sepam. ¦¦
bility of/ implies the separability of h. Therefore A ia birational.
too. Therefore by Z»ri ski's Main Theorem, ft is an isomorphism. •
Corollary 1, A separable isoQtny f:X ->¦ 7 is an (late morphiatn. -
Corollary ? Ltt J X -+ Y bt an iaogtny of order prime to p.-
The* Ifo kernel of J and the kernel of f**. Vic(Y) ^ Fie {X) are dtKtll
finite abeltan group) i
Pboof Apply Proposition 3 and Theorem 4 t
:!
ft. The dual abdian variety: char 0. We will use the hypo- <
thesis of char 0 only towards the end of this section.
Pii^ (X) \t the tubgrovp of Pie(X) contorting of fine
bundle* L *uc* that the komomorphism <j>L ia identically ztro.
By the theorem of the Bquare, the image of each <j>i, 1B oonttuaed
in J?le?{X), so we get an exact sequence:
0 +Vi<t>{X) >-Pic(X) »-Hom(J
4 ;
The main purpose of this section sto show (in ohar 0) that Pi^(JC) i
is naturally iaotnorphio to another abelian variety X, oalled the du*l
of J. We make some general observations about Pio°(X) 3
(i) LePio»(X) *s- T; LsL.aXlxeX I
on X x X
Pboof. By the See-aaw theorem, m*L®j>fL~l®p^L is
trivial if and only if it ia trivial on X x {a} and on {0} x X, But
it always is trivial on {0} x X and its restriotion tolx[«) is
jaomorphio to T\L®L~l
(jj) If L e Pio°(X), then for all aohemea S and all morphisms
f,g:&~*-X, (/ -|- g)*Ij -f*L&g*I*.
r. Consider the laatisomorphiem in (i) and pull it baoktoi?
(something in Pio°(i))
(iii) If ? e Pio°(X), n* L s V.
Proof. Apply induction to (ii).
(it) Eoroll?6Pio(Z),n^L
Pboo*. In fact, by J6, n* L-z L-'gi[L®(- lI)*?-']<li—'>«so
itiuffioasto prove that ?®(-lI)*i-'6Pie°(J). By translating
hy *, we get
by (iii)
by theorem of the square
(v) If L e Pio(X) has finite order, then h e Pio*{X)
Pnoor. If L" ia trivial, 0 = ^{x} = «&» — ^,(n*} all » e J
Since X is divisible, this shows that <j>L s 0.
(vi) For all varieties 8, and all line bundles h on X X S, tf
i. = ilixW, then L^Si-'ePio^}, (*„,»! efl).
Pboof Replacing jS by open sets belonging to a covering of
8, we can assume that L\mxs is trivial. Further, replacing ? by
Jj® P*!^1), we can assume that LH is trivial and we must
Hum prove that L,ePirf>(J) allteS We ahall show that

76
ABKLTAN VARIETIES
a line bundle M on XxXxS
In foot oonstrql
Then if is trivial oo X x {0} x.8, {0} x X x8 &nd X x X x^
Therefore by the theorem of the oube, M is trivial. But M reatri?
f
{Yii) If Z,ePio°(X) andiia not trivial, then H'fJ.LHfO) slli;
Pboo>. If JI'(L) ^ 0, then ? = <Pj(.D) for some non-negsti^
divisor D. Then lr*sr{ - \Z)*L s <M(-I,r)*J0).n8llM ffJ 3 ?® iU
stfj(D + t-l^)*D). Therefore D 4-1-li)*J3 =0, hence i>^|
and LstSz whiah oontradiots our assumption. This proves thj|
fl »(?) = @). Let it be the amaUest integer such that Sl[L) # (q|
Letx^X-i-XxXbe the map*^!) '^(x, 0). Uwng the foot ttii
m*Lxp*L® p*L and the Kiinneth formula, we get the diagram-.";
m ''.
-XX T-
X X m*L)
HI
4+i-*
i>
= i^, the dotted arrow is the identity. But if i +J-H
t > 1 then either t < i or j < i, hence in all oaaes 4
ALQSBKAtC THBOBY VIA VARIETIES
i] =@).
Therefore, the identity from B*(X, L) to H*{X L) factors through
We now oome to the really key point in the theory of Pio°
Thsobsm 1 Let L bt ample and M e Vt<P(X) Thin /or some
xeX
j g At map 4l
Pboo?. The whole idea is to look at the oohomology on X X X
of the line bundle
This oohomoiogyis the abutment of two Leray spectral sequences
asaooiated to the two projections of X x X onto X;
(I) H'(X,.
Sotice that on the fibres {z} x X of y, and X x {x} of p, K restnots
to the line bundles
Therefore, if if = r*I®?-' for any at, it follows that JT|wlcIi«
a non-trivial line bundle in Ho* for every x. But by (vii), this
menus that all the eofiomology groups of K ||3) „ x are @). Therefore
JSlp1,,(X)={0>for6liiby Corollary 2. |5. ThereforeH*(XxX,JT)
<={0) by spectral sequence A):
Now uae the other apeotral sequenoe. Since TfLQL ' is non
trivial and in Pic" if * f K{L), it fullows tliat
S nco
s a finite set, spectral sequence {2) degenerates to

ABELLUT VAKTCTtES
But ff*(X X X.E) = @), so jR%, (K) = @) too Therefor, |
Bk(X,K\x,M)'~[Q) for «11*, by Oor. 4,j6. But jK^ is th»i|
trivial line bundle, hence has a non-zero JJ° I Thus we have *
oontradiotion, go that the theorem must be true.
For a Beoond proof of a slight weakening of the theorem, set
Leng, p. 9B. This important theorem shows that as an abstract
group, Pic"(X) is isomorphio to the abelian variety X/.ff(L).. If J
is an abeljan variety isomorphio as abstract group to PWfZ),
what properties would we expect which would oberacterize thie
"extra structure" on Pic°(X))
(a) WewantalinebundlePonXxX the Poineori handle, snoh
that for all a e X. the restriction PB of P to X x {a} represents tho •
element of Pio° (X) given by * under the isomorphism pio° (X) s X,
Moreover, we require tbet P|(O)«J >" trivial. (These properties
characterise P by the See-saw theorem.) ,
(b) For every normal variety 8, and every line bundle X on
XxSsuoh that (i) K,=K\gxVt ie in Pio°(X), for one and henoe all
t e S, and (ii) K\mxS is trivial, the unique set-theoretio map
/¦jS-+X
«uoh that K, s P/(J) is to be a morphism and K is to be isomorphio
ALGEBRAIC THEORY VIA VABIBTIES
T9
It is easy to check that (a) and (b) uniquely characterize both X f
and P up to canonical isomorphisms. The problem is to construct i
anoh an X and P So fix an ample h on X, and as suggested by the ¦
theorem, take X to be the quotient XjK(L) constructed in JT.
Let jt: X -*X be the given morphism. To oonstruot P, we shall
use Prop. 2, J7. We want {lzK-n)*P to be the line bundle ¦
K*=m*L® $L~* %p\L~x onXxX. (This clearly should be ths
case: apply (b) with8 =X, K =m*L®p^L-i®PtI'-1. Then/ = » i
so K should be isontorphic to (lxXir)*P]. According to Prop. 2, we j
mu>t lift the translation action of Ker(lT x *¦} = @) x E(L) on I
X % X, *° BIi ant'on °f tne same group on the line bEBdle K But
^(i), compute the pull-back T^K:
e K{L). Therefore, there is an automorphism ^,: K -v K
the automorphism T^,A): X x X -+ X x X on the base
ipice. However, each ^, coald be changed by a scalar, so there
ja no reason why ^.'"h — lf»+* should bold if the ^,'b are chosen
Bibitrarily. However, if h~l(9) is the fibre of the line bundle
Jy > over 0, then notice that there is a canonical isomorphism:
(trivial bundle
i-1@) xX
Suppose we require that the automorphism ^ of K should restrict
on {0} x X to the product automorphism:
(A »)t—* (A, x + a)
i~'@)x X^L~\9) x X.
Ctesrly, there is a unique & which has this restriction to {0} x X
Since the restrictions then obey &s^| = 4>4+t> so do the ^,'b
thwnsehres. With this action of Kerfl^ x w) on K we construct
s i"on XxX such that (l^x ir)*PsK.
Notice first that for all a e X, if a = nix), then
i. *¦ P, represents the element faz) e PW (X) Therefore if X is
identified with Pio* (X) so as to make the diagram

BO
ABELJAN VARIETIES
commute, the first part of (a) holds. Moreover, P\mxi 'a
quotient of K\mtLX by Ker(n), i.e. of i-^O) x X by the
action of E{L). Therefore, P|wlxi a L-\0)x X, a trivial
Thus the second part of (a) holds.
To check that (b) holds, given 8 and K, consider the line
B=P&(K)®P,*,iP-1) onXxSxX Then g\XxM*K.®p^
and the subset of i9 x X: ¦'§
F={(*,«)|*|,KMtri™l) §
%
te Zariski-olosed ini&xX. But since $ \x*{,,«\ '* trivial if and only if
K, s; i*,, r is nothing but the graph of the set-theoretio map-0;
In particular, the projection f -+ J3 is a injection. Now since tlrf
charaoteristio is 0\ this shews that F and S are biratbn&Hj
equivalent varieties, and since8 is normal, F -+Sisan isomorphisijj
of varieties by Zariski's Main Theorem. Therefore F is the graph of
a morphism, i- e. / is a morphism. The last assertion in (b) follow
from the See-saw theorem. S
RIMABX3. A) For every line bundle L on X, the ma?
fa ; X -+X ia a morphism. This comes out of applying the universal
mapping property (b)to the linebundU)(n*(?)®ja]f(L~1)®p^(iri):
B) If X >¦ Y is a hotnomorphiam of iibelian variBtie», the inj
ducedniap Pio F-^Pic X maps T?ic"Y into Pio'X, and t!uia wo g4
a natural map /: Y-+X, and this ia a niorphisw. In fnot, if Qtstlis;
fThii ii tho on]/ plM6 when wo at ob»r, = 0* Howovor, it is r| tt eiwitii|
Tin Jt w» h»™ oortstmcted would definitely bs^'wrsng" in char )j. '%
ALGEBRAICI THEORY VIA VABJETIBS 81
bundle on YxY, (/xl}*(C) is a line bundle on X x 7
that fot ye 9, (fxl9)*Q\z*Girepresents /*fe) ePic°X,and
by the universal mapping property,/: Y ¦* X is a morphism.
13) If/ ¦ •^*+ ^ '9 ttQ '¦"S6^'80 's /• 7-*-X, and there is a
canonical duality (of finite abelian groups) between Ker/and Ker/.
paoor. We have seen in §7that Ker(/) and Ker(/*: Pio (F)
e dual. If JePic(r)is such that/*(j?) =- 0, then this
that y has finite order, henoe yePic^tT). Therefore
Kw(/*: P r-*-P'°X) =» Ker/! Knally, sinae dim Z = dimX—
dim 7 •= d>m 7, it follows that / is an iaogeny too.
The final point we want to make is that the relationship between
JiHidJ is in reality lymmetrie like the relationship between two
vector spaces set up by a bilinear pairing, We can see this as
follows.
Definition, Let X and Y be abeiian varieties A dwisorial
correspondence between X and Y m a linebitmSi Q on X X Y ichott
ralrictions to {0} x Y and X X {0} ore trivial
PnopoalTlON 2. LetXand Y be two abelian varieties of the tame
dimension, and Q a divi»onal eorretpondmee between X and Y. Tht
foitotnTtg ore tguivale.rtt
A) IfQ\wX? " trivial, U«n = 0,
B) if Q liIM « trivial, then y = 0,
// them AoW, then Isf with Q isomorphtc to (he Pomcari bundle
ProfY,and Y x X with Q tsomarphic totte Poincaribundle Pz of X
pHOor By symmetry, it suffices to deduce B) from A). If A)
holds, there is an fnjective morphism <f>:X •+ y suoh that
G = (^Xlr)*Pr, Pr being the Poinoar* bundle on Y X Y. Sinoo
dim X = dim 7 and <j> is injective, <j> is also surjeotive; since the
oharaoteristio is 0, this implies that ^ is an isoinorphism, i.e. Xsa Y

82
AJ9ELIAN VARIETIES
ALQBBRAIO THEORY VIA VARIETIES
S3
Now, let <ji: Y-+ X be the morphisin suoh that if Pz is
Poincare bundle on X xj, {1^- xf) *PZ = Q, To prove B), we ha*
to show that iji is infective. D' not, we oau find a finite subgroup ?¦
¦ and^faotoriieaaar1 > YjK—^+ X where t> ia I
natural homomorphism. Thus, if L ia the line bundle (lTx0)*i
onXxr/Jf.wehavethatQcOjXil^iJ.Now, ? induoes a honJ
morph sm a, X -+ (F/X} and the isomorphism Q s A»:
1
¦ (YjK)
means preoiaely that the composite X >¦ (YjK)~~'-**¦ Y is ttfl
homomorphism defined by Q. Thua this composite is an isonw
phism. Thus a is injective, and since dim X = dim {YjK), « u
7) ore both isomorphisms. But we know that t\ has a non-trivii
kernel, vii. the dual abelian group of K. 1'his is a oontradiotio
proving that *(> is injeotive -^
9 The ca»e i = C We want to link up the methods of ChapWt
1 and 2 in this section, Therefore weasaume that the ground fief'
i — C, and that X => VjV (Fa complex vector space, [/a lattice) \
an abelian variety over C. Recall that every line bundle on X L
laomorphio to ?(i/, a) for a unique Hermitian form if on F SKcfi
that ? = Im jff ia Integral on [7 x 17, and a unique map «: G -* (ff
satisfying 1
a) u by definition, the quotient (C x V)jV for the a«twjf
- (A a(»
We shall do twe things: (A) compute T%[HE, «)], (self
explicitly and hanoe interpret fap^ in the analytic case, {if
DeMnibe diviaorial oorreapondenoos an L(B «)'a and henoe oompv^
the du*l X aoalytio&lly
(A) To keep the picture clear, it ia oonveriient to generalize a
Jittle- Let b discrete group U act freely and diacreteiy on Vt and V,
and let I1: V, -+ 7, be a [/morphiem. Let X( = VJU, and let
7: X, ->-X, bfc induced by T. Suppose V acts linearly on C X ?t by
<*„(*, *) = (A.eu(a),tt(i)), *eP,, AeC «e[7
where ejfc) ia a multiplinative co-cj'de
tet ij = (C x f'j}/^. Suppose we now want to describe the
imc bundle ix = ?'*(t,). Then
» [^1 x r.<c X ^t)]/^ wiiere U acts or both factors
Therefore, i, = {Cx I'1)/G with the action
Now the co-cycle ^e.} might be normalized to have a special form
which foT} might not ha™. Then we might use an automorphism
ofC X V\:
(A t) 1 >- (A.j(z), z)
CxF,
-»-C x f,
and carrying over the action of V obtain a new description of Lt
as (C x Vj)lU with action
Apply all this to the case F, = P, = V, X, =^J
byo 6^,1" = T^,, translation by ir(a], and i,
that yj0)iffl, k) ia (C x K)/f/ with aotion
a =X, Ta translation
= L(H, a). It follows
To simplify thia co-cycle, take g[z) to be the non-Bero holoraorphic
function «-••««>. We get the new action
^{A, x) = {A.«{«).e*'H".">-fl(-^W.e-a<''ll)+t*HA''«[ j + u)
and shoe JI(o o) -ff(« a) = 2tff(a «) we conolude-

84 ABELIAN VABIETIES
Proposition T%J[L(H «}}= L(H a, y.) where /„<«) =
We get immediately lots of nice consequences.
W
(u) In particular, sinoe y^-y^ =V«1+.1i it follows that
is a bomomorphism; this is the theorem of the square
(Hi) We find
?{X(ff, a)) = [/ fV c V/V - X
whereto - (al^a.u) eZ, alUe 17}.
(it) Therefore L(B, «) is in Pic^X), as defined algebraically
by ?@,
*=> HsiO, i.e ?(/7,n)isinPiul)(X),fts defined analytioajly,41
(v) Moreover, S
if{L(ff a)) is finite ¦==> Lf1/!/ la finite 3
*=> t/1 is a lattice j'
«» ? is non-degenerate •
-s=> Jtf is non-degenerate. :¦
(vi) Not cc that if H is non-degenerate, e.g. if ?(H, «) ia ample.yt
theu every homomorphism G-j>B is given by u -+ E[a, •),!':
for Homo a 6 V. Therefore every homomorphism a : U •¦* CJ'.i'
is given by *
for some a e V Thus every element of V\s?(X) is equal to?;
¦?(°> •>*»), some oeV; thU proves the main theorem of (g-;.;
when Jfc = C t.
I
(B) LetX( = Fi/(/1(,i = l, 2, be two ttbelian varieties.
i{H,«) on X, x Xj ia a divisorial oorrespondenoe if Q is
on {0} X Xt and on Xt X {0}. This means that (a) the Hennitian*
form fl is 0 on {0} x Vt and on Fj X {0} and (b) « bs 1 on {0} X tf,|
and on DjXtO}. Define f
ALGEBRAIC THEORY VIA VARIETIB8
@
66
Then B is an H-bilinear form on Ft x F,, oomplex linear on F,, and
jnti-linearon Ft. Let Im [B) = jS Then )8 is integral on Ut x V%
juid we get
!. 0), fcy,, 0})
+ ^(@,*,), ^.
,, 0), @,
(Off,))
a,).
«««, «t» = *((»,. 0».«(@
Thus the divisarial oorreapondenoe Q ia detemoned entirely by ?
In order to find the map from X, to Xy induoed by Qr we next
calcinate the restriction of Q to X, x {»»(«,)}, o, e T,. Let 0' =
Vt)jVx. If 1 x w,: X, x F, -*XX x X, is the natural
C'(l)«<?de'| e
map,
p 1<?e|I
of tr, on C x ?! X F, ia given by
The action
Restricting to FiX {a,}, it foilowa tbet C|z
mth action
equals (CxFj)
Modifying this group action by the automorphism of C x F,
soalai multiplication by «-»*hu«p>, we get the action
Thus we have
Protobitiox. ez
In particular, we Me that b order that flonl,xl, satisfy
the equivalent conditions of Proposition 2, JS, it ia necessary and
aufieient that dim Xj = dim X, and that for all ^ ? Ft
«i e t/, -(=» ^(k!^) e 2, all Uj e P,. Thia implies that B is a
non-degenerate pairing of Fi and F,. Hence

89
ABEIJAN VARIETIES
, Xt ~ Xt and Xt = X\ if and only if
COEOLLIBI.
(i) B t» tKm-ikgtneratf
(ii) under J) Ux and Ut art dual lattwsa te
Ut ^feeP, |j}(i>1,x,}eZ, oJ(Uie t/Jond wee versa.
Explioitly, therefore, if X = 7/G, then the dual abelian variety
X oan be oonstruoted as follows, Let
T
}
Then X=?jV. The form B: Fxf^-C is aimply S(x, 1} ^l^j^.
and the Pbinoarfi bundle Px on XxX in simplj' ?(ff a) with-;
Moreover the line bundle on X corresponding to a point
»(() EljIeTjii then just L(o,«,) where
There anses a small question of compatibility : we just con-!
struoted X — TjV and showed that X a Pie9 (X) But in §2, via
the exact sequence
0 yZ y &z »- 0* yo .-
we ooDstruoted an isomorphism :
Pio»(J) s H'(X, &i)iH\X Z)
apdia jl we found an isomorphism ff'fX.ffj) = 5*. We would lik«
to be sure that we have resdly found essentially the same deacdptton
of Vv?(X) twioe. As we have just seen our seoond description off
P>o°(X) rests on the map ¦¦
T
t
Ho,*,)
ALGEBRAIC THEORY VIA VARIETIES
87
compute the composite map T—^> ff'fff^) >- Pio(X)
wbioh gave us the first descriptioQ. THb mep is given by
_
Using group oohomology of U we get a diagram
-r;—> H <u H*)
br it!.
KoraiV C) ?^= H 1{U C) .
iff
HomR( V, C)
'I
Tc 1 © T ¦
I
ff'{X C).
I
where ^*=«multiplioative group of non-iero holomorphio functions
on V, and where the square on the left oommutes according to the
compaotibilities verified in Jl. Therefore, if 1 e T, the first desorip
tion associates to I the C-oo-cyole «-> *•*"<**
But
where g[t) = e-1-"^ is holomorphic in z In other words the first
description rests on the mep
f > Pio*(X)
So the two mepe difier only by multiplication by & (experf
mental errort)

REMARKS ON EFFECTIVE DIVISORS
by M V NORI
Proposition Let f X -+ Y be a morphxsm of varieties
an abelitm variety. For each x eI, let Fr be the connected
off~lf{x) to which x belongs. Then there is a closed connected
F of X such thai Ft = x+ F.foraltxeX.
Psoor Fa. xeX Let ?: X x Ft -*¦ Y by
Since ^ is complete and connected and 4({e} x FM) = /(a;}, by tl|
rigidity lemma fa, ») = ?(*, x) for all z e X,« e l?r In partioiila§
/(« — a; 4- Ft) = /(*) for »11 * e X. Since ss — x + Fm is connected,
have J*. contains » — x + Fr Reversing the positions of x, z eX,
aotaally have F, = z - * + F, for all x, x e X. In particular,!
F =* F,, F, = % + F for all zeX, So it only remains to show that!
ib a subgroup of X. 1
Let it e F. Then F _„=-«+ f, »o e g J_u. It foltows that J"_ J
J, = JC. Therefore - u + F = F, for each u e F, 80 J* is a Bubgi«|
of X (by the very definition of Fx, F is closed and connected)
mow on, F witt be called the kernel off |
One checks easily the following |
Thbokbu. Let D be an effective divisor an X f
(l) Let H(D) = {xeX\Tl{D) = D) (equality of divisors, *§
equivalence). CUatly H(D) is Zantte, doted
(ii) Let L ^ L{D), since L1 Ao* no ba«e pointi
be the tomtponding morphUm. Let F(D) = ker/ o* i»l
oto propowtwn Tftwt ff(D)«-^(L)« = F(D).
Note that the above theorem implies Application 1 on p 60.
Ill
ALGEBRAIC THEORY VIA SCHEMES
JD The theojem at the Cube (II}. In this chapter, we shall
Iwuyt; metin by a scheme a scheme of finite type over an
glgebraicully closed field h, and a point will always mean a dosed
point of the scheme.
We begin with the following seemingly innocuous generalization
of Corollary B to the semioontinuity theorem.
Proposition. Lit X be a complete variety, T any scheme and L
a lint trundle mlxK, Tien then exists a unique dosed eubacheme
yx a. r Y havxng the following properties:
(a) if Li '"" the restriction of Ho XxT,, then is a line bundle
M on Yt and an itomorpkitm p*Mt s L, on X x Y%;
(b) if f :Z-+Y is any morpkistn such that there exists a (mm
hinife if mi Z and an isomorphism jf (JC) = (lx x/)*(i) mlx2
f ma bt factored as Z-^Y^t . r.
PbOOT. Tile uniqueness is immediate, since if Tj r « Y
jnd yj c » 7 are two closed subschemeB of Y satisfying
(».) and (b), each of these closed immersions can be factored
through the other, and they must coincide.
Note next that if p*^ s ?„ then J^ ^ J>^,(Lt) (by the
KUrcneth formula), hence to show that there is a, line bundle Mt
juch that j)Jifj - Llr it is eqaivalent to showing that Pi^fiJ = 3Rt
»»n invertible sheaf ajnd the natural homomorphism ^(SH,)-*-Jj,
it an isomorphism.
In view of this, we arc reduced to proving that there is an open
toimng {Vi} of Y Baeh that the proposition holds for X x Vt~* V(
Mid the restriction of L to X x V(. In foot, if we hove done this, we
obtain a closed subschema W, c *¦ F, such that (a) and (b) are
valid with Y replaced by Vf. Then clearly Wt n {?(n Vt) and
ffjfif^n Vt) are two dosed subaohemes of F( n Vt such that
(a) and (b) hold with Y replaced by Vf r> 7j, hence they ore equal
Thus we obtain a closed subscheme Y1 of Y such that Yt n V^Wp

ABELJAK VARIETIES
and (a) (because of our local reformulation) is clearly valid. As fo,
(b), the local version of (b) implies that for each >, /^(F,)-
faotoriusethrough fflhence/-1(P,)^7faotoriiesthrough Yx,
ALOEBBAIO THEORY VIA SCHEMES
SO Z ¦
¦ 7 faotorizes through 7t
Thus ire may uninw 7 = Spec A, and it suffices to find an ope^
neighborhood of each point of T in which the proposition is
By shrinking Y if necessary, ire may assume that we
free complex 0 -*¦ A'' ^ A'i -»¦ ... giving the direot images ofv
L universally as in {5 Let M be the aokernel of the tram.::
'4 «
pose '4 of 4 A?i *- AT'-*M-*-Q Than for any jl-algebra ?,;
EF'—-*¦&* -+ M ® AB -*¦ 0 is exaot and benoe oe u 0-J
Hom^Jf ®AB, B)-*Br>—^.ffi. This shows that for all/: SpwJCi
^•8peoJ*=.r,y^.({lJ. x/}*(?))*Homjl(.if, .B). Now let 7 be!
the set of points y e Y suoh that the restriotion Lt = L\X x {y}W
is trivial, so that J* is a closed subset of T by our earlier result
(applied to 7,^ and the restriction of ? to X X T^). If F >,
1* — F, the empty subschema Tx = 0 of 7' satisfies (a) and (b) ¦
with respect to Y'. Hence, it suffices to show that for any yeF,f-
has an open neighborhood in which the proposition holds. If y s/S
1 = dim H*{X X fe}, L,) = dim Hom^fJf, J/ffl,} = dim, =^, «$
1
L
that by N»koyam»'s lemma, there is on element of M wbioh gens-.'
rates Jf in an open neighborhood of y. Bestrioting oureelves to thk :
neighborhood we m»y assu me that MaAl%, where His an ideal of;>
A. Let T[ be the closed subaoheme defined by Hi ij the restriotion
of ? to X X Y[ and ? the Msooiated sheaf. Then f>^»(?i) is thai.
sheaf associated to Horn j(j!/8, AjV.)sAjHon T\, and is henoe ftsi:;:
of rank one. Consider the natural homomorphism j{(p^(^|))~
> ?, on X X 7t. Since both sides are locally free of rank on«, ¦
this is an isomorphism, at a point z el x 7[ if and only if the;;
induoed bemomorphum of 'fibers' :
w aurjeotive. Now, since Homj(A/a. At%)-* Hom4(^/K, AfSRt)
^jfllxljli i]Xx{jf})is mirjeotive and ?|Zx fpjis trivial, A U
in isomorphism at all points of Ix {y}. On the other hand
the aet Z of points on X x Y[ where A in not an isomorphism
being the union of the supports of ker A and ooker A, U closed
and does not meet X x [y]. Henoe its projection into Y is a
doeed subset not containing V- By restricting ourselves to an
sffine open neighborhood of y not meeting this projection, we
may assume that M s AjZ, and that if 7t is the closed subsoheme
defined by H, condition (o) is fulfilled on 7t. We claim that (b)
follows. In faot, since the oondition that /: Z -*¦ Y faotoriie
through Fi is local on Z, we may assume Z = Speo Baffin*, and B
becomes an ^-algebra via/. Further, we may assume K trivial on Z,
eo that (lz X /)•(?) = eXxx and p^z X f)*[L) = Pi,,(^K«) = <P*
(jinceX is a complete variety). Hence we have on isomorphism of
J}.moduiea.B:rHoiDi(jl/S, B), so that S.B=0 and j4-*3Iaotori
through J/B. Thus /factors through Tlt proving the proposition,
Under the.assumptions of the proposition, we shall refer to the
dosed enbscheme Yt of Y given by the proposition as the maximal
closed subsoheme of Y over whioh L is trivial.
We get the following strengthened varoion and direct proof of
the theorem of the cube.
Thborik Ld X and 7 be eompttU vanttttt Z a aonnteled
teheme and L a line bundli an XxYx Z u>\rm restriction* to
(H,}x Tx Z, Xx. [yD}x Z and XxYx {^ art tnviat for torn*
*»eX,y0 e 7 and x, e Z Then L « trivial.
Pboof Let Z' be the meximal cbsed subsoheme of Z over
wlooh L is trivial, so that Z' & 0 since *,eZ'. We have to show
that Z'=Z, and since Z is connected, it suffices to show that if a
point belongs to Z\ Z' contains on open neighborhood (considered
u an open subscheme of Z) of that point. Let us denote this point
again by %, by 3H the maximal ideal of 4\A and by I — IH the

ideal defining JJ'»ti,,»o that 7c HJJ We have to show that /=@),
If net since Cl W =• @) by Krall s Theorem, we can find saf
¦>«
integer n > 0 suoh th»t BP 3 I, BP+1 > J, bo that
[fflP+i + i/Sir+i] c[BB"/SR"+1]
is ft non-zoro 4-voetor space Houoe, if fl, = aH"+l+/, weoannndf'
«n ideal «, with 8,38,3 Of-*-1 Mid dim* 51 = 1. Hanoo Jt, -= R S
+ *.afcrsome« e «, and «! 3 / but «» $ /. Lot «, = SR. Let*,S
bo the closed subschemas of Z consisting of the single point i,,i
with structure sheaf 0*^1%, bo that: ' f
Z
o o
tot ^(»=>O,l,2) be the rwtriotion to Xxyx^, of the tht»t
L of sections of L, Note that Z,,, i, ars trivial on IxTx^.i
? X F X 2^ mspeotively, since Za, Zl c Z\ so that we have
isomorphisms i$ = sixrx*i (»=0. !)• Farther, unoe the utruoture
ohosves of 2^ Zv and 2^ are related by the ernet saqnenoci
mult, by <t
reetr
we also have an exact sequence of sheaves on the topologioJ
¦pace Zf
mult by a restr
Consider the section a eT(X x 7 X Zt,Lx) equal teA(l) tmderthe
isomorphism A $xxT*m, *-^i The neoessary and sufficient
ALOEBHAIO THEORY VIA H0HEMK8 90
. that Lt be trivial is that, a can be lifted to a section
if we can do this, multiplication by a' is a homo
A': i!l
which rednoed modulo the maximal
ideal of w>y point J of Z x T x fo} ifl an isomorphism * "•*>
i, &K *» henoe A' is an iBomorphjam (the ohoeves being locally
free). Conversely, if Lt h trivial, using the induced trivialutation
of i,, the map r(i,r-f r(i,) becomes the map V{Gtl) -*•!•( ff.J,
irbiob a onrjeotive. Now, fix an isomorphism L,, s SXxT. The
obetrootion to lifting a to V{Lt) ia then an element f e Hl{X x T
«JXr). Sinoe the restriction* of L to X x \gt} X Z and {at J x I" X Z
twee also to X x {l/,} X 2, and {x,} X T x Zt, are triTial, tha
natrietionsofitoZ xfv,} X ^iandfi,} x J" X Z, can be lifted to
i|i X W x ^i and -^If**) x 7 X #« mepeotiTely. This meant
th»t the intftge of f by the maps Bl(X x Y,6XlT) -+Hi{X, ffz)
tnd El(X X F ffixl.) ¦+B1(r, 6T) induced by *>—» (*,y,J and
y,—»• (iD, j) are mm. But by the Kilnneth formula, these maps
Therefore f=0, and a can be lifted to jtx 7 xZ,, and i, is trivial.
Tbi* i* a contradiction, so Z' oentains an open neighborhood of t,
11. flaac Theory of Group Schtmi*. We oontinue to work over
t Sxed algebraically olosed field it, with sohemes alweys of finite
type over k and with closed points only. Boh will denote the
category of schemes of finite type over h
One of the most basio tools in the theory of schemes u the
tcnoept of 8-valwtd poVnfe: if X and jS are schemes, an 5-vahied
point of X is a morphism from S to X The set of all mioh
u denoted
Uomt E, X) or X(8)
If I is filed the map 31—> X(S) U a eontraTuiant functor
X:8oh« *-8ete
The unportanoeofthU functor iethiB: if X Mid Fare two schemes
than (ft) * morphiwn /: X->-T defines a morphism from the functor

ABELIAN VARIETIES
X to the functor T (i.e. » map X{8) —t- _T(iS) for every 8
that for every g:8-*-T, the diagram
/(*)
I I
oonunutei), and (b) conversely, a morphjem from the functor Xt
the functor T is defined by a unique morphism of sohemes /: J-
T. (a) and (b) are really tautologies holding for any category: tb
reader Bhould prove them for himself if he has not seen them, Tb;
remark will torn out to be an excellent tool for oonstruotia
morphismB as we will boo Formally (») and (b) any that
ib itself a fully faithful functor from the category Soh to tfa|
category BW (Sot0, Sets) of all oontravanant functor* tna;
Soh to Sots, hence Sen it equivalent to b full suboategory (j;
?un (Soh", Sets). Cf- Mumford [U 3], Cb ! |
If AlgdenoteB the category of t-algebras of finite type, and jj|
Obj Alg, let u» put X(JI) = X(Speo if), Than X defines a oovariaol)
funotoT from Alg to Seta, which functor again we denote by tl^
same symbol X*The elements of X{R) are also referred to as j|
valued points of X. ltV = I*un (Alg, Sets), it is once again »nM|
matter to show that Horo^X, T) -^»- Hom^ (X T) is bijeotiT|
so that Soh can ha identified with a full suboategory of V i
Dbfitotioh. A group schemt u a scheme 0 togtthar vn& (»)^
maUtpliaition morphitm m: O x 0 -+ G, (b) on identity point, ti|
0 morjAwnt e: 8peo k-yO, and (o) on tm«r« mwyiMW »: 0 -»ft!
AkA that iht. JoUcwing axiom* hold.
ALGEBRAIC THEORY VIA SCHEMES
A) (Aswaaiimty) The diagram
Q xG xCt > 0 xG
9B
lff xm
GxG —
-*¦ a
0 x Speo h ¦
l\
G
l\
Spec k x G ¦
it commntotm
C) The diagram
-+GxQ
-*¦ 0
txl,
J"
e
f"
0x0
fe tammutoftee

96 ABXLUN VAMETIES
Now let G be a scheme, and let us interpret the oonditious 1
giving a structure of group scheme to 0 in terms of the :
which it represents (either on Soh or on Alg). In view of our earli
remarks and the fact that produots in Sch correspond to produ
of funotore, m, i, and e can be interpreted respectively as m
0(8) x G(8) -+ 6(8), G{8) -+ G(8). and as giving a diatinguiu
element of 0(8), funotorially in 8 in the obvious sense.
tions (l)-C) then simply say that Q{8) is a group for each 8 -\
m, i, and e defining the group law, inverse and identity eh
and that for all morphisma S' -»¦ 8 of schemes, the induced i
Q{8) -+ Ql,8') is a group bomomorphiam. We thus see that to |
a group scheme structure on a scheme 0 is equivalent to ]
the set of 3-valued points of Q into a group for every 8, fu
in 8. It would also be enough to make the set of JJ valued points t
Cf into a group, for every fc-»lgebra S, funotorially in it
If i ia an ordinary point of G then the morphism
X, 0 _Z+ Qx{x}cGxG ¦ ." > G
is an automorphism of 0 called right-multiplication hy x
generally, if x: 8 -*¦ G is an S-v&hied point of 0, x indnoes i
automorphism S, of Q X S over 8, whioh we oan call rig
multiplioation by x. We use mo(l0 x *j: 0 x 8 -*¦ G and tet
(mt(la x x), p,) so as to get a commutative diagram
.Ox*
IIkS <
It ia easy to oheok that with the group structure on 0E),
get Jtif^BfOR,. Interohanging the factors in 0 x G, we »i;
define left-multiplication 1^ similarly.
lax Aloibbas Let Z be a eoheme, and Qx the sheaf of Kah
differentials on X over k. By a. wclor fidd on X we shall mi
ALGEBRAIC THEORY VIA SCHEMES 97
' map of sheaves D: <S)X -> Sx suoh that the induced map
?: SZ(V) -* dz(U) is a derivatbn over k, for every open tet U
'JjiK is equivalent to saying that D factors
d
whore / is an (t^-hnear homomorphism of sheaves.
Bya tangent vector at x el we mean a derivation 8x^^-k or
equivalently an element of Homff ((lT|t, i). Now, it is well known
tbat if SDt, is the maximal ideal of @Xjl the natural map
is an isomorphism, Thus, »tangent vector is uniquely determined
by giving a linear form on SR^SttJ. Clearly, » tangent field in a
neighborhood of x determines a tangent vector at x (by composition
of the derivation 8IM~>- 6Zt and the evaluation map
whioh we shall call the value of the veotor field at x.
h),
Let Xand 7 be two schemes, and Da vector field on X. If J^
(i = l,2) denotes the Ith projection of XxY, we have a canonical
isomorphism p*(Qx) © rffflyt——'¦*• Pzxr- 0o tltat tIlere u *
onique vector field D ® I on X X 7 such that when factored
through Qj-xr, Z*® 1 agrees on J>f (Q*} with D And it is zero on
is said
Now let O be a group scheme. A vector field D on Q
tobeie/l fitmnant if the following diagram oonunutes:

ABBLIAH VAWBTIEB
ALOBBBAIO THEOBY VIA SCHEMES
the vertical maps being the natural ones induced by the multiply!
oation raorpiusm OxO > 0
For any tangent vector tatetoQ, there it a untqm&
left invariant vector field on G having the value t at e. i
Pbooi. Ifirst we interpret tangent vectors and veotor fields im
a slightly different wey. Let A be the fc-algebra Afcl^**) mil
)j: J^t]/{t*)->Jt the homomorphiun with tj(«) =0. If ji and fiiJ
are i-ahjebia* and it is an j4-algebra, the ^-derivations D of A fa*
B are in ous-ous oorrespondence with algebra homomorphiunti
fi-t^'s fl®» A such that tfa) = o.l + multiple of t.
oorrespondence is given by D ¦*—>¦ <j> $a) = a.l + (Da).t
we deduce that if X is a soheme and x a point of X, a
veotor at z is a raorphiam Speo A -+ X such that Spee (k)'
Speo-A ->-X is tbe point z; and a vector field D on X is «'
automorphism over A: '.
which restricts to the identity lx: JC -> X when you look at til,?
fibres over 8peo(i) t- ¦¦¦» 8poo(A). '%
Oua then sees easily that a vector field Dona group scheme (?¦¦•
is left invariant if and only if for the associated automorphism jj*'
the diagtam ?;
Ox Ox Spec A
I IDX 1
Ox Spec A
D
+ «xflx Spec A y
»»X 1 (*|:K
O x Spec A
j commutative. If D'=px*D: Ox SpeoA -*¦ 0, then in term* of
oints x, y of G and 1 of SpeoA, this diagram says
Bus dearly holds if and only tiD\x,l) = x.D'(e,l) for all x and f
In other words, if ( equals p^Do^V): Spec A->- Q then we want
5 to be right-multiplication by the A-valued pomt T of Q, Therefore
given any A-Talued point t of 0, we get a unique automorphism
5of ?7x Spec (A) such that (*)<»mmates andp,oSo(e,l)=T But
thus means eiaotiy that D is left invariant and has value t at e.
Thus we get a oanonical isomorphism of the i-veotor space of
left invariant vector fielda on G and the tangent space at e to O.
Now, given any two vector fields Dt, D, on a soheme X, considering
them as endomorphisms of Oz, we see that D=[Dlt i>J=D1Z),—
DtDi is again a vector field on X, called the Poiston brocket of D1
and Dt. Furthermore, if char Jb = p > 0, 2Jf is again a veotor field
If 0 is a group soheme, one verifies trivially that the PoiMon
bracket and p power operation take left invariant veotor fields
to left invariant vector fields.
The Lie algebra of a group ttieme it the ^vector
tpaei of lt.fl invariant vector field* together with At operation of
Pottton bracket defined on it, as veil as tht jfi power opemtien if
oh»rt =>p > o
We shall agree to denote the Lie algebra of G ff ff, eto
by g, §, g,,...eto We hare then:
A) The map fl X fl-(- fl, (X, T) t—+ [X T], is bilinear ovar t
and the map X i-^* X* satigfle* {XXf =• MX*.
B) lor X e g, [X, X] =- 0
C) For X, Y and Z e 9, we have
[j, [r, aj] +it, [z, X]] + [z, [x, rj] = o
(*) If char ? —p > 0, there is » certain universal noin-oom
mutative polynomial J, (depending only on p) in two
variables mob. that

ABELIAH VARIETIES
ad (X) = (ad
100
ad T)T,
where ad X ta the endomorpnlsm of g defined by ad X( T) = JX, J
For the actual expression of Ff, which is unimportant for o#
purposed, we refer to [JJ. It is however uniquely deternrinjjj |
hy the condition that if A is an associative algebra over k «$
we define [X, T\ = XT TX and X»to be they powering, §;
satisfies D), Thus, Fp may be calculated by taking A to be a f^ |
associative algebra on two generators X and 7 over i, and shawih' '
that(X + r)»oan be expanded as in D)in.ji. Since we are
interested in the cose when g is "abelian'*, that is, when [X 7J
for X, Y e(), it suffices for us to know that F. has 'no
term', that is, Ff{0, 0) =• 0. Thus, in the abelian oaM, X-+X>
just a p-linear map (i.e. an additive homomorphism of q into
with {XXr^VX*). In the case of the Lie algebra of an a
variety the pt* power map is called the Haase-Witt map.
We remark that if 0 its commutative, that is, if the dii
ALGEBRAIC THEORY VIA SCHFMEB
101
OxQ-
• 9x0
is commutative, where a is the 'ewitoh map' tr — {pt,p}) then,
is abelian, that is, [X, 7] - 0 for all X, ? e (j. To prove this,
start with the following observation. Let D, (i = 1, 2) be V
fields on any scheme X, Dx — [Dt, Z»,] and D^ X X Speo A
Speo A be the associated automorphisms, where A = &[<]/(<*). M:
A' = i[<, •']/(«', «'*), and let u(: A-»-A' be the ib-algebra hon>i>ma^
phisroe defined hy a^t) =tlat(t) — t' and <r,(t) = ««'. Then i|
induces a morphism ^ = Speo of: Spec A -+¦ Spec A A< t < s|
and we get automorphisms
taking fibre products with Speo A' over Speo A. via ^. One
n checks easily (by taking X affine} tbat & is the commutator
XJ^XtXtXi'Xi1- Now suppose O is a group scheme, let ^,:
0 (i = 1, 2) tangent vectors at Q, and let Dt be corres-
corresponding left invariant vector fields. Then fit is just right-transla-
right-translation with respect to t^, and if (jo^ = Tt: Speo A' -»¦ O, then ^ ia
ngjit translation of O x Speo A' by the point Tt 6 O(A'). Hfence
*1'8 riSnt translation by [3\, TJeGfA'}, Since G(A'}
commutative group, it follows that [?'1,T,] = 0, bence
'i =X» = Identity, and [D,, i)J = 0.
ThbOBBM. Any group scheme over a field k of characteristic 0 tj
smcxtth (and in particular reduced).
Proof. We show in fact that if Xisanyschemeoveritofcharao-
tariatii! zero and x a point of X auoh that there exist vector fields
J),,..., JD, in a neighborhood of x with n ~ dirot —| inducing
independent tangent vectors at x, then X is smooth at *. We aan
choose x{ A < t < n) in S(, such that thoy form a basis modulo
SH^of aK,/SR*and(i)(ai)(^) = Sti. Since clearly D^SKJ) c SK»-', D
eitend to derivations of the completion 6>x of tfr Now, we have a
unique continuous i-homoniorphism a: i[[( , („]] -*¦ 0t with
«(^) = Xf. On the other hand, define $: (P,-+i[[(,, .. (.]]
by putting
where
. v.1 and f
.„(,¦". By
Leibrui'a formula and induotion on a, one cheeks trivially that
? is again a continuous jt-homomorphism of local rings. Now

102
ABELIAN VAMEHBH
ALGEBRAIC THEORY VIA SCHEMES
103
« ib suijective since its image contains a set of generators!
SB, and ifp,,..., y] is complete. On the other hand, p
(mod([,, ...,(„)•) bo that ^) generate the maximal ideal!
*C[*i U]. and^isalsosurjeotive. Hence so is jSoet, so that J]
ib an automorphism of *([(„ ..., yj. Henoe a is also inje
hence an isomorphism. Henoe 0m and (?, are regular, and
smooth at x.
In positive characteristic, we will soon have plenty of ewnp
of non-reduced group schemes.
Subgroup smmas, xbunils quombhts
Let Q be a group scheme, and .ff a closed subscheme We i
that IT is a roAproup scheme if, denoting by i: R< tQ ¦
closed immersion, ».(ix<): H x H •*¦ Q factors through B;
This is equivalent to saying that for every Se Ob] Sch, H>_
subgroup of Q{8). Clearly B then becomes a group scheme in
own right, and t: H -*¦ O a homomorphism of group sen
(defined in an obvious way).
To any group scheme 0 over k of characteristic p > 0, one ci
associate in a natural way an increasing sequence On of olc*
Rubschemes (n > 0), all having support at the identity eleme
of <?, as follows. Let <P = (P, be the local ring at the identity to
Sffi its maximal ideal, and 3K" the ideal generated by {*»*|ie!
Put 0.— Speo [(P/SP*], so that Gf« is a closed subscheme of 0 ttj
support at«. Lst ff'=^^ bethelooal ring of (e,«) on Ox fl.i
that 6' is the localisation of 8®t (S> with respect to the maxuH
ideal ffi® ffl + <P® SK. The multiplication map m indnoes a lo§
homomorphism of ringswi*" 8-r-tP' so that ibr/eSR,«
, A a unit in 0'; hence n>
'>]. 0'. This proves that the composite C, x (?„-*¦
gx^—>- 0 factorizes through the subaoheme ff,, pro-ving that
^ (8 a subgroup scheme One has evidently Lie ((?„) = Lie 0
for n >'
EkammJ!3- (I) In any characteristic, define G,, the additive
group over k, as Spec ^[T1) = A', the group law G.xGB-*G,
being defined by addition, or equivalently, by the homomorphium
«•: *[n -+ k[T] ® t[T], m*(T) = T® 1 + I ® T, The group of
gvalued points G.(jS) is just the group T(S, t>B).
If char k = p > 0, we also write «,,» for the subgroup Fcfreme
(G,),. that isiyi = Spec [p^|. We have Lie (G.) Le (v) =
',»>!, and VE) = (/efffi, H)a) I/" = 0}.
B( In any characteristic, define Gm to be the multiplicative
group over i, coinciding as a scheme with A1 — {0], the group law
being multiplication. Writing GB = Speo k [ T, IJT], the homomor-
phiam m* : k\T, 1/T]^ i[T, IJT] ®t k[T, 1/T] h given by ¦»•(?) =
*. The group of S-valued points of GB is the group of units
If characteriatic t = j» > 0 we shall put juy. = (GJ, = Speo
P\ T-']/((T - I)'") = Spec ^T]/(T^ - 1). We have Lie Gn =
> I), and ^.E) = {/e r(
Now, let Q and /f be group aohetnea and / : Q -+ H a homomor-
pliism of group schemes. The fiber /~'(«ff) = Q xff{%} ovBr the
elo»ed point tB of H which is the identity element of if ia a closed
subschema K of Q. By definition, for any g-valued point of O
$:8-+G, $ faotorizes through K if end only if /o^ factories
through es : Spec ?->?f or in other words, K[3) is the kernel of
0(S)-+H(ff). Therefore K is a subgroup scheme of G.
A» an example, consider the homomorphism GH -¦ G,, defined
-Ti—f X" The kernel is denoted by ^,. IVir (n, p)= 1, ^, is just

1M
ABEL1AN VARIETIES
a discrete group (i.e. reduoed and finite group) isomorpbio to j
«* roots of unity in ?*. On the other hand, it follows from <
tions that ft^ is the same group defined earlier.
Next, suppose 0 and B are group schemes, and <j>: S
homotnorphism. A pair {OjH.iq), where QjB is a scheme
y.Q-r-GjH is a morphism, is said to be a quotient of Q\
B, if it is universal for all pairs (S,f) where 8 is a scheme,
/: O -*8 a morphism anoh that the following diagram comma
HxQ
¦+¦
It can be shown that quotients exist, that / 1b ahrays flat
surjeotive, and further that when fl is a normal subgroup i
in G(i.e. J?(#) ie a normal subgroup of Q{8) for all 8),
inherits a unique structure of group Boheme suob that q :Q-
U a homomorpbisin and lias kernel precisely H, We will i
need the result in this generality, but only in the Bpeoiol
when if is a finite group Mheme. We will consider this in )
limn IXVAKIAOT SrFVXBBMTIlX. OPHBATOBB.
We want to study tbe algebra of maps &>a -> ffo generated!
left-invariant derivations We filet introduce the
Hoffl:
k)
H
where Hom^^ means tho maps L: f>m -t- k whioh are
in the sense that L(SRf+') = @) for some N. Alternately,
H= lim
11 iUm\ mlwilMim
ICO
ALOEBRAIC THEORY VIA SCHEMES
106
If* means the t-veotor spaoe dual to a vector space W If
lies in the aubspace T(&z)+, we will say that L is
by Z. This definition makes it clear that H is an
¦io analog of the spaoe of distributions on a Lie
supported on a finite set, H has a lot of structure.
(I) We get an associative and distributive convolution
product
* H®H—»-H
by defining
Bore precisely, if L, is supported by Z{, i = 1,2, and if Z, c 0 U
i finite subschema such that the group law m factors :
then ?t * Ij, is to be supported by Zt and the equation (§} is
to be interpreted with / eF(<P(l} and with m as the restriction
of m to Zv x Z,.
B) Evaluation at any point st eO is a continuous linear map
%,;$,-*¦ k henoe an element ^ e H supported by the point x with
reduoed structure. B, U a two-sided identity for convolution;
Horeover, eveluatvon of elemants ? e H at the function X e
induces an augmentation
E.H-
¦k.
Note that if 0 is finite and reduced, the 8,'a are a basis of H
over Ic, and Binne St ¦ Sf = &n H is nothing but the group algebra
ifO} of 0.

toe
ABELIAN VARIETIES
C) The ordinary product of functions §r ® 0a ¦
. oo -product
ff, ind
This satisfies many identities, whioh we will consider later in jj
for finite commutative group schemes 6.
The interesting thing is that the elements L e H eit
uniquely to left-invariant operators DL:Ga-* (9g such that, rot
(¦Dc/)(e) = ?(/)¦ These operators DL are combinations of differ
tial operators (which for every open set V c O, map G>g{U) )
Qg{V)) and translation operators / i—t- f, /'(*) = /(*»), (¦»
map ffs(l7} to ei^V.a'1)). We need some definitions.
DarimnoH. A differential operator D on a scheme X is a
linear endomerphism of the structure sheaf 0X such that thai
era integer N > 0 having the property that if f e SRf+1, toi
ie! Am Df(x) = 0. r*e Jeo«f tucA ^ if mlini (A* order of D:
For example, the differential operators of order 0 are multijj:
cations by functions, and those of order 1 are the sums of derivl
tions plus multiplications by/'s. Now say L eH is supported 1
ALGEBRAIC THEORY VIA 8CHEMKB
6e which consists of i«
I
2= U
We then define an operator DL ffa ¦
of maps
whenever Y.ty c U, \ < i < », oompetible with restrictions,
define Dx to be the composition ¦
1® L
—,
It is easy to see that Dt is a sum of opetators D^, whioh are {
by differential operators of order <<^, followed by right transit
by O(. In particular, if L e H,, then Dh is a differential opersto
Moreover 2>? ia left-invariant, i.e. the diagram
107
¦+<!>,,
oommutes. In fact, substituting the definition of DL this diagram
becomes the outer rectangle of
m*®I,
-> So
The lsft-squace commutes by associativity of the group law m
while the right square obviously commutes.
It follows immediately from the definition that if/ e Qg{U),
•ad a, e tf, 1< i < n, then D(/ ia defined at e and DJ(e) — L{f).
The wrreapondence i i—*¦ DL generalizes the isomorphism T^z
{left-tnvariant derivations} used in defining the Lie algebra of 0.
In fact, the tangent spaoe T^o can be naturally identified with a
aubspace of H:
and if L e T^s, then BL is the unique left-invariant derivation
with value L at e. An important fact is that convolution of L'&
goe» over to composition of operators:
-0,
'L.L,
Tbs proof is (straightforward and is left to the reader. In parti-
goIm, (•) shows that the Poisson bracket of left-invariant deriva-
derivations can be interpreted by a commutator with respect to con-
TOlution product ia H and that the p* power on left-invariant

10S
ABBLIAN VAMETIES
derivations can be interpreted ae j) power in H. Note that ifl
is commutative, then m* : ff0 -* &B%g is oo-commutative, i.e.
switch faotors
ooramutes, and therefore oonvolution product in H will
commutative too
This gives us a second proof that if G is commutative, th
the bracket on its Lie algebra is zero.
To illustrate hyperalgebrae look at the simplest en
Q — p, and 0 = ot. Writing
^-SpeotfXJ/l^-l)
then in the fitst oose all left-invariant diffarential operate^
are given by f
and in the second case by
Thus H = t{D]}(D* - Z>) or = i[2>]y(H») where D = X
12 Quotients by Finite Group Schema. An actum of a
scheme 0 on a scheme X is a morphiem
ft: (J XI-+I
ALGEBRAIC THEORY VIA SCHEMES
uoEi that
(i) the composite
106
) xl
w the identity-
(ii) the diagram
GxQx X
*
¦> OxX-
QxX 1 > X
a commutative.
(Here, as usual, <s is the identity and m is multiplioation.) This
is equivalent to saying that Q(8) acts on X(8) for every aoheme 8
(or even every afflne 8), funotorially in 8. It ia also equivalent to
saying that for every 5-valued point z g 0(8), we are given an
tutomorpbism over 8:
Such that
(i> if*.
(ii) if/: St ->-3t ie any morphism, x: 8t -+Q is an jS^ valuad
point, so #./: 8X~+Qia an jS^-valued point, then
¦+ X x8t
commutes

110
ABISLIAN
Explioitly, the T, s are lnduoed by ft via the formula Tt =
(ftoooflj x %), Px) where cr:XxO-*-0xX is the switeh
morphiam; and conversely, if we let 8 = Q and take 3 to be the
<?-valued point le: 0 -+ Oof G, then ftoJ^iXxG-^Xis just ^
(except for the factors being reversed)
A morphism/: I->7ii said to be 0 invariant if the diagram
GxX
is commutative, i.e. in terms of S-valued points g and * of G and
X, f{p(g, x)) = }(x). In particular, taking 7 = A1, we get the
notion of a (^-invariant function. We lay that the action of 0 on
X in/r« if the morphiam
<tx x-
•X xl
is a closed immersion.
The action of a group Sont scheme X has a differential-
geometric as well as a set-theoretic aspect. Let H, be the part of
the hypeialgebra of 0 with supports ate, and let L eH,. Then L
defines a differential operator DL on X via
It is easy to check that (a) Dj^^^D^vD^
identity, and @) if L 6 T^g c H,, then J>t is a differential
operator of order 1, i.e. a derivation from 0t to 6Z. In particular,
the Lie algebra of 0 is represented by derivations of <SZ.
Let & be a coherent sheaf on X. A lift of the action ft to JP is by
definition an isomorphism A: p\{&) >- f>*(^) of aheaves on
Q x X such that the diagram of sheaves on O x O x X !
ALGEBRAIC THEORY VIA BCHEMEB
HI
<m X !,)*(*)
where l = /t ° (j>,. fj), 1 = /t » (m x lj) = ft 0 (]g x ft} and p(
is the ith projeotion of O x 0 >! X, is oommntfttive.
A more manipulable way of defining a lift of an notion A to & is
to reqqire, for every tf-valued point/of 0, an automorphism^of
the aheaf ^® ffson 1x8 covering the automorphism
XxS
¦> X xB
eooh that A) the V<> ore funatoriat in/, and B) A,., = A,o \.
Having stated these definitions, let us generalize the principal
results of §7 on quotients by finite group* to quotients by finite
group schemes. The following theorem is proved analogously
to the firat proposition of |7, so that we content ourselves with
stating the necessary modifications
Thiobhk 1 (A) Let Q *e a finite group icheme acting m a
itheme X such that the orbit of any point is contained in an afline
open tubset of X. Then them it a pair (T,v), whereY is a mh*m«
and n: X -*¦ Y a morphum satisfying tkt following conditions
(i) as a tapohgieat spate [T, v) w the quotient of X for the
aUton of the underlying jtntte group
(ii) the morphUm n: X -* Y %t 6-invartant and if vt{t>x)a
ienotts the tvbsheaf of n,{0x) of Q-*ntariant fmettant, tie natural
Imonorphism &r -* n,(Hj.)a it em ttomorphtm

112
ABEUAK VABIETIES
ALOEBBAIC THEORY VIA BCHKMKfl
113
The pair (Y, it) w emtquely determined up to isomorphism by these J
conditions. The morpMsm * is finite and smjective. Y will be denottd f
XjQ, and it has the fvnctoriai property: V G-invariant morphisntt
f: X -*¦ Z, 3 a unique morphism g: Y -*¦ Z such (Aot/ = o = ir.
(B) Suppose further that the action of G is free and G = 3peo( J()(
n — d\mkB. Then -n U a flat morphUm of degrte n, i.e. nt(t9x) u
a locally free Or-modvXt of tank n and the subseheme of X x X
defined by the closed immersion
w equal to the subseheme X xrXcX x X Finally, \f & u a
coherent Sy-module n*^* has a naturally defined 0-action lifting
that on X, and
is an equivalence of the category of coherent fly-modules {resp. locally I
frw tPy-modulesof finite- rank) and the category of coherent &x-modulu \
with Q-aetion [reap locally free t>x-modvles of finite rank with
Q-action) '¦¦
Pbooi 09 (A) Aa before we are reduced to the case when X =
Spec A isaffine. Let B be the ring of O, c: S -»¦ k the evaluation
map at e,m*: R->- R®t Ji a-nd p*: A -+X®llAth<i horaomorphismi ;
of t-algebraB induced by m and p. Let JB=jJo={oe4ifi*(a) = 10o} :;
be the algebra of CJ-invariants in A. LetNnij: JJ®^ -+A be the .:}
norm mapping (defined ainoe R®tA it free of finite tank over A),
bo that Nm is a homogeneous polynomial function over A of degree f
ti = dimt(Ji} which is multiplicative. Define it: A -* A by putting :
N{x) =¦ Um(ft*(«)), so that N is again multiplicative and it-homo, i;
geneouaof d^ree n. Weossert tbet N(A)cB. ;;
To prove this, we have to show that for any a. e A, :
ft*(Jtf(«)) = 1 ® JTa. For any t-&lgebra B, denote by Nmj ¦:
the norm mapping R ®kB -+ B, Define <f>; A-* R QtA and:;
^ X ®iiJ®«^ -+X®tS®tA by setting ;J
Uote that if/: B -»¦ 0 is a homomorphism of k algebras we have
/oNmJ We thua have
N ^* v Nm,,
Now, if we oonsider it ®tj9 ®tJi as an R ®^d algebra v a
the hamomorphism ft ®^4 -+ iJ®t Jtg,^ given by ij ® m—»¦
I ® ij ® a, i/i is an automorphism of the R (&tA algebra
so that Nm^o^lfm^. Thus we obtain
. A, ® ^) <. ^*
Thia proves our assertion
Now, Q also acts on X x A1, (by acting trivially on A1), so that
N\ A[T] -*A[T] ia also defined. For any as A, put XM{T)=N{T-a).
Then x,(T) = T" + s,!*1 + ,.,+t^ id 0-iD?ariant, and is the
characteristic polynomial of the endomorphism of the free >4-module
B 0t A defined by the element ft*(a). Since *®l:.B®f.ii-»-Ais
aurjeotive and (t® l)(^*(a))=o, ^*(o)-a defines the zero map on
the quotient A of StS^ jl(via «® 1), henoe detl^laj-a) =x«(a)
=0. The equation x.t«}= 0 snows that a is integrally dependent on
B. Thus A ia integral over B. Sinoe jl is finitely generated over k, A
is also integral over a finitely generated subftlgeb*a of B. Thus A
and hence 5 is a finite module over thia Bubalgebra, and so B it.
»lso finitely generated over *. Let Y= Speo Band let w: X~t-Y
be the induced morphism. Then a in finite and surjective. Using
the map S, one sees as before that w separates orbits, so that (i)
holds. Further iris clearly ff-b variant, so that we have an inclusion
6t Cwt{&x)G, inducing isomorphism of global sections, On the
ether hand, v^{ffz)l> is a coherent #r-module, being the kernel

HP1
114
ABBIJAH VARIETIES
of the <Py-homomorphism A : ir*i@z) -+ ir,(c^)®* <R, A(/) =e
fi*{J)—f® 1, and this shows that (PF aair^ftfyH. This proves (ii),
Pboojt oz (E) For the construction of the O action od **(^"}
tha basic fact is the following. If X y 7 Y Z are morphistoi
Mid IF a sheaf of <PM-module8, there is a natural isomorphism
•¦ (ff°/)*(#")
dition- if X >- F
/*&*(#'» satisfying the following oon
- 2 >¦ T are morphisms the square
19 commutative.
Returning to our special case the equality of the tiro oompoaitea
T enables us (by mnans of \) to define an too
O x X % X.
Pi
morphiSDi A: P*ty)-+ti*{$) Q =»»*(#") andonachecks that thuu
an action, using the aboTe remark.
On the other hand, let g be a, sheaf on X with a O ution cover
ing the action on X. A section a e jj(X) is said to be G-invariant
if A(pJ(tr)) => n*{<i). Localizing, we tave the notion of the snbaheaf
*V(g)8 of O-invariante of ff,(g}- This U clearly an Cjr-modold
which is coherent, being the subsbsaf of *» (g) where two <Pr-homo
morphisms *,(fl) fr w«(g®t-B) ooincide. At before, we h»s
oyident natural tranafbnttations 8{P);P: -*¦ vt{ir*(F)}g Mid
T(g):fft(ir,(g)(')-+g, and it is again suffioient to show that (whom
the action is freej« is flat, GxX ~> XXr X, and T(g) is *D
Isomorphism for every O-sheaf g on X (as in the proof of
Proposition 2, JT). In view of these remarks, sinoe all the dofini-
AiOEBRAjC THEORY VIA SCHEMES
US
tions loufvlizt), we may assume X — Spec A affine. Recall that the
freeneas of the action means that the morphlsm
: G X X
X x X
is a closed smmeraiou. Sinoe it ib (J-invariant, (p,pt) factors
through a closed immersion of OxX into X xrX. On the ring
level, this means that the homoraorphism
A A ®s A
fi®, A
is aurjecti?e. We have to show (a) that X is flat over ? = A" and A
jsinJBotive, (b) if g is a O-aheaf on X with g(X) = if, AQglP -*M
is an isomorphism, and (c) if M is projeotive over 4, M° is .B-pro-
jeotive. Note however that (c)ia an immediate consequence of (a)
and (b). In fact, it suffices to show that M" is ?-flat, i.e. the
functor N i—» N®d0 M° is exact on the category of .^-modules,
and since A is faithfully flat over A", it suffices to show
that N i—». (iVSjoif"}® A(,Az(N<S> Ao A) ®d(A ^^M^s
(N ®Aa ^) ® j ^ *s exact, and this holds in view of (a).
To prove (a), we may pass to the ring of quotients of A and B
with respect to S = B — SB, where HR is a maximal idea] of -B, so
that we may assume B looal and A semitocal.
If we eonsidar A0BA and B®tA as ^-modules through their
second factors, A is a aurjeotive ^-homomorphistu, so that JJ®tJi
u generated by /i*{A). Since A is semilocal and p*{A) generates
the free module B®tA, one shows easily that there are
{«,) A < » < n = dimtjf) such (hatr^*{ut) form a basis
over A, How suppose a A,,... \, eA I claim:

ABBLIAJJ VABTBTIES
is obvious by applying /i* and using the faot
if \ 6 B. To prove ¦*¦ use the fact that
l}(fi*o) in Jt<S>t -B®t A hence substituting
the expansion of /i* a, we deduce that !¦
The implication
that p*{\) =
ALQBBKAJO THEOBT VIA BCBEMES
H7
Since the fi*K) are linearly independent over J in U®, 4,
A®^*)^*^) ore linearly independent oybt S®kAinS®kB®kA, '
the latter being considered as an algebra over the former via the ¦;
last two factors. Hence the above equation yields that l®fi*\ =>
1®1®\ i.e. \eA" — B. Applying the homomorphism *®1 to '.:
the equation f**{a) = S(l ® A,)./t*(<ii)> -we get that o = SAi«i, «i ;
required. This proves (•), But(*) says that A ia a free S-moduk
with baaiis o,, ...,aB and that the homoraorphism A:^®^^ -+ ;
fi©,.ji regarded aa a map of free j4-modules via the aeoond factor,
takes the basis a,® 1 of tho module on the left to the basis fi*{a() of
the module on the right Therefore A is an isomorphism
We now prove (b). If Jf = g(X) ia the j4-module corresponding
to (j, we first interpret the action of 0 on Q in terms of Jf.
Firstly, M®SA and AQSM will be considered as A®tA-
modules, where the first and Beoood factors in A ® ^ A act on the
first and aeoond faotors of the modules. It is easy to see that these
are the two AQ^A -modules obtained from the ^module jtf bj
tensor product with respect to the two homomorpbiams A j
j4® B A,a*-*+ a® 1 and m—* I® a. In view of this, and the
fact that K<Z>tA is naturally isomorphio to A®BA, it ia easily
shacked that an action of & on Q amounts to an isomorphism of
j4® b A modules :
that
if M<g>aA$>gA, A®tM®tA, A®BA®aM denote tha
usual J ®B J®jj J-roodules, then the diagram
jlgpif (givl
conunotei.
t on the first third
factors
4 on the second factor
to be proven is that if Jf is the sub B module of M
N = {m eif JTtI®ra) = m®I},
then the natural map ,W® B A -+ M is an isomorphism. We may
rephrase the definition of N and say that N is the kernel of the
homomorphism of Jl-modules :
f Jf
Since A is flat over B it follows that ^®BJ is the kernel of the
komoinorphiam
In otber words
But the associative law (a), applied to the element
eA&A®M, says exactly that T(l® mj e j*f®j,d hao the
property described in this equation. Therefore regarding NQBA
and r{l®jtf) as two subsets of M<S>aA, we find N&BA 3 T(l®Jf).
Now both of them are modules over the subring 1®A cAQbaA.
Uoreover, N®^A is generated over this ring by the elements m®1,
neiV, and siuoe T(l®») = rt®l, these elements are inT(l®Jf).
Therefore N®aA=r{l<S)M) But finally the raeps

ABELIAN VARIETIES
AL0EBRAIO THEORY VIA SCHEMES
lie
M
axe isomorphisms, sinve ,d U faithfully flat over B, hence we obta
Mi isomorphism If ®BA sM. This is clearly the canonical
too, eo (b) is proven
DttmfmoN. A homomorphum f: X -> T of group schemes it,
eptmorphism if ft* nurjectiv* and f* :&T->-ft&z itinjeetivt.
Corollary 1 Suppose X itttlf t» a group scheme and 0 u
flntte normal* subgroup scheme, acting on X by right-tranalattoi,^
Then XjQ is again a group scheme, w : X -+ XjG is an
and0= ker(*).Convenely, iff:X-*TUanepimorphism of
schemes, and t/<? = ker(/) is finiu, then TzXjG.
In other words, for every group aoheme X we get a Galois-ty
correspondence between (a) normal finite subgroup schemes (jj
and (b) finite epimorphisms » : JC-+ Y. In foot, if the word "finite")
is dropped from (») and (b}, the oorrespondenoe is still oorreot;]
but we will not prove this.
Pnoor, First, say X is a group scheme and flcls finite i
subgroup. Let m: X x X -*¦ X be the group low and oonaider thij
solid arrowe in the diagram:
-+
XjQ x XjQ ¦
x/a.
Then XjG x XjG is the quotient of X X X by 0 x 0, and l|
claim that tr°m is a G x ©-invariant morphism In fact, i||
^i> Sv St a(* ^valued points of X and 0 respectively, thenf
where gt = (xilgtxt).gt eG(S). It is easy to oheok that the dotti
t Normal muni O(ff) la ¦ normal nmgnnip of i(ff) tar ivory 8. ;:|
irrow defines a group law m' on JT/ff in terms of whioh ir is a
jpniomorphiflm. Now since d xX = X xr X, it follows that if
x j;|eX(jS),thenw(i,)=w(x1)if andonlyif x^x^g some(reOC).
]p particular, if K = kernel (jt),
i! eif(S)*=* n-fi,) sijr^l-iss-i! =? some (f eQ{8)
thus C = kemol (ir).
The aeoond half of the oorollary is harder. Let ir: X-*- XjQ be
tie canonioal map. In view of the G-invarianoe of/, there is a
unique g: XjQ -+ 7 suoh that
X
commutes. By the first half of the oorollary, XjG m a group scheme
mdone oheoks easily that g is ft horaomorphism. In fact, pis also an
(pimorphism with trivial kernel and we are reduced to proving the
special case that an epimorphigm /; X -*¦ T with trivial kernel is an
iiomorphism, We recall that if g: 5 -* T is any morphism such that
}-l(t) is a finite set for every (e T, then there is an open dense
jet T,cT such that if j^ = (T'tr,) g)tf; St ¦+¦ Tt is a finite
morphism. In our case, say/lI#:.X, -* f, (a finite. Then for every
t e X, if Bt is right translation by X, we get a diagram:
res/
»/is a finite morphism from X,• se to Yt-f{x) too. Since the open
. »etB F,./(k) cover 7, / itself is a finite morphism. Therefore, to
;»how that/is an isomorptiisni, it suffices to prove that the homo

ISO
ABELTAS VABIBTIEa
ALGEBRAIC THEORY VIA BCBBMS8
121
morphisui /* ; <PP -*/,Slx is an isomorphism. But we know J*\
injeotive, and by Nakayama's lemma /* is surjeotive if for,
j/e Y the map
is sin-jeotive. But Speof/jffj® flySH,) is the fibre /"'(y), and
the fibres of / are iaomorphio by translation to the kernel of
This kernel is trivial, so we deduce that for all y e Y, S~Hy) is oti'
point with reduced structure Therefore /* is surjeotive, and jj I
an isomorphism
Corollary 2 Let Y - XjG at in tht theorem Let g be
coherent sheaf onX acted on by G. Then there \s a natural uomorph
Prooi1 Qiven the a'tnation:
and o sheaf g on X, the natural horaoraorphism/*(/1>(g)}-*-(ii;
'*//'h),oc what is the
V. fl /l{9(fl» /p
this is an isomorphism. In fact, the problem being local onT
Y', we may assume both (and hence all four of X, Y,X\ Y') affli
and the assertion is then obvious. Apply this remark with X =
f=g=7rl to deduce that "r"(ir,(fl))-j>ttpt{jj)wheref>j;XxrX-,?
are the projections. Denoting the t* projection of Q x X by ),, at)'
using the isomorphism ()i,Q,y.Gx X-*XxrXand the
on n, we get that »¦(«¦»(j))
proves the corollary
. ffe now want to prove a theorem on the Euler characteristic of
Averse-images of coherent sheaves for a class of morpiiisms Ear
this we introduce some definitions.
Let 0 be a finite group schema acting freely on a scheme Xsuoh
that the quotient X/O exists. Let F be a finite scheme on which G
iota. Then G aots on X x F in an obvious way. It is easy to oheok
that this action is again free, that the quotient P=Bx F)jG
eii«ta, and that we have a natural morphism U y V = XjO.
The morphism it obtained in. this way will be called the jtbration
with fiber F associated to the principal Q-hmuOe X->-XjQ. n is
finite and flat, so that ir,(<P[,) is locally free over Sr of constant
rank. We shall oatl this rank the degree of n.
Tbkobbh 2, Let v: V -»¦ V ie a fibration with finite fibers
auociated to a principal 0 bundle over V where Q is a finite
povp ichetnt For any coherent shtaf & on V wt have
It suffices to prove the theorem when Q sots freely on
V and ? — V/G since in the general ease, we have X x F-t-
Jx FjQ -*¦ XjG and the theorem would he true for the composite
Hid for the first morphiam, so tliat it is also true for the second.
Thus we assume ? — VjG for » free action of 0 on U. For a
coherent sheaf J^ on F, let./ be tiicaheaf of ideals nnuilnhtt ing jF,
aid sail the closed subschemes of V defined by J the aii[>|iur( of 9.
Then W is a coherent shoaf on thrs subaclienio. If the theorem
imct true, since V is a noetherian space, wo oan find H-n SF with
ilipport ?'c? for which the theorem is not valid, whereas for any
coherent g with Supp g J Supp f, the theorem is valid. Set
\1'<=• v~l{V). Then, using (B) of the proposition, one sees that V
i« ft principal G-space over V. Replacing O and V by V and V
respectively, we see that we may assume the theorem to be valid
jrhenever Supp y JT, that is, whenever Ann P ^@). It is further
cW that if in a short exact sequence of coherent sheaves on V,
lbs theorem holds for two of the sheaves it holds for the third
(tenwmbet n-isflftt).

ABEIJAN VAWBTJES
Now, if 7 were reducible, we can find a short exact i
O-*-(j1-»-^r-*g,-»-0 with g, having proper support 5 V, and t|
theorem would hold for &. Hence we may sqppose V irredudb
If J is the subsheaf of nilpotent elements of V and J ^ 0 we hftl
W\JJF have supports proper closed subsohemes of V. Thus we mi*'
assume V reduced and irreduoible. Let r be the rank of the gene||
fiber of & Then there is a sheaf of ideals J on V and an injjl
tive homomorphism J' -+ JF with ookernel having proper suppoj
Thus, the theorem holds for & if and only if it holds for J\ m
again by the exftct sequence 0 -*J -+ u?r -+ OvjJ -* 0, we see that
theorem holds for > if and only if it holds for <V We see there!
that it suffices to prove the theorem for one ooherent sheaf on V,
non-zero rank. But then, vt(ffB) is such a sheaf, since by
2 to the proposition,
=(degir)x(ff,<<Pp))
Rbmabk. The reason we have insisted on an associated
space, rather than confine ourselves to a space on which 0
freely is the following. Let /: G-*- V be a unite Stale
everywhere of degree n. Then / oan always be realised as the
oiated fiber space with finite fibers of a principal 2(n)-space, v
?{n) is the symmetric group of order n. In fact, in the n-fold
product UxTUxT ... x r V, consider the set P of points (ij,.
auch that Xf ikXf for i =?j. Then P is both open and p]
U x TU x r... X TV and is stable for the natural action of t[\
Further, P is etale over V, and V is the set-theoretic quol
P by ?(»). It clearly follows that V is the seheme-i
quotient of P by ?(n) and P is a principal fiber space
structure group E(n)over V. Let F be the reduced saheme
point* [], 2 n) on which 2(n) acts naturally. Define
P x F~*-7 by ((z, a;J,t)i—*x,. This map is invariant for
notion of S(») on P x F and V is the set-theoretio quol
Since P y, F -f V ia again 6tale, V is the scheme quotient
Pxfby
ALGEBRAIC THEORY VIA BCHEMB8
123
Thus, the theorem e applicable to any finite dtale morphism of
oonatant degree n
13. The Dual Abdian Variety in any characteristic. For a Line
bundle L on an abelian variety X, we had earlier denned a
closed subset K{L) of X as consiating of the points zsX for whioh
T*(L) fd L, and we had shown that it is a subgroup. We shall now
define a structure of aubscheme on K(L). Namely, consider the
etandard line bundle M=* m*(L}® p*{L)~*® yj(i)^1 on Xx X
and define ?<?) to be the maximal subscheme of X such that
H\K{L) X X is trivial. (See { 10).
We can interpret the fi-valued points of K(L) roughly as the
get of 8- valued points /: 8~+ X such that L is invariant under
translation by /. Namely, X, = X x S, ftnd let T't
be the automorphism of Xa induced by / {i.e.T/p, *)=(*+/(*)i*)
in terms of revalued points x, 9 of X, S). Let Ls be the induced
line bundle p*L on X3. Now when iS is a big space, the condition
Tf(L3) 2 Lt is too strong; for example, Ls and T*(LS) can be
isomorphic on the aeta XxU{ for {17,} an open cover of S, without
: being isomorpbio. The correct oondition to look at ia:
Tf(Ls) = Ls®pt N, N a line bundle on 8 (*)
Then I olaim that (•) holds if and only if/ is an tf-vatued point
¦ 71 i>,
Pboof Note that the composite 1x5 —'+ XxS —>¦ X
m just the oomposite IxS * XxX *¦ X, bo Jj{La)s
(lzxf)*m*L. Thus L^^ is trivial while Tj(L9)\mxa =f*L

121
ABELIAN VARIETIES
ALGEBRAIC THEORY VIA SCHEMES
]2<S
Now whenever (*J holds, the N wh ch ooours can he identified hyl
restricting both sides to @) X 8:
Thus (•) holds if and only if
But {1
if and only if (I* xf)*M is trivial, which means, by definition-!
that / factors through K(L).
An immediate consequence is that K(L) (8) is a subgroup of X(8),"Jj
hence K{L) is a subgroup scheme of X.
Our next aim ts to construct the dual abeltan variety of Jt,:;
imitating the procedure in characteristic 0. Choose an L which ?
ample. Then K(L) is a finite group scheme. We define X to be the
quotient abelian variety XjK(L), Let s: X - X be the natural
homomorphism. As before, we wish to define the Poinoare* line |
bundle P on X x X by deflning it as the quotient of the line bundle J
if on X x X by a suitable aotiott of K[L) x {0} lifting the transU-f
tion action on X x X {it is easily checked that the natural homo- s,
morphism X x XjK{L) x {0} >¦ X\K(L) x X is an isomorphism).^
Recall that an aotion of a subgroup El c X on any coherent »beaf |||
,F on X oan be desoribed as the giving, for each 8 e Obj Soh,
action of the (abstract) group gifi) on #"s = & ®t ?>a lifting
action on X,, this aotion varying funotorially in 8 in an obvious
sense
Let us agree to denote nil objects obtained by base extension J
to 8 by a subscript S. Now, K(L){8) consists of the subgroup-5
of x e X_(8) such that if Tt: X3 -+XSdenotes translation by *,^
TZ{LS) x L3® Lq, where Lo is the lift to Xs of a line bundle on iS.;
On X, xaXs = (X x X)a, MBamj(?s)® j«f (Zj.)-1*
:it ^vin^cMt^ to give ao. isomorphism of these line bundles on the
-ubschenie Xs Xs 0s, This ia beeituee tiny two isomorphisms, either
onXj X-aX-s or on -^s xs"a, differ by multiplication by a unit, and
t[ie groups of global units on theao aflhemas,flo((XxX}g, 0*{xtlX)a)
Wid H"iX3 Xs 0a, ^iaKSo^), are both iaomorphio to
bence the restriction map #°((X x X)a, ff*(IyIlg)-^»-JB'
P* x ) is an isomorphism. Next, let V lie the 1-dimensional
Victor space dual to the fiber hfHRffL of L at 0 on X, and let
y x X3 be the trivial line bundle over Xs with fibre V. Then, if
>; XB = Xs Xa 0a
Thus, J^fifj) = JfB, and to define a lift of I
to Ma or equivalently an isomorphism of T^m{Mt) with if
Vheie all the isomorphisms are canonical. Therefore, we can choose
s unique lifting of the translation T(J 0) to Jtf$ by requiring that
this lifting when restricted to Xs x3 03, becomes the map lr X TM
on K x Xs. It is then easy to check that the action of K(L)(S)
op Ma defined like this is a, group aotion, as required.
We conolude that there is a unique line bundle P on X x X =>
X\K[L) x X such that its pull back is iaomorphio, as a line bundle
MtedonbyJC(i), to M onXxX. The restriction* of Pto{0}xX
*nd X X {0} are trivial. Further, sinoe $L: X ~* Pic*X is surjective
»jtb kernel K{L) we see that there is an induced isomorphism of
abstract groups X--^> Pio°X, and if a eX, the restriction of
$ to fa} x X, considered as a line bundle on X, is nothing but the
element of Pie°X correspond'ng to a. Thus, to eheck that X is a
dual variety and P a Poincare1 bundle on X x X, we have to
prove the following
Tbioebm. Let 3 be any scheme, and L a l\m bundle on 8xX
nckthatL 8 x {0} U trivial and L | {«} xX <=Fic°X for every teS
Tien there U a unique morphism $:B-*X tuck (hat L = {$><. \X)*{P)
Vaaos. Consider the line bundle M = ji*,(P) ® pf,(?)-' on
8 X X x X and let Ps be the maximal subschema of 3 x X

I2S
ABBLIAH VARIETIES
ALGEBRAIC THEORY VIA SCHEMES
127
over which this line bundle ia trivial. The main point is to i
that if w: Fs -*¦ 8 is the restriction to Psof the projection S xX~
it is an isomorphism. For this, it is clearly sufficient to show tH
for any closed subschema ft0 of S having support at one point off
(iS0 X jFj) -+$„ is an isomorphism. On the other hand, by deflnittj
of Fj, if rSi denotes the corresponding closed subscheme.of St;
formed with respect to the line bundle L\SlixX on ?„ x X, wel
SjXjr^^F-. Thus for the proof of the main point, we may i
jJ=Spec.B, where .Bin a finite-dimensional local fe-algebra. Furt!
if i U the unique point of S, one sees easily that the statement j
be proved remains unaltered if we replace It by L<gpJ (?[{»} xX)|
(since 3 xeX such that L\{s] x X = P\?}x X) so that we
further assume that L | {*} X X is trivial.
Now, for all points (»,x)g 8 xX, the restriction of M to \$) x
{x} belongs to Pic*X (since this is so for (», 0)); and there arej
most finitely many points (#, i) such that M restricted to fs} ;
{x\ is trivial, since there are at most finitely many xeXsuoh till
m*{L)® j>*(?)"I®pf(?)"*|Xx {x} — T*(L)®L~l is trivial. Hens
all the direct images Jt'p13it(M) on 8 x X have discrete
so that by the Leray spectral sequence, Hr{S
B*(8 x X,B?pllJt{iI)}. On the other hand,
, so that we have isomorphisms of 5-modu
0 -*
Therefore these bo homo logy groups are free jS-modules, On
other hand, consider the direct images R'pat(M).
M | {«} x [x\ x X e Pic "X for all x and is trivial only for *"= 0, |
these sheaves JJ'j'u,* {M) are concentrated at the point
e8 xX. Let 0 -»¦ Ke -*¦ Kt -*¦ ... -+Kt -* 0 be a complex of :
modules of finite typa over the local ring A = B <%>t $t J
{»,0) e j5 X i given by the base change theorem for direct in
Then B'(K.) = [ J?* Pi,,^^)^) are modules of finite length ov«|
and henoe also over #,_?. Now we have the
Let 0 bt. a regular heal nng of dimtnston g, ani
-t-Kt -+.,,-*¦ Kf -+0 6e a complex of finitely generated free
over S. If the ^(K.) are artinian module) we have
Since there is nothing to prove for g = 0, we may
assume g > 0 and that the result holds in smaller dimensions.
Choose an a; in the maximal ideal 3R of 0 but nqt in SR1, bo that
5 = 0\®z is regular of dimension g — 1. Putting K.~G>§)g K., we
x
have the exact sequence of complexes 0 >¦ K >¦ K >- K
y. o, from which we get the exact sequence
This shows that the B*[K,) are artinian By induction hypothesis,
*.R*+l(K.) is injeotive
for p<g - 1. But since i*»+t(^.) U artinian, *" kills B'+l{K.)
for some n and hence H*+t(K.) = 0, y <ff — 1. The lemma
is proved.
Applying the lemma to our complex of A free (and hence
(F^j-free) modules above, we deduce that R^p^^M) = 0 for
0<»<S, and that 0 -+ ^0 -+ Kl ->¦...-*¦ K, ¦+ N •* 0 is an
exact sequenoe of A modules, where N — &pa,, (JW^ s
W(S xX x S, M), which we saw above is JS-free. Now, the derived
niodulesof theoomplex 0-*K,->- #,„, -»-... -*?,->¦ 0, where E{=
Hom^J^, A), are again artinian, so that by another application of
the above lemma, we get an exact sequenoe 0-*^,^.., -+jK, -+
K -»¦ 0, where K is an artinian A-module Since we hove the exact
sequence
k
II
we get that the ookernel of ?, ® t -*¦ -K, ® t is of dimension one
¦over it, that is, K®jk = KjWj,K is one-dimensional. IWefore

12S
ABBLIAS VAMETIES
4-module, for an ideal St of A, and the free com]
(St) resolves Aj%. This implies that the homologiea of its
complex (Kt) are also annihilated by 8, that is, «.JV =0f. Stool
N is jB-free,« n B® I = @), that is, 5-* AjVL is injeotive. For wsv:
SR-primary ideal & in J, if f (?)) is the closed subaoheme defined by:
6 in 8 X X, we have
from which it follows that V{"&) contains the maximal subschem^
raof SxXover which Jf is trivial. Oq the other hand, a*(F(«)x
Mln*)*z - Honijfji/M AflU) a Aj%, and the natural map of
line bundles
is suvjeotive, since reduced modulo the maximal ideal, tha Ifldo
map on sections
¦4/*)
n
is surjeotive, and the line bundle JftM,,m)tX is trivial.
M\rcnxx is trivial, which shows that T,= V{B). In particular^
B -* fl*(rg1 tfra) = A\% is injective. On the other hand, nr).
is a closed subschema of r$ n(()xl auoh that the restriction i
{*} X P to ir"'(j) x X is trivial. Since {0} is the maximal aubacb
of X over which P ia trivial (practically by construction of jj|
ir"'(s) becomes the reduoed point («, 0) This means th
Aj% +HRbj1 = t, so that B -*¦ AjV. Is also surjective, henoe
isomorphism. Thus, it; Fg ¦+ iS is an isomorphism
Now, for any scheme S and line bundle Lon8 xX satisfying t
hypothesis of the theorem, and any morphism ^* 8 -> J(|
denoting by r^:S -»-iS X X the graph morphism we have
t In («t, for.vory o.«, rtmw thrt th. wdmiorphism of <?,) gtvm
mnltiplinmtim bj a ii null-hamottipic htmM k I* th. mi tnllomorphi
ALQKBBAIG THBOBY VIA BCBEMBS
129
eqelvalence (^x lx)*{P) s L-e=> P^ Motors through V3. Bo tha
theorem clearly follows from the fact that Vs is already the graph
of a unique morphism from 8 to X
We fame
@)
kifi
1
mx
Proof. Using the notations of the proof of the theorem with
= k,h trivial, we hove established that K ~ it. i.e.
ia ft free resolution of it. On the other hand, any two free resolutions
of one module are homotopioally equivalent, and the residue
field b ot any regular local ring such as tf 0,5 has a well-known
standard resolution, theEosznl complex. Namely, let *„...,x, eJH,
hft a basis p,} of ffij/as;. Let Lk be the free ff^-raodule with
the formal symbols ^ A ... A %{K *i <»t <
bam Then
O-^-^i,,^ ^X,_
... Ai^ A A^
ii the Koszul complex. Then the dual complex
0 »¦ E% j-JEj >- vKt v 0
is hamotopio to the dual of the Kossul complex which it is easy
to we is still iaomorphio to the original Koszul oomplex. This
gives Corollary 1 by calculating oohomologies.
2 far on abdtan parttty X of dtmennon g
In fact H*(X 0) is isomorphic to the p oohomology
? of the oomplex

130
ABBUAN VARIETIES
<P yKt<& k >¦ %!<& k > > Kf ® fe.—> 0 s
which is homotopio to the Koazul complex tensored with h. Not,
the differential operators of the Kosiu! complex tensored with k -
are trivial, bo that we have dim^H^iX, 0) = rank of module of ¦
j)-oochaina of Kosiul complex — I ' I
Cobollaby 3' Thtre, vs a canonical isomorphism of the tattgm ,
$ptu% to @) oft X and H^X, 0z). 'v
Paooi1. LetS = Spec J^ , so that the tangent anaceto X atO
18 canonioally ison orphiu to Hom((/S, X), where Hom , denotes ths-
Bet of morphisms of ?1 into X tnappmg tlie unique po'ot se of 8'
onto 0. On the other hand, we have by the theorem, 5
1 X) = (Line bundles on 8 x X trivial on {«„} x X] I
But bow we have an exoot sequence of multiplicative nhcav«j
i—y l+.e^—k §*xz—t-e*—> i, I
and as sheaves of abelian groups, -1 +1 (Px =: 0X Therefore tht>.;
cohomoktgy sequence gives %
0 »- W(QX) y H*(8 x X.0Z,x) > H (X, (PJ) I
and this gives a natural isomorphism of the tangent space t« Out If
with BHX^i), at least as abelian groups. It can be oheoked thtii
this is actually an isomorphism of fc-vector spaoes. |
Now, let/: X -* Y be an iaogenjr of abeliait varieties. BytU*
theorem, we get a unique homomorphiam f;Y-+X of abelioot
varieties such that if FZ,PT are the Poinoare bundles on XX
and Y x, Y reBpeotively, we have
/
Henoe, if we denote this line bundle by Q, on applying the prop>|
aition of 518, we get that «
ALGEBRAIC THEORY VIA SCHEMES
131
xiQ) " deg /. xfPz) ^ deg ] xiPr).
and since X{PZ) -=xiPT) =(- W by Corollary 1, we get
4, For an isogeny /: X -+Y of abelian vanetit)
deg/ = deg/
. For eoery line bundle L tm X, the $tt-Oux>retie
: X -* Pio" Imltm morphtsn, and K{L)t with
iU tfJieme structure as above, i& tt* kernel.
. In faat ^ is the unique morphism from X to X
that
Now by definition K(L) is the largest aubschemeS of X such that
the bundle on the right is trivial on 8 x X; and in view of the
Universal Mapping Property of Z, ker(^L)ie the largest aubBcheme
8 of X such that the bundle on the left is trivial on 8 X X
Sy«lIBTBIO DMTNITIOH OF X
We wish to set up the relations betweeo X and X iu an obviously
symmetric way. Define a dimioriai comspondtnee between two
ibelian varieties X, T of the same dimension to he a h'ne bundle
on X x Y whoBe restriotions to {0} x Jf and X x {0} are trivial.
Pbofositknt. For a divuonal eorrwjwrtAince Q betaxm X and
7, Ae faflotntng are equivalent
H T\ert U no svbtekeme ZojX dtfftreat from @) tuch that the
ratnttion o} Q to Zx Y U trtmal.
(b) There is *o iubsdmm* Z a} Y different jrom (O)«oS(AafO
rutneted to X X Z' is trivial
(o) fhe absolute value of x(Q) u 1
Vitn these condittotu) hold, Y ta ottnonicalty ucmorphic to tht dual
«l I, and X t* eanmucaily immorphte to the dual of Y.
By symmetry, it suffices to Bhow that (b) «=> (o) Now
¦ Y »¦ Xsuoh that {lzxf)*[P)=Q. Now /

132
ABBLTAH VARIETIES
ALGEBRAIC THEORY VIA SGHHMIg
133
has to be a homomorphiam since /@) = 0. Clearly, (b) ia equiv
to saying that/ has trivial kernel. If this holds, then sinoe dim ]
dimX =-diroX,/iganisogeny, hence by Corollary 1, {12, / i
isomorphism. Thus we have to show that / ie an
If and only if \x(Q)\ = 1. By Theorem 2 of {12, if/ is an is
and the result follows On the other band, if/ hae positive (
aionat kernel we can choose a finite subgroup F c
arbitrarily huge order 4. The mop Jx x /: X x T * X)
factors as
bo that 0 is the pull-back of a line bundle on X X TfF. Then
<f|lx{C)l by Theorem 2 JIB and since this holds for arbiti
large dlX(Q)=O
CobOllabt (The duality hypothesis.) for any abiltan
X tht canonical morphism t: X -*¦ X defined by Out Poincari tn^
PonXxX (regarded at a family of line bundle* on .
by X) it an Mamorphwm.
Pbooj. In fact, the divisorial uorceBpondence P OB J)
fulfila (b) (or (o)), henoe also (a).
14. Duality Theory <rf Finite Commutative Group Scboms.'
ghout this section, tha ground field t is assumed algeb
oloeed, of positive characteristic p >0.
Let O be a finite commutative group scheme, so that O is (
henoe O — Spec (JJ), where if is a finite-dimensional
The group law gives us a map /i: JJ -*¦ JJ ®t B, the inverse (
us a mep i • B~+R, and evaluation at the identity t giv
augmentation 8; J{ -»¦ Jt. In the present case, the by
of Q ia simply the dual vector space JJ* of B, and we
the notation B* instead ofH. As in$ll, the group law f> duaB
to an associative multiplication
;'Since 0 is commutative, ^ is co-commutative and so /i* ia commu
Utive i.e. B* ia also a flnite-dimensional commutative i-algebra
-As in 511' the linear functional E is the identity element of JJ* On
jha other hand, if m: Jf ®t JJ ->• J? ia the multiplication of B, then
«•: B*~*B*®tX* will be a oo-asBopiative, ooHjommutative map,
Together with the dual i*: B* -*¦ B* of i, we get a group law and
jn mveree making the scheme O = Speo JI* into a second finite
| oornmutative group scheme. The identity point of 0 comsponds to
(be homomorphism B* -*-k gotten by evaluating linear funotionala
it leJJ- Thus, to every finite commutative group scheme Q we have
jSBXjiated in a canonical fashion another finite commutative group
--,joheine Q, which we shall call the dual of O. This oonstruotion U
;.dnotoGartier. WeihaEnowgive amore'geometric'definition (of
; : Oort tQLl)
For group sohomes 0 and B and any scheme 8, let Homa(G, H)
denote the set of morphisms f-8xQ -*8 x B auch that th«
diagrams
-Jxff
8
-t-SxO
f
(8xJB)xa[8xJB)
l.^lre commutative, where m^g and
the multiplications of (J
'j-|nd if lifted to 5x0 and jSfxJJ respectively. Such morphisms / will
l_:;iiU called d-homomorphisms from QtoH, One checks easily that
T is commutative HoiOjfff, H) o»n be made into a commutative
by defining/ + (f = mstH*{/,9). Given a morphism T-+8,
ahavean associated homoroorphism Homs((?, JE() -»> Homj,(O, H)

134
ABELIAN VARIETIES
ALGEBRAIC THEORY VIA SCHEMES
130
given by/i—t-fX3T, bo that for given G, H, we get a fi
8oh -+ Seta given by 8 t—* Horas((?, it}, and this is in
commutative group-valued functor when H is commutative.
Now suppose C 1=1 Speo jl ia a finite commutative group schoi
and B the multiplicative group scheme Gm. We then assert that i
funotor Soh -+ Ab, 8 i—* Hom^O, GM) ia represented by G, that-i
there ia for each 8 an isomorphism
0(8) a Homa(G GB)
which ia funotorial in 8. Now, if we restrict iS to vary in
open subsets of a fixed scheme, it is clear that both sides d«fi
sheaves on this scheme. Hence by standard arguments, it auffioesi
establish the above isomorphism as 8 varies through affine soheij;
8 = Speo ii. Let us denote as usual the objects, morphiams, <
obtained after base extension to ii by a subscript ii. Then the i
dinate rings of OB and OmB beoome bi-algebras over R, that j
they are algebras with 1 over it, with co-multiplication mapa At
A^^^Ag, etc., satisfying the nsual identities, and co-ideutijB
maps AR -*¦ It, etc The notion of homomorphiam of 1
then clear. We then have
B) =
where (AB)* is the if-algebra HomR{AK, B). Further,
where t is the inolusioa defined by i{<j>) = <j>( T). Clearly, an elemeij
a e Ax is in the image of i if and only if (i) a ia a unit in Ax, i
(ii) jtA{«) *=i® «. But in the presence of assumption (ii), (i),j
equivalent to (i)'*r(<*}=¦ 1, where fK: Ax -> S ia the homomorpbj^
given by the Identity. In fact, (i) implies that ta{a) is a unit, i
on the other hand {<ji® *h)(^W) = ?ji(a). tl'lat 'a> <*(«)* ="j
bo that (jgfsjsl; on the other hand, if (i)' and (ii) hold
t*: Ax
of the diagram
is got from the inverse, we have oommutativig
Spec ii
uld taking the pull-baok of the function a. by these niapa, we find
i.i*(a) = 1, so that a is a unit. Knw, since As Is Jidee of finite
tBnk we have a natural isomorphism
ind under this isomorphism, elements a satisfying (i)' and (U) go
* over into element / e HomR((Jj,)*, ii} auoh that/(l}=l and
¦'jIXY) = J(X).f{Y) for X, Y e {AK)* (since XY = nJ(X $ F)).
¦ ¦ i ¦¦ .**
ThoS we have set up a aet-tbeoretic bijection between Q(R) and
0pmB(O, G_) natural in ii. We leave it to the reader to check
thit this is an isomorphism of groups.
V In particular taking 8 = G in the above isomorphism, we get a
.morphiam flxS-* GB corresponding to the identity of G(G),
Refined by the homomorphism of it-algebras k[T,T~'] -* A*<&tA,
v/j1)—f. 8, 8 = the 'diagonal element' of A*®M A (which oorreaponds
'Mi lj?Homt(jl, A) under the natural isomorphism A*g>tA »¦
**'ifomt(jl, A)). One checks easily from this that this, 'universal
a-chfjttcter' Gx 0 -*¦ Gm is in fact a bilinear map of group scheme*
fin the obvious sense.
:SKEiamflss. (I) Suppose 0 ia a discrete (i.e. reduced) finite
ttnmp at order n prime to p. By the above, the geometric points
i| Q form a group isomorphic to HomjO, h*), which is of order n

136
ABBLIAN VARIETIES
On the other hand, if A is t he ring of functions of G, dim A = dim ill
= n, whioh shows that G is again reduoed and is isomorphio tu j|
diaorete group to Hom(G, b*). $
iLQBBRAIO THEOBY VIA SCHEMES
137
B) Next suppose O ia reduoed and isomorphio to Z/p"Z.
reduced 0 let lc\G} be the group algebra of G. If wa identify aoy!
jeO with the linear form A -+ k which is evaluation at g, we get?
an isomorphism of vector spaaes Jt[G] -^i- ^*, whioh is trivially see/:
to be an isomorphism of algebras. Henoe S has coordinate ring a?.
isoraorphio to the group algebra of Z/p"Z, i.e. A*s i(,r)/{.X*" -1$
where Jf corresponds to evaluation at the generator T g Z/y"?§
On the other hand, for f.geA, T(/.(r) = (/ff)(T) = /<T). ({!)=¦:;
(T® l)(/®ff), which shows that the co-multiplication on A* »¦'
given by X i—*¦ Z0I, Time we have an isomorphism ;:
hence also (jt^J = Z/jTZ.
Now, let O be any finite commutative group scheme. We
say that G is of type i(local) or r(redueed) if the underlying
of G is a single point or if G is reduced respectively. We shall eiy:
that Q is of type (I, I) (resp. (!, r), (r, I). (»¦.»¦)} if 0 is of type I and
G is of type Ifresp. 0 of type I, G of type r; etc.). We shall aho*
that any group admits a unique decomposition as a product
of groups of the indicated types. In faot, if G" is the connecteij.
component of identity in 0, considered aa an open and cloned
subscneine, and if <?„, is the reduoed group, the closed immersion**;
G° c • Q and (?„,, = • Q induce a homomorphisin G*x Ojf
-» G whioh iB clearly an isomorphism. Further, this deoomposU*;
tiou of G into the product of a reduoed sud a local group u"
clearly unique. Thus, it suffices to show that each local (reap.7
reduced) group is uniquely eipresaible as a product C^ xQ?;
(resp. O^xGrf). Now, if O ia reduoed, G is uniquely ezproaaibii|:
ag a produot (?j x Gt where Ql ta of order prime to p and Qt is a
o croup- By the above, G1 is again reduced and Ot is local, which
proves the assertion for reduced groups. When G is local, split G
into it-8 looal and reduced part*. Dualizing back, this implies a
upiqae decomposition of a local 0 into groups of type (I, r)
Hid ('. I) respectively. This proves the assertion for local groups.
It follows from our discussion that the only groups of type (r, r)
are those reduced groups of orders prime top, henoe direct pioduots
of oyolsc prime power groups, the primes being distinct from p; that
the only groups of type (r, I) arep-groupa, henoedireot produots of
Z/p"*Z's; and that the only groups of type A, r) are duals of
fl-groupa, henoe direct produots of un/s
There are plenty of examples of local-local groups. For instance
the groups a^> are local-local. In fact, since Spea -i— cannot be
decomposed as a prod not (the tangent space being one-dimensional),
it suffices to see that it is not isomorphio to ft,*. Sinoe a, is a quotient
of Ojji, it even suffices to see that a^ is not isomorphic to jtt But
now, if i is the linear form on A = jt[JC]/(JC) defined by
[
ire have f»(J») =(f ®^)(A®X + Ap® 1)') - 0 if t< p - 1
whioh shows that A* has ni[potent elements
So far we have developed the airole of ideas involving homomor
phisms from G to GB. Next we turn to homomorphisms from
<J to G, and related results. We fix the notations Q = Spec B,
G = Spea ]t* as above and we let roman letters x, y,... be elements
of JJ; greek letters a, 0, ... be elements of B*. Let i and «* be
the oo-mu [triplications in R and B* respectively. We recall that in
jll, we saw how R* operated naturally on the sheaf &tt. In our
affiue case this means that there u a natural inclusion

188
ABEUAN VABEBTTE8
. Homt{Jt, B)
Explicitly, if a: B •*¦ k \a an element of S* we get a
DMt B -*¦ M by the composition
B > Jci3f ¦» > Ji
In particular, if «A) = 0, «(SJIJ) = (OJ, then Dt U derivation
of B over i. The operators jt>. are all translation-invariant
the usual sense. Moreover the transposed maps D*: B* -+
are just the compositions
ALGEBRAIC 1HB0BY VIA SCHEMES
mult
T®t B* ¦
i.e. the maps 0 i—*¦ au/1, multiplication by «.
As an application, we can interpret Hom((J, G#) in a new
(sinoe a homomorphism 4 from k[X] to 8* is determined by
image n =* #X), and ^ is compatible with eo-multiplication if i
only if #*a =a® 1 + 1®«). Snoh o's are called primitive element;
of B*. If we associate to any «ei' the map D± which is
transpose of multiplication by <x, we see that primitive elementij
correspond precisely to invariant derivations of Jt:
sr D: B >B O an invariant derivation);
s Lie(O)
The oonclusion is that
Hom(G, G,)sLie(O).
Sforeover, the p* power operation in Lie(G) is obta ned by ta
D to D°,,.aD, which corresponds to raising at to its p°* poi
which corresponds to composing a map ?: 6 -* G, with
Frobenius ?• G,->G, (since F*(X) =X*).
Now look at the following type of group
139
A group scheme G is of height one if if comuU o/ o
e. »«( i/ *' = I), >r oil « 6 !ffi,. {G need nol H
¦j It is easy to aee that, as mtotne*, suob groups 0 are iwmorphic
to SpeoW-^ip ¦¦¦> ¦3^]/(^Ti ¦•¦> %*))¦ In fact, choosing JE^ ..., JB
gffi, which induce a basis of S(f,/1K*, and letting R = Ti(!e), we
ftiii that Jiisa quotient of t{Xv...,X,y{X* J»). On the other
hand, if we use the fact that B admits derivations D^.B-rB
gach that A(Zj) ^ S(i {mod TO,), then it ie easy to see that there
(Bri be no relation of linear dependenoe over k among the
pionomials
*i
0 < a, < P
We have already seen in $11 that any group soheme G has a
rnwimaS subgroup scheme G**' c G of height one, having the same
Lie algebra aa 0. Then the analog in characteristic p > 0 of the
olasaJDal equivalence of categories between Lie algebras and germs
of Lib groups is the following
Theorem. Thr functor Gi—»Lie G sets op an equivalence of the
categorku of finite, group wheinet of height one and finite-dimensional
p-Lir, algebras oeer k {i.e. a finite-dimeicsional vector space with
bfttc.tet and jf" power map, as til §11).
We shall prove this only for commutative Q which correspond
to p-Lie algebras with trivial bracket.
Proof. For any Jfc-vector space g with a j»-linear map « -+ a^'(
let t/(g) be the t-aU|ebra 8(q)jI, where / is the ideal generated
in the symmetrio algebra jS(g) of g by elements of the form a(p)
«', (teg. Note that if at, ...,«» are a basis of A, then I1^,O<T| < p
*~i
we a basis of &[$)¦ Define a oo-muitiplication«: ClgJ^-lTlgJigjyig)
bj putting for (teg, J«=l®« +a® 1 (check that this goes
Sown to U(q)), It is easily verified that G(g) is a commutative
ioite-dimensional bialgebra, and is a oo variant Ainctor in g.

ABBLEAN VARIETIES
ALGEBRAIC THEORY VIA SCHEkES
Put JJ(g> - P{g)* and 0<g) « 8peoP(g)» oonsidered as a grot|
scheme
We ahall prove that the two functors
wo imrersea of each other.
We prove first that for any veotor space g with a p-linear
G($) ia of height one and there is a natural isomorphism g
Lieg(fl).Wow, we have a natural inohmiony ¦ I7(f))offlintot
primitive elementsof tf (g), which giveaan injection ge . Lie fl(n|
To show that Ofgjia of height one, it suffices to show that for x
fl(()) belonging to the maximal ideal of 0 in G(g), st» = 0. If
identify G(g) with a subalgebra of Endifi(g) via D, it will ml
toshowthntif aeG(g), Dt(&)=*0, Sinoe (/(g) is generated by g
it even suffices to ahow that for a 6 g, -D.te*) =0, which
dear since D. ia a derivation. It only remains to show that g
Uo<?{g) ia surjective. But if »=»diaij( g
by onr remarka on a basis of G(g), p" = dimt^g^, and by ail
remarks on the structure of r(ff0Uj),y*= dimtJl(g). Thus n = tn.
2^
Seooudly, let G ha a commutative gronp scheme of height qjv
with coordinate ring JJ. Let <?; Lie(G) -+ B* be the usual ind
mep. We have seen in jll that ^tt} = <?(«)», so <f> extends toi
hemomorphism
? P(Lie G) -—y B*
Since for a 6 LiefO), « and <?(») ere primitive with respect to till
eo-multiplication in E7(LieG) and .B* resp., ^ is a homomorphisg!
of bi-algebras The transpose of <fi is again a homomorphism p
bi-algebras j|
$*:M y?(LmG) |
and passing to the Speo's we obtain a homomorphism of grou
schemes:
f: O(LieO) >¦ Q.
By the first part of the proof, Lie G is the Lie algebra of (JfLie O)
jnd it follows from our construction that the differential off
, induces an isomorphism from the Lie algebra of <J(IJe 0} to that
pf Q. Now by Nakayama'a lemma, any morpbism ftX^Yot
jobsmes with one point each, whose differential la inject ire, is a
closed immersion. This appLea to $'; in other words, ^* is surjeo-
tive. But if n = dim,Lte G, then p" = dim^Ji = dirotS(Lie G)
Therefore $* and ^' are bath isomorphisms.
Oobollaby. // G is a tiommutative group tckeme of hagju one
¦ fte homamorphtim pg (mult, by p in G) is zero
Phooi. In foot, multiplication by p kills Lie Q so the result
follows from the Theorem,
It 0 = Speo B is a group scheme of height one, then not only
ia its structure as a group determined by its ji Lie algebra but
also to give on notion of G on a scheme X is equivalent to giving
m action of Lie G on X, i.e. a Jt-Jiiiear map ti—t Dm from
lie 0 to derivations D,;{)x-t- 0x such that
I (I) W,. fy) = 0, all a, $ e Lie G
: (If) !>„-wmr, i
/We oan see this in several stops
: (I) To give a map a i—» Dt as above is equivalent to giving e
::,: homomorphism of algebras
¦.]:¦;; D; (/(Lie G) —»- TnS(&z)
- where D)B{@Z) ia the algebra of differential operators on
0z, such that D carries LieC into derivations.
(II) Interpreting D as a map
¦/ 5

141
ABKLIAN VARIETIES
and using the fact that the coordinate ring it is
dual of U(l?e (?) we sea that to give D is equivalent
giving a transposed map
(HI)
n*:6\
R<g>tQz
The oondition that D{1) = 1 a
easily into the conditions that
commute. It oera also be checked that the oondition
D take Lie 0 to derivations, i.e. D commutes '
oo-multiplioation means that p.* is a homomorphism ofj
algebra*. ;1
To give a homomorphism p.* is the same as giving 0,
moiphism fi: G xX -> J and the two commutative diagranygf
with fi* say exactly that p is an action of 6 on X.
Finally, let as look at the dseemposition into pieces of a qom^'
mutative finite group scheme Q of height one. We get
ALOEBRA10 THEORY VIA SCHEMES 113
I both &!,, and ®it arfJ aSain °f height one Siace G^ is killed
for aonje n. We have seen that Lie (ji,) is one-dimensional with a
generator e such that ew =«. Nowths decomposition (•) induces
1 deooinpoeition
(«<) Lie G — Lie Ctl ffi Li* <?jj »= gi © fj><
It follows that g, has a basis e1P ... ,?,Booh that «J*' =e,. On the
other hand, since Qlf is again local, and g, is contained in the
maximal ideal of Tffi^t, it follows that for all meg,, a" = 0 for n
[urge. In view of the theorem, we deduce a ooroltary in "p-lineor
algebra:
CoBOLLABt. // Vitang vector tpacewititap-linearfnapx*—+&*
¦Aen u a unique decomposition, invariant under xi—*¦ rfT):
wcilhai V, has a basis xlt ,,,,i, jot vihwh x\r =ij
n-1* m a niipofenJ mwp on F,.
F a?«i V, art called the senu-timpU and rrfJpoJenl
/, atuf in tA« case of Lie 0, toe haw
o/ K
(Lie 0),= Lie %
13. AppKcatioos to Abdian Varieties.
: TaaoasM 1 Let f: X -+ 7 be on wosenjf o/ a6el«w
wilkhemelK Let f; 7-+X btitte dual map, toilh kernel K' Then
^ere C» a canonical iaomorphitm of K' u>ttS the dval KofK.
; Proof. If L is any line bundle on Fand^ el, then
tepraaents the lbe bundle TffJ'L) ® /¦i, and sinee
it follows that

144
ABRLIAN VARIETIES
In particular, if $S.L ia the zero rasp, then ^ must be the
map too; i,e./*L el'iog F => Le Pic°X It follows then that
any scheme 8, we have natural isomorphisma'
) b Ker [Horn (8, Y) -+ Horn (8, X)]
line bundles on
8x7, trivial on
8 x @)
line Infinites u]
8y,Y
= Ker
=;Ker
line bundleb on
8 xX, trivial
Sx @)
line bundles oj
I
Ti e htst isomorphism is correct because if a line bundle on 8x1
becomes trivial on Sx X, then it must also be trivial on jSx @)
But now 8 X Y ia the quotient of 8 X X for the froe action of ?
Thus according to the results in $12, there is a natural
isomorphism:
Lifting^ of the
action of K on
Sx X to actions
line bundles\ — /''ne bundleB\l
™"" ™ " (. X )\~
(l
Now »n action of K on 5 X ¦? X A1 is defined by a morphism ^
fitting iiito a diagram*
A1
IxA1
Pit
where /ig is the translation action of K on X, with S thrown in.
If A=p,o^i ia the induced morphism from K x Sx JTx A1 to A1,
then in terms of T-valued points, tiie lifted action can be
described as
h U x a.)
(it x+lt, A(i, ( z a))
ALOBBRAIC TH10BT VIA SCHEMES US
= K{T),*e8(T) ieI(T) oteA^l"). Smoe the aotion should ha
linear on A1,
X(k, i. x, k) = a.A{ib, j, x, 1).
Since X is a complete variety, for any scheme W, V[ffwxxte^^ir)
hence all morphiams ITxX-* A' factor through W. In particular,
\ doea not depend on the factor x in X. Thus the aotion is given by
To be ao action, we need
Afifc, + k,, t, 0, 1) = A(i, a 0 1) A(i, s 0 1}
A@, *, 0, 1) = 1
In ether words, X is given by an 5- homo morphism ^ Ky,8-*-Gm
Cbnversely, any such x defines an aotion n via
fi(k, t,x a) = (j x + ? ^(^ s) a)
Therefore,
' Liftings of the action of K oa8 x X"! s Horns(K G,,)
to actions on S x X x A1 J
Putting alt til is together, K ? K
Definition. An isogeny f;X -+Y of abtlian varieties U mid
la be of height one if, denoting by h[Y) ami ?(X) the rapectiv*
jmxtum fields, we Aaoe Jt(X)*c lc[Y).
We shall show that / ia of height one if and only if ker / is a
group scheme of height one. In fact, assume/ is of height one. $It
a the integral closure of <Pro in k[X) and <9ra ia integrally closed
Therefore <PTfl =Sz/i n k(Y)j {f]ffeXfi}, hence JRri) 3 {f\fe,
j(w}. Since ker / = Spec [^z,i>l^rfi^i.o}> tllis "'">*» tll«t ker/
ia of height ono. Cnnversely, suppose K = ker / is of height one,
and let R* be the biftlgcbra of K. Let U be any nun-void affine
open subset of X with coordinato ring A. The notion of K on U is
given by a hoinomorphism of E* into the algebra of differential
flpemtora on A, aucli that a act of generators of S* gelB mapped
into rector fields on U. The elements of A invariant under the

146
ABELIAN VARIETEBB
notion of K therefore oonaiat precisely of those elementB of
which are killed by these derivations; in particular they
X<r> = y» \feA), This proves that t(Z)» c Jb( 7).
Theorem 2. For all abelian varieties X, there is a one-one
retpondtnce between isogr.nies /: X -*-Y of height <rae (uji to
yftwm) and mtb p-Ltt algebra* of Lie X
Pboov. In fact, isogenies of height one are uniquely
mined up to an isomorphism by their kernels, whioh are
schemes of height one, or what is the same, subgroup-schemes
the maximal height one subgroup-scheme Xp = Spe
But by an earlier theorem, subgroup-schemes of Xr are in natu
one-one oorrospondenoa with p-IAe siibalgebras of Lie Xp = Lie
Example. The above theorem enables ua to give an oxam
of an abelian variety X admitting an infinity of distinct isogeni
X -f 7 of height one.
In fact, for every prime p > 0, there is an ellipt.io curve' j
unique up to isogeny, suoh that the ^ power map in Lie X
{Douring; of. §21), But if E is sued a ourve, and X = E x jt
any I-dimensional subapaoe of Lie X ia stable for the pa
map, and henoe defines an isogeny of height one.
ALGBBBAIO THEORY VIA SCHEMES
147
and one deduces by induction that for any »> M
Nov. by Theorem 1 of this section, OB ia the kernel of (p")J, oo
jt follows that there is an integer » such that for any n > 0
(ff.litd = (^/P"^)' Thus tne decomposition of 0B into its pieces
M follows:
Let X be an abelian variety of dimension g in
p > 0, and n = jo^m an integer > 0, r > 0, m > 1, {p, m) =:
We want to analyze the structure of the finite groop
X. = ker nx. Now, Xm and X^ are subgroup sohemes of Xm,
¦we have a homomorphism X^ x Xft -*¦ X*. ThiB is, in fact,
laoraorphism since Xm ia the (r, r)-part of XM, »nd X^ is the
of the (r,I), (i,r) and (t.JJ-partu of X.. As we saw in §fl, J,,
disorete reduced group i»omorphio to (ZjmZ,)*' Thus, it
to study the structure of X^, which we rename Om.
now that {G^ ={ZjpZ)'. Since X is divisible for «ny
JP
0
we have an exact sequence
¦<?.
where (?H is tooal local. Sinoe 0, is of order jP™ we deduoo that
there is an integer t > 0 such that r + « + l = % and 0* is of
order y.
V We shall show that the integers t = tz, » = »x and ( = ij are the
i»me for isogenous abelian varieties. It suffices to prove this for t
und e, since r+4+f=2#. Further, since Jf^tj, it even tufitoes to
verify this for r. Let /: X -•¦ T be an isogeny, with kernel of order fc.
Since /(X^) c 7^,, we have that the order of (X^,)^ is at most
¦:'.? times that of (y"),^. that is, p < k.p"T for all », hence
f, < rt. Now, ker / is a finite group scheme, hence is annihilated
by an N > 0, whioh showe that ker N^ d ker /. Therefore, lfz
¦?&¦¦ i 9 9
¦fictoriies as X —v Y y X. Thus, 7 >¦ X is an isogeny
i-iaArT<.rI. This proves that r, * and t are tsogeny invariant. In
.^particular, since for any abelian variety X, X and X are isogeaoua
:!im deduce that rx =rg =*x. Ihus, we have that
i iritii (J° locaMooa) of order p"*, 2r +1 =¦ 2j. In partieutar, wa see
Itiit r < j. The integer r is Milled the p-rank of X, and is an
invariant.
Now, since pz induces the 0-map on Lie algebras, we see thai
X-Lie(kerpx) = LieO, =Lie(f.,)r© LieO;. Since Of is
l-local, the p power map on LieOJ is nilpotent, whereas
'(ft,)' admits a basis e,.. t, such that ef = «,. We thus tee

148
ABELIAH VARIETIES
ALGEBRAIC THEORY VIA SCHEMES
149
that the p-rank of X equals the dimension of the semi-simple part;
of Lie X with respect to the p power map. The same result hold/
for Lie X, sinco X and X have the same p-rank. On the other hand,
we have established a canonical isomorphism Lie X tt H*(X, <Sty
Let F: &x -*¦ $x be the Frobsntus homomorphism F{a) = a*, and
denote the induced p-linear map.H1(X,<!'r)^#1(X1 ff^} again byi-
We shall establish that under the isomorphism lie X ~ H1(X, Ot)t
the p power map in Lie X goes over into F. It follows that tbe
prank of X is also the dimension of the semi-simple part of
H'iX.Oj) with respect to the Frobenins map F. Thus, we need
to prove
Theobkm 3 l/wler the natural isomorphism Lie X~ Hl(X, dx),
the p** power operation in Lie X joe* over into the Frobenitu map
Proob. First we give a desoription of the p power operation
on vector lieldB on a scheme X, using the functor X, analogous to
the one given for the Poisson braoket in §11. Let B be a vector
field, interpreted now as an automorphism of X x Spec A over
Spec A, where A
X ' > X x Speo A
~z , whioh is the identity on the olosed fibre
!,...,^), and let Vi: A -+ if be the
k algebra homomorphisms defined by rji{e}=ti and let fa =
Speo i),: Spec if-*- Speo A. By base change using ^, D induten an
automorphism fl( of I x Spec M over Spec if, and hence
&' = JD1oD4o...oi)r is an automorphism of X x Spoo M over
Speo Jf. Let it, (l<i<p) be the elementary symmetric functions
in M of degree i in e, t^. Ona checks trivially that Jjf( =
(t + l)«i+1 A < i<,p ~ 1), so that thesubring M' in Jf generated
by *, «, ifl ^lPs,]. Let i/i; Speo Jtf -¦ Spec M' be the natural
morphism. We then assert that there is a unique automorphism
V of X x Spec if' over Spec if' whioh induces D' on tat;
extension to Speo 1?. Further, the only relations between tv
jf => j&fjj, ip) axe «J = 0, jj = 0, so that we have a homomor
phism J): H' ¦* A, tjfa,^} = 0, jj($,) = s Let $ = Speo jj * Speo A
,+ Speo Jf' We have defined the maps
A ~^> if 3 .if —^ A
<ji ifi tji
Speo A -< Speo if >¦ Speo if -< Speo A
Hien via ^, D" indnces an automorphism Dm of X x Speo A over
Speo A which on the closed fibre Xc » X x Spec A is the
identity. So D" may be thought of on a vector field on X. We
assert that D** = D^K
To verify those assertions, we may assume X=> Speo A Then
D, is obtained by the automorphism of M-algebras A$QtM-+A<QiM
determined by a (—¦ a -f- {Da)'t(, and D' by the automorphism of
A^)k M over if determined by
Our assertions can be read off from this.
How suppose O is a group scheme and D a left invariant vector
Geld, whose value at the identity is the tangent veolor f e Q(A).
Then the corresponding automorphism of O X Speo A is just
translation by 1, and Di is translation of 0 x Speo M by the image
of t under the morphism O(A) —= y Q{M) Hence D is
translation by ('= ft S(n,) {() efl(if). Now, ('is the image of an
i-i ~
element f e 0{M') by the map O(W) -*¦ <J(M) The homomorphism
fffif): G{if') -+ Q[A) maps t" to !<>•>
t
t
m
i
Apply these remarks to the situation where X is an abeiian
THtiety and X its dual, with 0 = X. Then for any t-algebra Jf
fl(it) is the group of all line bundles ? on X X Speo ? trivial on

160
ABELIAN VARIETIES
{0}x SpecBeuoh that for any point Pe Spec R, L\X X {P} belotj
to Pic°X, One sees by definition that if the l-oooyole {/^}fo^
oovering H of X with coefficients in the sheaf Ox represent s. cob
mology class f in #'{X, <Py), the corresponding tangent vectj
ijeLieX c X(A) is # »(X, <PJKi!pMA) ia represented by the 1-eooj^
{I +ifu] for the same covering. Hence, the element of X(j
cH1[X&) we obtain ia represented by tho 1-cocyi
K)' It- follows tlj
(?>> 6 Lie 5 c^(A)isrepreaented by the l-cooycie {I +f$t) «|
henoo comes from the 1-eooyole {/J} for 0z,
16. CoJwmology of Ltne Bundles. Our first ami in this aeeti|
18 to prove the two following theorems |
Thb Rismakh Koch thbobsm for all lint, (mndlu L a* J,|
m -
= cleg &
where (D1) u the g fold self-intersection number of D
Tkb ViKisaiKo thbobjm, // for a lint trundle L on |!
K(L) it finite, there it a unique integer i •= \{L), 0 < i(L) <j^
aucA *Ai« H*(J, L) = @) /ot p ¦? i and &&, L) ^ @). Furth^
Peoofs. (I) If ?[ and ?, are two line bundles on JC such Ihli,
Li®**1 6Pio°X, then xi^i) -xiL*>- la faot oonsider the lii|;
bundle pf (JV® P on X X.X. Then the Eider oharaeteristio §
its restriction 1^ to JT x {y}, ^ e X), ia independent of p. Bat Jj
and jC^ ore both isomorpliio to one of the if'a, so x(A)
Next, if ? is ayminetrio, wjf//1) = lmm', so
ALGEBRAIC THEORY VIA BCKBMBB
lot
any
«>n •« "written Lt ® i, where ?, is symmetrio
nij ^, (*) holds for any lino bundle L. Since %{lf)
in i polynomial in i, (*) iihowB tbat x{Zi*) (for any L) ia a homo
polynomial of degree j. Let
so that x(?) = —'. ""d *° h*™ oalY to W»hlish that a(L)
> Aasume this for the moment when L ia very ample
(i *= I. 2) be very ample. Then
let
d »polynomial in n, and»,. Since xti^*1 _ . . .... ..
j> is homogeneous of degree g. If Dt is the divisor corresponding to
^, sinoe Jj? ® L$ is veiy ample for »H, »i> '. ^ WB tuatr
P(ih, »«) =¦ «»r»i + "i^J*) = (? (?) n'i "S"'l^i **"'> if "»¦ "•
> 1, and it foLowe that the same equality holds for all nlt ft, 6 Z
in particular for % = !,», = —1. Bat now any line bundle LonX
(m be written aa i, ® if1 with i, very ample.
Thus, it only remains to show that a(?) = {D1) for L very ample.
We can then ohoose sections a0,..., at of L on X such that (i) <f,
hsre no common zero, and (ii) the divisor of zeros of it, <r,
inteoaot transversally at (D") distinct points. Because of condition
p), we get a morphism X yV defined by x i—»¦ (<70(a),..., tr,(x)),
»ndby(ii). theO-cycle^-'tfl, 0,..., 0))iu of degree (D*), hence 4 is
of degree [D*). Hence, by the proposition in the appendix to §6, a(L)
it (D>) times the leading coefficient of (f! x{0w(»)). '¦*- «W = <B')'
Next, we have to show that xW* = deB fe- Suppose first that
f (?) ie finite. Then, by definition of ^, we have
ob X x X. Arguing aa in {8 and §13, we see that m*L®
pflT1®p'i.&-1 has higher direot images on the flrat factor
concentrated on the finite set K{L). Therefore,

162
It follows that
ABELIAN VARIETIES
ALQBBBAIO THEORY VIA BCHEMBH
provea the Hiematln-Rooh theorem
We have the Cartesian diagram
2><
153
Since
tlxJis an isomorphism, and
- ™*L®p*L-\ we find
;i,s. the left top corner identifies itself to the fibre produot
loirer left and upper right corners), and m'Lgi^L-'gi
-(IX $l)*{P)- Sinoe ^ ia flat, we have by Corollary 5, J
of the
pJIr*
S, that
But we have aeon in §13 that -fi*pi>(.P)= @) if q & g and
is the residue field ?@) at 0 e X. Henoa, we deduce tliat
Sinoe X x X 13 the quotient of Ixl by tl e free act on of !
K(L), we deduce that
( I)' deg
deg 4,L,X(P)
Fisoily suppose fi'(i) is not finite. We oan choose a finite sib;
group F c K(ti) of arbitrarily large order /. The map lz x fc:
IxX-fJxf therefore factors m IxI-tlxI/J-flxl
oo that m'iigijjji being the inverae image by lj X ^ of
P®p^L, hss'Euler characteriatic divisible by /. Since this holds
for arbitrarily large /, m'Z®^^1 has Ealer charaoteiistic 0.
But as before x( ^
deg ^ = 0
is finite, by argumenta whioh we used in A), we may
replace m*L® J^® p|?" * by m'LOjj'i in (•). Taking
cotomologioa, we get that
In this formula we may as in A) replace vi*L®p*L~' by
jl^1, so using the Kiitineth formula, we ace that if
Sinoe all these h''s are non-negative, it is easy to see that this
only hold if only ant of the Anil's is positive and only one of
^'J's is positive, and that the sum of llicac two t's ia g.

164 ABELIAU VAE1BT1BS
Bbmabks. Consider the case k = C X = VjU where V
j-dimensional oomplex vector space and U a lattice in V.
L = L[H, «) be a line bundle on X and B =¦ Ira H so that ?jM
skew symmetric real bilinear form on V and S{U X U) c Z,
consider V as an oriented vector space by means of the
structure. If *,,..., «^ isa basis of U, we oall det{.ST(u(, «,)) th»
determinant of B; this is independent of choice of basis for U and
is always positive. Further, we have aeen that K{L) ia finite if
and only if B is non-degenerate, and that
; = Order K(L) = order {V^U),
ALOEBRAIO THEORY VIA SCHEMES
IBS
where U1 = {xe V\E{x, «)eZ,y ueV). Now, the elemental
defined by \[x)^B(x,vl) form a baaU of the dual, Hom([A,Z)(
of Ul so that we have
Henoe we obtain that |x(i)l =+Vdet(B(ii([«)}). Now, given tnj
skew symmetric matrix A of degree 2g, it ia welt known that
there ia a uniquely determined polynomial function pf(A), tbs
Pfaffian of A, in the entries of A, suoh that pf(JI = (let >4 and
=» 1 where Ba is the matrix
0 1
1 0
0
0
0
0 ]
1 0
0
0
0
Now, if X rune through SL{2g, C), pf(Z AX]* = det
det A =• pf(AI and ainoe SL{ig, C) is eoHJteoted, we deduce tUt
pf(JC'>iZ) = pf(X)if Xeifita^.C). Now, tiro different b*8M
wt u^ of 17 such that % A ... A Wv is pomKee differ byamatetf
in SUfyTL), so that for Buoh uit ....i^.pRJ^,^)) is independdiit
of the ohoioe of basia, i.e. it U determined by B, the lattice V
and the orientation of V induced by the complex structure on V
yfe then assert that we have actually
Bince both sides extend to functions on Q®iPio X which on any
finite-dimensional subspace is a polynomial function, we see that
either X(?) = pf^ftt;. «,)) for all L or X(L) pffJ^, y))
for all ?. If we show that for ? ample, pf(J5(«(, i^)} m positive,
it wculd follow that only the first alternative oan hold. Bat then,
I = L{B,a) with H positive definite hermitian. Thus, we have
only to show that if B is positive definite hermitian on a complex
T«otor space Fandu^u ,1*^ is any real basis with u, A ¦¦¦ A«%,
poaitite, pf(Im H(ii,, w,))>0. By our earlier remark, if this holds
For one suoh basis, it holds for any other auoh. Thus we may use a
It-basis «,,ttti,«i, i« , n^ where #(«(,,«() = Su. But now, the
matrix of imH with respect to this basis equals E9 and the
Pfaffian of this matrix is 1
This proves the assert on
Ta» IBDBX OF UNO BUNDLCS.
The purpose of the rest of this section is to prove that t(L)
on be coropoted as follows
Thiobim Let L be an ample line bundle on an obtlian variety
X, and if a nott-degtnerale tine ImnditK Let P{t) be the polynomial
itjiwtd by P{n) = x(?"® M). Then P has aU its g roots real and
At index of M tqnals the number of positive, roots, counted wiitt rnnUt-
flieity.ofPit).
The proof will be given in a series of steps. We make heavy
use of the next lemma. Before atating the lemma, let us introduce
•one notations For a = (olf.... Oj) e Z* we write |a| = Z la,)
and if Ll Lt are i-line bandies we denote the lino" -bundle
by IT.

1GS
ABELIAN VABCBTIES
Lvuha. Lei X is o projectivt variety of dimension r
i,, ..., Zfc fine 6«ndte« on X Then then u a eorwton* c
only on the Lt «i<* that
for » > 0 and o e Z*
Proof. We can easily reduce ourselves to the case when
jj, ate all very ample. In foot, choose a very ample ?i+1 snob
that ijgin.! ate very ample for 1 < j < t, and put I^ =
Lt g> it+i, it+i = -E>i+i- Then there is a linear automorphism T
of Z*+1 such that lbto = (i] altl) e 36*+1, ip®
(?')*• and cr'|a| < | Ta!< a|a| for a suitable « > 0.
Thus we assume all ?,- A < » < b) very ample. We proceed
by induction On the integer r = dim X + k. When v => 0, tha
assertion is trmal. Thus we may HBautne v > 0 and that the
assertion holds for smaller values of v. If now k —0, the assertion
is again clear. Thus suppose k > 0, and let L' be the system of
line bundles [Llt..., Lt_,} and for a = (o,,..., Bj) 6 Z*, set o' =.
(a,,.,:, Of^!) e Z*~". Then the existence of a constant o for all
with ot=0 follows by induotion hypothesis. For any
we have an exadfc sequence
where // is a hyperplane section for the protective imbedding of I
given by Lt, Suppose now that at>0. We have the exact aequonce
IP[L"'® 14) *-Hi(L'"'® ?j|+l) y HUH L'*'® ij+1 |H)
from wliich we have
<e(|of-' +1)
for 0 < )t < at and a suitable constant c, by induction hypotheau.
Summing over all p with G < ft < at, we obtain
® ^XC.I^KIor1 + 1)+ dim
ALQBBRAIC THEORY VIA SCHMEB 1«7
, if ak < 0, the exactness of
gives, for 0 > ft > <!», U»e lneqnalities
wd aumimiig over ^ with 0 > /t > a, we get as hefore the
required inetjuality.
Stsp A Let i be any non -degenerate tine bundle on an abehan
variety X and U a very *raple line bundle on X, and let P be
the polynomial in two variables denned by P{m, »> = xl^"® J^**>
IfP(l.<) ?iOforO<t< 1, then »(?) = »(?»,&).
Pboof. The following remark is eBBential to what follows
If/; I-f I" is an isogeny of abelian varieties of degree
prime to p and L a line bundle on 7 with x(?) ^ 0, »(/•(?)) = i{L).
In fact, we know by Cor. to Prop. 3, §7 that ? is a direot summand
@} ^ «'{/*{?)) # @). Also if A, e Pio»Z, then
In f!Wt. for some at 6 X, L® Lo r IJ?, hence
fl() ? {0} => H*(I??) t* {0) * iP(?® ?,) ?t @). In particular,
since »{{?) =X""® ?„ with L, e Pic»JC we see that »(?"*) = i(L)
for any n > 0, with p ^ ».
Suppose then that -Y is a large square prime top. If»(?) #
t(?®ff), then ;(?*} ^ t(L* ® J*). flo there is a least integer a
in 0 <«< If with t, =i{V®H") *i(If®U*-1) = i(Vl). (Note
that since P[l,t) has no rational leroes, 0< t< J, all the bundles
&QH* ara non-degenerate so i(Z/*®ff*) is well defined.) The
«aot sequence 0 -* i»® tf-' -* i*® if -+ZJ'o3>i^ | r -»¦ 0
wbero K is a hyperplane section of Z for the imbedding given by
H, gives us that
is exact so that

1W
ABEUAN VABlBTIES
l,O>dimfl(t( ?*«#•)
But there is a lower bound •:
P(l,t)> o> 0 if 0<t< 1, ;:
Mid sinoethis holds for arbitrarily Urge N, while Via of d inengion'!
g — 1, we get a contradiction to the lemma.
StbfB. If ij and?, aretwa line bundles on an abelian variety
and F{>, t) is the homogeneous polynomial defined by F{m, ») e,
X(L- ® L't), and if JT((, 1 - () * 0 for 0 < (<, \, then »(?,) j
Proof. Choose a very ample L, suoh that L,® ,?,
very ample. Set/(a, 6, e) = x(^I® ?j® J^), so that f(a, *, 0) =.
f (o, 6). Since jf((, I - () ?i 0 for 0< t < 1, by oontiauity, we o»n>
obooBe a square N, prime to p, so large that for 0 < r < W - l,;!
The first equation ooupled with Step A gives us that for 0< r<;
JT— 1, ran integer, ;.
and the second gives ua that
for 0 < t < if — 1, r an integer Taken together we get that for
t in the same range,
bo that we obtain
ALGEBRAIC THEORY VIA SCHEMES 159
IfLia rum dtgeniraU tmd n any integer > 0
C If ij L,, and ?j® i, are non degenerate
ilA®^} <t(?.) + '"(i,).
Lot t: Jt xl-+l be tbe morphiem (r, y) i—> * y
and set ? =jjf (A)® ff (ii)- Then p-'fO) is the diagonal of X x X,
and if we identify it with X, L \ v"l{0) beoomea isomorphio to
?j gi ?,, which shows that L \ _, is non-degenerate for any sc s X
and has index t = t{Lt(gi Lt). Henoe, the direqt images JPi{L)
Taniabforj<», and by tlte Latay apeotral sequence Hr{XxX L)
= 0 for p < i. But now,
ao that iftj + »(?,)>i =t(L,®?1l)
Stip D Let ?, and Lj be non-Regenerate line bundles on an
abehan variety X such that ?1@I^-1 is ample. If /(m,n) =
^(?T® iS). Buppoae /((, 1 -1) has a unique zero r in [0, 1] of
multiplicity X. Then
0< *(?,)-•(?,)< A
Proof. By Step C, we have
»(?,) =i(?,® ?fl® ?,) < «(it® ?,») + »(?,) =t(L,)
which proves the first inequality.
Since / is homogeneous and t(?"} = i{L] for n> 0, we may
leplaoe ?, and L, by suitable powers to suppose L,®Lf' veryampla.
Let us denote by H,H*,..., respectively a liyperpbnc section
of X, a hyperplnne section of this section, etc. for the projective
imbedding given by ?,® L^1
Let ff be any largo integer, which wo suppose coprime to the
denominator of t if t is rational. Then tlicrc is a unique Integer
ruitho <*¦<
and i(?,J -
rJ <t</. Pot * V-r, HLt)
di V

160
ABELIAN VARIETIEB
We first propose to show by induction on « that for k < r — i <|>
J = IT — t, we have \
The assertion is obvious for a =• 0, sinew by Step B,
Suppose then that a > 0 and the above holds for a — I instead of ¦
From the eiaot sequence
o—*-i4-'®4+1 |^-i—
we get the exactness of
and the assertion follows by induction hypothesis. But now, tho
exactness of
Boupled with the above foot gives us that the map
l* injeotive for p < ia — «. Taking p =«„ we deduce that
is tnjeotive. Thus we deduce that
=H/(;. 4,
By the first lemma, we therefore deduce that there is a constant
e > 0 suoh that tor all large N (prime to the denominator of t if
r is rational},
,.^ <
Now, /[I, 1 — t) = (* — t)V@ where g is non-vanishing at t Thus
for all large N as above, there is a <s' > 0 such that
ALOEBRAI0 THEORY VIA SCHEMES
161
N
<e'N
If r were rational and N ooprime to the denominator q of r, we
have
for ail r, so that we must liave
Nq "" ™" " "~ *
<l,henoe»,<tI +>. If t were irrational, by Kroneoltor's theorem on
the density of fractional parts of if t for t irrational, for a sequence
N( of integers -+ oo, the distance of N^ from the nearest
integer tends to t so that we again deduoe that-i-—1 < 1 henoe
t, < »i + A-
Thts complet-aa the proof of Step D.
Pboof of thi theorbk. We are given an ample L and a non-
degenerate M. Let P(»,») => j((i"®Jtf"). We must prove that
all the ? zeraes of ,P((,1)=O are real, and that \(M) is the
nnwber of positive ones (t = 0 is not a aero, since Jf is non
degenerate). Choose a positive integer q so that the real
! (t t^ of P{t, 1) are divided up as follows'
Let \ be the multiplicity of the root *,. Then by Steps B and D
we conclude that if t(r) = i(i'® Jf'). then
....= i(r.) < t{r, - 1) - «>,) <
1) = US)
and
»(r,-1) -«(r,}< V
Bnt for large rt J7® Jf ifl ample, hence t(r) = 0. And for large
ntgative r, (V® Jf') is ample, henoe t(r) = Q by the last
statement of the vanishing theorem Therefore

192
ABELIAH VARIETIES
ALGEBRAIC THEORY VIA SCHEMES
183
very . . very . f * -i », |
j=i( negative )-»{ positive 1= ^t(r(— )) i{r,) < > A,, f
But ?\, the number of real leroes of P{t, 1) = 0 is at most g, m ¦
equality holds everywhere, and i@) = t(iO " just the sum of the
\'s for which the corresponding root I, is positive.
Cobollaby. Let V be a complex vector tpace of dimenaon g and
V a lattice, fit il such that X = VjU is on abelian vanety. Let L^
L{H, %) bt a non-degenerate line bwuiit on X, so that H it a non-
degenerate hermitian form on V Then >"(?} equals the number of
negative eigenvalue* of H
Piioor. Choose u basis
, uu of V over Z such that
ti A**-- A«i, defines the orientation of V. Let Lt = L\Bllta!t)
be mi ample line bundle on X, so that Htt is positive definite.
We set S = Im H and .fi, = Im Ht. We then have
Choose ef such that
Then nEt + S has the matrix
and
n X
where #,, and B are considered as skew symmetric matrices, using
the above basis of U. Now, if we use any other positively oriented
real basis of V. the Pfaffian gets multiplied by a positive scalar.
Thus, if e, e, is a C-basis of V, we can utilize the basis
e1,ti1,et,iet,,..,et,iel to compute the sign of the above Pfaffian.
S d H{) XSXR*
and the PTaffian of thiB matrix U H(n+^), (aiuoe its aquafe is the
determinant and it takes the value A!... A, for n — 0, hence the
value 1 if all \ = + 1). Thus, the index of L is the nnmhar of
negative \,
This corollary can also be deduced from the results of Andreottt
and Grauert [A-G]
17 Very ample hue buddies. The object of this section is to
prove the following theorem.
Tbbobbm. For any ample Kne bnndU L on. an abeltan, varttty X
L" it very ample, if n > 3.
Pboo». For simplicity, we shall prove L* is very ample If
n>3, the same proof works. Since dimfl°(X, ?} = v(L)>0, we oati
ohoose an effective divisor D such that <PX(D) is LBomorphio to the
sheaf of sections of L, Further, D can be chosen bo as not to have
any multiple components, for if kj$ occurs in D with if irreducible
t t
and A > 1, kE is linearly equivalent to "LTffi) for Si^^O
<-i i
and for suitable choice of the *¦„ the T^ffijare all dtatinct and
distinct from the other components of D,
Thus we assume D without multiple components. Note that
for any *, y e X, Tt{D) +T?fp) + Ttm_t(D) 613D | We have now
to establish the following statements,
A) Given f,,*^! with io^a!|. there is a O e | 30 | such
that k, 6 Supp D', x^ 4 Supp D'.
C) Given any tangent vector I to X at *,, there is a D e|8D|
such that z0 e Supp D' and t is not tangential to D' (i.e. if 4 =• °
is a local equation of D' at xt, <t, if^> ^ 0).
Since we are making these assertions for any ample h, it suffices
to prove A) and B) for any ample L with xt = 0, ainoe the general
oase follows by applying the result to a translate of ?.
Thus, if A) were not true (with tt — 0) we would have that
for »ny D as above and any x, y € X

104
0g8upj>D
ABMJAN VARIETIES
: =* *! e (Sapp i) — a) u (Supp i) - y)
Since we may clearly choose y such that s^ does not belong to the
lost two members, we deduce that x e Supp D implies x e Supp h
— xy, that is, Supp D = Supp D — i,. Since the divisor D has no
multiple components, this means that T^D) = D. In particular,:
*! e Jt(L), hence x, has finite order. Let i, generate the finite
group jP. We then have an etale morphism it: X -* XjF, anrf
!>! = ir(Supp D) is a closed nuhset pure of oodimension one in XjF,
which we may consider as ft divisor with all components of
multiplicity one. Since w is etale, u*{C1) is again a divisor with I
all components of multiplicity one and has the same support &b
D, so that D = ir*(D,) and L =: jr+f^,,!!),)) But note that
. (Order fj-dim fl'fJT^,
Since the set of a)) divisors DL such tbat
fall into a finite set of linear equivalence classes, this proves that j
all sections * e V(L) either define multiple divisors, or lie in one ¦;;;
of a finite number of lower-dimensional subspaoea v*F{t>z flD])). ;¦
This is a contradiction, bo A) holds, ;1
Similarly, suppose B) is not true for a non-zero tangent vector;;
t at 0, and let T be the invariant vector field denned by f. If C) >
is false for all the divisors TJ{Z>) + Tf{D) + Tt^,{I>), it follows ^
immediately that for all x e Supp/), the vector TM is tangent J
to O at at. Since D has no multiple components, this is equivalent
to the property: [
V U cX open, V local equations ^ = 0for .Don U
(*)
In terms of the *[*]/(**)-valued automorphism of X defined by T,
(*) just says that the divisor D - a zubsobeme of Z is inv-
ALOEBBAIO THBOBY VIA BOHBMBS 165
.*¦¦
.; .,iint. This imphes that the %]/(<*)-valued point of X defined
t lain the subgroup K(L) of points leaving L invariant. Now
aharaateristio 0, all group schemes are reduced, so K(L) is
and discrete and this cannot hold unless 1 = 0. On the
other hand, in characteristic p, let H be the smallest subgroup
of ?(?) containing (; then H cJ'1 and wilt be determined by its
Lie algebra Ij whioh will be the span of t and itB p powers. It
i« easy to see that D will be invariant under translations by all
points of E. [In fact, if if = Spec(^), the action of H gives a
jiomomorphism of B* into the ring of differential operators on
X, mapping elements of (j into the corresponding invariant
derivations. Since b, generates it*, and the sheaf of ideals
$z[-D) is stable under E) by (¦), it i* also stable under B*
we get a hamomorphism M* -*¦ Diffftf^), i.e. an action of
D.] Let X - XjE, D' = DjH. From the results of $12, we
fad that w: X -*¦ X' is flat and surjeotive, that D' is a closed sub-
utheme of X' and D s D' XrX. Therefore if J' is the sheaf of
ideals of D'
Sinoe D is a divisor, <Pj( - D) is a looally free sheaf, ao by Part (B),
Tfceorem 1, {IS, J' is a looally free sheaf, i.e. D' is a divisor too.
Sow D " it*(O'), so we compute, M before:
d = deg
Exactly <u Wore, thui implies that all sections * ? T(?) either
define multiple divisors, or lie in one of a finite set of proper sub
¦paoes-a contradiction.

CHAPTER IV
X) AND THE 1 ADIC REPRESENTATION
Etak covennp. The main result is the following
Thbokbh. (Sena-Lang.) If X id an ttbtlian variety; Y a variety
andf: Y -*¦ Z ** an itole covering then Y hat a structure of oielian
tiafieiy twh that f bteomu a separable wogeny.
Pboo*. Let rHbe the graph of the multiplication m:X xJT +
I in X x X X Z, and I" the inverse image in Y x Y x Y of rm by
fxfxf. Sinoe (l)F' -+ rB ia an itale covering, [2)pu: rm-*X a X
it (in isomorphism, C) we have the commutative diagram
r
YxY
fyf
Z xl
Mid D) / x/ ifl in etale covering ft,,: F' -* F X Y « an stale
oovering too. Choose a point yt e Y «uch that /(y0) =¦ 0, and let
F be the connected component of I" containing (j;,, yt, y0) (which
belongs to T' since/(y0) =0 and @, 0, 0) eFn). Then the restric-
restriction p: T -> Y X Y of plt is again an 6tale oovering, so that the
degreeofjaequalB thenumberof pointsof any nbre of p. We want
to show tbat p is an isomorphism, or equivalently that there is one
point of Y x r whose inverse image in P is again a single point
Leta,,.ja:r->-r be denned by <My) = {s», y. »).¦»«(») = (S>»» 9)
(Since (r.(r)cr' and (tfo. ?,,. J^eo,!!"), it follows that o,G) cT.)
Then the restriction of p to ot(Y) ie a bijeotkin of ott,Y) onto
7 x {&,}. It therefore guffioes to establish that p~'{Y y {&}) =
wjr), or equivalently that if q: T ->¦ Y is the restriction to T of
*,: 7 x r x F -+ r, g-'tyo) °" "i(f)' s«"»"ti1") '" "> "rreduoib16
component of 9-1(jfe), it euiEoes to show that ("'(if,) is ircednoibU
Kow r i» non-singular being Male over Y x Y and hence Z x X

168
ABELIAN VARIETIES
and siiiee it is also connected, it ia irreducible. Further, the mor.W;
phism q-.V-i-Yia smooth, being the composite of the etale mor-?-
phism T-+Y X Y and the projection Y x Y -^ Y.
»,'. Y-t-T is a section for q. Now the assertion thftt 9~'({f0)
irreduoible follows from the
ie( /. X -* Y be a proper smooth morphism 9
tiarieitaj such that there it a section a: Y -+J faB=s\
Then ail fibres of f are irreducible.
We may assume Y = Speo ,4 affine, let B 1|
ao that H is an jl-algebra which ia a domain since X ia irreducible.:
Mid a finite ji-module since/ is proper. The morphism / factories.
S h
as X ¦ >¦ Spec(fi) >¦ Y where Spec(B) is again an kredu,
oible variety. But go a is a section of h, and since dim(Speo 5)^
= dim Y, fa is fluTJeotive, hence A ia Mi isomorphiatn, and A=B.
Since / is smooth, its fibres are non-singular and it suffices to
show that they are connected. Let 0 -viT, ~+Kt ¦*... beacoraplei
of free finitely generated A -modules giving the direct images of Qz
universally, so that by the above, we have an exact sequence;
0-*.^-* JT,-*-^,. Let y beany point of Y, Wl its maximal ideal
in A. Since completion with respect to the Ufi-actio topology ia M
exact functor we have an exact aequanoe 0 -*• A -> Kt -*¦ if;, so'
Ker
ao the natural map A
lim H"(/
is a
isomorphism. K/"'[y) were not oonneoted, ^otf-1(y)^=Zi u Zt,Zt
closed, Z, oZ, = 0 and Zi^=f~1(jf). We can find a unique
SmeHa{}~\y), <Pi/9H"<Pj) whioh reduces to 1 on Zx and 0 on
Hom{X,I) AND THE 1 ADIO REPRESENTATION 10»
BBd {/»} defines an element / e lim B*{SzjWr6z) s A with /» -=/
and / ^ 0 or 1 This is impossible since iiaa local ring
The lemma is proved
Returning to the proof of the theorem, we have shown that
pit;V -* Y X Y is an isomorphism, so that f^ft'ffi': Yx7-*Y
is a morphism. Sinoe, as we saw, T d <r,{7) and cr,G), it follow*
that r[y,yo) =y = f(y«.y). Therefore, from the theorem proved
in the Appendix to J4, T is an abelian variety with composition
!»w * and zero element ya. Since /(y4) = 0 / ifl a homomorphism
of abelian varieties.
B«makk, If X is an abelian variety and /: F-+Iu tn
isogeny, then we can find an iaogeny g:X->-Y with/op^jt, for
acme h > 0. In fact, sinoe ker/ is a finite group soheme, it ii
killed by some integer n > 0, hanoe fcer/? ker(»r). Since
X x F/ker /, it follows that nT f&otoriEes as nr — go/ for a homo
morphiam f. X ¦* 7. Bnt then/op^^ too since for all at eX
t =f(y) for some yeY, and therefore
/«V(«) =/(?</(»)}) =/(»») =V(Sf) »»
We can interpret our results in terms of the fundamental group
<rt(X) Recall, that if X is any non-singular variety, and xt el
is a base point, the group njX.z,) is constructed as follows'
oonsider the set of all morphisms
together with a base point y, e F lying over a:0 such that
A) a finite group QY aots freely on T and X= r/(?r and
B) J" is oonneoted, henoe 7 ia again a non-singular variety
Given two such :
iMitU*, of. [OS] pp. SO-ei

170
ABBLIAH VAMETIBS
Hom(X.I) AND THE I ADIO
171
recall that there is at most one morphiBm / T' -*¦ T' such that'
(i) »•./ = „', -<i
<ii) M)= ? %
When/exists there is a unique anrjeotive homomorphism T
p:Qr y-Qy. |
auch the,l/(«-y) ¦=/>(»)•/(*>• all <,eQrt y el". We order the*
triples (y, ft, w> by saying (I"', jj, ir')> A*. jj, w") if such an /,;
exists. Then the set of [Y, j/(, ir)'i forms an inverse system, ind;'*
we define ¦¦
»,(Z, *,) « ^. 0r w
"** . i
Now suppose I u m abelian variety and xt — 0. Then all >:
such 7'e are abehan varieties, and OT is just the kernel of ir
acting on F by translations. In particular, we see that tts(X) il ;:
abelian. To describe it more explicitly, it is convenient to break j;
it up into the product of its t-primary piece for different printei i. J
First suppose I & p. By the remark following the theorem, th< i
set of itale oeverings >
J y J ?.
u ooftnal in the set of all Stale sovennge
Y-^-t-X #(Ker«r)«J» somem ;*
Therefore, the i-adio component of irt(Z) is the inverse limit of'
. This is catted the 1-adicTatt.grfmp of X :;
— lim X,, wftere th« inner** «#»tom «< :
an inverse limit of finite abelian 1-torBbn groups, T,{X) has
the structure of a module over the I-adic integers Z,, Since Xp, s
(Z/rZ)*, it is eaay to see that T,(Z) a Uf as a Z,-module
Secondly suppose 1 ¦= p In this case, break up
Ker(pJ) = ZJ, x J^.
where ZJ» is local and ZJ» is reduced. Than hy the remark
following the theorem, the set of Wale coverings ir. in the
diagram"!
-+Z
il coBnal in the let of all (Stale coverings of X whose degree is a
power of p. But Kerfjr,,) = GR(ZJ») s X^*, bo the p-adic component
of irttX) is again the p-adic discrete Tate group.
Dbwkitiok Tr{X) = lim ZJn (fha ttuxrte system an before)
, T,{X) is a Zj-roodule and if r=;y-rBnk of Z, then clearly Tr{X)
Tbe foil fundamental group is then given by
Now suppose fc = C and Z= VjU where, as usual, P is a
oompler veotor space and V is a lattice. Then, in addition to the
algebraic fundamental group as just denned, we have the usual
iopologioal fundamental group w,10*^^ which, as we&aw in §1, is
(¦nonioally isomorphio to V. On the other hand, since

172
it follows that
ABELIAN VABIETCIEH
* Jim p 17/G
1 ' I
VjV > p VjU
with
T,(X) a lim
1
with maps ir/r+i17 >¦ U{VV
In other words T&X) is the I-adic completion of V = irj1
= lim OMO
= fnll pro-finite completion of V
19 Structure of Hom(Z, J}- For two abelian varieties
and 7, we denote by Hom(X, 7) the group of horoomorphiami'
of X into 7 and by End X the ring Hom(X, X). Further;!
we shall put HonAJC, Y) = Q®zHom(X, Y) and Bnd°(Z)
Q®zEnd J(End0X is classically called the algebra of oomplejfe
multiplications of X) :'i:
Composition of homomorphisms extends to a unique Q-bili:
map Hom»(X, 7\ x Hbm"(r, Z) -* Hom°(X, Z). so that we
form a category whoee objects are abelian varieties, and morphiama'r
from X to F are elements of Hom^X, 7), the so-called categorjf
of "abelian varieties up to isogeny". We have seen that givei*
any isogeny /: 7 •*¦ X, there ig another iaogeny g: X -+ 7 snot:: i
that f°g=.nx, and this proves that in the new"oategory, tab-:
gsnies are isomorphisms. Thus in future, whenever we
an isogeny /: 7 -+ X, we shall denote hy /-*¦ its invwae
ADD THE 1 ASIC REPRESENTATION
173
Hom°(X, r). It ia also clear that we can give the following
more fancy definition of Hom°(X, Y);
Hom°(X 7) = lira Hom(X 7)
1. (Poinoare' a complete reduoibibty theorem.) // X
u an abelian variety and T an abelian subvariety there m an
abelian nAvmUty Z such that 7 n Z it finite and T + Z = X
In other worth, X w uogenous to Y x Z
Fbooi. Let »: 7 -> X be the inclusion, and 7; X ->¦ 7 its dual
homouorphism. Let ? be ample on X so that fa. J-s-Xis
an iaogeny. We take Z to be the connected component of 0 of
^'(ter i). We then have dim Z = dim ker i > dim X — dim 9 =
dun X —dim 7. Further, by definition of t and ^, if *e 7, than
tefcHker T}n 7 -e=» f*?® L~X\Y U trivial
Since ?|r is ample, X(L|7) and hence Z n T is finite. This
means that the natural homomorphism Z x 7 ->X has finite
kernel, and since dimB x Y) = dim 2 + dim Y > dun X, it ia
aiao snrjeotive.
Bsmabk. Over the complex field, the complete reductibility
theorem in very aimple to prove. In fcot, let X = YjV with V a
complex vector space and U a lattice, and H a positive finite
herroitian form on V which is non-degenerate with E = Im B
integral on O x V. Then any abelian subvariety r of X is of
the form 7,/f n Vx where 7t is a complex subspace of V with
VtnU a. lattice in Vv If 7,ia the orthogonal complement of V
for H, then (a) F, is also the orthogonal complement of F for
E, hence tbe lattice 17 n Fs is of maximal rank in 7,: and
(b) Vt n 7, — @) sinoe /f is poaitive definite. Thus, if
Z = 7,/ F, n f. 2 iB a complex subtorus of X such that Y n Z is
finite. The restriction of ,ff to V» gives a Eiemann form on Va
which shows that Z is an abelian sabvMiety.

174
ABELIAK VABMTIBS
AMD THB i-ADIO RBPRBBBMTATION
178
In &noy language, the theorem show* that the
of abelian varieties up to isogeny U a "semi-simple abeuan gt
category all of whose object* have finite length". More con- ^
oretely we get the
memts
e[/ otwi
Ooaouisrl
«. x X XJ*
Ac X, o
(proof standard )
For a
„«( not i
icty Z, i/ X = X? x ... x X?
[ffere *,(*) = r&W * * mofrww over B.} |
Pboo* For X simple, any non-iero endomorphism of XM
U » i-ogony, heDoe an invertible element in Bnd*X, wh.oh:
proves the first assertion. As for the second, HomtX,", X?)
fori#j, so End«jr= 9 WHW And ^ad^J
the algebra of matrices 'of ordw n, on the division algebra Dfl
We shall say that a function 0 defined on a vector space'
F is a polynomial function of degree n if restricted to anjj
fin.te-dimeneioi.al aubspace, it is a polynomial function of degree,
» or, equivalent!?, if for any »., 0, 6 F, ^a,«, + *,».) IB *{
polynomial in x, and ., of degree ». Thus for instance. —
have *een that X(L> extends to a homogeneous p
function of degree g oil the vector space #S{X)®iQ
End'Z
Pboof Binoe for <fa e End X and iteZ,
deg n^ = deg nT. dog ^ ^
it suffices to show that for ^, 16 e End X, the function P{n)
=<deg(»^ + 0) is a polynomial function. If i is an ample
line bundle, we have that
deg
Therefore it snffloes to show that x(("^ + ?)'(?)) is polynomial
in n. Putting i,rt = (»^ + ^}*(L) and applying Corollary 2 of
the theorem of the cube to the three morphisms tufi + i/r <j> i/r
respectively we get that
from whioh it follows by induction on n that for suitable line
bundles 1^, L%t and Lt on X,
1B R P°'y
* polynomial function of L
nominl in n.
To ge further and prove, in particular, that dim(jHom*(X T)
is finite, it seems to be essential to use some entirely now
method. If t = C we can compute Hom(X, F) very quickly
like this.
Let X1^'FJU1,gt^4imX1
X, = VJUt, gt =¦ dim X,
F, complex vector spaces U( lattices
Then every algebraic homomorphism /: J, -»¦ Xt hfta to »
ootnplez-analytic homomorphism f : V\ -*¦ Vr Ab is well known,
snoh /'s are limply the complex linear maps from V,to F,.
OonveTOely a complex linear map L: Fj •+ F< induces an analytio
homomorphism/:Xt-»-Xt if and only if L(Vy)cOt, and by
Chow's theorem (of. §1) all analytio homomorphisms from Xl to
X, are algebraic

176
ABELIAN VARIETIES
Tbis proves
Hom.b-|U (X^X^slL 7i-*-7i L compkx-ltntar L{VX) c IT, I ;*:
T#J) itself ia a canonical horaomorphiBtn
In partioular L is determined by its restriction to Vx bo we get - j
an injection !
T Horn j»H.n (X, X,) > Homz(t/! 17,) •¦!;
vuiatla 1
Sinoe fj is the topological fundamental group ffjftXj) (or tbe I
homology group ?Fj(X()), the map T is just the funotoriil repre- ..
sentation of maps between spaces via maps between n1's (or;"'
i/r'fl). If we introduce bases, we have a faithful representation
of Hom(XlP JC,) by 2pt x 2fji,-integral matrices. In particular, i
Hom(X,, Xt) is a free abelian group on at most 4fft0i generators.'*'
Even when the group field h is not C, an analog of the aboTe : ;
method works. This consists in using the free Zj-module T/X)' I
instead of the free Z-module V = jrjftX). In fwrt Tt{X) is just ¦
the I-priraary ooroponent of the algebraio fundamental group' j
jtj(X), and when k = C, T?X) is nothing but the l-adio completion
of U. If X] and X(are two abelian varieties, every bomomorphism
/:Xi -+X| restriots to maps /: (-X,),* -> (Xt),", and benoe it
induces a map
The map/i
known an the i-adte «yt-ft>ew(o(to». In fact, In terms of bases
of Tt(XJ over Zj, this represents homomorphiams / hy 2^ X 2f t 1
matrices with ooemcientB in Z,. We now prove ;.
Th«obsk 3. for on]/ pair of afalian vanities X and Y;Y
Hom(I, F) t» a finitely generated fret abelian group and fy
natural map
Hom(X,X) AND THE I ADIC REPRESENTATION
177
,\ Hom(I 7)
io»y pnnw
psooT, Note that ainoe Hom{X Y) is torsion froa we have
an inclusion Hom(X, Y) c Hom°(X, Y).
Step I. For any finitely generated subgroup M of Hom(X, T)
Q M n Hom(X, J) = {^ e Hom(X r) | n^ e Jf some n ?t 0}
ia again finitely generated.
To prove this, choose isogenies DX^"* -* X and Y -*¦ nTjt
where X{, Yt are simple abelian varieties. Then HomfX, Y) gets
mapped injectively into JIHomfXf, Yf), so that it suffices to
prove this result for X and Y simple. If X and Y ore not
isogenous, Hom(X, T) — @), eo that we may assume that they
ore; in tbis cose using the injection HomfX, Y) -*¦ End X
induoed by an isogenj Y ~* X, we ore reduced to the case
X = Y and X simple. By the earlier theorem, there is a homo
geneous polynomial function P on End"X suab that for ^ 6
EndX, P(^)=deg^eZ, Sinoe any ^ # 0 ia an isogeny
P(fy > 1 if <? e End Jf and i? ^0. Now QM ia a finite-dimensional
apace, and I P(^)| < 1 is a neighborhood V of 0 in this space
Therefore Ur,#ud{X)= @), so DndXnQ-Af is discrete in
Q M and hence is finitely generated.
Step II. For any I =? J> the map (*) is injective.
In fact, it suffices to show, in view of Step I, that for any
finitely generated (hence free) submodule M of Hom(X Y) such
that Jf = Q.lf o Hom(X, Y),
Is injective Let A,...,/, be a Z-base of M. If tlus map is not
injective, sinoe the right side ia Z,-free, we can ftnd (^ e Z, with
at least one Ki a unii such that S aj/,>—*• 0. Hanca we can ftnd
Z,®zHom(X F)
TJIT))
integers n( A < i < j)) not all s 0 (mod I) such that 3*,(i B(/()
t
maps T^X) into IT,{Y) By the vary definition of T,[f) tbis

178 ABEUAN VAB1ETIES
means that ? ti( /(— / maps Xt nto 0 But then / factories &«
1
I g
X y X »¦ Y and gince g e Q,Jf n Hnxn.(X, Y) - M
g = Jlm(/(. Thus Sn(/( = iSm,/,, and /,, ... /# being a basu
of Jf, I)"* fur all t, a contradiction.
The theorem now follows. In fact, because of the injeotivity of
(*), Hom%3f, Y) is finite-dimensional over Q, and because of
Step I, Hom(J", 1'j ia finitely gonerated and being torsion tree t
is ftlso free
CoBOiAARr 1. Hom(ir, Y) ~ 7/ vnih p < 4d m X dim Y
Vsaov. In foot, the rank of Hom( J, F) is at most that of
Honi^ayX), T,{7» which U 4dim Xdim JT.
Coboiaabt 2. for twvy ofciuti tmnefy X (to (Croup JV?(?) =>
FtoXfPio'X isfne offitiiu rank («atfed tht bait nwnber ofX).
Fnoo*. In faot, the homomorphlBm ? i—t- ^ induces an
injection of ifcS(.X> into HotnfX, X)
Cobo^Uht 3 End'Z m a/inti*-(itm«jt«»oiwJ scniwtmple oipei™
over Q
Let jl be a finite-dimensional associative algebra over a field
r whioh, for simplicity, we assume to be infinite. By a tract form
on A over r, we mean a IMraear form
8: A ».r
such that 3{XY)=^8{YX) for X.YeA AnormformanAovetF
im a Tujn-zero polynomial function
^:j! *V
(i.e. in terms of a basil of J over F, ff(a) can be written as •
polynomial over T in the components of a) suoh that W(jr)=i
JT(X)JfG) for X,Y 6 J. The following lemma is well known
but we include a proof for the take of oompleteneoe.
1TB
AND THE (-ADIC RFPMiaENTATION
a field V (assumed infinite) with atnier A, *eporflftle <wer T,
w a co«o»ioai norm /orm Jf0 a«d a conofltool trace form Tr*
o/ X over A *«eA Da( ony norm form (resrp. (face /orm} of A over Ta
U of the type {T&mAtr°N'1)* with k an integer > 0 irtsp. ^>Tr° where
^:A ->¦ T is a r-linear form]. If [A: A] = rf*, JV'w SwnojenewM
o/ degree d
Fboof. When r = A is separably closed, A nan be taken to be
a matrix algebra Jf^{r). In this case, the elements XY— YX span
the vector Bubspaoe of matrices of zero trace, and any norm form
gives rise to a rational hoMomorphtsm of algebraic groups OL(d)
-+GB. This shows the validity of the lemma with Tr = matrix
trace, N =- matrix determinant.
In the general case, let V be the separable cIoboto of F, at
A-+F {1 < »<[A:T]) the varions imbeddings of A in Fovert1,
and I^ttie field F considered as a A-algebra through at. We have
an isomorphism of f^algebrao
Denote the image of a 6 A®rV under this isomorphism by {¦&(«)}•
If jV is any norm form on A over P, it extends to a norm form of
A^>VT over T, and defines a norm form N( on A® rr(() by the
equation NJg) = N(l, 1, ...,?, I,..., 1). By what we have seen
N( a {Nfy* , where JVJ is the reduced norm of j*® jj*,,, over r{()
«o that we get
We shall show that tbe norm
of j4 g)rF over
_
F comes from a norm of A over P by base extension if and only
if all the itj are equal. Since F is the separable closure of T, N
domes from F if and only if for any automorphism a of P over V
we have W{A® tr)a} = oGfa), that is to say

ISO
ABKUAN VABIETIES
THK IADIO EBPBKSEMTATIOH
181
n
1
n
i
Not, thare is a permutation it of tbe integers from 1 to [A P]
such that oro(r( =-<r^(), and we have the commutative diagram
Ar<f
so that (since the second, vertical arrow is an isomorphism of
simple algebras over the isomorphism a ai separably olosed base
fields) we get ff^[^,({l® «){*))] = tr J^(^(m)), and on substi-
substitution, we see that we roust have n<i} — n, for all t. Now, tbe
O»loie group of V over T acts transitively on the inibeddmgs over
A in T, so that we must have all the fl, equal.
Thus we see that we may take jV»(a) = tl^^(a)) in the
i
lemma, and then N{a) = NmAff(iT^(a))"* The assertion about tbe
trace is even simpler.
DirnrmoH. 'SmA/r"Na wiil be called the reduced norm of A over
T and Tr^poTr0 wdl be ealUd the rtdnctd trace of A over F
We ou now prore the foilowvng important
Thbobbk i Let f beanendomorphism of anabttian variety and
T,(f) this induced automorphism of T,(X) (lj= eharacttrittit). Then
hence
where P{i) i* t*e eharactenstic polynomial, det(( - T,{fj) of TJJ)
The polynomial P it manic of degree 2f, hat rational integral
, and P{J) - 0
The functions / i—»¦ deg / and / i—»• det T({/) beth
eitsnd uniquely to norm forma Nt and Nt reapeotively of degree
jj on the semi-aimple Q,-a!gebra Q,®z End X, where Q, is the
quotient field of Z,, If f | denotes the J-adio absolute value, we
assart that j N,« | = |JT,i»| for all <*€Q,g>z End Z. In faot, it
(uffioes to verify this by homogeneity fora e 2, ®z End X, and by
oontinuity for s e End X. Tbua we have to show that the power of
[dividing deg/equals the power of I dividing det Tt[f). Now, the
power of I dividing deg / is the order of the kernel of X,, —*¦ X&
for n large, or what is the same the order of the ookernel of this
map fcr » large. On passing to the limit, it is also the order of
the cokernei of T,(f), which is !', where v is the power of J
occurring in det T,{f)
r
let Q^jEndZs; 11^, be the decomposition of
Q^End X into a product of simple algebraa. The norms J^ and
lft go over into norms on 11A,, i.e. into power products
2)
where Iff are the norm forms on A] over Q, of lowest degree
by the lemma. On taking a, = I for j ^jt, we deduce that
|#j,(S,) f^t^'h = 1 for all ^ g A^. Since Nit is homogeneous
of positive degree, we see {by multiplying a^ by I) that ^ = ^
and since this holds for all j0, JVj = JT,.
This proves the first statement of the theorem. The second
fellows an substituting n.lz f for / and using T,(n.lz~f)=-
"•'fKZ) ~ Ttifl- Now i1 has to be monio of degree 2o and, sinoe
P{n) is an integer for all n, its coefficients are all rational
Further, sinoe End X is a finite Z-module, / is integral over
Z, so / and henoe ?*,(/) satisfies s monio equation over Z
Henae all the eigenvalues of the matrix T,(f) are algebraio
integers and its oharaoteristio polynomial has ooeffioionts
whioh are algebraio integers. Sinoe the ooeffiaients are also

182
ABELLUt VABIETTE8
rational, they are rational integers. Hence P[f) is a well-define) •:' '•-
element of EndX and we have finally T{{P{f)) = P(T,f) = n j
so that P{/)=0.
D*FlKmon. The above polynomial P(t) {which belongs to Z[l]
and it independent of I) U ettHed thi tharaderUtic polynomial off /j,
constant term and mintu the ooefficient of f-1 are called the norm
and trace respectively of f.
I
By the lemma proved ember, we Bee that if End°JC =
j4] X ... X A,, where Af are simple algebras over Q, and m
denote the components of an / e End°X in Af by /(, and th«
reduced norm of A^ over Q and the reduced trace over Q *
hy Nm* and Tr" respeotivejy we have
t
Nm/ \\ <MmV<r<
t
Tr/ Jv^Vo
\
where nt are integers > 0
Cohollaby. ?ei jT &? o nmpic oieiion txtriejy 0/ dirmmion j,
rt« center o/(*e o^eftro End0Z, [ifjQ] = e, [End"Z: K] = d»,
de divtdee 2g. •
Pnoor We have Nm / = (Nm'/f f°r eorao n. But Nm is a I
polynomial function of degree 2g, and Nm' is a polynomial
function of degree de
Rkhabk. When the chKracteriatio of k ie sero, with assump- y
tionsasinthe above corollary, one can say even that d*e divide! :
2g. In fact, we may asanme (by the LeEaobetg principle) that k i&
the oomplei field. Let X = V/U as usual. Then the division ring |
End*X admits a faithful representation in the rational vector ,
space UQQ, bo I7®Q becomes a veotor space over End(J. (
Hence dim<)f7®Q = 2g mnst be divisible by dim0 End*X = d*e
This is definitely false in positive characteristic. In fact, for
any characteristic p > 0, we shall see in $22 that there exists an
AND THE 1-ADJC BEFREBENTAT1ON
183
jljptic ourvB Z with jj-raiik 0 and that for such a curve, Rtid'J
¦ non-commutative of rank 4 with center Q. Thus, in this oase
PsriKiTIOM. A simple, abelum variety X is of (CM)-typz »/
fa = 2g where d1 it the rank of End"X over its center K, e is the
fcqrtt. of K over Q and g the dimension of X.
Now, in a division algebra A of rank d1 over itfl center K it is
known that all maximal commutative subfields have degree d
oTer K. Tliua, a simple abelian variety X is of (CM)-type if and
only if Bud'JT admits a subSoid of dogreo '2g (since in any case
it < 2g)-
20. Riemann forma. Let I be a prime different from the
characteristic of fc, and 1st ^ be the group of P-th loots of unity
in k*. We have homomoTphiBnia fi,.+1 -+ n^ given by f i—^ f1, and
this makes {p^} a projective system. I*t us put Jf, = lim p^
if has the structure of Zj-module. Since evidently we oan choose
isomorphiKma ^ •- Z/i"Z suoh that the maps fi^+i -* ftp. go over
into the natural maps Z/l*+lZ ¦+ Z/("Z the projeotiye limit is
(non oanoiiically) isomorphio to the Z, itaelf.
Now, let n be any integer prime to the characteristic. We have
set up a canonical isomorphism of ker nj- with the dual of ker nz,
th»t is to say, we have defined a pairing which we will call i.:
X, x (Z)» -* rt,, where m ia the group of »-ih roota of unity in
it* Reoall the definition:
Take »el, and AefZ),, and let \ correspond to the line bundle
I Than ?¦ is trivial, so ftj? ia trivial too and
L s A1 X X l\ aotion of J.
fcr a oharaoter
**¦ Then

184
ABEUAB VARIETIES
Hom(XJCj AMD THB l.ASIC RBPRB8BNTATION
185
In other words, we take the canonical action of ZB on nj? sn(j
carry it over to an action of X. on the trivial handle, where it u
given by a character ^. It is useful to have an alternate
definition of «, using divisora instead of line bundles. Let D be a
divisor Buoh that
Since i* and nJX ire trivial there are rational funotions / am] j
on X Buoh that
Then
so for some constant n,
fr» =«./(*».«;), all* eZ
It follows that \g(x)}g{x + o)]» = 1 for all « e J, i.e. g{x)\g{z + 9)
la a constant n-th root of unity, and we can prove
Lbbiu. *,(«,*) = -.?(ic>.
Pbchw Let
sheaves
* t)(a) Consider tha diagram of mape of
mult, by g
It follows that for all affine open sets U c Z, if V — n^'{IT) then
we get a diagram
mult, by g
and this identifies F(IT,u?r(.D)) with the subspaoe of F(F (t^j of
functions /(*) auoh that
fix -{-u).g{x -fu) =S(s).p(i
On the other hand if we let if be the quotient of A1 x X by the
i
j then V{V, M) is identified with the aub»paee of F( V Sz) of funotions
| ftp) such that
i.e. /(* 4- u) = i(«)./{a;). all « e Z.. This ia tlie same condition
u before, so M = BX{D), i.e., Jf = i Therefore t) must equal x.
' We want to paaa to the limit over n, by means of the
Lei m, n be (wo integers copnme to the cfteroe
y e
PBOOF Let T be a ouraplste variety, G a finite group
acting freely on V, H a normal subgroup of O, La line bundle
ca VjG which beoomes trivial when pulled back to V/?T. Ttmn
we get an assooiated homomorphiem x of GjH into i* as above.
But now, L beoomes trivial also wlien pulled back to V, and we
thus get a homomorphism x' °f ^ into ^'^ I* >s ^^en clear
'-Q -+QJH is *be natural homomorphiam.
Let us now apply this remark with V =Z, ?J =
The quotients Z -+ XjG, X -*- X/tf and XjH -> Z/O identify
themselves to the maps(fnn)z:X -+X.mz:X -t-Zand nx: X -+X
rsspeotively, and the natural homomorphism 0 -> QjB beoomea
X^ —^- Xu. Hence, for any line bundle L on X suoh that «?(L)
ia trivial and any x e XM, we have by the above that
where A e Z, corresponds to 1/ Writing A — my with
the proposition folio we

186
ABKLIAN VARIETIES
In particular taking n — I* and m = I we get the ©ommutatiys
diagram
f-th pewer
and henoe by passage to the limit as i ->¦ oo we get a natural
pairing
V r,(J) x r,(?) > if,-
One oheoks trivially that this pairing is ^-bilinear and non-
degenerate, Further, if /: X -+ T is a horaomorphism of abelian
varieties, / itB dual and T?f) and T,lf) are the homomcrphisms
induoedon the Tate modules, we have for ? e Tt(X) a d i)eT,(Y),
'ATl{f}tihv) = *t(t>T,{?)W)- (I)
This follows from a corresponding equation for «. whioh follows
readily after writing out the definitions.
TOM R3«*AJ*lf FORM OF A IttVlSOB
Dbfmitiok. Let Lbia line bundle on on oidtan uon>(j/ X, and
I a prime dUtiw-t from iht cKaraUeriatic of k. We then define tie
Jtamann form gP of L to be the Z,H&tK«ear map SL: T,{X) x T,(X)
-». Jf, j/ivm by EL{x, y) = «,(*, r,(fe){jf»-
Thbobbm 1 Tht. Siemana form SL of any lvu> bundU L u
tban-tymmeinc
We will give a sheaf-theoretic proof of this in §23 Bather
than ohase through oonfuBing diagrams here is the proof in the
language of divisors
Plioor. It auffioes to prove that \{a, $L(a)) =1, all a e i.
Lot the divisor D represent L. If a e X,, and Q utiafies
AND XHK (-ADIC REPRBBEMTATION
then «e must prove that q(x + a) = g[x), all x e X Choose b such
that «* = <!, and let JB = ti^'Z), so
Then
E is invariant under Tv
»i
Therefore A(a) = TT a(>; + »6) is a constant hence
f-0
¦-t
T~f ?(*+*+»*)
n
S(«)
Thus Li—>Zf1 induces a map
This is injeotive sinoe ED = 0 » <fiL = 0 since e, is non-degenerate
How, if /: X -+ r is b, homomorphism of abelian varieties, and
? is a line bundle on F, then
at y e TX (II)
*, y) =
Kent, we can compute EF, when P is tho Foincnr4 bundle on
IxX Identifying T,{X x X) with T,(X) x T,{X). then

188
ABELIAN VARIETIES
Proof. By skew-symmetry and linearity, it suffices to chow that
.»'«*, 0), ($, 0)) = JP-KO, 2). <0, y)) = 0 and**<(», 0), @, y}) = e,(x, y).
Using the funotoriaJity property of ? for the inclusion of X x @)
in I x I and the triviality of the restriction of P to Xx@),
a follows that Sr{(z, 0), (y,0)) - 0; similarly Er{@. ?), @, J.)) =o.
To prove the last assertion note that we have an isomorphism
(I X J) *¦ X X Y whioh ia given by the map of line bandies
Lt—*(L\X x @) H{0)x7)foT Le?iD»(X xY). In particular
taking T = X we have am identification of (X x X) with Ixl,
Now, for any (*, x) e X X X. ^.({x. 3)) ia given by the line bundle
-1, and thia is determined by the pair of bundles
Therefore <f>F{{x, x)) = (z, i(z)) where i; X -> X a the natural
homomorphuun. Thna, we obtain
*'{(», 0). @, y)) = ^(x, 0), (y, 0)) = «,<*, J)
Theorem 1 has the following partial converse
Tkbobim 2 LdXbtan abtUan variety and $: X ->¦ X a honto-
mofyAwnt Then the bilinear farm (x, y) i—> t,(x, <fa) on T,[X) u
tkew-tymmttrie if and onitf t/ Mere u a itn* intndte LonX such Oat
Proof If 8# =¦ 4, !!«,(*, «y)) = «,(*, ^(y)) is skaw^gymmetno
by Theorem 1 henoe ao is «,(*, fa). Oonreraely suppose this form
is Bkew-aymmetric, and let L be the pull back of the Poiaoar*
bundle P by the homomorphism A, ^|:Z-+Xxf. Then we claim
that 24 = ^. It BufBooe, because of the non-degeneracy of e,, to
show that 2e,(x, ^) = e,(*,#Irj() forany »,ye T,(Z). Now we have
)t*), A,
AND THE I-ASIO REPRESENTATION
IB*
by formulas (II) and (III) and the akew-symmetry of e,{x, <fa)
guuBK. We shall see in {23 that if 2^ =- fa for some line
bundle i, we most have ^ = ^. for another line bundle ?', Thus
the above theorem would then give a neoeaaary and BurHaient
oondition for a homomorphism ^' X ¦+ X to be of the form ^
Thj Bowti nr?OLnnoN
We fix an ample line bundle i on the abehan variety X so
^ i* an uogeny
2%e Jtoxiri inw)l«Jion on the algebra EntfX wtfil
M^ed to h uOk «.tp
Om has tha following properties of this map.
(I)
These are clear,
B) Extend the bomomorphism of nogs 71,: End X -*-
End^T^J) to a homomorphism End^-^-EndgjlQ1®^ Tt[X)) and
denote the extended map again by Tt. Then for any 4 e End*X
T0) is the adjoint of
form SL that is, we have
In particular f = ^
Pbooi We have
i.e..
for the non-degenerate bilinear
mTolution
«,{*,
whioh proTos the equation.

ISO
ABELIAN VABtETIES
C) Identify Q®ZN8{X)
, X) by the map Jtf t—*-^j,. Then, under
Hom°(X, X) -^ End'X given by tfi + fc* *$, the
goes over into the subspaee {^ 6 End°X | ^' = ^} of symmetric
elements of End"X fo* the Rosati involution. ?
Pboof. In fact, by the last theorem, an element Ji e End'Jf ¦
belongs to this Bubapaoe if and only if for ^=^°^, we bars
«i(*. +|d = - «!(».#*). Bat this means EL(x,$y) = -?%?(),
that is, ?<¦{*. tfy) = ^(K j,) = Jif^x, fy), for all t,p I1,! X), The :
result follows since BL is non-degenerate and ^ i—•+ T((^i) i| :
faithful.
Thkrbm 3 iet Z be an oielian mndy 'Tfen there u a
gtneraior
v 6 HoinIi(A*3',(Z). if*')
twtt (fe following property: for all divisors Dt,.... Dt o» ^,
let L{bethf.linelnindiet LX(D,) and kt Et=ELi be their Bitmap
forms Then
Sinee both the left and the right depend in a poly-
polynomial fashion on the images of the Lt in N3(X), this formnU
results by polarization from the formula with Li = ...=>Z^
Z>, =...= D, Using the fact that x{?) = (J>)/fj! we are reduced to
proving
1A
Fix an laomorptuem M^Z,, and ohoose a basis for T,(J) over
Then A ^(-I) beoomea uomorphic te Z, by using this basis, and so
that [EL]M beoomes a soalar. Farther, by using this basis, Ml
becomoB a matrix. We assert that
In fact, thia aqaation remains invariant under change of
basis for Qt® Tt{X), and it follows by a simple computation
on choosing a basis so that EL takes the standard form ::
AND THE I-ADIC KEPRBSBNTATIOH
0
0
0
0
0
0
smce both sides of the equality of the theorem are poly
in L, we are reduced to showing that %{1j? =e. det BP,
where o is an ^die nnit. By the Riamann-Booh theorem this is
equivalent to: deg 4% =e.det BL, where* is an i-adio unit. Now,
gi{Xi y) =e,(i!, &,(*/)), and ainoe 6) is a non-degenerate pairing over
Z, we see that detfl* = detT,{^) if we define the matrix
representation of T,(&) using a dual basis of T, (X)).
Choose and fix an isogeny ^: X ¦* X. We then have that
Thus to oomplete tha proof of the theorem we have only
to observe that deg ^/det T,{<f>) is an I-adio unit, and this
follows from the fact that the largest power of J dividing
deg ^ is the order of the kernel, or equivalent^ ookernel,
of ^lip.: Xp -+X,. for » large and this is the same as the
largest power of I dividing det r,{f)
Let X be a etmpU abelum vanetg of dtmtntwn
g,and KcRvd<>X a Q-svbafyebra evch that </,=<!>' for
Pboof. Since ^^'fH#)'=# for f ^ eK, R m asubfield
of End* X. further, since K consists of symmetric elements
it is oontained in the image of Q ®8 HcX/Pio^X by the map
^. Nawx(=*O depends only on the endomorphism

m
ABELIAK VABIETIE8
4l'm *a& ]t extends to ft homogeneous polynomial funotiog||
of degree g on the apace of symmetric elements of End'*
We assert that the restriction to ? of the function ^
is a norm fonotion on K. Now, it is easy to oheok that
it is already polynomial), if its square iJ?-' is multiplioatiwS
on K then it is multiplicative too. But ¦V.. '
deg fcrI"#ir. which is multiplicative in <fela$tt. Thus we get |!
norm function of degree { om ? and Jf being a field of
[K- Q], the oorollary follows.
21 Podtivity ol the Roaati involutioti.
Thiobxh 1 Let H be an ample divisor on on abdian »,¦«;;
i™ij{fl) <Ai associated line bundle and ' tht. involution o/Ead'I
jwon 4ji ?. Tien /or any ^ e End X tot ham
tohtre( ,) denotes intersection nttmbem. In particular
>* a jwjtitue definite quadratic form on Knd'X
Pttoor. The first assertion clearly implies the second, artoo fcr;
any eSectire divisor D and an ample divisor H, (H'* *. D) > 0.
It suffices therefore to prove the first statement. :
Choose and fix bases for Tt(X) and Jf, Applying Theorem }
{19 we get
= c. fJ5T"
for some I-adic nonotont o. Therefore
Since we have ?**<L> =
equation
x ^) we ere reduced to proving th*
ASD THB I-ADIO RBPRB8EHTATI0N
1
163
whdte ^' i> the transpose of ^ with respect to Jf1. This is purely
0 problem on linear algebra. We may utilize a basis «!,<„...,«w
of Q,®^-1) »uoh that JBI(«w_,,ew)=l and B^^^.e,)^
&{**>**! =0 if> ^2t or 2i ~ 1. Then the leftside becomes (by
definition of exterior multiplication)
Application I. Stbttotuiui ot End*i was aoiPLB X
We have seen that for a simple ahalian variety X, D — End°X
is a division algebra of finite rank over Q with an involution
x i—t- x' Buoh that if x ^ 0, Tr(ast') > 0 where the trace is tha
reduced trace over Q (or any positive multiple of it).
We shall now give the clasufioation, due to Alhart, of all pairs
(?),') where D is a division algebra of finite rank n over Q and
x i—* x' is an involution such that TxD)(i{xx ) > 0 for x e D
j # 0. We shall consistently one the following notations. The
center of D will be denoted by K, andK^ = {xeK\x' => x) is the
let of elements of if fiaed bj the involution. We put [D:K] = <P,
[Z:Q] = eand[ifo:Q]=e0 (so that n = e.i1 and « = % or e =. 2^
according aa the involution. U trivial on K or not). Without
further mention, we make use of the fact that the restriction of
I^fl/o to any simple subalgebra of K®Q J> ia a positive multiple
of the reduced trace over R of this subalgebra
jL.

194
ABBLIAN VAJUKTTES
St*p I. Now, lot at: Kq ->¦ R(l < i < r J be the set of distinct
real imbeddings of ?,,, and <r,t w: JC0 -»¦ C A< j< r ,)» set of oomplej
imbeddings such that any non-real complex imbedding of KB ii
C is either a certain trri+J or a oomplei conjugate of gome trr +f
Thus, »i + 2ri = V We then have an isomorphism of R slgehrst
Since the involution ie the identity on Kt for a e JCJ, we mm||
have Tr iE* > 0, and the same must hold in R®0Ara also {i|
fact this quadratic fonn has to be positive semi-definiteon Rg^JL
by continuity, and its null space, being the orthogonal oompkj
ment of the whole space for this quadratic form, must be t
rational subepaoe. But then, Trfi.i') > 0 for * 6 KB, x ^ 0, m
that it has no null space). But this implies that r, =0, as V
trivially seen, so that Kq is totally real
If now K jt Kq Ka-K^a.) for some *eKf,,\/zt-K0, and
(V«)'=-v'« Now, KSQjratB®^)®,,!:™ fi R^O^t
where R,o is R considered a» a ^,-aJgebra via a,. Now, R^iSk, 1
in inoraorphic as an E^aJgebra to either R x R or to C aooording m
c;(a) > 0 or oj(«) < 0, and the restriotion of the involution inter-
interchanges the factors in the first oaes and i» oomplex conjugation in
the seoond case. Again from the positive definitenean of Tr(atjr/)
on S.<SqK, one deduces easily that R x R cannot ocjonr aa *
factor, i.e., K la totally imaginary, and at(<z) < 0 for all t. Yfe
¦hall say that the involution ia of the fim bind if K = Ko, and
otherwise we say it is of the second kind !
i
Step II. If the involution is of the first kind, the involution
defines an isomorphism of D and its opposite algebra over the cental
K, so that its class in tho Brauer group Brfi:) off ie of order 1 or !.
If this order is one, we must have D*=K, Next assume that tb»
order is 2. Since by a theorem of Hasee-Braner-Noether, the rank
of a central division, algebra over a number field is the square of
its order in the Brauer group, D must be of rank 4 over K (i.e. •
AND THE {-ASIC REPRESENTATION
195
so called quaternion division algebra over K), In this case, there
is a canonical involution x i—*¦ x* of D over K given by
x* = TrJ(Ka! — x where Tr* is the reduced trace. (To check this
is an involution, extend D to the algebraic olosuTe of K, so that
ye are reduced to the case of a 2-by-2 matrix algebra over a
field, when thia ia trivial to check.) By the theorem of Stolem-
Noether, there is an oeJ)-{0) such that x' =<tx*a'x and the
condition that x' = x gives us that o* = t. a with t eK*. But
now, o = n** = (*.»)* = **<* bo that t ¦= ± 1.
Now, if« = l, a*~Ti,,IK
We have an isomorphism,
» a bo that a c K and x = x*
(*)
There R,j, is, as before, R considered as a JC-algebra through the
tth imbedding cr(, and each R<()®II> is R-isomorphio to either the
matrix algebra Aft(R) or to the standard quaternion algebra K
over R. If factors of the type ifj(R) occur, we wonld have that
for any A 6 Jf,(R), A^O, Tr((TrJ - A).A) > 0, that is, (Tr^I
> Tr A1, and this is false for A = I 1 Hence all the faetoru jn
the above decomposition of R^QD are isomorphic to K and the
involution restricts on each factor to the canonical involution.
Since for the standard involution on K, we have Trfc.a*) > 0
if i =i* 0, the conditions derived (when » = 1) are necessary and
efficient.
Next oonflider the ease when e— — 1. In the decomposition (*),
letffjbe the image ofa iaR(J)®xi> = Dt, so that on thia laotor, the
involution takes the form x\—* ^(Trjj^a— xfa1, and we have
also <? =Vr^mat — at = ~ i%, ao Trjj^a, = 0 Suppose now
that D, is R-isomorphio to K. Since <\a* ia real and positive, at
utiafies an equation x* + )? =0, \ eR*, and hence by fikolem-
Soether, we van choose an isomorphism of ffm with K such that
o, goes to At eK = R + Ri + Rj + Ri. But then, if * = i0 +
we have Trfx.x1) = 2(iJ + of - s^ - a?) which

19B
ANKLIAN VARIETIES
is not positive definite. Hence, eaoh D{ is isomorphio to Jfs(R).
Further, K\a\ is a subfietd of D stable for the involution suoh thst i
«' => - a, which shows that a' g K and a* is negative in every re&i:
imbedding of JC. Thus, eaea a, satisfies a minimal equation a} =¦
^ e R in Jf,(RJ with \ < 0. Again by Skolem-Noether (or trivial
ohecking) we oan choose an isomorphism R,(, (&J.D =; Jf,(lt) euoh
that a, goesto^J . ~ 0 J with ^, >0, ft, eR. But one checks
that in this easts, the involution on this factor is nothing but the
transpose (in the sense of matrices), and we certainly have
Tr(A.'A) > 0 for A e if»(R), ^ # 0. Thus the condition!
derived are necessary and sufficient for the positive definttenwi
of TM*.*1).
Stbf III. We oome now to the oaae of an involution of the
second kind. We summarize the results of class Reid theory con-
concerning the Brauer groups of on algebraic number field and p-adie
fields in the following theorem
Thbobbm. A) The* Braver group of a p-adte fieid « canonico%
uomorphu to Q/Z. If Lz K are two p-adic fields with [L:K]=n,
the induced mop BtlK) -t-Br(i.) goes over by means of tht. abm
isomorphisms into maltifiiooHon by n til Q/Z
The Braver group ofR is cyclic of order too and we identify tt
vnih the wtitptt cyclic subgroup of order S of Q/Z
The Brouer group ofCU trivial.
B) For any central simple algebra D over an algtbrmt nvmba
fieid K, and any finite or infinite' place v of K, if K, denotes tin
completion of K at v, let Inv,(?) denote tht, element of Q/Z cor-
corresponding to the das* of D®MKt m Br(Z,) Then me have on
txaat teqnetux
¦Bt{K)-
Q/Z.
v&ere the tteond map is gotten by forming tht rant of tht «Ienwnti
tuQ/Z
Hom{iJC| AND THE I ABIC KSPBESENTAITOH
187
Let us now look at the liwolutoriai division algebras of the
jeoond kind over Q. Let a be the restriotion of tho involution to K,
to that a induces au automorphism of Br(.ff). Tne existenoe of an
involution of the second kind implies that tt(ol(i))) = — ol{B), or
equivalently, using the above theorem, that for any place v of K,
Inv,(D)+Invw(D}=0. {A)
ginoe we have shown that K is totally imaginary, this condition is
always fulfilled for infinite o. Suppose then that (A) holds, so that D
Is isomorphia to the opposite algebra to the conjugate algebra Dw|
This means that we nan find a map D -+D,x i—y x* such that for
¦ f* d (y)* = y*x* By
Skolem-Hoether, any invohition inducing a on K must be of the
form x' =(W*a~1 for some a e D, a # 0. Since * i—t- x** is a
jf-automorphiam of D, we must have*** =axa for some« e D
and since
«*¦« l = («*)*•={*••.)* = ((w«-»)*=«*-1i;*ol*1 x e D,
we deduce that <x*a s K, and since (n*i)* = a*a, n*« e JC0. In
ordei that xi—t- x' = ojc*a~ ' he an involution, we must have that
t l il
dei t
*~ Hxar Io*o~' = * for all xeD, i.e.,
aa Hxar ooel, or equival
eutly, a" lo* = (ia for some p. eK. If we put ?(i) = «" '*• for x e i)
then $ is o-lbiear and the solvability of f (a) = fia with a ^ 0
impliea that
»o that «*« e NmX(ItJK*. Conversely, if this holds, let (a*«r * —
Hm^j^A, so that for any x e D, if«=A* + </•{%), we have
Tbus, under assumption (A), with * and a denned as above, the
necessary and sufficient condition far the existence of an involution
is that K*a e tim^^K*. Since JC/.K, is a quadratic (hence cyclic)
extension, this holds if and only if a*i is a norm in each (Ko)^
from K^, v0 being any place of ff0, and Ku being the direct produoi
of the oompletiona of K at all planes of K lying over va. II % in

I9S
ABELIAN VARIETIES
infinite and a^;Ka~*VL is the corresponding imbedding, D®, R
= D®I(X®itR)~D®xC=iir<(C) and • extends to a map of
MJC) onto itself of the form X*=AX? J-», AeQL{d, C). Hence
X**=A.A'~1.X.A^.A~1, so thattheimage of a inZ)®* R U A^^?
for some AeC*, and a*« has for image \\\* whioh is a norm fromC,
Thus, it suffices to look at the Arohimodean tr,. Again, if there
are two extensions of ^, to A", K^ is the direct product of two copies
°f {^oJn, astt i^t)*,-algebra, »o that the norm condition is vacuous.
Thus, we are left with the case of a e of K suoh that av = ti. In
this case Inv,(D) = 0 or \ by (A). If Iov,(D) = 0, D®K K, is a
matrix algebra over A", and vl i—*¦ t^A)' ia an involution of D®E E
inducing o on Kr, ao that, by the previous reasoning applied in the
looal case, «¦« is a norm. Suppose now that Inv,{D) = \, so that
D, = D<%)LKt is a matrix algebra over the quaternion division
algebra Q on K,. Since a induces the identity on Br(?,) (see the
theorem above), condition (A) gives us a vlinear map Q ->Q,
Xi—>X, suoh that (XY) =• YX. If we pnt X = fiXp-lhi
some ^e Q andX* —AX1 A~* for all X e B, and some A e ?>,,
we see that upto a factor whioh ia an elemant of the oenter, a
equals A A'~1^, and a*« differ! (rom
by a factor in Nmj^JCJ). Hence a*« is a norm in K^, if and
only if #) is, hence if and only if Q admits an involution inducing
a on K,, Soppose ' is suoh an involution. Then we have (by the
fuuotoriality of traoe) that Trr* ¦=• <r(Tr *), so that if i: Q-+Q
in the canonical inTolution of 0,t(i') = »W- Thus it—» «{*') w «
mitotnorphism ^ of order two of 0 induoing a on if If we put
G. = {* €01 *(*) = *}. % i» » A-0-subalgebra of 0 and Kw Ox,, % •*
Q is an isomorphism. But now, Q, is of tank four over ?»,
henoe a quaternion algebra and since BrfJT,,) ->¦ Br(K,) is by
ASD THE tADIO BEFBE8ENTATION
199
means of the canonical isomorphisms with Q/Z, nothing but
multiplication by 2, Q = Qa®XnK, is a matrix algebra over a",
which is a contradiction.
Thus, it Ko ia a totally real field, K a purely imaginary quadratic
extension of K and .0 a central division algebra on K, the neoes-
eary and snffloient condition for the existence of an involution of
D induoing the non-trivial automorphism a of & over Eo is that
besides (A), we also have
Inv,(U) = 0 if <ji? =»«.
(B)
Btbp IV Suppose then that (A) and (B) hold and let x t—*¦ x*
be an involution. We shall then show that there are positive
involutions too and we will classify them. For this, choose an
isomorphism
q
Then the given involution has an extension to the right side given
by{XltX XH)t—.MJ^r1 ii.^5^1) with 3J =i)tvi,,
ij(eC*, A( eGL(d,C). We mnst have |jj,| = 1, and on replacing
Af by a scalar multiple, we may assume ^ « 1, so that A\ = ^
Hence, if^ = (A 4J, wehaveX'^-J. The set of/lei)®,^
with jl*=jli8of the form V®QTL where Fis a Q-subspaoeof D,
so that we oan and an a e F «ueh that a® 1 is arbitrarily close to
i?D©QR. The mop xi—> x' = a'1**! is again an involution
of D whose extension to Jtf,,(C) x ... X Jf^fC) ia arbitrarily
eloee to (X, XJ i—»• {5*!,,,,, 5*,). Hence for « a good
enough approximation to A, 1tDn[x. z') > 0 if i ^t 0 x e D
On Mt{C) x ... x -Mj(C), this involution is of the form
with A( hermitian and close to /, so that the 4, are positive
definite. Let ?( be a positive definite square root of Ai, Modifying
the chosen isomorphism D®Q R "*, ^(C) X ... x Mt{C) by the
inner automorphiBm given by ? = (?, ...,^,), we see that we

200
ABBLULN VARIETIES
may Hsnme that the extension of the involution to M4{C) X .. x
is the standard one i
which la certainly positive. '¦
Thu*, when (A) and (B) hold, we have found one positive in, ;
volution on D and an isomorphism r>®QR^>if<{CJx... yJf^C).;
suah that the involution goea over into the standard one written)
above. Suppose * is any other positive involution, so thatt
x* = ax'*-1; a,' = Act for gome A e Z. Since A.A' ^Hin^A = l^|
we can write A = — for some p e Jf, and when a is replaced bj>|
jut, the involution is unchanged whereas the new a satisfies «' = i,:i
Henoe a goes over into (A, A^)eMt[C) x ... x Jfj(C) with^
^f hermitian. Fosttivity of Tr(x s*) gives ua the oondition thaif
Tr(X Af X^Af1) >0 for X e Mt(C), or equivalently, for aoina'
unitary V and any X 6 Mt[C), J
so that U'
is real diagonal
A, 0
0 A,
We must then have Tr(X D j? D ») > 0 for all X e Jf^C}.;
A,
¦o the oondition is that D la positive definite or negative definite.;1:
Sines we may raplaoe a by — a, this proves that all positive invohi^
tiona are of the form xi—> «a;'«~l where the hermitian matrices
At aje positive definite. I)
Summarissing, we have
Hom(X^E) AND TBK (-ADI0 BBPRBBBMTATIOH
991
Tkbobim 2. Lei D be a division aiyebra of finite rant over Q mth
o» involution ' tueb that TrDlf)(xx') >0forxsD,x^0 Ltt K bt
fa center of D and Ktthi subftdd of elements of K fixed by Then
iD,') i* one °f the following types
Tyfb 1. D = K = Kt is a totally real algebraio number field
Dad the involution is the identity.
Tyfb II. K = Kt is ft totally real algebraic number field and
X) a quaternion division algebra over K (i.e. a central division
tlgebra of rank 4 on K) such that for any imbedding a- K -*-R
let x* — Tr x — x be the standard involution of D and o 6 D suoh
that «'eJf and a* is totally negative. Then the involution is of
the form x' = a x*a~ », and conversely, any such map is a positive
involution. For any such involution, we can choose an isomorphism
K®QD-^MJ(R) x ... X Jf,(R)(e = [i::Q] faotora)
ouch that the involution extended to the right aide by S-linearity
k given by (X,,..., X.) -* {X{ X>).
Tvph III. K = Kt is a totally real algebraio number field and
D a quaternion division algebra over K suoh that for any
imbedding <t:K -*R
where K la the standard algebra of quaternions on R In this ooae
the involution ' is the standard one, x' =Ti])jKx~- x, and there is
so isomorphism
H®uDAKx xK,
carrying the involution into the product of the standard
involutions in each factor K
Tyfb IV. f, m totally real algebraio number field, K a
totally imaginary quadratic extension of Kt with oonjugation a
over Ka, Then B is a division algebra with oenter K such that (i)
if o U ft finite plaoe fixed by a, Iuv,[D) = 0, and (ii) for any finite
place v of K, Inv,{D) + Invrt(I>) = 0

202
ABEUAN VANTJ5TIES
In this case, there exist totally positive involutions *t—+ x' i
isomorphisms
R®QH -^- JtfjfC) x ... X M4(C)
which carry the involution into the standard involution (Xlt.
i—* (XJ,..., XJ,). Given one suoh ', any other positive involution '5
of D is of the form x*-=ax' a'1 with o e D, a' = a and suoh that;
the image of 1 © o by the above isomorphism is of the form;
{Ai ji^,) with At hermitian positive definite i
The following table gives the numerical invariants in al],
four types, and also indicates the restrictions on these invariant!;
when D*»E!nd*X where X is a simple ^-dimensional abeb'an;
variety. The symbols «, et, and d have the same significance
as before and 5 «={*eD|ic'= *} and i)= -^9^.
dmi0 D :.
Hom(i 2} AMD THE I ASIC BKPBEBBNTATIOM
893
Restriction in char 0
Type e d ij when D = End»X,
dimX = 3
ReBtriotfcin in aha p
>Owheni)=End"Z
dim X => g
I
II
III
IV
u
N
2ea
1
2
2
d
1
1
i
e\g
EzoepUng the indioated restrictions, all the assertions oontftined
in the table have been proved. As for the restrictions, they
ftrs immediate oonsequencea of the three divisibility reeuK*
established earlier, viz. (i) in char 0, dim D|2 dim X, (ii) in ohar p
>0, «i|2dimX, and (iii) if L a a nubfield of D whose elements
ore fixed by the involution, [L: Q]|j.
One might ask to what extent the conditions derived abov« i
on the endomorphism rings of » simple abehan variety are
complete that is, given a division algebra of one of the
four types and an integer g fulfilling the restrictions imposed;
> whether there exists a simple abelian variety of dimension
f having the giran algebra as endomoiphiam algebra. In
characteristic zero at least, the answer is known and is
due to Albert. The result is that there always exists euoh
an X, excepting when D is of type III or IV and the quotient
gf2t in the first case and p/e, cJ' in the second case is 1 or 2.
gren in those exceptional oaseB, it is known what farther
restrictions ensure the existence of an X (of. Shimura,[Sh], esp, {*).
On the other hand not muoh seems to be known in positive
characteristics,
APPUCATKW II, THB RtEMASN HYFOTHBBIB
We first prove the
Proposition. Ltt X be an abtltan variety,' the Botati invo
Mion on End*X defined by tome ample line bundle and a e
EndX such that a.'* = a eZ Ltt«,,..., aiu be the root* (tn C( of
At characteristic polynomial P of a. Then the subalgebra Q[a]c
End X generated by « w semi-simple and
(i) W(|' — a for all i;
O) the map mi -f — is a permuttitton of the root* a,
Fboos. Note that (ii) is an immediate consequence of (i) since
uj, and P is an integral polynomial. Next, let Q(X) be the
minimal polynomial overQ ofn(asan element in End X}. I claim
that P and Q have the same complex roots. In fact, Bince P has
integral coefficients, and P(<t) — 0. <J|P. But also P ]a the charaote
riatio polynomial of Tt(&) in the matrix representation
Tt: End(X) »- End(T,X).
If «i eQ[ (algebraio closure of Q() is a root of /', then <o is an
eigenvalue of T,{«), hence Q(w) is an eigenvalue of T((Q(a)). But
T,(Q{a)) = 0, so Q{tu) = 0, U, all roots of P in Q, are roots of Q
Therefore P\Q" for aorno n and P mid Q have the same complex
tools too.

204
ABEUAN VARIETIES
The restriction j9 of the traoe on Bnd*X to Q[a] is a tiaoj
form on Q(aJ satisfying 8(X.X') > 0 if XeQ[«], X #0. Further]
since a. is invartible in ?nd*X and Qfa] is finite-dimensional, itl
follows that a" e Q[a]. Hence «' = a/a e Q[a], so Qfa] is stsbl|
for the involution. If 0 cQ[ot] is any ideal in Q[a], and 6 is ita
orthogonal complement in Qfa] for the quadratic form 8(X.X'},
is again an ideal and H n 6 =» @), « © 6 = Q[s]. Thus Q[i] ii
semiBimple, hence isomorphio to Kt x Ktx ... X Kr where Kt
algebraio number fields. The involution, being an automorphisnv
of Q[a], permutes the faators K(. But since S{X.X')>0 for every!
X ^ 0, the involution must take each ?, onto itself, and therefore;
8 is a trace form on each Kf over Q with S(X.X') > 0 if X ?
Hence each ?, is either totally real with identity involution or is sj
totally imaginary quadratic extension of a totally real subfietf
with complex conjugation for involution. Now, the roots u of
the minimal polynomial of a are precisely the images of auntie
the various imbed dings ^ of the iC, in C. Since <^(i') = ^(z
for all x e Q[a], it follows that
We shall apply this proposition to obtain a proof of tha
Biemann hypothesis on abeiian varieties over finite fields,.
Let F =F( be a finite field with q =p? elements, and 1
a scheme of finite type over F. (We do not consider X,
as a variety whose points are geometric points with values in an
algebraically closed field, but as a scheme in GrothendieokV
sense.) We define the Frobeniw morphism on. X«, v0: Xt ¦+Xil:
to be the identity on the underlying space together with the-'
homomorphism <Stt -+®zt of structure sheaves given by/i—\
Note that this is a homomorphism of sheaves of F-algebrss
since )f=\ for AeF, so ir0 U a morphism over Spea F. Now
let h be the algebraic closure of F(i and let X be the Jfe-aoheme
X = i©pXr The morphism n:X->X obtained from v, by-
base extension is called the Frobtnius morphitm on X, relativ<
to F and to Xo. Lot us see what this looks like on the geomatrio
(or olosed) points of X Suppose that X, => Spec A whert
AND THB tADIO REPBESBKTATION
206
jFfZi, , ^l/a, so that X, is embedded as a closed subsoheme
id A", &nd the closed points of X can be considered as elements of
the set it*. The morphism ir0 is defined by the homomorphism of
F-olgebros A -*¦ A sending Xi into jSJ, ho that if («,,..., ajj U a
geometrio point of X, v maps it into the point (zj a^). In
particular, a point («„..., x^j is filed by n* if and only if af = ^,
i,e. if and only if ij is a rational point over the field Ff« with g"
BleoientB. Further, the Bfobenius morphism has the functorial
property that if/: Xq -t- Tt is a morphiam of F-sohemes and wttrjrg
and no,r, are tne Srobenius maps of Xt and To, respectively, n^,"/
= /°'ro,i,- Koilly, it is clear that the map induced on tangent
spaces bynat any point of XisO, since D(y)=0 for any derivation
D of a ring A of oharaoterUtio p and feA
Thsobbm 3. (Lang.) Zef JCj 6e o jcAcok ofw F, micA that
X = i®FX0 " OB «W""» «on«ly rftwt JE^ *<m o* feuf o»ts po»n(
raticww/ oner F(.
Pboof. If v is the Frobenius morphism, then tr must have the
formir(i) = xt +/B) forBome olosed point TBeX and some endo-
morphism/ofX. Then 1— / isanendomorphismofX. Sinoe-uand
benoe / induce the xero map on the tangent space at 0, 1 — /
induces the identity on this tangent space. Therefore ker(l—/)
is 0-dimenaional and I—/ is surjeetive. Then if A — /) t*i>= ar0,
it follows that », = afc+M) = »t*i), hence a^ is rational overF,.
Therefore, if X is an abelian variety, by choosing an appro-
appropriate origin OeX, we can always assume that 0 is F-rational.
Then each ** fixes 0 and is therefore an endomorphiam of X
Moreover, 1 — n* induces the identity on the tangent space at 0
so it is also a separable endomorphism. Henoe we obtain:
Nt = Number of F^-rational points of X =#(Kw(l - n»)f
But if uit, ,.,,ai^ are the roots of the characteristic polynomial of
ir, then the characteristic polynomial Pn(t) of n", for all n, is
110 - tJft Since deg(l - if") = P»(l), it follows that
4

?06
ABELIAN VARIETIES
*. - n <> «?)
•-I
Wa now wish to show jut,] =¦ \fq: this is the Riemann hypothesb.
Sinoe it suffices to prove that |«7I = y/tf* for some m, we may
replace F, by F^, JT0 by F^ ® F Xo and it by »* if necessary. By
doing this, we can assume that there is a line bundle ?„ on X,
such that L =. h®rLt ia ample on X (shoe any line bundle on X
ia of this form for suitably Iwge m). Denoting by ' the Rosati
involution with respect to L, we shall prove that
(i) it ° -a = q,
so that the proposition applies But by the definition of this
means that
La a ?] There-
ThereBut jt0 aots on &It by /i—yf, go it follows that
fore w*L = L* and
(iii)
Slnoe the line bundle on the left represent! w^^fi))) and the
Une bundle on the right represents g.^x), (ii) and henoe (ij are
oorreot.
We summarize our oonolnsions in
Theobbm 4. (Weil.) Let X, be a scheme over Ff stick that X—
i<3fX0 u an abtlian variety Let N, = the numAer of points ofX
rational over F Then
*-- n {<# - !>
<~i
oj, e C and they satisfy
@ Ki=v?i
(ii) in,, = g/tuj /or torni petmutaiion it
Hodi(XX AND THE I ADIO REPRESENTATION 107
CoaOLLART For some constant C tfK (f*\ < G g*' 1 for
iln
Another application of the proposition is
6. (Serre.) For any n > 3 and any L ample ott an
variety X, Iht restriction
ae AutJT
a L®
something\ I
in Pio" X } I
Proof. If ¦•? = L® (TnTic^) ' ihen *>'i-^' henoe
,,^.1=^, Tliifl means that for the Roaati involution defined
byt, a'«=l. Hancju by the proposition the roots of the charao
teristic polynomial of a are algebraic integers alt of whose oon
iugates have absolute value 1, and hence are all roots of unity.
Suppose now that a restricts to the identity on some XJn > 5)
T^ien the restriction of «-I to X. is 0, so that {ot —l)=n0 for somo
8 g End X. We deduce that if id ia any characteristic root of ¦
m _ ]. ~ nij where tj is an algebraic integer. We now have the
Lemma. If mil a root of unity such that tt = I +ttij where ntsa
fjfiojiol iidtger > 3 and tj aft atyefrraie integer then <a = 1.
Pbockf. If not, by raising to to a suitable power, we may assume
that a is a primitive p-th root of unity for a prime p Taking
norms over Q in the equation w — 1 = nij, we obtain
1^A «.')-»• 'if
where N = (— IJ'^'Nuli) is a rational integer But the left side ll
the derivative at X = 1 of X»- 1 = 'fl (X w1) that is j»
t-o
Renco n'~' divides p, which is impossible if n > 3
Applying the lemma, we deduce that the characteristic roots of
a are all I, so that 1 -« is nilpotent- But by the proposition,
Q[«) la Bemi-simp!e, so it has no nilpotent elements Thus « = 1

208
ABBIIMT VARIETIES
III. STBUOTtma OF
Let X be an abeKan variety and let NS*(X) =N8(X}<g)Q. Aa\
in S 20, if we fix an ample L on X, then we oan identify ;
N3*(X) -Z+ {« 6 End'Z | «' - «}.
p
In particular, N8*(X) has a nstural structure of Jordan algebra
over Q if we define
using composition in EndeZ. What oan ire say about this Jordan
algebrai Kiet of all, the foot that Tr{p(<tf) > 0, all ¦ E
a 9^0, implies immediately that JW°(Z) is/ormoHj real I.e.
(of Braun Koeoher, fB K] Ch. 11, $3). Now the formally real
Jordan algebras over R have been classified: of. Braun, Koeeher,
Ch. It, }S. In our aata, we do not get alt possible oooh algebras
by fanning tfSfZJSR.In faot. we have :
Tesokim 6. NS°(X)(g) R u taofimrpAtc to a product of Jordan
algebnu of lie type*:
Jfr(R) =fxr eymnefrtorealmatrxue
Jff(C). » rx r flermtiton complex iMtrustt
Jfr(E) = r x r Hermitutn qaatamionie tuatneet, i.e.
<Z - Z, wfora i-^i u tiU Jtandonf
tneoi«(io» on K.
i*BOO». Dooompose End"(Z) © R into a product of oopies of
JLf((R), if.(C) and MJYL). Then ffS»(X)«iR is uwmorphio to ths |
¦et of fixed points here under a positive involution. But it is easy'J
to oheak that every such involution (a) flies eaoh of the factor* |
M.{K), K = R, C or K, and (b) by Inner (wtomorphifla of eaoh %
factor, can be put in the standard form X -*'X. [Of. the proof of :|
Theorem 2 Step IV*, for the case K = C; the other oases are
analogous. One checks first that every positive involution of
Af.tiT) is of the form Ii—> A-'S-A~X where 'J = A. One thenf
AND THE I-AIHC BEPHEBEHTATIOK
209
cheoks that A = U D- V'1 where D ifl diagonal with real entries
either all positive or all negative and 'V — U~\ Knally, solve
^P^iPwid use the inner automorphism defined by UEU~X to
put the involution in standard form.]
Fix isomorphisms ^ and $
(I)
What happens to the polynomial function x: NS^K)® R -> RI R>t
any * eEnd*(X)®R, let ^(as), A.iW- ^t*) denote the oom
ponents of ${x) in the above decomposition. Then we know that
the function "degree' oan be written
(II) degOt) = ndetW^sp. nidetW^jf* nNm^as)"
where Nm: J/,(K}-t-R is the reduced norm {the multiplicative
polynomial of degree 2t). Now on N8{X), deg(pz) = n.xf1)' for
Boras constant a. It follows that the o( are even and that the
function x oan be written
is Hennitian, so its complex
determinant is real. Here "HNm" is the "Haupt norm" of Braun-
Koeeber (Ch. 2. §4), a polynomial function of degree * from Jf ,(K)
to B. If Ao e N&>(X)<&1L is the point defined by the ample L on
Junedto Betup/., then p(A,) = ], so ^J{A,)=i, so the constant
in the above formula is X(K)- Using tho results of J16, it follow*
finally that:
(IV) IfX{z)*0, then
;. eigenvalues
x

310
VARIETIES
(The eigenvalues of a quaternionio Hermitian matrix H are defined!;
M the entries of a diagonal matrix D suoh that H «= U*D*U~l
and 't/= ff.) Since ample line bundles L are characterised by;
X(?) =? 0, i(L) =>0, it follows from (III) and (IV) that the image*
of the ample line bundles in NS{X) are exactly the totally
positive elements of the formally realJordan algebra iW(X)@R, .
22 Exampltf.
FlBST KXiWLE, ABBLIAS YARIBTIBS OF CM ITIB OYSB C '
Let X be a simple jr-dimensional abalian variety, Z> = End°(X),!
tf = oenter of D,d*~[D:K] and e ==[#: Q]. Recall that «d|fy;
and that we hare oalled X of Cif-type if td — 2g. We wish tft
classify these when k = C. A glance at the table in 520 giving th
types of division algebras D tells us that we must have K = D, JC
a totally imaginary quadratio extension of a totally real field K^
of degree g over Q. ^
We pose the problem a little differently. Suppose we are given!
a totally rasl number field Eo of degree J over Q and a totally!
imaginary quadratio extension K of Kt. We consider all pabal
(i, X) where X is an abelian variety over C of dimension g and i: ?|
-+ Enci'X is an imbedding of the field jC in the ring End*X We|
define two such pairs (t, X) and (j, T) to be equivalent if thsrtl
isanlsogenyaiX-t-rsuohthatif a:Bnd»X-^>Bnd»r ia thf
induced isomorphism, we have ««< —j. It is easily ohocked thai
this ia an eqaivalenee relation. Our objeot is to exhibit a complete
set of representatives for the equivalence classes. i
Let (i,X) be any such pair, V the tangent spurn of X at 0 and|3
U the kernel of the exponential map from V to X so that wot
have a nstural teomorphiim 7/P~^*-X Than BndX Mt*|
faithfully at a ring of C-endomorphisms of the veotor space V^|
leaving U stable. Thus, if we pnt«- '(End X) « A c K, A is »h|
order (i.e. a finitely generated subring of maximal rank) in Kg
Mnd V beoomas an ^-module Thus, Q. U c V becomes a veaktir
<3iz^Ea? and oinae both ?andQ U are of dunen-J
Hom(Jt,X) AND THE J-jUDIC BEPBEBENTATIO1*
211
sion 2jf over Q, Q.Uiso one-dimsnsional X-vector space. Hence
if we choose a non-zero element «,eP, the map ^: A -*¦ V
defined by at—» a.«0 is aninjeotion of A into U, suoh that the
inde* {V: $(A)\ < + oo. Changing X by an isogeny, we oan first
tbrink V so that U= ij>{A), and then increase U so that V = >j>(At),
At= ring of integers in K. Next, the map <f< extends to an R-linaar
map which we still denote by <j>;
It follows that <j> defines an isomorphism between the Ttal tort
Note that if a 6 Ao, then this isomorphism has been set up exactly
bo that the endoraorphism i{a):X -+X corresponds to roulttplioa-
titm by 1 ® a in R$$iQK.
Next, let <D denote the eomplex structure on the real veotor
spaoe R<8qX obtained by pulling bank the complex structure on
V via 4>. Since multiplication by l®a in R®gJC(aeJt) goes
over via <j> to a oomplei-linear map from V to F, it follows that
in the oomplei structure *, multiplication by 1 ® o is complex-
linear too. In other words, 4 actually makes the R-algebra R ®q?
into a C-algebra as well as a C-vector space. We now invert this
whole construction
Dmihition. // K t* a* abovt., Ao = tntefeM wi K and » ts a
jfrueCure of C-algtbra on R®qK, Men let
») =lfe compltx tana R
iy 1 © a.
We have Bhown that for given K as above, and any pair (t, X)
tbere is a structured of oomplex algebra on the real algebra R®qJ?
raoh that (t, X) is equivalent to (i,, X(JC, O}). Our next aim is to
ihow that (i) for any structure 0 of complex algebra on R®qK
X(K,«) — R®Q?/1 O At is an abelian variety, and (ii) for

212
ABEXIAN VANISHES
different complex structures *, and *, On R®q.K (»»
and ((,,, Z(?, *,)} are not equivalent.
X{K
To prove (i), let us look more olosely at a structure * of comply
algebra on R®Q?. Giving such a O is equivalent to giving »
homomorphism 4 of R-algebras, <J>; C -+ R®qK. Now, if
d;A < i < f} are the distinct embeddiags of JC0 in R, we have
an isomorphism of R-algebras
A®
(A®« A ® «,... ,A®«)
where R((l is R sonsidered as a JS^-algebra through at Thus,
giving an R-algebra homomorphism 4: C -»¦ R ®qJT is in turn.-;
equivalent to giving B-algebra isomorpAwro* ty: C y R((}®i,i"
for !<«<#. For each i, there are clearly two such possible S-
iBomorphiBmsC-f-Rtj,®^^. Thus, we see that there ore exaotlj
2* possible <t on R <3IqJ?, and each $ is uniquely determined
by giving the corresponding R-isomorphisms *,; C->R(H(SI(,ff
A < »< g). Let t(: R(j)®Ii.K-+Cbe the inverse of <&t, bo that
t( restricted to if sr 1 ® Z c R^,®,^ is on imbedding of K in C
extending the imbedding <rt of ?o in R. We can choose an element
a e ^ suoh that a1 s JT0 and Ti(«)=tfi, fteR, ft> 0. Infect, if
ifo(VS) then t((-/S) = »n
e R*> "id we can find i) e
such that <h(ti) has the same sign M ylP and we can take a =- W'-
We may further assume a to be an algebraic integer. If t;(«) =¦
tft(l<»<j) we define a Hermitian form S on(R®gJT,4)
by putting
E(x
This form is dearly positive definite, and we shall show that Im B
is integral on the lattice A^. In fact, for x.yeAg, we have
lmE(x y)=
Boa(ZtX) AND THE 1ADIO SKf RBHENTATMMf
— 2 ^ Ee
SIS
where for any y e K, "y denotes its conjugate over K<,. Thus, for
(my oomplex algebra struoture * on R®Qf, and the lattice A^
in it, B denned above is a Eiemann form and X(K,<b) is an abeiian
proving oisertioa (i).
To prove (ii) suppose there is an equivalence of (i^, X{K, *[))
^nd {», , J(iC, *,)). Then we deduoe an isomorphism of C-veotor
epaoes 'a= (» ®(^, *,) -^ (R ®q&, *i) "W* th»t X(l ® J) =
1 ® X and A is an isomorphism of ^-modules. If A{1 ® 1) = 1 ® x,
xeK*, by replaoing A by (igiar'J.A we may Buppose further that
= 1® 1, Since A is both K and It-linear, we deduoe that
=a@>xl oeR, xeK. to that A is the identity. Since A is
C-linear we must have <&x =*i, whioh establishes (ii).
We have therefore proved the
Thcobbm Let K„ be a totally real number field of degrtt g over
Q and K a totally imaginary quadratic extension of Ka. Goiuide*
aBpairt (X,») where X m em abelian vaneiy overC and t: K -*¦
Eod'J on eisixddiim. with the equivaknct relation defied abovt.
Then there arts exactly 2» eqvivalmae dense*
<mplex*tni<>iureatmR®QK which make R®<^ff a C-atgtbra, (Ae
fain (JT(K,»), t») fffne a complete tyttm. of representative* in the
distinct equivalence classes
BiMABKfl. A) It is not true that X(.ff,*) ib always simple. It
oan be shown that in order for X{K,&) to be simple, it is neoes-
ssry and suffloieut that there does not exist a proper subfield X of
K satisfying the following conditions:
(i) ? U a quadratic extension of Z n JCf

211
ABELIAN VARIETIES
Horn XX) AND THE I-ADIC REPRESENTATION
215
(u) if 4 is given by the set of imbeddings tx iy of K in C
and if t, | L n Kt = ij | in X,, then t( | L = rt | L.
If such an h exists, X(K, <t) in isogenous to & power of X{L f),
where T is given by
B) Let us specialize to the ease of dimension one, that is, the '
ease of ellipfcio curves ovei C. IF X is an elliptic curve over C, j
either End°X = Q or End°X = QW -d) for some square free !
deZ,<J>0. Moreover, given any imaginary quadratic flsld :
Q(y _<j), there is an elliptic curve X with End(X= Q(\/-ity
and upto an iBogeny, X=f C/{n +mV- d|n,m e Z},
SwjOMD KXAKPIB: ELLTPTTO OITRVZB IN OHiJUCTEBIBTIO JI > (.
We begin with recalling some basin facts concerning
varieties of dimension one (or elliptic curves). These facts are :
immediate consequences of our general theory, as the reader mij I
verify for himself.
Let X be an abelian variety of dimension one. We shall denote
the divisor corresponding to a point Pby[P]. Then, for any diviwr
J) on X, we have xt^zf-^)) =* degD, and if further degD>0,
xfPxi1*)) = dimff«(<5*(-D)) and #»(«,{*))) = @). A divisor D
belongs to Pio°X if and only if degD = O. A divisor Z) = Sn([J1t)
of degree 0 is linearly equivalent to zero if and only if EjIj-P, = 0
on X. Any divisor D of degree > 3 is very ample
Suppose now that the characteristic is either 0 or greater thin
2 Let 0 be the identity element of the group X and let Pt,Pt, and
P, be the points of order two on X. Since dimflB((!'xC[0]} = a, to
o»n otioose a non-oonatant function x having a double pole at 0 &nd
regular elsewhere. Subtracting a constant from x, we may assume
x(Pt)= 0, and since the sum of the zeros with multiplicity is 0
and there are exactly two zeros, we deduce that Pt is a double zero
of a; and there are no other zeros. Thus, by dividing by a constant, ,
we may assume that x(P%)= landi(P,) = X eh*. By applying tin
above argument to as —I, we deduce that X#0,1. Sinee H°{@t[3[6\ti
is of dimension 3, we oan find a funotion Jf having a triple pole at!
and regular elsewhere. By subtracting a suitable linear combination
a*+6 from jf, we may assume that y{Pt) = (f(Pa) = O, andsinoe the
number of>oroa La 3 (taking multiplicity into account) and the sum
of the stcros is 0, we deduce that y has simple seros at P,, Pt and
P,= — Pf~ P,. Both the functions y1 and *{aj—I)(a—A) bave poles
of ordor 8 at 0 and double zeros at Pu i1,, and Ps and no other
aeros or poles anywhere, so that they differ by a non-zero scalar
factor. Replacing y by a non-aero scalar multiple, wo arrive at an
aquation
Z,:s'=¦-=«(*-!){*¦-A) <#,)
for X {0} in A'(A). Conversely, tho projeetive curve ^t = x(x- t)
[x — il) has no singularities and is of genus 1 for A ^ 0 or 1, and
hence defines an elliptic curve X, inP'{Jt).
We wish to find all possible values of A for which Xt is of
p-rank 0, for characteristics p > 2. We know that the y-rank is Q
if and only if the Frobenius map in Hl[X,€>) is trivial. The
mesomorphic form dx on X is regular in X — {0} and vanishes at
P, = @, 0), Pt = A, 0) and P, => (A, 0) to the first order, and no-
nowhere else, ainco d-x{P) = 0 and x(P) = a implies that x — n
vanishes to the second order at P, henoe 2P must be 0. Thus,
the form in = dxjy.ia regular and nowhere vanishing in X -~ {0}.
It follows that o> must be r^guiar and non-vanishing at 0 also. If
we put t/e = X - {0}, Vt = X {Pi), « = (Ua, UJ is an affiwe
covering of X. A 1-cooyole for this covering is a regular function
/in Ue n Vt — X — {0} - {Px}, and this i» a ooboundary if and
only if / = ff — h with g rogvihir on Vt and ft regular on Ux
Consider the linear form on C1^, 0) = F{Ut n Ut, &) defined by
/)—> ResPi(/<u). Since tho residue at any point of a meromorphio
form with a pole {of any order) at a single point of X and no other
poles is zero by the residue theorem, we see that ReBj, IJm) =0
if/is a ooboundary. On the other hand, the funotion yjx is regular
on U,y n Vl and has a simple pole at Px, so that Be»Pl (y/«. ui) ^ 0.
Since dimJT'tX, (!) = 1, we deduce that the above linear form
induces an isomorphism H^X, <P) ¦ ~fr k, and also that yjx
e V{U0 n Gj, 4P) defines a non-zero oohomology class inE^X.S).
Hence, the ProbeniuB map on H'(X, t>) is trivial if and only if

AND THB I ASIC REPHBSENTATION
217
216 ABELIAN VARIETIES
1 e T(lTeri Ux @) ib a eoboundary hence if and only if
ooefiioient of it*""' it
coefficient of a; a in
)*
(x
A)
= ±2
where
Now,
¦Iw
. of a**-1"* in a*
", an
1
. sothat*@)
^ 0, <t(l) ^ 0. Thus, every toot of 4 defines an elliptic curve
Xi of p-rant 0, and these arc the only elliptic curves of p rank 0
upto isomorphism
Let us call an elliptio carve in characteristic p > 0 *ujjst-
angular if its j)-rank is 0, We have then shown that in charac-
characteristic p > 2, any supersingular curve is iaornorphie to one of tkt Xt
when A w a root oj *{A). If p = 2, it ia not hard to see that
there is exaotly one supersingular elliptio curve, namely
y* -)- y = %*. We omit this. Therefore, in any positive oharaote-
riatio, there is one and there are only finitely many superamgulsr
onrrea upto isomorphiam
We now study the algebra EndM of an elliptic curve over a
geld of oharaoterifltio y > 0 Let h be the algebraically closed
field over whioh we work. We shall say that an abelian variety X
over h is defined over a aubfieid J^, of A if there is a scheme -X,
over ha such that Xn h <3itX. We can then easily eltabliah that
there ia a finite algebraic extension *! of i0, a rational point 0
in jfcl<&t<JC0= X, and a morphiamm!: Xj Xt)Xl •+ 3I1 over *j (raoh
that on base extension, we get (upto an isomcrphiam) the triplet
(X, 0, m)- In future, when, we speak of abelian varieties defined
over various fields, we will assume that 0 is rational and m
defined over this field. Another remark in this connection is that
if/: X-+ y ia a separable isogeny and if either X or 7 is defined
over an algebraically closed subfield ?, of k then /, Xand T are
all defined over *,,. This is dear if we assume X defined over *„
since T is the quotient of X by its kernel whioh is a reduced finite
subgroup of X, and all points of finite order Id X are ^-rational
Suppose on the other hand that 7 is defined over *„. By induc-
induction, we may assume / of prime degree I. If I & p, then there ia
• separable isogeny;: T-+X, defined over I, such that fog: Y-tY
is lT. so that we are reduced to the first oase, Suppose then that
\^p, and let 0 be the infinitesimal port of pT, and Y' = Yjff.
Then, Q and 7 are defined over V Then there is a separable
isogeny g: Y -+X, defined over le, such that the composite
Y -* T1-* X -*¦ Y is pT, so that we are reduced to the first ease
again.
Thsorbm. (Deurmg.) Let X bean ethpitccurw i»chtraettnn
(top>0 We, have the following equivalences
(a) X oamwt be defined over a finite pid
(b) Suppose X it defined otter a finite field
(i) End'i « imaginary quadratic over Q
1 «=> «• ,t j5 /or *uitaik tntej«r* n,
Ainn over i,.
//, ioteeiwr, tte p-nwfcwO, t»e»EndDX ilttefujKo womor-
, iuwgse) ftwtemion diouian oV*™ K^)ow* Q vAmA «a(t«/t««
o »/1 u finite and J> p and
End'X = Q
Then,
p-ranh of X a
(«)

218
ABKLIAW VARIETIES
nom(X,X) AND THB t-ADIC BEFRBSBSTATIOW
210
Finally then exists at least one. and upto isomorphisms at most !
fimUly many X *n eoeA characteristic for which (ii) holds. j
Proof. First consider the following three statements ¦'[
(A) X is of jt-rank 0. i
(B) End*X is non-oommutative
{C} X is defined over a finite field and if w is the Frobeniua
morphism over this field, u" «= i§ for some n and m > 0
We shall establish that (A) ¦<=> (B) and (A) ¦<=> (C) in that
order.
Suppose then that (B) holds. A look at the table of $21 tellg
us that End°(X) is a central simple quaternion algebra over Q,
henoe Q,®q EndQ(X) is also a oentral simple algebra over Q,. If
the jj-rank of X were one, Q,®^3^{X) would be a one-dimen-
one-dimensional Q^-vector space in which Q,®QEnd1? admits a represen-
representation, whioh is impossible. Henoe X has p-rank 0, proving (A).
Next, suppose (A) holds and suppose End X were commutative,
so that End'X =: K is an algebraic number field. Knee every
elliptic curve isogenous with X is again of j>-rank zero, and there
are only finitely many isomorphism classes of curves of j>-rank 0,
if JB is tbe ring of integers of K, we oan find an integer N>0 suoh
that for every X' isogenous to X, we have JV.JI c End X'. Choose a
prime 1 not dividing pN such that JH is a prime ideal in JI. (It is
known that suoh I exist.) Let« be a non-zero element of Tt{X) not
divisible by I and let Kt be the cyclic subgroup generated by
the image of a under the natural homomorphism Tt(X) ~*-Xfi.
Then JE» jj JE.+1. Again by the finiteness of the number of iso-
isomorphism classes of curves of jp-rank 0, we oan find integers m > »
sachthati^, JJ^ and there is an isomorphism (:XIKm > XjK^.
Thus, if 17: X/K^ ->-XIKm is the natural homomorphism induced
by the inclusion it. c K^, we get an endomorphism 1 = ij 0 ( e
End X', whare X •» XjEm, suoh that a has oyalio kernel of order
I* k > 0. Sinoe degree a and henoe Ndi^qi is a power of I and JB
is a prime ideal in 3 we must have a = t. u where u is a unit in S
wd r > 0. So Na — F.Nv, and N« e End X'. Now, the degree of
jit is JV» since u is a unit in B, so that the 1-priinary part of
ser(r.iW») is (Z/J'Z)». On the other hand, the [-primary part of
te'f.tf'O •» isomorphio to ker«, whioh is oyotio. This contradiction
proT^a (B)
We how prove that (A) -4=3- (C). We have already shown
that (A, implies that X ia defined over a finite field. I*t v be
the Frobsnius morphism over this field, Sinoe EndJf is finitely
generateu, we can find a finite extension of degree n, say, such that
erery element of End X is defined over this extension. Hence «•
commutes with End X, so if belongs to tbe center of End X. Sinoe
(A) holds, so does (B), so that EndeX is a quaternion algebra with
oenter Q. Hence, n" is an integer, and by consideration of degree
^=±^ for some m > 0, so ¦*• =pa". Conversely, if «* =pj for
«nde » and w > 0, then since ir is bijeotive pz is also bijeotive
snd X has p-rank 0.
We have thus shown that (A) *=• (B) *=> (C). Next if End»Z
is Bon-oommutative, it is a quaternion division algebra over
Q, and since for I * p, Q,®0End°X -+ End^Q,® z,T,(X)) is
injeotive and both sides have dimension 4 it is an isomor-
isomorphism. Therefore InV|(End°X) = 0 if I is finite and I =? p. Since
SInv,(End°X) = 0, the sum being over-the finite and infinite pla-
<m of Q, and InvB(End»X) = 0 or 1, we deduce that
Inv,(End°X) =Inv«,(End»X) =t. This establishes (b){ii).
Next, we show that if X is defined over a finite field, EndfX
# Q. We have proved this for X of y-rank 0. Suppose then that
p-rank of X is 1. As before, w is an endomorphism of X which is
bijwtive and of degree a power of p, so that it cannot equal mz
for any integer m. Thus, v e End X, v ? Z, It follows from the
Uble of possibilities of §21 that then End«X is an imaginary
quadratic extension of Q. This pToves (b}(i).
We have also established therefore that if EndfX = Q, X
o»nnot be denned over a finite field. Suppose finally that X is not
defined over a finite field. We may assume X is the normal form

230
ABKJJAIT VARD5MBS
(Nr) in characteristic p>i, with A transcendental. (If p = 2,
argument still works if a somewhat different normal form is m
Let us call this curve X,. Since any tiro transcendental elem
A and X' over toe prime field are oonjngateover the prime flel<
see that if EndXj = A, then EndX^s-tl for any other trans
dental p over the prime field. Now, Bnd'X must be either Q o
imaginary quadratic extension of Q, sinoe the only other possib
ia that of a non-commutative division algebra, in which oaee,
the implication (B) =* (A) above, X, has p-rank 0 and A i
be algebraic. Suppose then that End'X ia an imaginary qnadi
extension of Q. Then we can find an element «in A such tha
~S eZi, jV < 0. Hence for any transcendentalp over the pi
field, there is an a, eEnd X, with «J = #. Suppose I is a pr
not dividing ptf and (e T,{Xk). Let J, bo the oyolio gn
generated by the image of ? under the map 7f(XJ -* JU
and let p: Xj -*¦ XJK, be the natural map. By our reman
preceding the theorem, X^/JT, ia also not defined over a finite field
and is therefore of the form X, for some n transcendental o>
the prime field. With a, as above, since the map Bnd*X(l
End*Xj given by n—* p'1 o&<>p in an isomorphism we dedi
that {p~*•'*).<• p)* = NIt and sboe «*=>S^ and End'Xj it
commutative field, pa«^oy *=,± «, »nd sj, <• p*» ± P"*x- Thug
ai(keip)ckerp, that is, a1(JS,)c?m. ginos this holds for every
we deduce that «j(f) ««af for some oe2j. Thus, for ^ acting as i
endomorphism of Tt(Xx), every vector is an eigenvector ao that«,
act* as a scalar on T,(XX). Since the characteristic polynomial of (j
haa integer coeffioieutB, this scalar must be rational, and its gquari
cannot be negative. This contradiction shows that if X is not
defined over a finite field, Bnd*X =» Q, thereby proving (a).
A far-reaching generalization of part of tbia theorem to higher
dimensions has been proven by Tate and Grothendieok. Suppose
k has oJiar jp and X is a simple abelian variety defined over i.
AND THE i-iXHC HEPBKSENTATTOM
221
^Tx^;E^(oa"z -^--r.jh-ijwi,
was proven by Tate : [T3]; -a wea proven by Grothendieok
lai]). Tate shows further :
If X is defined over a finite field kt> tt *s the Frobtntus
over i8, and End (X, t») i« ^e rintf o/ it-ratonal tndo
Xsorfhkms, ihen
Eod(X
Q, =
The group *(i). There is a second approach to the Biemaoa
form of a line bundle, via a technique whiah is important in other
contexts suoh as the theory of moduli, and the obaer study of
linear systems on an abelian variety. We firet make a gronp-
theoretic digression to explain the claas of group sohames that we
trill need.
DirtmrtON A tketa group mil be a system of group scheme* and
Aomomorpnunu
¦ a-
that
(a) X m OHiuBuiattM (but G need not be),
(b) 3 an open covenng [Ut] of X and sections at of i
8
-A
(b) tua dosed tmmerston mahmg Gm into the kernel of v
(J) Gmccenter of G
When X is a finite group scheme, there is a global section
i: K-y 0 for tr, and then as at scheme, (J s G, x JT (i.e. define
t:GHx K ^>- G by <j>{<x, h) =i(a).o(k)). Having made this splitting
he group law on 0 can be carried over to a "twisted" group law
en Gm x X There will be a morphism

223
ABKLIAtt VABIETIES
such that the twisted group law le
where e, a ore S-valued points of GM, t, h' are ?~valued pointed
of K, and Kis written additively. /muat be » 2-oo-oyoIe : f
and changing the section a has the effect of altering / by n§
ooboundary: |
f*(*>.h)~f[k,k).glk + k) g{k) Kg(b) ' |
Conversely, given any suoh/, (*) makes Gm x ffinto » theta-group. |
In other words, the «et of nil theta-gronps over a fixed finite JC'|
ta JHomorphio to the oohamology group H'(?, Gm), computed vi*;|j
morphism ooahaint 1
The deviation of 0 from oommutativity is easily measured by S
taking the oommutator. For any two S-valued points x, </ of 0,s*
A) xyx~*y > is an S-valued point o{ GB fcnd B) it depends onljl
on ir(i), ir(j/) and not on «, y. Therefore there is a morphiem |
(ilxf >G. I
¦uoh that *
AND THE (-AD1C REPKB6BNTATIOH
SSJ
foot, if we regard X x K and GB x JT M groap-oehemta over
via J>t ^ben {«, ;>,): iTxf^CxiCiBa iC-homomoiphiam
e, a X'-vsJaed character of K, or a moxphism y,K-yK If
,):JCx ? -+ Gw is the universal pairing than In terms
f 5-valaed points k k' of K, y is given by
It us easily checked that e is a akow-synunetrio bihomomorphism :
(a)
(b) < |
(o) e(Jfc, t) = 1 |
If O admits a global section of K and so is deaoribod by a 2-oo-oyole I
/, nornuU»ed by the ooadition/@, 0) = 1, then ;|
(d) t{h.V)=![k,v)if(hl,h). I
In case K U finite, the bi*homomorphiam e alao can be expreBMd: I
asymmetrically m a homomorphism : -J
y ¦ K »¦ K
Pbofobitiok. If Kit finite, y:K -*¦ K a* above, titen for every
S-Mhted point x of Q, Wi) 6 ker yj *=^. [a; «* in the center of G] i.e.
far all tehtmes 8' over 8, and aUS'-mlami fointu y of Q, xy<=y*s-
FBOor. In fact, y{*(x)) & 0<e*- the character y(*(z)): Kx.8
¦+Gm x 8 w non-trivial <=> for some (S'-valuod point k of Kt
y[irix))(k) =jtl-^> tot some i, <t, y(w(*))> # 1 **¦ for nmt t
t(i, ir(a)) # 1 -«=* for some S'-valued point y of Q, xj/ & yx
Oqkollasy. The-following are equivalent
(i) y ij an womoiTiiijm.
(a) i(GM) it exactly the. center of Q, i.e. every 8-wtutd pomi x of
Q commuting totth all S'~vtdutd points, ali 8'13 itinHGj
Such thefea-groupa will be called nan-degaaerate
At the other extreme we need two facts about when anoh 0 B an
trivial,
l. (i) // K isfintte and Q m commutative, JAeit 0 s
^,x?«a frnup, i.e. «n Ik oafeffOfy o/ commutative ftoup
tdttmt, K finite » Eit^X.G,,) = @)
(ii) If K is finite of primi order, then G u commKtativt
Pboof. (i) ia a standard fact for oommutative algebraic
gtonp schemes. Tor instance once one sets up the long exact
Kqoenoe for Eit'e ia this category, one takes s> maximal chain of
mbgroups of K and rednoes (i) to the speoial oases K=ZjlX,
ijpZ, nT, or «,. In tbe first two oases, one lifts a generator of K to
» point of <? of the same order (uning the fact that fc* is divisible);
m the second two oases one obeoks by a direct computation that the

224
ABKLIAtt VARIETIES
p-Lie algebra of G moat aplit. Por details, of. Oort [0], or Semtnairt
Heidelberg-Strasbourg. To prove (ii), if K = Z/!Z or Z/pZ, li
generator of K to any point of O and note that if a reduced group
scheme is generated by a central aubgroup and one element, thenit
is commutative. If K=nr or a,, let g be the lie algebra of Q;
then Q =k.x-\~ k.y where at geneiatea LieGK and y lifts &
generator of Lie if. Since Gm U central in Q, [x, z] = 0, all >q.
Therefore (j is an abelian Lie algebra, which implies that (f*
is commutative (of. Seminaire Heidelberg-Strasbourg or [G 3).
Since, as a aoheme iJsG. X K, any iS-valued point of Ct is th»
product of S-valaed points of GB and of A', hence of 3-valued
points of G. and of <?°*. Since G,, is central and &*> is conunnts-
tive, this shows that 0 is commutative too
The natural idea for proving (ii) would be to show tbat when K
has prime order there ate no non-trivial skew-symmetrio bi-homo-
morphia ms
t. K x K 1- Gm.
But this is false in ohar 2, If 1 = *,1 This has faaomiiting con-
a: cf. [Br].
it »¦ JU
Moreover onoh a t induces an isomorphism t . L
and any such isomorphism t' induces an automorphism
oovering a.
We now return to abelian varieties. First of all, to eliminate
any possible confusion, let me atate dearly that if L w a Une bundle
overX, with projeotfon p: L-+X, and o: X -+X is an automor-
automorphism of X, then by an automorphism t : L -* L oovering o wt
always mean a linear automorphism fitting into a diagram
h a'L
t of i
Hom(X,X> AND THE I ADIO REFBGaENTATTOM
22B
Secondly, recall that if X is an abelian variety, S any aohetno
»nd /: S ¦* X 'a on S-volued point of X. then Tr, tranalation by /
denotes the 5-isomorphism of schemes (jij, tno (/ X lx)) :8x X -f-
where m ia the multiplication on X
here ia how theta groups arise
l Let L be a line bundle on an abelian mrttly X. For
tdu-me S, let Ant (LjX){8) be the group of automorphisms of
covering a transition map of 8xX. A«t(iJ/X) « a contra
variant group-valued functor on Soh, Then there u a pnmj) schene
f(L) and an MomorpAifni of group functors
Aut(L/Z) si ST(L)
For any scheme S the natural homomorphiams of groups
XI
)S-valued pointa /: 8 ¦
such that
¦1
i induce ho isomorphisms of group schemes
* 3
mak ng *{?) into a theta group
Pbocw. Let ?* be the oompletnent of the 0 section in L;
this is a principal fibre bundle over X with structure group G,,.
Ki a base point Po ejr'lO) n ?*,.andput &{L)=p-x(K(L)) n I>*.
Far any automorphism a of 8 x h covering a translation Tf of
SkI, where /gX(S), we get a morphiam 3 ; 8 -+L* defined
by « =• ^i^eosoi where *» is the map S z 3 Y. {Pt)-C * 8 x L
end pt: i5f x ? -*¦ L U the projection. Sinoe p°a is just the
morphisra /, and / e K(L) {8), we deduce that 5 /actors through
Then a h—* n defines a map
Aut

ABELIAN
AMD THB J-AD10 BBPSESKNTATIOH
227
whioh is functorial in S. I olaim that it is an isomorphism. Snppoi^
A, fi e Aut (?/Z)(S) oovering Ts aud T, respeotively, are suoh t(u|
Z = 0. We then have / = p*% = jx>p = ff, and y = 0°or' isiij;
automorphism of S x L over the base ^8f X X auoh that tbi:
oompositeiS-X-iS x i -—>¦ S x ? equals *0. Now, y is given by^
multiplication by an element Vl of ff^iSx X, ffj,,*) and
*,«>•<»«,. y1|Sx{J>0} = l. But JJ^xX.ffJ.^lsHtS,^}, m
y,= 1 and y = ^a*L- Tni* proves that the map cci—f « is injectivt,
To prove that it is surjeotive. let <j> 6 9F(?)(j8), and let / =
that/eX(i)(i3). Then by definition of
for some line bundle M on i9. We can cover 3 by open sets {$
aneh that if |p, i« trivial. Then over 17, X X, T*f{8 x ?) and 8xi0
are isomorphic, bo there exist automorphisms xi of G( x i
oovering T/(,/( = r«strietionof/to ff(. If ^ is the restriction oj
^to Vir we now have two liftings of the morphism/, to &(L)\ f
K{L)
Since 5(L) is a prinoipal fibre bundle over K(L) with group (v'
there is a unit ^eT(Pi,tf5) suoh that ^ = ^,)_B |
automorphism, mult, bj i,, of Vt X ?, then A^xi ^
Pinally, wnw ^ agrees with ^ on G, n fj, the two automorphism;1
\Xi and A,ft agree on A7, n U}) X L (using lnjeotivity of«i—*«?
oo there is an automorphism x of S X L extending \xe
~X = j> so « i—» * is a Burjeotlve map
This establishes an isomorphism of functors Aut (tjX) =
and dnoe the left side is » group funotot, 9{L) beoomea »
group eoheme. DeSne homomorpmsms t: Gn-
-* K{L) by the homomorphisnis of functors:
and j 9{L)
11—»¦ mult, by e
9 (?) (S) s Aut (i/X) (iSJ —* ff(i)(S)
a i—^ | the S-valued point fofX
[ such that a covers Zy
Then i is olsarly injaetive J ib Just the projeotion p, and Im(»J «=
Ksr{j). Sinoo there are sections locally to p: L -*¦ X, there are
also sections locally to j : *<?}-* ?(L). fuiallv, if ( eT^,^)
then the automorphism, mult, by t, clearly commutes with all
other automorphisms « e Aut {LjX)(8), BoiFM) is in the center of
This proves that 9(L) with t *ndj is a theto-group.
G,, t« (Ae *iew-^y
associated to the commutator in EAe
Look at the case L e YuPX. Then K(L) = X and the morphism
e» takes the oomplete variety X X X to the affine variety G*
Therefore e1 ™ I and 9{L) U a oommntative group Boheme, whioh
is an extension of X by Gm. It can in faot be shown that this mep
_+ Ext> (X Gw)
Li—*¦
ia an isomorphism (Theorem of Serre and RoBenlieht),
Suppose next that the line bundle L arises from a divisor D
Ls0M(D). We leave it to the reader to check that the discrete
group &(¦?)* can be described as follows.
This works amce

230
ABMJAN VAIUKTIBB
ADD THE J.ADIC BBPRE8KNTATION
22*
I
The subgroup k* = (GJt e *(?)* corresponds to the pairs * » 0
/ = «ei*;the projection #(i}t ^ JC(?L corresponds to the map
<» /)_* a;
FvsoTOBiii. 3?BOP*BTrfia ob **¦
In the following formulas, the symbols se, y eto are to be
understood as it-valued points for any jfc-algebra S. One oould
equivalently interpret the formulas as oommutativity of oartsm
diagrams of morphisms. With this speoiGo understanding, we shall
often omit R from the statements and proofs and speak as though
we were just dealing with ordinary points, but all the assertion!
are to ha understood in the stronger sense mentioned.
A) If /: X -*¦ T is a homomorphigm of abelian Tftnetioe and
L a line bundle on T, we have
on X
B) Vot any tine buudles Lv ?, on X,
«*••*"{», V) - **>(*, !f). *h{x, V). at,
C) Ear tilgebraioally equivalent line bandies Lv
D) Ew a) eJS:{?) and jre
E) Botaej
(n any integer with
A) We mey assume f(x) —i(fl, /to) =jfo).
j: ?(L)-*-K(L) is the natural homomorphiiun. (When x,g are
Ji-valued points, wa can find f, if after localizing on Spec it.) Wft
of /*(-&) oofeting
then lift f and i) to automorphisms 4>,
31, reapeotively, and than ^H-I ^"'
precisely A).
B) Again we may assume that there ore automorphisms
(BBpeotively of i|, i = 1, 2, covering T. and 31,. Then ^i ® #»
and ^ ® ^» ore automorphisms of I/! ® i» covering J!,, 3*r
respectively, and the commutator of these two automorphisms
jg the tensor product of the commutators of ^ and ^ (t = 1,2)
whioh proves (8)
Write L, = ij©!^1, ao that ?, e Pio'X and ?t —
(= X. Apply B) to the line bundles 1^ and Lx
\ (!) By replacing the ring R by a ring R d R if neoesuary we
may assume * =• m for some z e it^1 (K[L)). (In fact, we havo
only to take Spec R' = Speo R Xjc^n^^tL)) which is finite and
flat over Spee.fi, bo that it is affine and S'D R.) We then
nave^veni'^Wt^^t^}. so that by A) applied to nj(?)
and B) applied repeatedly, »nd making use of the algebraic
equivalence of nj(i) and W*, we obtain
t*{nz »sr) = e'^V. V)
(«*"(**))"
which is formula D).
F) As in D), we mey assume that y = «* for some z and tl e
equation to be proved assumes the form
uitioe ?*' is algebraically equivalent to n^{L) and z 6 K{V') =•
K{n*{L)). The foot that «e ?(»*(?>)) means that we have an
automorphism a of »?(?) covering the translation J1, (looalixe
3peo ? if need be).

230
ABBLLUT VABEETtBS
Hom(i,X) AMD THE tjLDIC BBPBESENTATION
231
Let us agree to denote (temporarily), for line bundles M,
em X, the line bundle associated to the locally free sheaf of gemu
of homomorphisms of M into N by Hom(Jf, N), so that we hav^j
a natural isomorphism HomfJf, N) (*— Jf®^. Note th»t
there is a natural action of X* on any pull back n?(Jf) covering
translations, hence also natural actions on tensor products, Homs
and translates of pull-backs (this last, since any translation;
commutes with any other}. With this understanding, we have
natural isomorphisms of the following line bundles on X cost
muting with this X, action: •:
Bj<I^?®?-1)«»lB^?)}l8>t>*<L '}« *1<
But what is %{; &,{jf)I nJlT^L® ?"') is isomorphic to the trivial ..
bundle, and ^{..^(tf)) is given in the usual way by the naturtt-;
action of X, carried over to the trivial bundle. Equivalency,; !
n*{T*L® L~l) luus a nowhere vanishing Motion, unique up to ,
scelars, and «(.{¦, ^sto)) w given by the action on Xn on this Bectioii| j
Now make this computation on the bundle on the right instead of |
theoneon the left. Anowhere vanishing section o( the line bundit |
on the right ie just an isomorphism n^{L) >¦ nj(L) coverinj
Tt, and tbe natural action of 3 e X, on the right maps this seotioii
into the section «V: «J(?)-vnJ(L} defined by f = ^<l^l"f1i;
where v nJ(Z) ¦+ »J(i) " the natural isomorphism coverinj
Tm. We must therefore have th,0^0^1 = e,,(i, &(tt))-f Applyin|
this in particular to the automorphism v covering T, chosen earlier,
we get that ^'unifi1'"! =««(*, ^,ty)) But this means by
definition that f^ix, z) = t%{x1 foy)}. i
As a corollary we get a second proof of the skew symmetry of
the Riemann form of L:
how »U the e* s, on points of order J* cen be computed from the
Biemsnn forms.
Thmoehm S. Suppose*. X-*T%tan Hogmy of abtlianvarieties
and Lit a line bwxMt «n X. Then (here U a natural one-one, earru
pondenee between
(a) isomorphumclatstsofhno trundles M on T »uch that ir*Jf a L
{b) homomorphismi a: ker w -+ »{?) lifting tAe tndumon
ker w e ' X
Bttoof. This is just a restatement, in a special oase of the
general descent theorem in $12 for coherent sheaves with respect
to quotients by finite group schemes. Use the foot that
Honjjfkerw 9(L)) s I actions of ker won L covering 1
I its translation action on X \
if x,peX,. The formula F), ooupled with D), shows how tlji
Riemanti forma can be computed from the e?'s and cenverasly.
Corou,aby Given v:X~*Yand Lot about. Then
Jf on r «wA that „*# k L extsts if and only if ker{w) c K[L) and
Peoot Let p: 0{L) -+ K{L) ba the projection and let O =
p *(ker v). Then 0 is a theta-group over ker -a, and by part (i) of
the lemma at the beginning of this j, eEl*«. xtat. ™ 1 ** 0 is
cammutative -s* fl;C,x ker w as group-scheme «=s- 3 a
bemomorphiam a: ker u-t- 0 onoh that p*<t. = 1 ¦*>> if eziBts
We bow prove the following theorem promised in }20
Tbsobxk 3. If Lisa lint bundle on an abelian variety X and
keZ L~ M* for tome line Inndits M if and only if K{L)o X*
Pbqof The 'only if1 part follows from the equation ^ =n^a
Assumeoonversely that K{L)d X*. To show that L s JT" for some
line bundle M, it eumoea to show that If s »{(W) for some Jf
Indeed, we would them have that (?® ff-Te Pio'X, hence also
igltf-" gKo'I, and if we choose P e Pio"X sooh that L®If ¦
0 P\ wo would obtain L s (tf® P]?1

SSS ABKLIAS VAMXTIKS
To show that V s n%(N) we merely compute for any
points x y of -X.
AND THE l-ADIC REPRESENTATION
333
The desired oonoluaion then follows from the corollary
Theorem 2.
Our last result concerns the non-degeneracy of <H{h) We ae
gome preliminary results. - \
Suppose e: K x K -*¦ Gm is a skew-symmetrio bi-homomorphisa|t ;
on a finite group K. Lety:Z^JtbetheaBsooUtedhomomorphisn?; i
Then if.flc.KU a subgroup, I olftiro that there is a second subgroujt
H*- oharaoterized by the property:
if fc is an fl-valued point of K jj j
*eHM<9) -s=> {for all S'/S, all 5' valued points kotH, e(fc,fc') = 1}| j
In fact, Testrioting oharaotoru of K to H dcftnen a morpbiain ;
q: K~*H, andfl-'-is oleatly the kernel of qoy; K^t-B. Now Buppwf;
v: JL-y Y is an isogeny of abelian varieties, M is a line bundle pn i
F and L = v*M. Let Jf = ker(w): then as we have Been H ist
subgroup of -ff(L) suoh that e'Jjrjia ^ 1. Iu other words,
The result we require is
Luu 2 ^{JO s HLIB
Pboos Let i: 3-*-X ha an iS-valued point of X. We must show;
that I is a point of H>- if and only if nx is a point of K(M). ftwuT
enffloe to prove this when jS = Speoli?}, fi a local ring, in whict
case 8 oarrie* only trivial line bandies. Then using the descent
theorem of §12, *
M)sS>s. M '"[
3 an isomorphism JJ(Sx?)sSx?
oommuting with the action of ker n on
these two line bundles
Jf suppose e:fl-»-S?(i} ia the homomorphism giving the action
of if on L for which Jf is the quotient. Then we oontinus our
quivalence 6:
3 an jS-valued point u> of 1f(L) suoh that
p(ui) — x and w commutes with «(B)
'd, A)= l.allS'valued points A of if}
The same argument shows in fact that
but we do not need this foot We now apply the lemma to
Theosbm 4 Lti Lbta non-degenerate lino bundle, on an abtlvin
tarietyX. If B cK(L) is a maximal subgroup trtch that e*1nxB ** l
|Aen B = RX and order (HI = order ?
Let H be a maximal subgroup scheme of K[L) euoh
Uiate|ff)io =1 LeU =JC/iT, and let w: Z -* Y be the natural
homomorphiam. By the Cor. to Th. 2 there exists a line bundle M
on Y suoh that v*M = i. The faot that H h maximal means that
there are no further Uogenies w': Y-* Y' for which il = it'*(M')
eicept the identity.
Lcuma 3. iet L be a non-dtgintraU Itne otoxile on an aftriton
X. If there are no wogtnie* w: X-*-7 (^ dejwe > 1 *ucA (Aai
Y.then \
Pboot. If | X(L) | > 1, then K(L) is non-trivial. Then there
eiists a subgroup ? clji) of prime order, i.e., B-z'ZjlZ, Z/pZ,
(i, or «,. Look at the inverse image ? in 9(L); this is a theta-
gwup scheme O over E. By the lemine at the beginning of this
Motion, Gis commutative, hence QxGmxB, henoa there exists a
homomorphism:

ABELLAH VARIETIES
so order (B)* =• order K(L),
• order (K(L));
1 Buerj/ afehan oorwty X w woyenoiu to a
pally polan&d abeitan vanity T, »j! one which tamet on ampU 5
line bttndfe ? tntk x(?) = 1
Proof. Apply the theorem to any ample L on X, and ]«t ¦'
r =XjH,S maximal in ^(L) with*1 Ijtxf " 1 Then L=ir*(J0,.:
and if U ample with ^(«) = 1. '[}_
CSobouubt 2 If Li* a non-4egtneraU Itne frundfe on X (A«i^
AND THE 1 ADJC REPRESENTATION
H *K(L) ¦¦¦ >HjD
Therefore by Theorem 2, L descends to XjE contradicting the >j
assumption. So |*<L) I =¦ 1. f
Returning to the proof of the theorem we deduce that s j
jr<jV)«(O)»id|x(iOlsl- Thereib»byLemma2,fl = flJ-(and* ;
by the results of S18, S .
Pbooi Let y: K{L) -+K(L) he the homomorphiam assooiated/'i i
to e1 nod suppose Z) is its kernel. Choose an B cK{L) with B =;si I
B1 and order (fl)* — oider?(?) as in the theorem. Now
@), we find
235
< order H. order
= (order H)*/order U
This provea that order D = 1.
The next step in the development of the theory of theto
groups is to show that A) all representations of non-degenerate
theta-groups which restrict to the identity oharscter on the center
GB Me completely reducible, B) that there is only one irreducible
representation with this property, and C} that when L is non-
degenerate, » =i(i), then 3f(L) acts naturally on fl*(X, ?) and
that this is the irreducible representation in B). for these facts
and their application, of. [M2]
24. The case tc = C. The purpose of this last section is to tie
together the algebraic approach of this chapter with the analytic
methods of Chapter I. In particular, I want to relate the
analytia and algebraic RLemann formB of a line handle L
uid I want to show how the positivity of the Rosati involution
fellows immediately from the poeitivity of the analytic Biemann
form in its guise as a Hermitian form when L is ample.
As always let X = 7IV V a complex vector space and 17
a lattice. Let L~L[H,a.) be a line bundle on it, where H
is a Hermitian form on V such that S=Imi? is integral
ao V and «¦ ff^Cf u > function such that
Therefore DcBx, so Dcfl and aleo allohamoters y(i) annihilate ,
D »UiE K(L){3) Now by defitutbu 51 is the kernel of. -;
the homomorpfaiam K(L) * K{L) y B It follows thst
Im(qoy)c(SjD) and that we hare an exact seqaenoe -
Consider the group 9 of analytio automorphisms <jitll of C x V
gifen by
Then

236 ABELUN VAREETIBB
p = a. t. «-«••">
so 3f is an extension analogous to the theta-groups of j 2S: i
The data a defines a lifti ig t of G into 9
.1/
n
-*- v
AND THE [-ASIC RKPEEBBHTATION S3!
gives us a natural homomoTphism &0-t-9(L(H,tt)}. Bntwe
saw in 19 that K(L{B,a.)} s V-jU, so we get in foot Woorphio
uitensions:
i ».C' —
!• 1'
1 y C* y 91MB, «.)) >- K(L{E a)) > 1
It follows that the commutators in these two groups are equal,
hence we have proven
Im S, and a : V -* X i» the natural map, thtn for ail x, y eV-
sothet?{H,a) is, by definition, the quotient C x Vji[U). Ths
eommntator in?, as in the theta-groups of §23, is given by4
bi-fcotnomorphUm. e: V X F -+ C^ aa follows:
Sinoe onr ground Bold ia C, there is a oanonioal primitive ft* root
of 1 for all « namely ?, =• e**" Therefore the module
if, =. Urn ft.
baa » oanoiuoal basis element ( too, given by the sequence
fi^ We can now relate the Riemann forma
Therefore, if XP- =.{ue Fl^fu.w') eZ, all «' e I/} as before, it)
follows that the group §FQ = j" '(f1) = {it,, J 0 e t»} i« the
oentraliier of i(V) in 9. Therefore, all the antostorphisnu i,,
ee !7A deaoond to automorphisms ^,(, o
Cx V-
->-cx
f...
BUY. V+Z, where E = lmH,
JP:T,i X r,X-»- Jf,, defined in j 20
Let ffj denote the natural map from 17 to TpZ, i.e. «,(«! is given by
the sequence n, = w(u/f) e JE)*, with to,,+1 = n,. Then if u, v e 17
^(BiK, jt,o) = the seqnenoe e,«(«^t &1J,}
= the sequence ^(t^i «V> (S 28, property F)}
= the sequence e-W"<"»"''«'> (Theorem 1)
== the sequence (J^,) "•"•^
Thus, except for sign, ^ is the Z, linear extansion of B fram 17

338
ABELIAM VARIETIES
Hom(JCJT) AND THE J.ABIC REPRESENTATION
230
Applying this to the case where X is Y x Y and L is the ;
Poinoard bundle jP, we see that the canonical non-degenorato J
integral pairing of the lattices U and V corresponding to 7
and 7, of the analytio theory (of. {9, part (B)) is, up to sign, the
same as the canonical non-degenerate I-adio pairing e, of Tt 7 and,'
Next, consider the Rosati involution of End%30 Analytically,
we use the interpretation:
f set of complex-linear endomorphiams T V -*¦
\ enchthat T{Q.U)cQ.U
Then the natural involution is the adjoint with respeot to M
H(T*x y) ^H(x,Ty) aU at y s V
Thus T* = T
Now for any oomplex-lineM operator T: V -*¦ V, if T* is it«
adjoint, then T*T is a positive Jielf-adjoint operator on the
Hennitian vector spaoe V, heno» all its eigenvslaea sxe positive,
hence the complex trace, Ti(T+T), is positive. If Te End^Z), w
that IT maps the rational vector space Q.E/ into itself, its
trace here is just twice its complex trace; and its I-adio trace in :
TtX x V® Z, is equal to its rational traoe. Therefore for any of
these traces
j]uia the poiitivity of the Bosati involntion is obvious from the
existence of the positive definite H with Im H = B. In a sense,
^e bars shown that over any ground field oue can reverse thia
wgument: namely, using the positivity of the Bosati involution,
IB haVB realised N8*{X) as a formally real Jordan algebra in
ample L'a are the poBitive elements
Since if igQ,!/, then for all ye Q.V, E(T*x, y) = E(x, Ty)z(},-;
it followB thai T*x must be in Q.V too, i.e. f'gEnd'fZ). It T"
is the image of T under the algebr&io Hoaati involution, then
for all x y e V I
- T)x y) = Im H{T*x, y)
= Im H(jc, Ity)

APPENDIX I
THE THEOREM OF TATE
By C P RAMANUJAM
Wi havb seen that if X and T ore abelion varieties over a field ^
(algebraically closed) and I a prime different from the charaa
of k, the natural bemomorphism
T,
Z^HomfX, Y) y Ham^T^X), T,{Y))
is injeotive, It is obviously of importance to know the image
By t. field of definitional X we shall mean a aubflekl i0 of i ovsfl j
which there is a group tclitme Xo and an isomorphism of ffroiii^
tthtmes XD ®lf k^X. We also say that X is an abelian variety define}!
over ij. Now ohoose a oommon field of definition i0 for X and X]i
we may and shall assume io to be of finite type over the prime fieUl
APPENDIX I THE THEOREM Of TATH
241
Let i,, be the algebraic closure of i,, int.
. and nr(ne Z)ai(|
|
morphkma defined over l0, their respective kernels XH and Y'|
oonsist entirely of ^-rational points, or in other words, the points off:
finite order in X and Y are io-rational (i.e., they oome from iq%
rational points of Zo®^ ^ and Ya®ltia reap,) Thus, if Q = Q(k,lk^
in the Galois group of i, over jfc,,, 0 acts on the groups X, and yj|
m m $
compatible with the honiomorphisras XB. > JfBandrwB *-Ttfy
Hence Q acts continuously on the Z,-moduIes T,{X) and ^
where we give O the Krull topology and T&X) and T,(Y) ;
i-adio topologies ^
Furthermore let y be any homomorphiam of X \ntoY, Since the '
points of finite order in X are ^-rational and i-dease, and <p mapi'
points of finite order into points of finite order, f is defined over i, *
and hence over a finite extension kl of Jtg in k\ (i.e., comes by baMk
eztension from > homomorphism of group schemes XB®ltk, -*:~
ro®i, *i)' I* tnen follows that if H = Gfio/^) c G is the subgroup.^
of 0 fixing i,, for any ^-rational point x of X, we have hf(x) — <({h)';
for all heH. Thus, if we make O Mton Hom^T^X), T,(Y)) by
putting (ffAMs) = giMs')) for g eO, 3ie Homjl^X), T,(y)) «<»
»6T,(X), weBeethatkr,M = T,(ip)forfceH, In other words, for
any 9 ? Horn (X, Y), there is a neighborhood H of the identity in 67
fixing I"i(?). We see therefore that if for any 0-moduIeJf we denote
by Jf*U) the subgroup of elements filed by same neighborhood of*,
the image of A) is oontained in Hom^r/X), T&Yft"*, and we
obtain a homomorphiuin
Z, ®x Hom(Xt Y) > Hom^THX). Ttfr,) ° . B)
Tate conjeciurea that B) is an isomorphism (or equivalent!/ (?} is
mrjcctive) for any field of definition t, of finite type over the prime
field. He proves this for abelian varieties defined over a finite field.
It has also been proved when kt is an algebraic number field and
X = Y is of dimension one (Serre). We shall now reproduoe Tate'»
proof in the case of finite fields
Theokek 1. Let X and Y be abelian'Varieties defined over a finite
ftld *,. Then lilt, homtmorphiam B) is an isomorphism,
The rest of this section will be devoted to the proof, which we g ve
in several steps.
Step I. It aufftcts to thaw thai the homonwrphtsm
, Y) ** Q ®Q Hom»(X, Y) —»-
C)
wktre VAX) - Q,®ItrXX), U bijective.
In foot, if this were so, the image of the homomorphism A) w°u'il
be a ZrBubmotlule of maximal rank in Hom^7lt(X), T^X))W. On
the other hand, if this image is if, Hom^T,{X), T,{X))IM has ue
torsion; for, suppose9 e Hornz^T,(X), T&X)) and I<re3i. We can
find ^.e Hom(X, Y) such that Ttf.) -+ I<p, so that for all large
»)r,tfj=i9.,i..eHoBi^2l(X),((r))MlJtf.Kl(()
and i|t, Vftnishes on X,. Thus </>, admits a factorisation f, = X« ° ^ —
, and the j-. converge to a, cerUia x "' %®z Hom(X, f) with
=1>. so that f e if. This proves our assertion, and establishes
X r)'

241
ABELIAN \AKlKTIEfl
APPENDIX I : THE THEOREM <>F TATK
343
STEP II. It «ijp«s to sham that for asy ttbeU/iit nirietg X defi^
over a finite
is an isomorphism.
In fact, if D) is mi isomorphism with A' >; )' instead of X sm<»
we have the direct Burn decomposition!! uh f f-module*
w r in
I 1
im lyx » n> -— , «k,
the above diagram is commutative and the lir.it vertical arrow jg »o
isomorphism, C) is ui isomorphism.
Thus, henceforward we shall restrict ourselves to a single abeliia
variety X defined over a finite field, and prove that D) is aii isomor-
isomorphism
Stsp III. ft tuffiu* to dtau that A w an isomorphism far one I
and Ihit dimQ| End (F)(X))<ll'> i* indrpmdent of 1 ^ char. t.
This follows from the fact that the dimension of the left member
In D) is independent of I and A is always injectivrt.
Step IV. To utabtith D) for an i, it suffices io show tht fcllowiaj
Let S be the image of X, and F the initrstttion oeer all neighborhood)
of tin G of Ika imbaigebrae generated in Eih\Qi(V,) by thus neighbor
hood*. Then Futht commvtant of E in Knd( V,).
In fact, since S in semi-simple, if the above wore true, it would
follow from von Neumann's density theorem that K is the ooinniu-
tantof F. Farther, since EmlQ|( I',) is of finit* dimension overQ,, /
actually equals the aubsJgebra generated by so rue neighborhood,
and all smaller neighborhoods generate the Kline subalgcbiu. Hence
the oommntant of J is precisely 1'-ih1Qi( V,)™,
Now, Steps I-IV are valid for any field of definition ka, and do not
make use of the fact that *D ** Anite. The next step makes use of »
hypothesis which is enaily seen to be true for finite fields,
jnd probably holds for any field of finite type over its prime field.
jfa therefore state it as a hypothesis, verifv it for a finite field and
deduce its Consequences
Let X be an abelian variety defined over a field k,, 1 a prime We
than state the
Hypothesis (it,, X, I): Let dbt any integer > 1. Then there exvd
njito isomorphism only a finite number of abthan vantltet T defined
ever t,, imk that:
(a) then w an ample line bundle L on Y defined over *„ wi(A
(bj there exists a kt-\»ogen-y Y-
X of i-power degree
,(fc, a line bundle L on Y is said
(If X m Xo ®tJe and T ~ Y,
to be defined over k0 if there is a line bundle L, on Y, such that
h~ Lt%Itk, and on iaogeny Y-t-X is said to be defined over kt
if it arises from a homomorphfom Yt-+ Xt by base extension to k.)
For finite fields *„, we beve in fact the following stronger
LkbmaI Ifkt it a finite field d>0 and g>Q then oreupto
isomorphism only finitely many abelian varttttt* Y defined over kt of
dimension g and carrying on ample lint bundle L defined over t, wi(A
xiD^d.
Peoop. The line bundle 1? on Y is defined over i, and gives a
protective embedding of Y as a tj-closed subvariety of projective
spsoe of dimension ^(L|-1 =3* d- l.The degree of this cnbvariety
ig the v-fold self intersection number of L*, that is, 3'.(if.<i = V.q\&.
Thus, there corresponds to it a J^ rational point of the Chow variety
of oyclea of dimension 0 and degree &.g\d in P**~*. Since j^ is a
finite field, there are only finitely many J^-rational point* cm tbJt
Chow variety, which proves the lemma
Stkf T Suppose X is on abettan mraty defined over kt and L an
ample hut bundle on X defined over i,. Suppose for a pnras

244
AUK LI AN
char. t,. Hyp (*,,*,/) kaldt. Ul WcV,(X) be a tubtpace of )
it Gttabit and is Maximal iiotropif. for the »&tw-tynaiuttit
form S1. Tim tktre it a* dtmeni ue i. vni\M(V^X)) = W
Vwaat Set T T,{X), V =?V,(X) and for asch integer » > o
APPENDIX i . THE THEOREM OP TVTE
245
If i^.TJX) ^Xf is the natural homomorphism, set X*-
r, = JCfifJ,»nd let»#:X -+y. be the natural hotnoroorphism.
ir. A,
il')x factors as X > V, y X Since
obtain that ir, o A, = if)i-m- so that
o 1, o
J* », we
Furthermore, for wy »>». *.«r.W3 (*.ow,)(X(.)-JTi- - w
that Wl«r.Hir r,(J[), 71,(AJ(T,(r,)) = T,.
We now verify that Y, in an abelian variety defined over *, ud
that A, is a ^-morphUm. First uete that if X ~ X#®,tt whei* X,
h a group scheme ovet kt, the point* of A>, are separably algebrtiq
over tr In fact, since U")x: X -* X is etale. no is (I")r>: Jf, -* Xr
sotliat (J*txJ(O) consists of points whose residue fields are separably
algebraic over iy Further, since W and henoe {T n ff) + PT an
ft stable bv nssumption, K. is also » stable. Thus the fact that Y,
as a. variety is cteHneri over Jfc, aud »,: X -+ Ym is defined ovwi,
foltou from the following general
Lemua i. Let X hta <ftuuri-projectipe vanity de/tiwaj over i^ <uU
tl a finite group of avtomorpiufat of X tuck Hurt
(i) titrt it a tpanMy alqtbraic tiltn/noti of ta over wiutl lit
automorpkuHu cf H art dtfintd;
(ii) for any automorpAwm a cj tbt algebratc damn ia af i, over
Thtn XjXl it defined over I, and tkt natural map X -* XjU u
defined over kt
Choose a Galois extension ij/ij over which the antomor-
of II are denned. If J/isalg-affine open subset of X, fl XU is
led
*gsm Vafline open and II-stable, by assumption {ii) of the lemma
Thus, we may assume x affine, X = SpBo A, A = Aa 0^ k, At
bring a jfcg-atgebra. If Ax —¦ At &it tlP II operates on At, and
if ®»* = -4". Further, 6 = Gal (i^lfe) operates on J,, and
¦tnoe by assumption A II X~' = II for any XeO.it follows that A?
is 0-itable. Let ? be the *„-algebra of G-rorariaiita, and {«„} a
bMts of kjkr If 2«, ® «(ejlf, o(e jf0, then i (S «, ® fl() =
SdjgAfS,)e^i1, and nn det (i(S(»j.o,(^0 «,® 1 6 A? n J, c JJ,
which proves that A? => B (g^ tr
This proves the lemma
Thus, as a variety, YA in denned over fc» and n,: X ~+ Yn is
defined over i0, so that m,@} is irrational. Since the addition map
« F# x K, -+ 1", and the inverse /: Y* -*¦ Ym are defined over
» Rsparably algebraic extension of i^, and are invariant under the
aobun of Gal( j^t,,), they are again denned over ?,. Hence 7, is
deAned over *(, as jui abelian variety. Now, (J"Jj = X, o it. and w,
an defined over *„, hence A, is definecl over k0, since for a Vregular
function <p on an open subset of X, <f o A, ie ^-regular for a Galois
«xtension Jb of i-() and is invariant under Oal(i(/t,,) since
Let d be the degree of the ample line bmnllo L on Jf We shall
produce an ample lino bundle of degree 'X.d defined over *, on each
r. We have
y)
j, since (K isisotropiofor «*. Since «,:
it follows that ?j.,,,,(r((JC)) ct*Tt{X)
F

?46
ABELIAX \ARIETIES
It follon-e from the theorem of §23 that ? = <ytw for some
bundle L, on T, defined over the algebraic closure, hence over some
normal extension of kg. We may assume that dissymmetric. Hence
if p denotes the characteristic of i,, if this is positive and p = j
if the characteristic U zero, for a suitable integer N > 0 Lf j,
defined over it Galois extension fc, of k0. We now have
Lemjia :i. Lit Y be an abelian variety defined over ktand LePi&y
n Hut bundle defined over the atgtbrait eloewrt fc, of ko in k. Ut
ire Gal (i-y/io), and denote by <r[L) the line bundle on Y defined over 1^
obtained by pulling back J, by the ntorphism l^xSpeco- F, 0^ t,-+
Ya®tt !(,. Then o\L) e Pic" 1".
Pa OOF. Let M be an ample line hundle on Y defined over kt, ud
consider tho line bundle A" = m*{M) ® p*{M)~l ® pJ(Jf) on
Yx K.We can find an algebraic point ye Y such thtXN\{y}x Y=L,
It is then easy to see that X \{ay\ X Y - a(L), so that a{L}eVi<?Y,
We shall make use of the notation a{L) introduced in the lemnu.
in future. Further, if hx and Lt are two line bundles on Y with
Z,® L,-'ePie" K. we shall write L, = 1^.
Besuming the earlier discussion, it a 6 Gal {?,/?,), we see that
o(LJ*f U also symmetric, and o(Lj^V -n \*{V^), so that since
ATS(K,)«torsion-freeo(LJ2') eh Lj'forevery oeGal {kjt,). Hence,
o(i.J*) and ij* differ by an element of order 2 in Pic* Y. Thus, if w»
put Mm = m", we have oiM,) ~ M, for every »eG»l (kjkj end
M*m Atfi*'*) -i L*»*'? We now have
Lemma 4. Lef Y be a amiptett vaneiy defintd over *, wiiA a t,
rational point, I, a Galou ftrttiwtos o/ it( and L a line bundle o* Y
defined over k, tutk thai for every <r 6 Gal^/fc,), a{L) ~ h. Then L
can be defined over k,.
Pitoor. Put X - JT.O^t JC, = JC,®tit, andlct w:X, -»Xt
the natural morphism. Since projectile modules of constant nnk
over »emi-kical rings are free, we fsmn find an aiHne covering U —
((/,),„ of X, such that Llv-'{U() is free. I*t {•(,} be the 1
APPKMJIX Is THE THEORBM OF TATE 247
vering (ir~'((/()} with values in ffjl defining /
Our assumption implies that for any <teG = Gal ikjkj, we have
a er(*'1(U|). "*,t sue*> ^at.
Oi«!i) __ ft^
«u "" ft.- '
If ?t is a ^-rational |Kiint of Xt with yt^Ult mid iy, - """'{?»), hy
dividing the ftiO by ft,.(//,), we may assume that $ttlJyt) = I. Now,
ft;
)"' ft",' in ^l "
"fc
w that for anyo. tcC, i.jef,
ft,,'(ft.,)-1 ft'..' = ft,-
since Ap is complete, ft,, o (ft,,) ^r,1 = <?..* for aI1 *-
j = t, and evaluating at y,, we see that
"(ft..) ft;.1. A--I-
Grant for the moment that ir A is any total ring of
jl®ti*,, and if B* is the group of unita of B, Hl @, J*) ¦= (li- We
deduce that there is a covering {W,.,}«,, of I/, and >ver{*r'(U,,)
tbet
Ths oooycle mj^with respect to theeovenng{[7tl} lfl oohomo-
lagans to | a^} and we have
It snly i«mains to show that if A is the local ring of a point on
r,and B^ky^A.H^Q, B*) = (I). Let fl be tho quotient
field of B; we then have the exact sequence
B°(Q, B*) -* H»{0, R*IB") ¦* H'[G, B+) -*- H'fO &*) >= {1}

24 8
ABELIAH VARIETIES
APPENDIX I THE THEOREM OF TATE
219
An element of flT°((?, R*jB*\ is represented by an element /g .#»
such that Zj e B* for all aeG and wo shall show that we can wnt«
rt^anduefi* Writing / = ? with /, e fit
weBeethatwemayaasumethet/e.fi Now,
efl» fOT
all ir ? Gf, the ideal B/ is <? invariant, hence of the form B %, % being
an ideal in /l! But now, since Bf ~ i,<8>t, St, S! is /4-project! ve, henw
principal. Thus, Bf = Bg for some geA and/*= gu, gsA, tte B*
This completes the proof of Lemma 4.
It follows that the line bundle Jf, on F, is denned over k,.
Now, the g.c.d. of 2p" and 1* ia 1 or 2 according aa I ia 2 or not.
Hence, we can find integers a, b such that Sop" -h 61* = 2. Define
jv\ ~ if;® AJ(Lf, so that #„ is denned over *„. Further,
bo thattf, is ample with *(#„) -2» ;((?,,) =S". I
^nd since Endz (T,) ia compaot, we can select a subsequence (%Vj
which converges to a limit u'. Since E is closed in End (Vt) and
tt' also belongs to A'. Since 2'. is compact, u'(TH) consiata of
of the form x = lim xt whore zteu^Tn) — Tt, and since
gets Ti are decreasing, it follows that
An alternative way of producing an ample line bundle of degree j
&d on K, is to observe that (i) if ? ia an abeliau variety denned |
over i, and we construct Y aa Y/K(L) for a line bundle L denned
over ka, ? ia defined over t, and the Poiucarfi bundle P on Y x Y
is defined over *„; (ii) if Ti^M = J"?, since A;(L) is defined over
Jc0, so is fi;,?) and hence also f; and finally; (iii) if P, is the Poinoari :
bundle on r,x 9, and x = A,^): Yx -> Y,x ?„, then X*(P.) =N,
is n line Iiiiintli' ilHineil over kt with ?jf ~ 2$ (wr S l:i). «o that
x(^.)* = (leg ^ = 2* deg ^ = a*!"*1" deg A^(L) = 2* I *"x
(deg A.J'xf-t)* ~ 2*x(^)*
Anyhow, we deduce from Hypfi,, X, I) that there is an infinite
set / of natural integers with smallest integer n and iaomorphiaitiB
i\: Yr + Yt for all i e J, Consider the elements tt( = ^ », A^'eKnd" A'
and then-images tt|eEndo(F ) We have u'^TJ = ?'( c T, for ieJ,
This completes the proof of Step V.
Step VI. Suppose that for any finite algebraic extension t, of kv
KyV{kx,X,l) hold,, and thai FU Uomerpkic at a Q,-algebrato a direct
ptodtet of capita of Qs. Then {i) it o» iaomorphiam.
Proof. Replacing it,, by a finite algebraio eitonaion kt over which
all elements of End X are denned and which is such that Gal (i,/^)
generates F in End(F,), we may aaaurae that h0 itself has these
ptoperUes. Let D be the eommutant of E in End{ V,), so that D 3 F
We first show that any isotropic subspace W for EL which is F-stable
u tiao ?>-stable. We proceed by downward induction on dim W
If W is maximal isotropio, i.e. if dim W~g, we can by Step V find a
«etf such that «( V) - W, and hence DW ^DuV^itDV=uV= W
¦which proves the assertion. Suppose then that dim W=r <g and the
assertion holds for f stable isotropio subspaces of dimension r + 1
The orthogonal complement WL of W for EL ia also /-etable, since EL
ia invariant under the action of Gal (ij/i,). Further, since any simple
f-module is one-dimensional and dim W1—dim W = 20—2 dim W =
i(g— r) > 2{g — g + 1) =* 2, we can find f-stable one-dimcnBional
subspaoes Lt and Xj of If-1- such that the enm W + Lt + i, is direct.
By induction hypothesis, IK + L, and fF + L, are Datable, hence
bo is their intersection W. This completes the induction. We deduce
that any eigenvector for F in V is also an eigen-vector for D. It
Mows that Dc F. (The decomposition of V into factors V( eorres
ponding to the simple factors of J'.reduoea this assertion to the evident
statement that an endomorphism of Vt for which every element of
Vt is an eigen-veetor is a scalar multiplication). Hence F — D
completing the proof of Step VI

250
AII ELM X VARIETIES
APPENDIX I i THE THEOREM OF TATF
2G1
Step VII. End of proof of Iheortm.
We assume from now on that t'c is a finite Reid. By replacing it by
a finite extension if necessary, we may assuirie that every element
of End JC is defined over i4. Let it be the Frobenius morphism over k^
Thun ir belongs to Che center of End" X, and hence Q[ir] is a com-
mnUtive semi-simple subalgebra of End0 X. We shall first show that
there arc an infinity of primes I for which Q,SQQ[u] is isomorphic
a» a Qr algebra to a direct product of copies of Q(.
In fact, writing Q[ir] = #, x ...xA', where K( are finite extensions
of Q, it suffices to show that for an infinity of primes (, eachQ,0tf(
is isomorphic to a product of copies of Q], Lot K be a finite Galois
extension of Q in which all the Kt are embeddable. Then it suffices to
show that for an infinity oft, /f®oQ, splits as a product, of copies of
Q, as a Q,-algebra. It suffices for this that there is one simple factor
ofA'gQQ,isomorphiBto 0,. In fact, if K®qQt =! ltx ... x Lt, the
Galoia group » ofKIQ permutes the factors Lt, It also acts transi-
transitively on the simple factors. For, if not, suppose L x ¦¦ X L, ii
jr-Stable; then the element A, 1 1, 0, 0, 0) 6 l^ X ... X L, is
rllmn
Tr-st&ble. On tho other hand, since it fixes only the elements of Q in
K, it fixes only tho elements of Qt in Q,®QA', that is, element)
of the form («, «,..., a) e L, x ... x Lk with a. e Q, This proves
the assertion.
Now, choose an algebraic integer a of if generating K over Q
and let F{X)eZ.\X] be its irreducible monic polynomial over Q.
Since K 8>QQ, ^ Q^Jf]/(/"(JC)), it is enough to find an infinity off
for which F(X) has a zero in Q,, Let A be the discriminant of F{X),
and I any prime not dividing & such thai; F(X) a 0 (mod I) has »
solution n in Z. Then J"(n) p^ 0 (modi), so that by Hensel's lemma,
n can be refined to a root of F in Zj. Thus, we are reduced to proving
the following
Lemma S. Lei F(X) 6 Z[X] bt a nan constant polynomial. Then
ihert are an infinity of primal i for which F(X) ss 0 (mod t)
in Z
Proof, Let f(JC) = a0X* + o^* + ... + (t,. The lemma
being trivial when a, = 0, since X is a factor of F(X) jn this case,
we may assume a, y= 0. Further, by substituting aAX for X in f and
removing the common factor a,, we may assume a, = 1. Let & he a
Bnitesetofprimesjp. UN = IT p. then
5
bo that no prime of 8 divides *'{ vN). On the other hand, F( <>N) f J- >
for r large hence has a prime factor t not bolonging to S.
Next, we show that for alt I =? char.it,,, the dimension of
End(j V,IS) is the same. Again, assume that every clement of End X
il defined over &„ so that the Frobenius ir belongs to the center of
End»X, hence to the oentre of Q,® Q End°Jf. Then Qfa] is semi-simple
and Vt is a Qj[tr]-module. so that the image n' of ir in End^jfj is
¦emi-simple. The characteristic polynomial P(t) of ir' in End0|K,
has coefficients in Z independent of I. Further, the closed subgroups
of Gal (V**' generated by n form a fundamental system of neigh-
neighborhoods of e in this group, so that End^Fi05 is the commutant
of w in End0 V, for n large. The characteristic polynomial of w'
has for roots tfj',.--, 6? where 9lt ..,, ffB are the rootn of F{1) repeated
with multiplicity. For all n large, the number of distinct elements of
91',..., *JJ as well as their multiplicities is the same, Thus, our
assertion is a consequence of the following lemma, applied to an
algebraic closure of Qi
Lskma 6 Let A and B be absolutely temi-simpU endomorphismt
of two vector tpaees ? and W of finite dimensions over a field k renpec
lively, vntk characteristic polynomials PA and PB Let
hi the feumposilions of PA and PB as product* of powers of distinct
irrtdveibti manic polynomials p. Then the vector spare

252
ABE LI AN VABEET1E9
APPENDIX I i THE THEOREM OF TATE
2G3
has dimttuum
and this integer it invariant under any extension of the bate field k
Proov. Make V (resp. IK) into a i [X|-modul6 by making X act
through A (reap. B), Denote the ifJCI-module t[X]/(f<X))by if,,
Because of our assumption of semi-si mplicitity, we have isomor-
isomorphisms of if Ar]-iuodules
V a n Mf*
r
W a jl if**1.
y
Now, the li, are non-isomorphio simple *[X]-moduJes for distinct p,
uul?MnoUiiiig1>ntHaiutl.(.,(IMtr). Sinoedim^Hom^xifjr,,. MT)~
&na.k Mr = degp, and since A* clearly 'commutes with base exten
skin', the lemma follows.
The main Theorem lifl a consequeoce of whet we have proved,
eombiued with Steps III and VI.
Rbmabk. The theorem can he stated n the following seem ngly
stronger form
Theorem 1'. Lei X and Ybtabelian varieties defined over a finil*
field k0, and let Homt< (A', Y) be the group of homoniorphistm of X into
Y defined over kr If Q is the Galois group of the algebraic ctoeuA of It,
over kt, we have an isomorphism
Z ®z Hom^tA", Y) -^V Hom^ (T,(X), T,( Y)f
Pkoof, Take (?-invariants uu botli sides of the isomorphism (?).
Appucaiioxs. We give some easy consequences of Theorem ]'. We
ahall make our etatementa over a. fixed finite SeH. The 'geometrio'
statements over the algebraic closure are easily obtained from these.
We shell consistently use the notation r{f,g) for two polynomials /
and g, introduced in Lemma 6. Further, oa we have shown above th»t
if 'X is an abelian variety defined over a finite field i, with Frobeiuiu
rr then the Frobenius morphiam ir* over a finite extension
{if jbginduoeasemi-simpleendomorphisniaof V,(X) overQ(, it follows
linoe Qi is of characteristic xem that it itaolf induoes (absolutely)
ismi-iimple endomorphisms of Vt(X) for all I =? otiar. Jb0. Thus, the
itruotuie of Vt(X) as a module over the Galois group of h, over i,,
|j uniquely determined by the oharacteristic polynomial of w, as
jiptained in the proof of Lemma fi
We now have
Thsorbu 2. Let X and Y be abeltan varittiet defintti over a finite
ttid i0. and lei Px and Pv be tit eharaeteristic poiynomtats of their
ftdbenxvs endomorphisms relative to i(. Then
(a) tie inn*
rank (Homlt(JT, Y}) —r{Ps Pr)
(b) tbt following statements are equivalent:
(bj) Y m kfyMogenous to an, abelum subvartety of X defined
(bit V,{Y)isO-i3omorphtctoaOmttnpaceof V,lX)for
i (b,) PT divides Px\
' (a) the following statement* are equivalent
(cj) X and Y are k^-Uogenow
(c.) PfPr
(a,) X and y have the same number of kl rational potntt for
every finite extension k of hg
j Phooi. (a) follows from Theorem 1' and Lemma 6 since the
i ftobeniue morptiism generates the Galois group in the topological
: nn«e. The implications {b,)=>{bt) •=¦ (b,} axe clear, in view of our
earlier remarks. We show that (b,)=-(b,). If(b,) holds, we can find
M inject!ve C-homomorphiam if of T,(Y) into T,{X). Then <p is in
the image of Zt ®zHomti(X, Y), so that we can find f 6 Homt((X, Y)
woh that Tt[ifi} approximates arbitrarily closely to f, in partieuUr
with T^iji) injective. If ^ ia not an iaogeny, we can find an abelian

204 AliFLIAM VAK1KTI1W
Bubvoriety Z of Y in the kernel of ^, and the submodulo
of T,{Y) would be in the kernel of T'(^). Hence $ is an isogeny
proving (b).
The equivalence (c,) ¦» (ea) is a special case of (b,) o (b,) when
dim X = dim Y. Further, we have seen during the proof of the
Riemann hypothesis in |20 that if ut[iel) and oy'^jeJ) aro the
roots of Px and Py respectively and A*n and N'K are the number of
rational points of X and Y respectively in the extension of degiti
* of i, we have
APPENDIX I ; THb IHKfJRBM OK -PATE
235
w, - n (i .;ii
thus, we have to show that Px = /•,.« II A — u>J) = n A— «*/)for
* j
every n > 0 The implication =- is obvious. To prove the other
implication, pote that
where ] S |, | T | denote the respective cardinalities and <ua — n »;,
«r ^ II uj. Multiplying the given equation by I", where i is a
variable, and summing as formal power series we obtain
Since |«(I = | ui') | =¦ ij, comparing the poles on both aides on the
circle |(| == q"*, ive obtain that there is a bijoctiou ct: f-t-J sued
that w; = u'^j. Hence, Px = Pr.
Before we come to the next theorem, we need some preliminaries.
Lot X be an abelian variety denned over k,, bo that X = A',,®^ for
some group-scheme Xt over t0 Let Y be an aliclian Bubvuriety
of X, which is a ^-closed subset so that if A: A" -+ Xo ia the nntunl
i, there is a closed subset F0of Xc with A"I(ro) = Y in th»
iet-theoretic sense. We give ro the structure of a, reduced subscheme
tfX,. If m,:^ x(iy Jlo-+,Yo ia the multiplication morphism, our
bypothesisimplieathe jeMAeoreftcincIiwiotiMidiy,, Xi Yt) cY. Hence
^restricts to a morphism wi,: (l^x^ Tj)^, -+ Y^. If we can assert
that J*=* yo®t(it and Y, x tt K, are reduced, it would follow that Y is
,n abeiian variety defined over i(. Both of these are consequences
of the assertion that the function field RtJi To) is a regular.ex tension
of it,, or equivaiently that Jiti( Yt) is a separable extension of it,. This
ii always true (vide 8. Lang, Abelian varieties, Chap. 1), but we
shall not prove this, since we shall need it Only when t, is a fin te
(hence perfect) field, so that this is trivially satisfied.
I Next, suppose X is an abelian variety denned over kt, and Y an
: nbelian subvariety which is fc0-c!oBed. We want to show \afc there
i» wi abelian subvariety Z of X defined over ltt suoh that Y + Z~X
and Y n Z is finite. We know (vide JIB, proof of Theorem 1) that if L
; is m ample line bundle -on X, we can take Z to be the connectod
! component of 0 of the group Z' ={zeX\ T$(L)<g> L~}\r is trivial},
so that if we can ensure that Z is definoct over ka for a suitable choice
of L, we are through. Now if we choose L to be a line bundle denned
. orar *,, Z' is defined over the algebraic closure Io of kt and is stable
for all automorphisms of k^ over kr Hence Z' is t0-«losed. Further,
the conjugations of fc0 over ka permute the components of Z', and
ance Z is a component of Z' containing the (^-rational point 0, Z
it also stable under these conjugations. Hence Z is fc,-closed, and it
follows from the comments of the earlier paragraph that Z is an
ibelmti variety defined over kK
, It follows from this by repeating the arguments of $ IS that if «e
i mfl an abelian variety X defined over t, to be iysimple if it does not
«ntain an abeltan aubvariety Y defined over it with 1V|O), l'#Ar,
ttien (i) any abelian variety defined over i,, ia t0-Lsogenmi8 to a
product of ij-mmplo abelian varieties, and (ii) if X is ^-iaogenous
to a product Xp x ,.. y, X?', »-her« Xt are *„-simple mid X, imd'.Yj
ttenott^isogenousift #j, then End;, A' = J/.^/J,)* ...xJInriDt)
»1imb Z>( s= Eiuil!t {Xt} are division algebras of tinito rnuk over Q

256
APPENDIX I THE THEOBM OF TAIE
26T
ABEUAN VARIETIES
We now have
Theorem 3. Let X bt an abelian variety of dimension g definsd
over a finite field kt. Ltl it be the Frobenitt* endemorphiitn of X relative
to k0 and P its characteristic polynomial. We then ham the fotktwinj
statements;
(a) The algebra f — Q[«] u the ctnier of the aemt-nmplt algebra
? - Enii;,(AF);
(b) EndJ^JC) contains a letitisimple Q subalgebra A of rank 2j
which is maximal eomtBU/alttx;
(c) the folkncxng statement art equivalent
(c,) P has no multiple root
<°J ^ ¦**
(c,) ? tj commutative
(d) ttc joUounitg statement! are tqmvafoni
(dj [*:Q] = Bff)'p
(d,) J* i'j a power of a linear prjyti/muol
id,) f-Q
(d,) ? u MomorfiAtc to M' ttlqtbra of g by g matrices ovtr tit
quaternion division algebra Dtover Q(l'---- char ka) whtck
splits at all primes I -p p, so,
(d() Xis k^-uogenout tolhe glhpowtrofa super-singular cunt
alt of tfJaje endomorphisms are defined over it,
(e) A' is k0iaogeno>is toapowtr ofa tt-simple abelian variety ifand
only if P is a power of a Q-irredttctble polynomial. When this is the
com, Eisa central simple algebra over F which splits at all finite primes
v of F not dividing p, but does not split at any rail prime of F',
Proof It follows fromtlio mnin theorem that F1=QiQqF ie the
center of A', — Q &(, E, winch proves that F is the center of E
Suppose ? = j*, x .,, x At is the expression of E as a product of
p algebras A{ with centers K^ Letf/tj: Q]—a(. and [A(: Kt] = b\.
can choose subringe L{ of At containing Ki with ij semi-Bimple
maximal commutative, [L,: KJ — 6f. Then L — i, X ... X Lr
Bubalgebra of ii which is maximal commutative
and [L ¦ Q]— E <i(i|. Now for any ( we have
where K ®QQ, - II Xy with /fy fields On the other hand if
s
P *= ft ^T* is the decomposition of P over Q, into a. produot of
t
powers of irreduoible polynomials over Q(, and if we consider Tt(X)
w * Oi [71]-module by making T aot via ir, we have an uomor
ptusm of Q,[T]-modules
p.)
W that ?®qQj being the commutant of n in EndT(X) ii
faomorphie to
Comparing the two fioUnisationa of S®q 0, and keeping in mind
I that K'fj is the center of A{t$t K'(i,-we deduce that (i) for any prime
i^p and any prime oof Kt lying aver I, At splits at v and (ii) there
l>it partition of [1, s] into r disjoint subsets /, I, such that

2i>8
AUELIAN VARTET1UK
APPENDIX. I : THE THEOREM ( K TATF
2S9
It follows thatmF = 6, for ve/ and ? [8 01= S [XjjjQ^
ell J-l
[Ar : Q] = «(, »o thnt
-1 rtjj
': Q']" 2 ffl'[S':
» Jeg P - deg P 2,
This proves (b).
Since F is the center oC E and A1 contains a maximal commutativo
subring of rank 2g, (c,), {c,) and (c() are equivalent, mid since R
commutative oE± commutative s-m, = 1 with the above notation*,
these are also equivalent to (os). This proves {a}.
Now, [?: Q] - Bp)s if and only if Q,®,, B = Jtf»(Q|), henee if
and only if » *= 1, 5. = Qi or equivalently, JJ in a power of a linear
polynomial In this case, Q, is the oenter of Q,®t)E, so that Q
ib the center of E, and conversely, if this holds, Q, ®q? in the
commutant of Q, in End V,, »o that it is the whole of End !•',. Thu»
(d,), (d.) and (d,) are equivalent. If Q,®off - MV{Q,). E in &
central simple algebra over Q whose invariants at all finite primes
I ^ j» ure 0. Since its invariant at the infinite primo is 0 or { and the
sum of invariants at all prime* is 0, E is either MU{Q) or Mti,Dr)
where ?>,, is the quaternion algebra over Q splitting at all finite primes
1 # p. The first possibility is ruled out, aince X cannot be a product
of 2pabelian varieties. This proves that (d,) o (d,,). In view o{ our
remsrkB preceding the theorem, (d,) is equivalent to saying that
X ~ C, where C is an elliptic curve with En^C =: Dp. We have
then shown that C is mipersingular (§22). This proves (d).
Lot Q be the product of the distinct irreducible factors of P. Since
F = Q [it] is semi-simple, and P{w) = 0, we ha-veGM =0. Further,
ir acts as mi endomorphism of V,, any itredticible factor over Q,of the
eharacteristic j)olynomial P divides the minimal polynomial of it so
thst pistlie minimal polynomial of wOVerQ. Kow, X is 1,,-isogeiiOiM
to • power of a tu-siinplc nbelian variety if and only if E is simpfe
; hence if nnd only if the center F = QO] of ? is a field. Since
'] f a Q[X]I[Q(X)). F is a field if and only if Q in irreducible, or
eqiiivttleiitly, P is the power of an irreducible polynomial Q. If F is
the center of E, we have shown earlier that E splits at any finite
1 prime v of F not dividing 1. Suppose v is a real imbedding of F, bo
(hat f(it) is a reni number. Since v(n) satisfies f (t>(ir)) — D and the
roots of P have absolute value \/q, we must have «(w) = ± ^q. If 5
ill square, v<it)eQ and F — Q, so that the equivalent condition of
(<t9) and (dt) implies that ? does not split at 00. If g is not a square,
/= Q(Vp). I*1 '1 be the quadratic extension of i^, and it' ~ jt1
the Frobenius overt,. Then w'eQ, so that the center /"ofE' —
End^ X is Q. Appealing to (d), we conclude that E' — Mt{Dr). On
the Other hand, we have F'cFcEcE', andM is the commutant of F
m E'. By a known result, on central simple algebras, we soo that E
and F®rE' define the same element of Uie Brauer group over F,
that is, ? is the image of E' under the natural map Br(F') -s- Br(F}.
Since both the real primes of Q(Vp) lie over the real prime 00 of Q
: ind E' has invariant J at 00 with respect to F' = Q, E has invariant
j at either of the real primes of F = Qfv'j"). This completes the
proof of (e).
Any two elliptic curves defined over finite fields with
uamorpkic algebras of complex multiplications art isogenout (over the
vtqtbraically closed jictd k).
In particular, any two super singular elliptic curves are isogenous
PROOF. Suppose X, Y are Biipersingular elliptic curves. Wo can
choose a common finite field of definition lev such that Knd^X and
End^y are quaternion algebras over Q, so that they have Q for
! center. Thus, their Frobenius morphiams vx and ity lie in Q. Since
they must both have absolute value \Z? where q = card (k,,), we see
thit tt\
-- q. Thus, if k1 ie the quadratic eitennion of t0, there
is an isomorphism T,{X) ^. 1\{Y) carrying the action of ti'x into n
where ir'v — it*- and n'y = it\ are the Frobenius morphisms ovor'
By Theorem 2, X and Y are isogenous overt,.

260
ABKL1AN VARIETIES
Ne*t suppose End'X ~ JT End* 7 ~ K for some
quadratic extensionJCofQ. Choose i common finite field of (Jefini.
tram of X and I7 over which all their endomorphisros arc defined and
all the points of order p ate rational. Now, Q,®QEndllX admit*
a one-dimensional representation in Tf(X). Hence, p splits into t,
product of two distinct primesp andp' in K which are conjugal^
and Q,&9 Knd* X =:?, X K^,. Suppose for instance that ^®0End*Jf
act* on TJX) via Kt. By what we have said, it follows tb«
tj is I (jo), md sinoe ffmwj inapower of p, (wj.) has to be a pow«
of p' in the ring of integers of Jf. A similar assertion (possibly witk
p replacing p') hold* for *Y. By altering the isomorphism End0 Y~K
by the conjugation of K if neoeBsary we may assume that (»,.) i,
also a power of p'.Sinoe tfum, ^ Wmirj. = 3 = j/, we see that in tho
ring of integera of K, (ir^J = (jrv) = b'^ao that ns and j.,-differ by »
tmit, i.e., a root uf unity since A' is imaginary quadratic. Thui
11J = «J in K for suitable n, and they have the same minimal equ*-
tion over Q of degree 2. Sine* this has to be their characteristic
polynomial X and Y ate isogenous over an extension uf degtea
APPENDIX II
MOBDELL WEIL THEOREM
ByYu I MANIN
I Suumcnt and tketch of the proof. Let K be any finite
extension of the field of rational numbers Q, X — an abelian
variety defined over K. We shall prove the following result.
TaxoRBH The grtwp X{K) of rational points of tie vaneiy X m
finitely gauntltd
Poinoari had already conjectured thu result for elliptic curves. It
ru Mordell who proved this conjetture of Poinoare, over Q, and
Wail generalised tbia proof to the higher dimensional oaae, by intro
dwing a series of new ideas and technical toots. Both the theorem
tod the method of its proof play a central role in modern "Diophnn-
,-tine Geometry".
The generalisation of Mordell-Weil theorem to the case of the baae
field being a field of finite type (over the prime field} is given by
3 Lang*. No essentially new ideas are required for the same.
The proof of the theorem consists of three steps of completely
different kinds, We shall sum up these remilts in the form of three
Msertions.
Profositjok 1. Let « 1 be any integer Then, the groip
This is known as "Weak Mordetl Well Theorem audits proof is
given in 52.
Proposition 2, There ixisU on X{K) a symmetric bilinear taaior
ffoduci X(K) x X(K) > R
{x,y) ^v1 ^ (*>?) mthlhtfallowingprope ties'
(a} <*,z> > 0/oro«*e.Y(K),
(b) thtttt {xbX{K)\<,x,x} <C)t«Jl»We>i'afIC > 0,
t S Lug!

282 ABKLIAN VARTOTIIIg
The conduction of this scalar prod™* i, based on
of heights of Weil-Tate. This is oontamed m J3 and {4
TU-. '11 , • „ .* - . _ .
APPENDIX II: MOHDELL-WEIL THKORRM
283
!
we shall see now.
tT TT ¦"* form^ *^
ite generate of the group *<*) fron, the above property
see now ^
M.
A) rinTisfinUt for aUn>li
(S) (teri! ez««a wF
I' is finitely generated
Proof. First, we choose a system of representatives i, x, of af
the classes of r/»r. .
We observe further that there exists a constant G with the (ailov\
ing properties
<*, x) >(?»<*-«, * ~ x > < 2 <* i> for all » - 1
In fact
* A.
ond
*i a;~a
and henee (x x x~^> inoreasw eoymptoticallv as <a: ^> ft,
i
tioTo^ftr™ i*1^1'7 B) ,fo,uows from the fact tha* *• quad™-:
fop |
<*(. xi> <m, «> and is non-negative for aU integers m, ».]
We set now if = fe...,,
constant above and prove that the finite set M is a set of generator,
for the group T. °
^«^.ifitwerenottheca5e,thereeIistsanelamcatJ:erwhicl •
does not belong to the subgroup ro generated by M with <*, «> j
(for these values form a discrete subset of R in view of the
„„,., ,, , ,, Obviously <ar, K> > C. Besides *— xt ~ ny for some
i( e if and jeF, By using inequaJity {2) we obtain
1 .2
<Sf »> - ~i
xx
Therefore y e To and then # = ny -f x e r», in coutradiction to the
choice of x. The proposition is proved.
This part of the proof of Mordell-Weil theorem is sometimes called
'method of descent". In fact, the proof by contradiction gives a
method of representing any point lETua linear combination of
elements of JIT by successively "descending" from a; to x' =
^
*/ and so on until the heights1
*" >
snd from * to*
of the new points no longer decrease
2. Weak McrdcH-WtU Theorem The symbols X, K have the same
meaning in this paragraph as in the previous one. Let K 3 it be the
algebraic closure of K, We consider geometrical points of X over
the field K and its subfieldn. We fix an integer n > 1, and denote
by X,, the groupofgeometricalpointaoforder»ini'andby ft, the
group of ith roots of unity
We shall prove betow Proposition 1, under the additional assump-
Uott that Xo c X(K) and jk, c K. This can always be achieved by
passing from K to ft finite extension und it is obvious that the strong
form of Mordell-Weil theorem for the extension yields the theorem
for the field K.
?s< L =
'X{K))-tl
of all fields i» K obtained by adjoining iT '«,.* eX(JC) T*e» Im
finite oWton «xtcn«i'on of A'.
Deduction o» Proposiiion 1 *bom thb Fuwoaxbmtaj, Lbkma
Let 0 be the Galois group of L\K Consider the mapi X(K) ¦*¦
Horn (O, X,): a-w—>A

264
A B ELI AX VARIETIES
APPENDIX II: HORDE!.LHTU. THEOREM
2fl5
It is clear that/, is independent of the choice of n 'x, since X.cXfK).
It is easily verified that /, is a homomorphism and that /, ^-,;
/,,-/„¦ further
Therefore the map .r'vv »/, defines an imbedding
MK)jnX(K) c—* Horn (O, X,).
By virtue of the fundamental lemma, the latter group is finite and
that gives the required.
We shall divide the proof of the fundamental lemma into several
steps. All the necessary algebraic geometric considerations «r*
concentrated in the proof of Lemma 1, the rest U purely arithmetic,
LjQItu 1. Let A be any ting of integers in tkt field K Tkett
txtsts on open subset Y =* Specie Spec A, a scheme X-t-Y protec-
protective over Y, Y morphisntm = X x X ~* X and a tectton 7: Y-t- Xwith
At following properties
(a) the fibre X over a generic point of the scheme Y (together uiih
Ike restrictions m and e of the morphiums m and 7 to this fibre) it
tsomorphie to the abelian variety X over K
(b) X it a group scheme over Y and all fibre [schemes) over doted
point* ye r [together with the restrictions of the morphisms m and e
to these) are abelian varieties;
(e) the mapptng in (a) induces an tsontorphism of grcv.pt
XiY) ^ X{K)
Outunb 01 Proof. Suppose more generally, we are given an
algebraic variety defined over K. By taking a finite cover by mean*
of affine seta, we select a finite number of generators of the co-
coordinate rings of each of these sets and a finite number of generators
of the corresponding ideals. We obtain thus a finite presentation of
the variety by meana of a finite number of polynomials and rational
functions"tn a finite number of variables. The coefficient* of these
polynomials lie m a subring A$. c K of finite type over K. Thus
our variety can be considered as a generic fibre over Spec A^ defined
by the "tame eqantions", An analogous construction "model over
4 ring" is possible for any diagram consisting of a finite number of
varieties and morphisms over the field K. By applying this argument
to the given defining abelisn variety A* over A', we can meet the
requirement in (a).
In order to establish assertion (b), it can be shown that one has
necessarily to reduce still Spec A# by puncturing at a finite number
of points. The most important requirement consists in that, for y e
Spec A& the fibre scheme X, be a smooth variety. We omit the proof
of the fact that this holds for all but u finite number of points if the
generic fibre is a smooth variety. The other requirement in (b) can
be secured in an analogous manner, if one applies this result also
' to the diagrams of relevant morphisms.
;j Finally, the condition (c) is automatically fulfilled if we choose
Spec ^3 so small, that As be a principal ideal ring (this is always
possible, for the group of ideal classes of A is finite). In fact, the
unique invertible sheaf over Spec Av will be the structure sheaf.
! Since Jf -* Y is a project!ve morphism, X is a closed subscheme in
I P* and it is sufficient to check that P*(y)-zFll(K). This follovre,
! for example, from Proposition 3 of Lecture 5 in Mumford'a book*
' Indeed, according to this Proposition
: P?<rH{(«., -.•»). •lei*8l»ioOpriiBe}/JI
! where R id the equivalence relation (*„,..., «„) ~(u»g,.... «,),ueAs*
{invertible elements). But since Ax is a principal ideal ring, it is
' dear that all points of PjJ(A') lie in the above described set. This
pond ud«B the proof.
i We observe that now considerably more subtle and precise facts
! on the models of abelian varieties over rings and in particular on the
behavior of "degenerate fibres" (Neron'B theory) are known
From Lemma 1 follows immediately
CoROLUJir 1. LetJcGX(Y).C<toMderxasacL>Md8vbschemcofXand
¦ morphism nz multiplication by n. Then the natural projeciionn~l{x)-i-
\ Spec (As) i* Hale over all points y e Spec ASI whenever char Hy) Jfti.
D. Rtumftirfl I Lecliire* on rurv™ on an algubmLb surf»o*, Antoai* of Math
fltt, Princ^K^U University Pn

w
268
ABELIAN VARIETIES
APPENDIX II KORDE1X WEIL THEOREM
287
Proof. This follows from the faot that every component of n~1(l;)
maps onto y and the number of different geometrical points in the fibre
X over tlie point y is equal to the order ker {Xt —>- Jfj) i.e. »*•, g a,
dim ,Y, if char )ti,y))(n. Since this number coincides with the degree
of the nrojeetivc morphram, the definition of the etalenoss gives tho
required.
Reformulating this result in the classical terminology we obtain
Corollary 2. There, exist* a finite art E of prime ideals of Ikt
ring A, such that for any point r.eX(K) and y e n!!)^X(K), thi
extension. K{y) 3 K is unramifitd outside E.
(In E, it is necessary to include Spec A\ Spec As and divisors of n.)
Wo return now to the fundamental lemma. The field K{n~l x) ig
normal over K, because the conjugates of the point m~' at are of the
form n->x+ y, ye X. C X{K) and Jfftt" '*) = #(»»-»* + y). This also
gives the ahetian nature of the Galois group and its action on n~'x
defines an isomorphism with a subgroup of X,. Hence the oomposite
? = A'(n~' X(K)} is abeh'an of exponent n over K and is ramified
over only a finite number of prime ideals of the field.
Consequently the fu damantal lemma will he proved if we verifv
the following fact
Lemma 2, The maximal abelian extenston of exponent n of an
algebraic nv,7nber field K, which is unramified outside a fixed finite Set
E of prime ideals of the field, is finite.
Proo* Let FoK be the maximal abelian extension of exponent
«, H, its Gatois group Consider the exact sequence 1 -+ ii^-t-K*-*
H'-r-l. The exaotness of the sequence of cohomology groups of
Gal(*/Jf) gives K"I(K*)"-^r- Horn (Gal (i/if), (ij = Horn (H, p.).
(We use here that B'{G»l(KIK),K*) - 1 and B'{Oti (K/K), ^) =
Hom(Gal(^/if), ft.), since ft,cJC*). In other words there is a well-
defined nondegenerate bilinear pairing:
(H<w«tn« group H hu Knill topology ttidK'jK**, discrete topology)
Jo any eubfield KchcF we obtain from Galois theory a subgroup
Bj,cH which leaves its elements fixed. Let BcK* be the inverse
jniage in K* of Uie subgroup, orthogonal to HL with respeot to the
puring C)- By making use of this point of view, it ii not difficult
to conclude that
L = K(VB) = X(..., V*t, -). 6tE B.
This is clearly the algebraic part, the so-called Rummer tbeory
We now make use of the arithmetic of the Held K. It is necessary
to explain when the extension K(-fa) is unraroified outside S
(divisors of h are contained in E), Local considerations show im-
immediately that the necessary and sufficient condition for this is rt(a)
a O(mod »}, for all y $ E. The group of elements with this property
m K*, splits into finitely many classes mod K**. This Mows easily
from the theorems on the furiteness of the number of ideal classes of
Eunits(Diriohlst-NQnkowgki ¦ Hasse).
Thus, we see thet to obtain the extension K{n X{K)) it is
rafficient to extract nth roots of a finite number of ©lemonts of the
field K. This concludes the proof of the fnndamantal lemma and the
weak Mordell-Weil theorem
3. Height ot points of projccUve apacci. As before K denotes on
ilgebraic number field of finite degree We teoolleot souse eb-
mcatary fuits front the arithmetic of these fields By points of Ike
JieldK, wemeanobjectsofoneofthefollowingthreekindB:(a)priine
idealB of the ring of integers of the field K (non-arohtmedean points)
(b) imbeddings K -*¦ R, (c) pairs of complex conjugate imbeddinga
K-t-C. Let « be one auoh point of the field. This uniquely define*
normeJised |... [, of the field K satisfying the following properties.
Let xeK; then
|k|, = \x'\ where x' is the image of* undar the in bedding of ?
inRorCifvisan arohimedean pouit. ^
\p\r= I ifitis non archiraedean and the ideal dmdea the ideal
P
If) peZ a prime

288
AH ELI AN VARIETIES
Al'PKNUlX II MORDELL-WEIL THBORLM
269
In the sequel, we will often make use of the following simple
formulas. For all x, yeK
log xy[r = Jog icl. + log y\,
1°8 I * + S I. "- m»* <1(tg I * If. loR I »1J + « D)
, log 2 if p ia arehimedean,
e 1
I 0 if v is non-archimedean.
We denote by A'r the completion of the field K under the topology
induced by the normalised |... I, and net n, — [A>',:(M- For any
element i e A" x ,? 0
w
(the sum makes sense, for, |*|, -¦= 1 for all except finitely many u).
Por the sake of convenience, we set log 0 ™ — <x>, tenn than any
arbitrary real number.
I*t (i0>... ?„) 6 K" be any system of elfmenls not all of which
are zero.
Definition Lemma 1 The numher
I
*(*„ O
max (log |i-,-|f)
(8)
w uwtt defined and has the. following pr tperttes.
A) A(Aa^,..., Ai,J = A(j-,, ij/or «U \e X*.
The* h can be emmdertd as a function oh the K-poinU P"{K) of the
protective space P" provided with a distinguished system of co-ordinates
This function is called tht height.
B) A(«) > QforaUxeP"{K};
C) h{x) does not depend on the choice o/ the field tohere the projec
five coordinates of the point x lie.
Pkoof The first assertion follows from the faot thet
max loglJ&il, = max log l*; |, + log | kc\,
and product formula." {*) applied to A
For the proof of tlio second we remark that, dividing all the
; toordinates of the point by a suitable non-xero number, we may
. assume thet one of the coordinates iB a unit; then max log | xt |, > 0
for v non-archimedean no that h{x) > 0,
: For the third astertion, it is sufficient to prove for pairs of fields
?p K, one contained ill the other. Let to run through all paints of
the field L which are extensions of v. Then /f,®j, L = © A, from
> where [L-K] n, — V n^ bo that focMxeK
Since the valuation I |, on K coincides with I , this proves
the third assertion.
Example. Let K - Q, xt e Z with g.cd. ~ (^,. xm) =¦ 1 Then
In particular, the number of points of bounded height in P"(Q) is
Suite.
More generally;
Pbofobition 4. Let C > 0, d>0, deZ P" pngtcttve space
talk a fixed system, ofcoordinates. Tien the set
[xeV(K)lh(x) < C, deg Kin) < d) nfimte
- Proof. We reduce this assertion to the case K = Q with the
: help of the following transformation. Consider points x = {x^ xj
| of degree < d and denote the corresponding point (Xa,..., Xm)
I of the projective apace over Q of forma of degree deg Hx) in n + 1
variables. The coordinates of thu point would be coefficient* of
Ibe norm form Km (S xk Tt), the norm taken over Q. The mapping
[x0 x,} -w ^ (Xv ... ,Xm) defined on the set of points of fixed
degree, has finite fibre, since one and the sahie image is possible
only for conjugate points (the norm form splits into linear factors,
the factorisation being unique upto ordering). Thus it U sufficient

S70
AB ELIAN VAK1KT1KS
APPENDIX 11 M BBK1J.-WBII WBOBEM
271
to prove that points of bounded height go over to points of bounded
height. This is almost obvious. Making use of the inequality D), we
find that, for some constant if, (where v is any point of the Onloii
extension over Q generated by ^af)},
max log [a-,!, < d max, loi(|j- | id,.
i i.r*
where rf, may bo assumed to be iern for al! non-archimedean po 11«
of the field and (Trims though the conjugates of e overy. Hence
*<A"n JT.) < <TA (r,,... ,x.)+r
where d' and d" can be evaluated in terms of * an 1 rf
The assertion ia proved
We elia.ll analyse now the dependence of height on the system of
coordinates. Let/,,/t be two real-valued functions on some set. We
call them equivalent {/( ~/t) if |/, - /, | is bounded.
Proposition s. M\,, ht~ heights in P"{K) defined vnthruptd
to turn tyrtrute of coordmatt*. Then ft, ~ *,,
Proof. Let A = ((i(J) be the matrix describing the passage from
one coordinate system to the other
B\ means of formula (8), we find that
max log I x," |r < max log !*[', + max fog | (t,, , +¦ log (* + 1)
if v "s archimedean, and
max log |ar* |, <maxlog | »¦';!, +
it via lion archimedean
Hence A,(t)'< ft,(r) + h{ ...o,,...) +e for all * e P"(^). Making use
of theinvertibility of j4, we can show analogously AL < ht + C. Thj)
proves the assertion.
Taking into auoount this result, t« /uJur* toe tnli Constdtr
ptiiy ujrfo efuttufenee and calculate them for same convenient
coordinate system
4. Height ttaocmted with an inverbble thcal. Let now A" be any
variety defined over a field K, 9: X -* P" * mornhism b the projec-
ttve space over K, The height hr is ¦'eBned on the geometric points
X{K) by the formula, hf{x) =¦-¦ h{<e{x)) where Aisas before the height
in the protective space P".
pBOVoamoB 0. IM X be any ampUtt iwiety over K and
V: X-*P*, $'. X-*P> two K-morphitnu.
If <p*(«Wl» = ^*(M»)) aver X, then *,~fc+ in X{R).
?noor. Let i = <p*(<V(U) = f *(ffi^(l)). Then from the defini
tionofpxAj(
It suffices to show A, ~A.; then by symmetry *t -j)kt.
We may suppose that <?(*) is not contained in any proper linear
subspaceofP*, Choose a basis(T, Tt)cT*(r(F, ^(l))cr(X,ii)
and extend it to a basis (Tr .., TJ of the whole »paee V[X, I). The
hwghtA,, can be computed in the system (T1,, ...TJand A, in the
system (T,,...,Tt). I^etze X(K), (xr .,., *„) ~ coordiontes of the
point x(s); [xa, •¦¦ ,#t)-—coordinates of 11A).
Firstly
t * < *
max log
max
The estimate in the other direction is somewhat less trivial
Since X ia a complete variety, the image xW c P* is closed Let
J c K{TV ..., T,] he the defining ideal in the homogeneous coordinate
ring Jt = K{Tt,...,T,yi. Since the sections Tt,..., T» of the defining
tnorphism y do not all have a common zero on X, their images in
it gencratean ideal, the radical of which is called "irrelevant" ideal
H+1 generated by forms of degree > I, In particular, there exist* on
integer 5 >0 and forma Ft)zK\Tv ..., TJ of degree 9 1 such that
t
Consequently fot any point (i0,... ,jt,) =
j-u

272
AB ELIAN VARIETIES
By applying the estimate F} to the right side, we obtain
-l) mar log | ac41, + mw log 1^ j,
i j
where C, ia a set of constants depending on Fti such that C, 9* 0
only for finitely many u, Henoa
max log [sj, < mailog|x,|, + C,so thet A, < A. +C
isn fit
The proves the assertion.
Now we are in a position to prove the fundamental g&nenol theorem
on heights, due to A Weil.
Theosbm. Let X be a protective variety over tht field K for each
element L e Pic X we can make eorrupond a wett-defintd height
function. hL {unique npto equivalence) on X(K) having the following
properties:
w t^-^ + K
(b) ijX = P", then k ^-height defined in J3.
(c) for any K-morpkitmy: JT-+-Y wuf igpior, *t«(t('~At«<p Aobfo
Proof. Firstly the uniqueness of the height follows from the fact
that Pio X is generated by very ample line bundles, the height of
whoae classes, defined upto equivalence has properties (b) and (c) by
virtue of Propocition B,
Par the proof, it is essential to verify firstly property (a) for line
bundles ?, = <p*W)), L% = 0*(ff(l)) where9: X-+P\^;X -*t*.
Denote by a: P* x P* -»¦ Pw the Segre morphism which can be
described in terms of homogeneous coordinates aa follows:
«afc ...**>. W - V,)) = (-«,»,...) e P*+'M'+'i-i
Then with the obvious notation ¦
?1® i. - X*(<W1)> where *:X
iite morphism. Consequently i^.i
raajlog|*(jrj|,
it follows immediately that A^ ** kr +
P*xPl J0+ P"
A,; but from the formula
mailogly, |
i
At + hi j
APPENDIX IX MOBDELL WEIL THEOREM
273
Let then ? be any line bundle. Setting L = L,. ® l^ whete L,
and L» are very amplo, we define AL = Ai,-*v Prom the assertion
proved just now, it follow* at once that hL defined npto equlvalenoe
does not depend on the oholoe of Lx and ?, and that the height hL
*a a function on Ho X satisfies properties (a) and (b)
For the functoriftl property @), it is sufficient to check only on line
bundles of the form L = ^*(<P{1}) since they generate Pie T. For
inch Jine bundled hL *-*«„•*¦ by definition and h,.M ~AW1)*^¦? by
definition, whence A^, ^htfy.
The theorem ia proved
S. T«te hdght on abdiaii vanetk* LetnowXbeanabeJuuivariety
over K, L e Pio X. It seema J. Tate made the simple and beautiful
observation that in the first place kL is an "approximate quadratic"
function on the group X{?) and in the second place there ia a unique
ly denned function %L equivalant to hL which is in fact quadratic
We hegin with a lemma concerning arbitrary abelian groups
Lmiiu 3 Let T be ony abdian group A r-»-B a funeUonoa tl
tetufying the condition
(8)
TxTxT) Then tkttt
Wtlt a tt/mntiric bilinear pairing
TA«e mmuMkmm o^/Iim i onrf 1 uniquely.
Proof We let firttly
Then $ is symmetric and is "approximately liilinoar".

174
ARBUAN VARIKTIEA
and analogously in the second Moment. Since the proof followj
from the relation (9) in terms of A, it is enough to verify the equiva-
equivalence of this with the corresponding equation (8).
We »et now
b(x,,xj = lim'4 •
One obtains the existent* of the limit and the equivalence h —
rimulUneoiuly, if one notes that
«-<»+">^B"+ij,l,(¦+'»,)_ l-'^S-ar,, 2"!C,) + 4 "+ 0,
whew 9, = B.ixt, x,) ~ 0 *o that
From the formula {9} follows immediately the biline&nty of 6
Wo net now Mx) = A(x) - i b(z,«). Then
*(*, + *,J - A<ar,> - *{*,) = /»(«,, *,) — &<J=i *J ~ 0
Tbemfore, by »pptyUg the same averaging process to A we obtain a
horoomorphiun
Summing up,
The uniqueness nft and I m obvious
Defihiko!! — Lbmma 2 Lei X be an gldio* txmtty over on
algtbmic number fitli K, L e Pic X Then the hafht \L on X{K)
tcUUfitt Ike condition of Ltmma 3. Consequently the function! bb Jt
ami St are uniquely defined and tAe latter oniony (A«nt m eaifed Tale
the paint o:\ X {attocmted U> the lint bundle L)
Apply the theorem of the cube (to be exact Coroll»ryB)
of it) to the projeotions Pt; X x X xX-+X. Then we obtain
AfPKNDlX II HORDE LI WEIL THKOItKM
nr,
By calculating the height of the point («„ *„ xJeXxX xX with
respeot to the line bundle on the left side with the help of the theorems
on properties of Aj, we obtain
2 2
This concludes the proof
Rbmabk. Lettp: -S*^r,beamorphismof abelian vaiiaties, Luny
line bundle on Y. Then At.(M ~ *i o f whence \.(M = At o 9 and
in particular 6»w = tLoip, l,.(l) = IL.<p. By applying the latter
relation to the mapping ?{x) = — «, and setting f*{L) = L~, we
obtain 6^- = 6lt lt- = — lt. Let now ? be a aymmetrio line bundle
i.e L~ = L, Then Ij, = 0 and At(i) = i ^(a, a;J.
& Proof of Proposition 3. We select a symmetric very ample line
bundle ?011 X and choose < s, j ) = bh [x, y). Sines L i> Tery ample,
O, x> > Ofor all x6f (?). In fact, otherwise, A^nx)^ (n1 < x,x>
-» co aa »-+ 00 and this would contradict the fact A^far) <-- kL{y)
and in the olw» of funotioni equivalent to hL[y) there exists a
non-negative one {second property of height—compare Definition
Lemma 1)
Further, from Proposition i it follows tbet
the set {xe{X{K)jh^x) < c} U finite and therefore the same m
true of the set {x^X{K}Kx, ir.) <O).
This complete* the proof of Proposttiou 3 and that of the theorem
of MordeUWeil.

BIBLIOGRAPHY
[A Q] A. Akbxbotti and H. Gbauibt ; Thioreaieg de finltude
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[B] WAE.TBB BaiLT : On the theory of fl-funetions, the moduli
of ftbelion varieties, and the moduli of onrvea, Annalt
of Math. 76 A902), 342.
[B K] H. BKiUH and H. KowjHitn Nonfat) Aijeinm Springer
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[B H] Abmasd Boeki, and Qsones MoStow : editors Algebrau
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in oarattflristiua po»itiv», Anttali deBa 8c, Norm. Pita
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1962-64 (Sahemtta en groupea)
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varieties over funotion fields, American J. Math. SI
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David MmrronD Oeometne invariant theory Sponger
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INDEX
Abelian v««y
Action of a group «hcmo on
aaohmnet '¦'-
App.l1.rlumbwt; theorem of
number,'
C.K.typ» ; ¦"'
t oonstriaotion ol...
Curlier dual
108
20
40,178
18S
ill
131
,ic polynomial of
of abelian
Chem olaM (Irt)
Chow, theorem of...
Compler homomorphlenu
Complex multipiioattoMt
algebra of *-* ++hJ+'
Cube, theorem of.
181
1*
11
1
na
m. si
of ¦ nnite morphijim
tnaeparaMe—•
Uu Blmii-n1'
DS.lt!
of
olliptie oarvea.r* -¦¦*
Differential operator—
[on A BchsmeL.n
order of. ** ***--*,
Divtoonal oorrmpumlene*.
Doibaault-reBoiulion
Dual abolian variety—
ary ehiraciarintki
I abetlm virility—
til
10*
108
SI
4
Bpimorphtom nf R™up-
¦ah«
Xtde
Haaw-Witt map
Hypsralgobra (of a gtoup-
acherAo) .++ *..T.,+n-
Index of Un» bnndte
Involution, of Hnl and M«md
Lind
Iiogenjr
pt j of height 1 *-
Jaoobian—(never u«tl)
Jorrian .algebra rinKtuw on
Neion-Sewn-i group
I map* .,¦*.*-¦¦ *
Flbmllon awjolaliiu to prin-
cip.llHinilkj HI
4«
p of variety Kt
>adte repromntation
&ngt thrown of—on ox
t/inca or rational point*
LafwhoU: theorem of,...*+
Lie-algebra
Hllpotmt partofjo-Uni'Br map
Non-tlogcnoratij Hnn Iiondle...
Hortn: of pnilomorphium of
abolian variety --,
Horm: roducod *.
Pf»Auui of ekov-fiyinmetrw
matrix....+ +
Poincari-buniliti
Poinfit* coinplute rwluei.
hility thw.mni
1B* j^rankof Ahclian varinty
Quatlratib runction +
Quotient hy a finll* group
Quotient of a group ttbo
I ly «i>|. group scheme,
Riemann-forin of a divitto
Uiomann hjp»lhmm
ltkimann-Roeh tlMornm
100
1M
14S
pigMity lemma
iti involution
: g^j.Bftw theorem ++ +
fl,,ni|.continuity theorem
^ini-Minph part of p-linear
map +-
ajrre and P.o«nlitiht;
tlwerem of -* *
¦: Berrri tlicofem of— on auto-
inorphiwma of aoulian
; gimplo fttiiilion variety
j gqusre; theorem of
,„. i Sub-group »ohome
xl" ; n t normal—
nt j Supsniinguuir olliptio curve..
* : I --^.valued point *>**-
88
A, ITS
155
178
181
ISO
II
IS
111
141
55
n
104
SI*
110
INDEX 219
ij Tat« group! 1-ailio ITO
ISO Tate group; Ji-ddic diecreti... 
rheta-funetfon *^
M Thefcigroup ML MS
60 ^ non-degenerate !2S
Trace form lTS
118 Trace of endomornVii»m of
abelian variety 182
227 Trace; r«luc™i «»
fypo of n Unite commutative
jrroup flchemet local,
tm j V. , 118
reduced, ,.. +
5» VauWiing theorem ¦ Is*
104 Vootorflold "*
]j§ H , loft.invariaht I"
2!' . II
0*( WeieTptrirta-funotwn -,... ""