PUBLICATIONS OF THE MATHEMATICAL
SOCIETY OF JAPAN
1. The Construction and Study of Certain Important Algebras. By
Claude Chevalley.
2. Lie Groups and Differential Geometry. By Katsumi Nomizu.
3. Lectures on Ergodic Theory. By Paul R. Halmos.
4. Introduction to the Problem of Minimal Models in the Theory of
Algebraic Surfaces. By Oscar Zariski.
5. Zur Reduktionstheorie Quadratischer Gormen. Von Carl Ludwig
Siegel.
6. Complex Multiplication of Abelian Varieties and its Applications
to Number Theory. By Goro Shimura and Yutaka Taniyama.
7. Equations Differentieller Ordinaires du Premier Ordre dans le
Champ Complexe. Par' Masuo Hukuhara, Tosihusa Kimura et Mme
Tizuko Matuda.

COMPLEX MULTIPLICATION OF
ABELIAN VARIETIES
AND ITS APPLICATIONS TO NUMBER THEORY
BY
GORO SHIMURA
AND
•r3 The late YUTAKA TANIYAMA
ев
f-..-ч
е- . as.
О
I
i
THE MATHEMATICAL SOCIETY OF JAPAN
1961

Copyright ©. 1961
by The Mathematical Society of Japan
Printed in Japan
by Kenkyusha Printing Co., Ltd., Tokyo

PREFACE
The history of complex multiplication began with the works of
Gauss and Abel on elliptic functions. It would be right, however, to
call Kronecker the initiator of number-theoretic investigation of the
subject. The main theorem of Kronecker's theory asserts that the
abelian extensions of every imaginary quadratic field are generated by
th.e special values of certain elliptic or elliptic modular functions; as
Kronecker left the work unfinished, the accomplishment needed the
efforts of the later authors, Weber and Takagi. A similar and simpler
result, which is also due to Kronecker, holds for the field of rational
numbers: the abelian extensions of the field of rational numbers are
generated by the roots of unity, the special values of the exponential
function. Hilbert conceived an idea to generalize these results, namely,
to construct abelian extensions over any given algebraic number field
by means of special values of analytic functions; he took this up as
the 12th problem in his Paris Vortrag and emphasized its importance
in number theory, it should be also mentioned that Kronecker had
already thought of the problem. The first essential progress in this
direction was made by Hecke, by following the idea of Hilbert. He
succeeded to construct unramified abelian extensions of certain biquad-
biquadratic fields by means of singular values of Hilbert modular functions
of two variables. This was the last work, as well as the first, till recent
years, which attacked the problem successfully. On the other hand,
a new development took place in the theory of complex multiplication
of elliptic functions. First H. Hasse perceived the connection between
complex multiplication and the Riemann hypothesis for congruence-
zeta-functions, which was later proved by A. Weil in a fully general
form. This observation led M. Deuring to establish a purely alge-
algebraic treatment of complex multiplication of elliptic curves. He could,
moreover, along the same line of ideas, determine the zeta-functions
of elliptic curves with complex multiplication. The definition of zeta-
function of an algebraic curve defined over an algebraic number field

vi PREFACE
is originally due to Hasse; and Weil is the first contributor to this
subject.
Now the advancement in algebraic geometry of late years, especially
in the abstract theory of abelian varieties, due to Weil, enabled us to
approach the problem in a fairly general form, as was shown in three
papers, published in the Proceedings of the International Symposium
on algebraic number theory, Tokyo-Nikko, 1955, of Weil and the
authors of the present monograph. It is the purpose of the monograph
to provide a full exposition of the results announced in these memoirs.
Our chief object is the arithmetic of an abelian variety A of dimen-
dimension n, whose endomorphism-ring is isomorphic to an order in an
algebraic number field К of degree 2n over the field of rational num-
numbers. The first task is to show that the field of moduli of A, whose
definition must and can be given by virtue of the notion of polariza-
polarization, and the fields generated by the coordinates of the points on A of
finite order, are'class-fields over a certain algebraic number field K*,
corresponding to'the ideal-groups determined by the arithmetical struc-
structure of A (Main Theorems 1, 2, 3 of Chap. IV). The number field К
can not be taken arbitrarily; it must be a totally imaginary quadratic
extension of a totally rell number field. K* is the algebraic number
field determined by К and the representation of К in the linear space
of invariant differential forms on A. If и = 1, we have К — К*, while
if n > A, both the cases К = К* and К =£ К* may occur. The abelian
extensions of K* thus obtained from A do not provide all the abelian
extensions of K* unless я = 1 ; at any rate, the classical results of
Kronecker and Hecke are included in our main theorems as particular
cases. It is noticeable that the prime ideal-decomposition of the N(p)-
th power endomorphism Kf of A(fi) is fundamental in our whole theory,
where A{p) denotes the reduction of the variety A modulo a prime
ideal p of a field of definition k for A. The above result is in close
connection with the investigation of the zeta-function of the abelian
variety A. In fact, a more precise analysis of zf shows that the cor-
correspondence p —► jtj, determines a Grossen-character of the field k.
We are then led to the expression of the zeta-function of A by the
product of several Hecke L-series attached to Grossen-characters (Main

PREFACE vii
Theorem 4 of Chap. IV); this is a generalization of the results of Weil
and Deuring mentioned above.
We now give a summary of the contents. Chap. I is an exposition
of more or less known results on abelian varieties, which are mostly
given without proofs; the only exception is § 2, where we have given
a detailed (but elementary) treatment of invariant differential forms on
abelian varieties. § 3 deals with the analytic representation of abelian
varieties, their homomorphisms and divisors by means of complex tori.
In §4, first the notion of polarized varieties is introduced and then
the definitions of field of moduli and Kummer variety are given. Chap.
II is devoted to the algebraic part of the theory of complex multipli-
multiplication. §§5, 6 contain a necessary and sufficient condition that an
algebraic number field К of degree In be realized as the endomor-
phism-algebra of an abelian variety of dimension n. § 7 is the study
of mutually isogenous abelian varieties in connection with the ideals
of the endomorphism-rings; § 8 concerns the phenomena ■which are
essential only in the case of dimension n > 1 and related to the def-
definition of the number field K*. Chap. Ill contains the theory of
reduction of algebraic varieties modulo a prime divisor of the basic
field. We shall prove in § 13 the fundamental theorem concerning the
prime ideal-decomposition of JV(|i)-th power homomorphism. Our
final aims are achieved in Chap. IV. The first step (§ 14) is the in-
investigation of the relations between abelian varieties, of the same type
of complex multiplication, whose polarization are also of the " same
type ". Then, in § 15, we prove the first main theorem; an unramified
class-field is obtained by the field of moduli. A similar argument
together with the analysis of the points of finite order gives us also
class-fields, whose characterization is the object of § 16. These results
are obtained assuming the endomorphism-ring to be the principal order
of the number field. In § 17, the case of non-principal order is com-
completely investigated. The last § 18 is devoted to the determination of
the zeta-function of an abelian variety of the type described above.
The large part of the contents was prepared in collaboration of
both authors during 1955-56 and published in 1957 in Japanese as the
first six chapters of the book with the title " Kindai-teki Seisu-ron ".

viii PREFACE
The English version was then planned; but, owing to the sudden
death of the second named author in the autumn of 1958, the work
had to be completed by the person left behind. The present volume
is not a mere translation, however; we have written afresh from be-
beginning to end, revising at many points, and adding new results such
as § 17 and several proofs of propositions which were previously omit-
omitted.
The present monograph owes much to the idea of Weil [54],
though we have not necessarily indicated explicit references in the text.
I take this opportunity to acknowledge my cordial gratitude to Pro-
Professor Andre Weil for his constant advice, suggestions and encourage-
encouragement. I wish to acknowledge also my thanks to Mr. Taira Honda
who read the manuscript and contributed many useful suggestions.
University of Tokyo,
February, 1900
< Goro Shimura
The author wishes to express his deep appreciation to Mr. C. Sudler Jr.
who gave a financial support for the publication of this volume.
December, 1960.

TABLE OF CONTENTS
CHAPTER I. PRELIMINARIES ON ABELIAN
VARIETIES
1. Homomorphisms and divisors 1
2. Differential forms 5
3. Analytic theory of abelian varieties 19
4. Fields of moduli and Rummer varieties 26
CHAPTER II. ABELIAN VARIETIES WITH COMPLEX
MULTIPLICATION
5. Structure of cVo(-<4) 38
6. Construction of abelian varieties with complex multiplication ... 44
7. Transformations and multiplications 52
8. The dual of a CM-type 66
CHAPTER III. REDUCTION OF CONSTANT FIELDS
9. Reduction of varieties and cycles 77
10. Reduction of rational mappings and differential forms 84
11. Reduction of abelian varieties 94
12. The theory " for almost all p" 99
13. Prime ideal decomposition of iV(p)-th power homomorphisms ... 110
CHAPTER IV. CONSTRUCTION OF CLASS-FIELDS
14. Polarized abelian varieties of type (K; {<pi}) 115
15. Unramified class-field obtained from the field of moduli 124
16. The class-fields generated by ideal-section points 131
17. The case of non-principal orders 136
18. The zeta-functions of abelian varieties with complex multi-
multiplication 142
Bibliography 152
Table of notations 155
Index 157
[ix]

NOTATIONS
We denote by Z, Q, R and C, respectively, the ring of rational
integers, the fields of rational numbers, real numbers and complex num-
numbers. I being a rational prime, Zt and Qi denote the ring of Z-adic
integers and the field of Z-adic numbers, respectively. Rn and Cn
denote the vector spaces composed of all matrices with n rows and 1
column, with real coefficients and complex coefficients, respectively.
If jc is a matrix ■with complex coefficients or a function with values
in C, etc., we denote by Re x, Im x and x the real part, the imaginary
part and the complex conjugate of x. The transpose of a matrix M
is denoted by lM. We denote the unit matrix of degree m by lm-
Terminologies and basic notations concerning algebraic geometry will
be the same as those of Weil's trilogy [44], [45], [46], with a few ex-
exceptions. In particular, k' being a finite algebraic extension of a field
k, we denote by [k': k]i and [k': k], the inseparable and the separable
factors of the degree of k' over k. If a is an isomorphism of a field
ki into a field k2, we denote by z", for every z e k1; the image of z
by a; furthermore, У being any algebro-geometric object defined with
respect to klt Y" will denote the transform of У by the isomorphism a.
If k is a field and ж is a point of an affine (resp. a projective) space,
we denote by k(x) the field generated over k by the coordinates (resp.
the quotients of the coodinates) of the point x. We use also the
notation k(x) for the points on an abstract variety (cf. [44]).

CHAPTER I. PRELIMINARIES ON
ABELIAN VARIETIES.
1. HOMOMORPHISMS AND DIVISORS.
The purpose of this § is to recall briefly some of the basic con-
concepts on abelian varieties defined over arbitrary ground fields. For the
general theory of abelian varieties, we refer to Weil [46] and Lang [26].
1. 1. Let A and В be two abelian varieties. By a homomorphism
of A into B, or an endomorphism when A = B, we shall always under-
understand a rational mapping X of A into B, satisfying X(x+y) = X(x)+X(y);
if Д is birational, we call it an isomorphism, or an automorphism when
A = В; the image of a point x on A by X will be denoted by X(x) or Xx.
We denote by Ju^{A, B) the set of all homomorphisms of A into В
and put ^jd(A) = J(f(A, A); <Ж(А, В) is a finitely generated free Z-
module. We put JCT^A, B) = J6T{A, B)<g>Q and ^/„(A) = *jd{A)<g>Q.
If A e с^о(Д -В) and ^ ё J^^B, C), we can define the product /Л in
a natural manner as an element of J^b{A, C). <jtfo(A) is then consider-
considered as an algebra over О and ^f(A) is an order in the algebra ^fb{A);
the identity element of ^jdu{A) will be denoted by 1^. We shall
denote by д(л) the kernel of a homomorphism X, and put Q{m, A)
= Q{m\A) for every rational integer m. A and В are called isogenous
if the}' are of the same dimension and there exists a homomorphism /
of the one onto the other; such a I is called an isogeny. A and В
being of the same dimension, let X be a homomorphism of A into В;
let k be a common field of definition for А, В and ?., and x a generic
point of A over A. We put, if X is an isogeny,
v(X) = [*(*): А(Ле)],
v.(J) = [feW : ft<7*)],, Vi(X) = [k(x): A(ic)]tl
and otherwise v(X) = и,(Л) = Vi(X) = 0; these numbers do not depend
on the choice of k and x. For every isogeny X of A onto B, we can
find an element X' of ^0(-В, Л) such that X'X = \A, IX' = 1B; ;.' is
[1]

2 PRELIMINARIES ON ABELIAN VARIETIES [I]
uniquely determined by these relations; we denote X' by A'1.
1. 2. Z-adic representations of homomorphisms. Let A be
an abelian variety of dimension n. Let / be a rational prime; put
!
If I is different from the characteristic of fields of definition for A, then
Qi(A) is isomorphic to the direct sum SDJ of 2и copies of the additive
group Qi\Zb We call any one of the isomorphisms of Qi(A) onto 2R
an l-adic coordinate-system of Qi(A); we shall consider every element
of 5Ш as a matrix with 2n rows and one column, with coefficients in
QijZi. Let В be another abelian variety of dimension m and ?. a
homomorphism of A into B. Choose l-adii coordinate-systems b of
%i{A) and to of Qi(B). Then there exists a matrix M with 2m rows
and 2n columns, with coefficients in Zt, such that, for every t e Qi(A),
we have to(^f) = Mb(t). If we fix b and to, the mapping ?. —>• Л/ is
uniquely extended ?to a representation of jy^A, B) by matrices with
coefficients in Qlt which'we dkll the l-adic representation of J&\{A, B)
with respect to » and to. Let f be an element of ^jdb{A). Mt being
an Z-adic representation of ^jaf0(^4), let
be the characteristic polynomial of Mj(f). Then, the at are rational
numbers^ and we have
P@ = £2n+aif2n-1 + -+a2nb = 0.
P(X) is determined only by f and independent of the choice of I and
/-adic coordinate-system. We call P{X) the characteristic polynomial of
с and the roots of P{X) the characteristic roots of f. Furthermore, if
с е cj</(v4), the at are rational integers and
A) 4f) = det M,(f).
We shall put, for even' f e *j*fo(A),
B) tr(f) = tr M,(f).
1. 3. Picard variety. Let A be an abelian variety; let 5fa(A)
and &i(A) denote respectively the set of divisors on A algebraically

[1.4] HOMOMORPHISMS AND DIVISORS 3
equivalent to 0 and the set of divisors on A linearly equivalent to 0.
Then there exists an abelian variety A* canonically isomorphic to
&a.{A)j&t{A), which is called a Picard variety of A. Every divisor
Y contained in &a(A) defines a point of A*, which we denote by
C1(Y). Let В be an abelian variety and B* a Picard variety of B.
For every homomorphism X of A into B, we obtain a homomorphism
X* of B* into A* such that
C) X*(Cl(Y)) = Cl(X-\Y))
whenever -?-1( Y) is defined. The mapping X —> X* is uniquely extended
to" an isomorphism of ьМУъ{А, В) onto J&f^B*, A*); we denote by la
the image of a by this isomorphism and call it the transpose of a.
If a <= Жа{А, В) and /9 e ЛУй(В, С), we have '(£«) = 'a'/9. Let X be
a divisor on A; we shall denote by -Xu the transform of X by the
translation зс—* x+u on A Now define the mapping <pz of ^4 into A*
by the relation
D) <px{u) = C\(XU-X)
for и е A. Then <px is a homomorphism of A into ^4*. The divisor
X is said to be non-degenerate if px is an isogeny. For any two
divisors X, Y on A, we have <px = <py if and only if X and Y are
algebraically equivalent (Barsotti [3], Serre [31]). Assuming X to be
non-degenerate, put for every $ e udo(A),
E) £' = сох [?9>х-
Then, it can be proved that £ —> £' is an involution of cjrfu(A) and
we have, for every £ ^= 0,
F) tr(ff') > 0.
We call this involution the involution of ^^{A) determined by X. Let
X be a homomorphism of A into В and У a divisor on В; assume
that X-\Y) is defined. Then, putting X = X-\Y), we have
G) wx = 'X<pyX.
1. 4. Z-adic representations of divisors. Let a be an integer
and Y a divisor on A such that аУ is linearly equivalent to 0. Then
there exist two functions Ф and ¥ on A such that (Ф) = aY, Ф(ах)

4 PRELIMINARIES ON ABELIAN VARIETIES [I]
= V''(x)a, where (Ф) denotes the divisor of Ф. For every point и on A
such that аи = 0, put
ea(u, Y) = ¥(x+u)F(x)-i;
then ea{u, Y) is an a-th root of unity. Now let X be a divisor on
A; and let u, v be two points on A such that аи = av = 0. Since
a(Xv—X) is linearly equivalent to 0, we can consider ea(u, Xv—X). Put
ex,a.{u, v) = ea(u, Xv—X).
Let k be a field of definition for A; and let 1 be a rational prime
other than the characteristic of k. Let Ui denote the set of roots of
unity, contained in the algebraic closure of k, whose orders are powers
of /; then Ut is isomorphic to Qi\Zb Take an isomorphism of Ui
onto QijZi and denote it by lg; choose an /-adic coordinate-system b
of §i{A). Then there exists a matrix Et(X) with coefficients in Zt
satisfying ("
lg; ex, i" (s, t) == I» ■ 'b(s)El(X)b(t) mod Zt
for every point s, t on A such that lus = l"t = 0. We call Ei(X) the
l-adic representation of >X with respect to b. We have Ei{X) — 0 if
and only if X is algebraically equivalent to 0.
1. f>. q-tb. power homomorphisms. Let A be an abelian varie-
variety defined over a field k and a an isomorphism of k onto a field k".
Then we obtain in a natural way an abelian variety A', defined over
k°, taking the transform 0° of the origin 0 of A as the origin of A'.
If В is an abelian variety and ?. is a homomorphism of A into B,
both defined over k, we denote by 1" the homomorphism of A' into B",
whose graph is the transform by с of the graph of Z. Now suppose
that the characteristic p of the universal domain is not 0; let q = pf
(/ > 0) be a power of p. We shal denote by X* the transform of any
algebro-geometric object X by the automorphism z —»z* of the uni-
universal domain. We can define a homomorphism к of A onto Aq by
for x e A. We call - the q-th power homomorphism of A. If Л is
defined over a finite field with q elements, A* coincides with A, and

[2.1] DIFFERENTIAL FORMS 5
hence tz is an endomorphism of A; we call then tz the q-th power
endomorpkism of A. All the characteristic roots of the 9-th power en-
endomorphism have absolute value q1!*; this is the so-called " Riemann
hypothesis for congruence zeta-functions" proved by A. Weil. Let
1 be a homomorphism of A into В; let izA and tzb denote respectively
the q-ih power homomorphisms of A and B. We have then
In particular, if A is defined over a finite field k with q elements, we
have
for every endomorphism a of A, defined over k.
2. DIFFERENTIAL FORMS.
2. 1. Definitions. In this §, the varieties are all assumed to be
defined over fields contained in a universal domain Q which we fix
once for all. Let V be a variety and k a field of definition for V.
We shall denote by k(V) the field of rational functions on V defined
over k and by @{V) the field of all rational functions on V. If jc is a
generic point of V over k, the mapping k(V) Э/—»/(#) gives an iso-
isomorphism of k(V) onto k(x). We denote by J@{V) and jg?(F; k)
respectively the set of all derivations of Q(V) over Q and the set of all
derivations of k(V) over k. If V is of dimension n, ^S{V\ k) is a linear
space of dimension я over k(V) and J2}{V) is a linear space obtained
from J2)(V; k) by the scalar extension Q(V) over k{V). We shall de-
denote by ЩУ) the dual space of 3>(V) and by rrD the scalar product
of ij e ®(F) and D e J0{V); then A7, D) —»-q-D is a bilinear mapping
of S)(F)X^(F) into £(F). Now, by a differential form of degree m
on F, we shall understand a homogeneous element of degree tn in the
Grassmann algebra defined over ®(F). If / is a function on F, the
mapping J^(F) эВ-> £•/ gives a linear mapping of J2}(V) into .G(F),
and hence defines an element of £>(F), a differential form of degree
one on V; we denote it by df; then we have df-D — Df. We see
that S)(F) is generated over Q{V) by the forms # for /e i2(F). If

6 PRELIMINARIES ON ABELIAN VARIETIES Щ
V is of dimension n, then there exists a set of и functions {gi,-~,gn}
in k(V) such that k{V) is separably algebraic over k(gu—, gn). If
{gw,gn} is taken as this, dglt---,dgn forma basis of S)(F) over Q{V).
By our definition, every differential form со on V has an expression '
M
where the fw яте elements of Q{V). We shall say that a differential
form со on V is defined over k if a» is written in the form
со — S
W
with the e(i) and the ^>4 in £(F). fei>", £»} being as above, a dif-
differential form '
is defined over ft if and only if the fm are contained in k{V).
Let F' be a sjmple subvariety of V. We shall say that a differen-
differential form a; on V is _^яг'ге a/o/^g (or at) V if uj is written in the form
a> = Hf{vdgil--dgir where the /(i) and the gt are functions on V
which are all defined and finite along V. If that is so, denoting by
the /'(i/and the g't the functions on V induced by the fw and the
gt, we pbtain a differential form и' = T,f\i-idg'il---dg'ir on V which is
determined only by со and V; со' does not depend upon the choice
of the /,,) and the gi. We call w' the differential form on V induced
by u>.
2. 2. Local parameters. Let К be a field and щ,---,ип be n
independent variables over K. If K1 is a separably algebraic exten-
extension of K(ub---,Un), there exist и derivations Du---,Dn of i^i over Я"
such that
= 1, A"/ = 0 for г i=j.
The Z)t are uniquely determined by these relations; we shall denote
Dt by djdui for each г.
Now let V be a variety of dimension и, defined over k, and у а

[2.3] DIFFERENTIAL FORMS 7
simple point on V. We call a set of n functions {^i,•■■,?■„} in k{V) a
system of local parameters for V at у defined over k, if the following
conditions are satisfied.
(LI) k{V) is separably algebraic over k(rlt---, rn).
(L2) The rt are all defined and finite at y.
(L3) For every f in k(V), defined and finite aty, the function dfjdn
is defined and finite at у for every i.
Let x be a generic point of V over k; let Va, ya, x, be affine repre-
representatives of V, y, x and S the ambient space for Va; let N be the
dimension of S. Then, by Koizumi [23], we know that и functions
Ti,---,Tn in k{V) form a system of local parameters for Vaty, defined
over k, if and only if the following conditions are satisfied.
(L'l) The Ti are all defined and finite at y.
(L'2) There exists a set of. N polynomials Fi{Xu~-,XN, Tu—,Tn)
■with coefficients in k such that Fi(xa, т(х)) = 0 for 1 <: i g N and
detidFtldXjfa, r(y))) + 0.
We shall prove this in § 10. 3 in a more general case. If {ri,---, rn}
is a system of local parameters ял~у, then dxu--,dx-a form a basis of
S)( V); and, by (L3), every differential form
is finite at у if and only if the fw яге all defined and finite at y.
2. 3. Let V and W be two varieties and T a rational mapping
of V into W; we denote by the same notation T the graph of T, and
by Y the projection of T on W. For every / in Q{W), defined and
finite along Y, we shall denote by /° T the function on V defined by
f°T(x) =f(T{x)) with respect to a field k of definition for V, W, Г and
/, where x is a generic point of V over k. Assuming that Y is simple
on W, let ш be a differential form on W finite along Y. Then w is
written in the form w = J^fmdgi^-'dgir where the/m and the gt are
elements of Q(W) which are all defined and finite along Y. Put
«' = S СЛ« • T)d(gtl- T)-.d(gir° T).
Then w' is a differential form on V which is determined only by w and

8 PRELIMINARIES ON ABELIAN VARIETIES [Г]
T, and does not depend on the choice of the /«> and the gi. We
shall denote the form w' by щ°Т. If V, W, T, w are all defined over
k, then w«T is also defined over k. If T is defined at a simple point
у on V and if T(y) is simple on W, then for every differential form
со which is finite at T(y), w°T is finite at y.
V, W, T, w being as above, let U be a variety and S a rational
mapping of U into V; suppose that the image X of U by 5 is simple
on V and T is defined along X. Then we obtain a rational mapping
T°S of E/ into PF, defined by T°S(x) = T(S(x)). Suppose further
that the image Z of U by 7"»5 is simple on W and a> is finite along
Z. We obtain then a differential form w°(T°S) on [/. It can be easily
verified that the differential form (w<>T)°S is also defined and equal to
<o«(T'S).
2. 4. Differential forms of the first kind. Let V be a com-
complete non-singular variety. We say that a differential form ш on 7is
of the first kind if w is eyerywhere finite on V. Let W be another
complete non-singular variety and T a rational mapping of V into W.
Then, for every differential form w of the first kind on W, co°T is
also a differential form 6f the first kind on V. This is easily proved
by rneajns of Proposition 5 of Koizumi [23].
PsfoposmoN 1. Let Vl and Vt be tiuo complete non-singular varie-
varieties arid pi the projection from ViXVt onto Vifor i=l, 2. Then, for
any differential form o> on V^ x F2 of the first kind and of degree 1, there
exist differential forms щ on Vt (i = 1, 2) of the first kind and of degree
1 such *hat os = a>1i
This is a restatement of Theorem 3 of [23].
2. 5. Let V be a variety defined over k and о an isomorphism
of k onto a field k°. Let / be a function in k{V) and Г the graph
of /. We shall denote by /" the function on V" whose .graph is Г':
then, /—>/• gives an isomorphism of k(V) onto k'{V'). Let a» be a
differential form on V defined over k. Then w has an expression
dgiy-dgir with the/(i) and the gt in k{V). It can be easily
(»)
shown that the differential form w' = J2fa)°dgii°~-dgir' on V' is deter-
@

[2.6] DIFFERENTIAL FORMS 9
mined only by w and a, and does not depend upon the choice of the
/,ii and the gt. We shall denote the form w' by W.
2. 6. Invariant differential forms on group varieties. Let G
be a group variety and t a point on G. We shall denote by Tt the
left translation x —* tx of G. Tt is obviously a birational correspond-
correspondence of G onto itself. We call a differential form w on G a left in-
invariant differential form on G if a>°Tt = a> for every t on G.
Proposition 2. Let G be a group variety and w a left invariant
differential form on G. Then w is everywhere finite on G.
Proof. Take a field of definition k for G and ai.'and a generic
point x of G over k. w is of course finite at x. Then, the relation
w = w°Txv shows that a> is finite at any point y'1 on G.
Lemma-1. - Let G be a group variety, w a differential form on G;
let kbe a field of definition for G and w, and t be a generic point of G
over k. If the relation (и«Т( — w holds, w is left invariant.
Proof. Let s be anypgeneric point of G over k. Then there
exists an isomorphism с of k(t) onto k(s) over k such that f = s. By
the relation w — w°Tt, we have w = w° = (a>=Ti)° — w»Ts. Now let x
be a point of G. Take a generic point у of G over k(x). Then,
since xy and y'1 are generic on G over k, we have ш=Тху = a>°T,,-i = w.
so that
This shows that ш is left invariant.
We shall denote by ©oCG) the set of all left invariant differen-
differential forms on G of degree one and by ^{G\ k) the set of elements
of S;0(G) which are defined over k.
Proposition 3. Let G be a group variety of demension n, defined
over k. Then 5H(G; k) is a linear space of dimension n over k; and
we have 2H(G) = 3H(G; k) <g)* Q and 5>(G) = ©„(G; k) (g)* Q(G).
Proof. It is obvious that SH(G; k) is a linear space over k.
Denote by e the identity element of the group G. Let {гь---,г„} be
a system of local parameters for G at e, defined over k. Then, every

10 PRELIMINARIES ON ABELIAN VARIETIES Щ
element w of ©((?), defined over k, can be expressed in the form
w = J^ftchi with the/i in k{G). Let x and Z be independent and
generic on G over k; put ft = ft«Tt-\, r'i = rjoTVi. We have then .
wTt-i = ZfidzU, ft(x) =/i(r»at), r',(*) = г«(Г«ж).
Since {rfrb---, <frn} is a basis of ®(G), there exist я2 functions
g'yA gign, tgy^n) in £?(G) such that
dx'i = S «*i/rfrj A S i £ n).
The g',7 are contained in k(t)(G), since the z't and r» are contained in
k{t)(G). Let gtj be the element of k(Gx G) defined by gy(i, x) = gltj{x)
for each i and 7. If у is a point on G where the 14 are local para-
parameters and if the r\ are all defined and finite at y, then the gltj are
all defined and finite at 3', so that the gij are all defined and finite at
txy, and we have gij(t, y) = glij{y). Applying this to the points t
and e, we see that the, gtJ- are all defined and finite at txt and txe,
and we have
gij(t, t%= gUj(t), gait, e) = gcij(e).
Now suppose that w is left invariant. Then, the relation w = w°Tt
implies Xfidu — Zfidr'i = T1figtijdrj; hence we have /, = Zfig'tj,
i i i,j i
nameK',
A) . Л(«) = Zft(t-lx)gtJ(t, x) A ^У S я).
Conversely, if n elements f\,---,fn of &(G) satisfy this relation, then
ш = J^ffdzt satisfies the relation w°Tt = ш, and hence, by Lemma 1,
ш is left invariant. Assume that w is left invariant, namely that the
fj satisf}- A). The fj are defined and finite at e since ш is finite at e.
Hence we can specialize (t, x) to (t, t) in the equation A). We have
then fj(t) = Zlfi(e)gij(t, t) for 1 SjSn. Let hit be the element of
i
k{G) defined by htj(t) = gij(t, t) for each i and /; and put a>i = T, hijdzj
j
for each i. Then we have w = J^fjdzj — £,fi(e)a>i. This shows that
j

[2.6] DIFFERENTIAL FORMS 11
every element of ^>a(G; k) is a linear combination of the a>i with coef-
coefficients in k. Hence the dimension of the linear space ^>0(G; k) over
k is not greater than n. We shall now prove that the wq A :g q ^ я)
are all left invariant. To prove this, it is sufficient to show that the
equality A) holds if we substitute hqi, hqj for ft, /,-, namely,
B) h^x) = 2 hti(t-^x)gij(t, x) (lg^n, Ui^ n),
or, as we have A,<(:c) = gqt(x, x),
B') д^х, x) = S fe(rlx, r»x)ft/(t, x) A g ? S n, 1 g У S я).
Let s be a generic point of G over &(i, x); put
Г"< = r^T.-I, X'"i = T'ioT,-l = ri.Tuo-l (lg!g Я).
Then we have
i
Define the functions gsi;- and gSIi; on G by
() ) )
we have then dz"i = Z g'ijdzj, dv'"i = Z g'^jdzj- for lgi'gn. These
У i
У
are obtained from the relation dz'i — 2 g'ijdtj by isomorphisms be-
i
i
tween the fields kit), k(s) and k(st). It follows that
2 rt^y 2 {gir)i 2
i i з'Л
this implies g"»; = 2 (M'ij°Ts-i)gsji for ISj'Sh, 1 g / g я, or
j
C) £„(**, яс) = 2 gy(«, J-'x^iCJ, ») A S i ^ n, 1 ^ Z S я).
Specializing (s, t, x) to (t, t~xx, x), we obtain B'); hence the су; are
left invariant. Our theorem is completely proved if we show that
the (ot are linearly independent over Q(G). To see this, it is sufficient
to prove det(hij) =jt 0, since dru---,dzn are linearly independent over
Q(G) and a>t = 2 hijdzj. As i is generic on G over fe, the zt are
j
local parameters at t'1 and consequently the г'< are local parameters at e.

12 PRELIMINARIES ON ABELIAN VARIETIES [I]
Hence, from the relation dt'i = 2 g'tjdrj it follows that det(g'{J(e)) ^ 0,
i
namely, det(ga(t, e)) ф 0. Specializing (s, t, x) to (s, t, s) in the relation
C), we get
ga(st, s) = E gij(t, e)gjt(s, s) (Hi|n,H/S «)•
Hence we have det(gij(s, s)) =/= 0; this completes the proof, on account
of the relation htj(s) = gtjis, s).
Now "specialize x to e in the relation B); we see then that the
h(j are defined at e and
D) detfAyfc)) *= 0.
We need this fact later. <
2. 7. Invariant differential forms on abelian varieties. As
an abelian variety is a commutative group, we call a left invariant
differential form' on an abelian variety simply an invariant differential
form. !<
Proposition 4. Let 'A be an abelian variety. Then every differen-
differential form of degree 1 опл A is of the first kind if and only if it is
invariant.
Proof. By Proposition 2, we have only to prove the " only if"
part. Eet w be a differential form on A of degree 1 and of the first
kind. Considering A as G in Proposition 3, we use the notations гч,
Шг, hij in the same sense as in that proof. Then, by that proposition,
there exist n functions at in Q{A) such that w = 2 а.ш*. As the wt
i
are invariant, we have, for every x on A,
Since со is everywhere finite on A, w°Tx is also everywhere finite on A ;
in particular, ш°Тх is finite at e. Therefore, recalling that the r,- are
local parameters at e, we see that the и functions 2 (а*° Tx)hij A ^j^ n)
i
are finite at e, so that, by D), the Oi«Tx are finite at e. As x is an
arbitrary point of A, this shows that the at are everywhere finite on
A. Such functions must be constant, since A is a complete variety.

[2.8] DIFFERENTIAL FORMS 13
Hence w is a linear combination of the ш; with constant coefficients;
so w is an invariant differential form.
2. 8. Differentials of homomorphisms. Let A and В be two
abelian varieties and / a homomorphism of A into B. If w is an
element of S)o(-B), then <o°X is defined and contained in ®0(^4). The
mapping w ~* w°2 gives an i2-linear mapping of ®0(i?) into ®0(^4). We
shall denote this linear mapping by d2, namely,
(dtyw = wX
for <d e ®o(-B)- If k is a field of definition for А, В and )., then a/.
gives a fc-linear mapping of 2H(-B; fc) into ®0(Л; k). For every
homomorphism ц of 5 into an abelian variety C, we have
Proposition 5. Z-e( у4 and В be two abelian varieties; and let ?. and
ft be two homomorphisms of A into B. Then we have
d(X+ft) = dl+dfi.
Proof. Define homomorphisms a, /3, у, 1а, /л0 as follows:
a(x) = xxx, p(xxy) = Z
20(xxy) = 2(x)xO,
We have then 2+p = y$a, ylaa = 2, yp^a = fi, so that д(Л+и) — дадрду,
дХ — Sa5105y, о/л = Sadfiooy- Hence our proposition is proved if we
show 5/3 = dX0+dfi0. Let pt and pz be respectively the projection from
BxB onto the first and the second factors of BxB. By Proposition
1, for every aiGS,(BxS), there exist two elements a>u cu2 in ®o(-B)
such that w = a>i°pi+a>2°p2- It follows that 0,801 =
We see easily
Pl°P = Pl°*0, />2°0=/>2%, ^>2°/9 = p2°Po,
so that we have

14 PRELIMINARIES ON ABELIAN VARIETIES [I]
ok0((o2°pz) — W2°2>2% = а>г«рг°О = О,
fyto(<Ui °£i) = a>i°pi°fio = Wi'p^O = 0.
It follows that dfio) = 520<o+dfi0w; this completes the proof.
Lemma 2. Let k(x) be an extension of a field k and s the smallest
number of quantities ut in k(x) such that k(x) is separably algebraic over
k(u). Then there exist s and no more than s linearly independent deri-
derivations of k(x) over k. Moreover, if the characteristic p of k is not 0,
гее have [k(x): k(xp)] — ps and [k(x): k(x*)] g qs for any power q = pf
zcith / > 0. (cf. [44] p. 14).
Proof. Let (и) — (щ,•■•, м,) be a set of quantities in k(x) such
that k(x) is separably algebraic over k(u). By our definition of s, for
every i, k(x) is -not separably algebraic over k(ult--, Uj_b щ+и---,и,).
Hence there existg1, for each i, a derivation Dt of k(x) such that
DiUi = 1, DiUj = 0 (г^;1); trie derivations D( are obviously linearly
independent. Let D be a derivation of k(x) over k; put Dtu = yt and
D' = D—2УгА» then Df is a derivation of k(x) over k(u). As k(x)
is separably algebraic over k(u), we have D' = 0; this proves the first
assertion. Now suppose that k is of characteristic p == 0. As fe(x) is
purely inseparable over k{x?) for any power q = pf with / > 0, A(x) is
purely inseparable and separable over k(xfl, u); so we have k(x) = k(x^, u).
It follows that we have [k(x): k(x^)] g qs, since the up are contained
in k(x?); in particular, we have [k(x): k(xv)) g ps. Suppose that we
have [k(x): k(xp)] = pr < p*; then there exist r quantities Vj in k(x)
such that k(x) = k(xp, v). By the first assertion of our lemma, the
number of linearly independent derivations of k{x) over k(xp) is not
greater than r. This is a contradiction since every derivation of k{x)
over k gives a derivation of k(x) over k(xp). Hence we must have
[k(x): k(xp)] = ps. This completes the proof.
Theorem 1. Let A and В be two abelian varieties and 1 a homo-
morphism of A into B; let k be a field of definition for А, В and X, and
x a generic point of A over k. If the linear mapping 32 of 2>o(-B) into-
is of rank r, then:

\2.8] DIFFERENTIAL FORMS 15
i) k(x) is separably generated over k{hc) if and only if dimi-(&:) = r ;
ii) assuming that A and В have the same dimension n, zee have
Vi(l) — 1 if and only if n — r;
iii) я being as in ii), if k is of characteristic p ^ 0 and if k{hc) Z> k(xfl)
for a power q = p' (e > 0) of p, then zue have v{2) = vt(X) ^ qn~r.
Proof. Let n and m be respectively the dimensions of A and B.
Let F denote the subfield {f°X \fek(B)} of k{A). Let О be a deri-
derivation of k{A) over k. We shall prove that (o/m)-D = 0 for all w e ®o(-B)
if and only if ZXF = 0. Take a basis {a>u---, wm} of ®0(-B; &) over
k; then, for every / G ^E), there exist, by Proposition 3, иг functions
gi in &(Б) such that df = 2 £**>«• If {Slai)-D = 0 for all ш G ®0(-B),
we have
D(f'X) = d(f°2)-D = 2 (г<.Д)(«о»()-£) = О;
this shows DF = 0. Conversely, suppose that DF = 0. Every <o in
2H(U) can be expressed in the form a> = J^fidhi with /, e O(B),
We have hence
2) = 0.
Thus we have proved that (ok<o)-D — 0 for all су g S>o(-S) 'f an<i only
if Di1' = 0. Now, by our assumption, cM[©0(-S)] is a linear subspace of
S)o(^) of dimension 7-. By Proposition 3, any linearly independent
elements of ^H{A) over Q are linearly independent over Q(A). Hence
there exist exactly n—r linearly independent derivations of k(A) over F.
If x is generic on A over k, the mapping /—►/(*) gives an isomorphism
of &(v4) onto k(x), and F corresponds to k(Xx) by this isomorphism;
so there exist exactly n—r linearly independent derivations of k(x) over
k(/x). By Lemma 2, we can find n—r elements ult---, ып~г in k(x) such
that k(x) is separably algebraic over k{hc, ult---, un_T). If dim^-bc) = r,
the щ must be independent variables over k(hc), so that k(x) is sepa-
separably generated over k(Xx). Conversely, if k(x) is separably generated
over k(?.x), then b^7 Proposition 16 of [44] Chap. I, the dimension of
k(x) over k(JLx) is n—r, so that we have dim^-fo:) = r. This proves

16 PRELIMINARIES ON ABELIAN VARIETIES [Ij
the assertion i). If m = n and щ(Х) = 1, k(x) is separably algebraic
over k{?M); consequently, by what we have just proved, we have
n = dimt(?jc) = r. Conversely, if rank 52 — n, there is no derivation
other than 0 in k(x) over k{?x), so that k(x) is separably algebraic over
k(Xx), namely, vtB) = 1; this implies ii). Suppose now that k is of
characteristic p i=0 and k{hc) Z) k(xfl) for q = pe with e > 0. Then, by
Lemma 2, we have
[k(x): k(hc)) = [k(x): k(hc, *«)] ^ qn~r;
this proves iii) of our theorem.
Corollary. Let В be an abeUan variety and A an abelian sub-
variety of B. If a denotes the injection of A into B, we have
This is an easy consequence of i) of Theorem 1.
Proposition 6. Let A, B, X, k, x be the same as in Theorem 1.
Suppose that the 'characteristic p of k is not 0. Then:
i) we have dX = 0 if and only if k(hc) с k(xp);
ii) assume that A and В are of the same dimension; if vt(P.) = 1, we
have k(x) = k{xfi, he) for* every power q — pe with e > 0; conversely, if
k(x) = tyx*, hS) for some q = pe with e > 0, we have vt(X) — 1.
Pr6of. The proof of Theorem 1 implies that SX = 0 if and only
is DF<== 0 for all derivations D of k(A) over k, the notations being as
there. Hence we have 5/L = 0 if and only if F С k{A)p-k; the latter
condition is equivalent to k(he) С k(xp); this proves i). If vt{X) = 1,
k(x) is separably algebraic over k(hc); as k(x) is purely inseparable
over k(xi) for every power q = p' with e > 0, we have k(x) = k(xt, ?jc).
Conversely, suppose that k(x) = k(x*, he) for some power q = pe with
e > 0. Then there is no derivation of k(x) over kBx) other than 0, so
that k(x) is separably algebraic over k(hc), namely vt(X) — 1.
Proposition 7. Let Abe an abelian variety of dimension n, defined
over a field of characteristic p ^ 0. Then, v^p\a) is a multiple of pn and
the order of q(P, A) is a divisor of pn.
P
Proof. By Proposition 5, we have д(р1л) = дAл+•••+1Л) =р8\л = 0.

[2.9] DIFFERENTIAL FORMS 17
Hence, by i) of Proposition 6, we have k{px) С &(xp), where k is a
field of definition for A and x a generic point of A over fe. It follows
that щ(р1л) = [k(x): k(px))i ^ [k(x): k(x")] = pn. Since the order of
§{p, A) is equal to v,(plA) and »г(р1л)щ(р1л) = v{plA) = p2n, we obtain
our proposition.
Let A be an abelian variety of dimension n and k a field of def-
definition for A. Denote by cjrf(A; k) the set of all elements in cjrf(A)
defined over k and by <jd0(A; k) the subset ud(A; k) (g) Q of <jdo{A).
For every Л e ^{A; A), cW gives a linear transformation of Ъй{А; k).
We have seen above that the relations 5(X+ft) = дк+др, d(?.ft) = dftd?.
hold, so that the mapping ?.—>5X gives an anti-representation of
cjrf(A; k). As Ъ0(А; k) is a linear space of dimension я over &, we
obtain, with respect to a basis of ®0(^; &) over k, an anti-representa-
anti-representation of ud(A; A) by .matrices of degree и яя£й coefficients in k. If &
is of characteristic p =jt 0, we have 5(/>lx) = 0; so our representation
is not one-to-one. If k is of characteristic 0, we get a one-to-one rep-
representation. In fact, if 5?. = 0, the rank of 52 is 0, so that by i) of
Theorem 1, we have dimk(hc) = 0; this implies X = 0. In case of
characteristic 0, we can extend uniquely the representation to a rep-
representation of tjafoC^'. k). We shall call this anti-representation a
representation of <j*fo(A; k) by invariant differential forms.
2. 9. Differential forms on a curve and its Jacobian variety.
In the sequel, we denote by 2>o(F) the set of all differential forms on
an algebraic variety V, of degree 1 and of the first kind.
Proposition 8. Let С be a complete curve without singular point
and J a Jacobian variety of С and <p a canonical viapping of С into J.
Then, <o —* (D°<p gives an isomorphism of ®о(У) onto ©0(C).
Proof. Let g be the genus of С; denote by C, and /„ the prod-
product Cx--:XC of g copies of С and the product Jx--xj of g copies
of /, respectively. Let k be a field of definition for C, J, and <p, and
Xi x - • • X xg a generic point of Cg over k; define a rational mapping W
g
of Ct into / by W(xu---,xe) — 2 <p(xt). Then ¥ is everywhere defined
on Cg\ and putting z — ¥(xlt---, xg), we see that &(xi,-•■,*,,) *s separably

18 PRELIMINARIES ON ABELIAN VARIETIES ffl
algebraic over k(z). Let a and r denote respectively the rational map-
mappings of Jg into J and of C, into Jg defined by
a{zu—,zg) = Zi + '-'+Zg, T(xi,—,xe) - y>(*i)X —X<p(xg).
Denote further by pt the projection of Cg onto the t-th factor and
by qt the projection of/, onto the г'-th factor. We have then ¥ = <r°r,
<7i°r = <p°pi and 5<t = 5qt-\ \-dqg by virtue of Proposition 5, so that
we have, for every w e ©0C/)>
= (Uef^orH + Ш»9°à = <D"(p'p1-\ \-(D°<p°pg.
Hence, if шор = 0, we have <o°W — 0; as the mapping W is sepa-
separably algebraic, <o°W = 0 implies ш = 0. This shows that the mapping
w—Ko°(p of SoCT) into ®0(C) is one-to-one. Our proposition is thereby
proved, since 2HC/^ and Ъ0(С) are of the same dimension g.
The notations-C, / and tpt being as above, let С be another com-
complete non-singular curve, J' its Jacobian variety and <p' a canonical
mapping of С into J'; and let & be a field of definition for C, J, <p,
C',J',ip'. Let X be a positive divisor of Cx С rational over k. X de-
determines * homomorphism of / into J' as follows (cf. Weil [45, 46]).
Take a generic point x of С over k and put
then, there exists a homomorphism 1 of _/ into /', defined over fe, and
a point 6 on У, rational over k, such that
^ and 6 do not depend on the choice of k and x.
Proposition 9. The notations being as above, let Co be a complete
non-singular curve with a generic point z over the algebraic closure ki of
k such that k(z) = k(x, yi,---,yn); and let p and the q* be the rational map-
mappings of Co into С and into С defined by p(z) = x, qv(z) = y, with re-
respect to ki. Then, for every w e ®0 C/0> we have

[3.1] ANALYTIC THEORY OF ABELIAN VARIETIES 19
Proof. Define the rational mappings a, yS, у as follows:
where the numbers of the factors in the products are both equal to
я, and
a(z) ~
r(viX--XVn) - Vi-j \-vn.
Put /?! = 2+b; we have then
Let ai.be an element of S>eC/0'> as e>--is an. invariant.form, we have
woXi = w°2. Denote by ?-„ the projection of J'x---Xj' onto the ч-th
factor. Then we have у = y±-\ \-yn, so that a>°y = tu»^iH \-w*yn
by Proposition 5. As we have yu°fi°a(z) — <p'(y,) — <p'°q,(z), we get
w°yu°P°a = <o°<p'°qt. Hence we have
2
v=l
3. ANALYTIC THEORY OF ABELIAN VARIETIES.
In this section, we shall' recall some of known results from the
classical theory of abelian varieties; a modern treatment for this sub-
subject can be found in Weil [57].
3. 1. Theta functions and Riemann forms. Let D be a dis-
discrete subgroup of Cn of rank 2tt; then CnJD is a complex torus. An
.R-bilinear form E{x, y) on C" with values in R is called a Riemann
form on C"/Z) if it satisfies the following conditions.
(Rl) The value E(x, y) is an integer for every x e D, у е D.
(R2) E(x, y) = -E(y, x).
(R3) The form E(x, V—ly) is a positive (not necessarily non-degen-
non-degenerate) symmetric form.

20 PRELIMINARIES ON ABELIAN VARIETIES Щ
A meromorphic function / on О is called a theta function on
O/D if we have
f{x+d) = f(x)exp[ld(x)+cd]
for every d <= D, where ld(x) is a C-linear form on Cn and ca is a
complex number, both depending on d. Then we can show that
there exist two .R-bilinear forms H, Ho and an 2?-linear form b, with
values in C, such that
A) f(x+d)=f(x)exp{2nJ=l[H(d, x)+±.H0(d, d)+b(d)]}
for d<=D,
H0(u, v) = H0(v, u),
H{du d2)-== H0(dirdt) mod Z for db d*.eD. ..
Putting ";
E(x, y) = H(x, y)-H(y, x),
we call E the alternating form defined by f. If / is holomorphic, E(x, y)
is a Riemann form on CnjD; we call then E the Riemann form defined
byf. A theta function/is said to be normalized if И is skew-hermitian
and b, is real valued; if that is so, we have
B) H(x, y) = -%[Щх, У)~ V=T E(x, -/=l
Conversely, let E(x, y) be a Riemann form on C"/Z>. Then there
exists a holomorphic theta function / on O/D such that E is the
Riemann form denned by /.
If / is a theta function on CnjD, then the divisor (/) of / is de-
defined on O/Z), which is an analytic divisor of Cn/Z). Conversely, if
X is an analytic divisor of Cn\D, there exists a theta function / on
O/Z) such that (f) — X. We can prove that the alternating form E
defined by / is determined only by X and independent of the choice
of /; so we call E the alternating (or Riemann) form defined by X,
and denote it by E(X). A complex torus CnjD has a structure of abelian
variety if and only if there exists a non-degenerate Riemann form on Cn\D.
Let A be an abelian variety defined over C. Then we can find

[3.2] ANALYTIC THEORY OF ABELIAN VARIETIES 21
a complex torus CnjD and an analytic isomorphism в of A onto Cn/Z).
We call the pair {C^-'jD, в) or simply the isomorphism в an analytic
coordinate-system of. A. If X is a divisor on ^3, then 0(X) is an analyt-
analytic divisor of CnjD, and conversely; we write E(X) = E@(X)) and
call it the alternating (or Riemann) form defined by X with respect to
в. We have E{X) = £(У) if and only if X and У are algebraically
equivalent.
3. 2. Analytic and rational representations of homomor-
phisms. Let A^ and A2 be two abelian varieties defined over C; let
(O/Di, #i) and (CmID2, 02) be analytic coordinate-systems of Аг and
■/42, respectively. Consider now a homomorphism /£ of At into ^42-
There exists a linear mapping Л of C" into Cm such that
Л must satisfy ЛСДО С D2. Conversely, every linear mapping of Cn
into Cm satisfying this condition corresponds to a homomorphism of
Аг into At. With respect to the coordinate-systems (zt) in О and (zu*)
in О, Л is represented by an mxn matrix S = (st/) with complex
coefficients as follows: regarding the Zj and the wt as functions on C"
and Cm, we have
The mapping Л —> Л (or >t —» 51) is uniquely extended to a representation
of Л^0(Аи Ая), which we call the analytic representation of Juf^Ai, At),
with respect to the analytic coordinate-systems Oi and 62.
Put now
u)j = dzfdi, rji = dtUi°6z.
Then, we see easily that {wu-■ ■, а>„} is a basis of ®0(-^i) an(l {>7ir"i 4m)
is a basis of ®0(^г); we have obviously,
У=1
This shows that 5 = (s»y) is the transpose of the representation of
dX with respect to the bases {a>j) and {^}.
Let {ult---,u2n} and {»!,•••, «г»} be respectively bases of Z^ and ZJ

22 PRELIMINARIES ON ABELIAN VARIETIES Щ
over Z. Since Л maps Dt into ZJ, there exists а 2»*x2n matrix
Af = (r»y) with coefficients in Z such that
2m
The correspondence i-»Mis uniquely extended to a representation
of ^JC^O{AU A2), which we call the rational representation of J^^A^, A2)
with respect to {щ} and {vi}. We can easily verify that the rational
representation of <jda{A) is equivalent to /-adic representations defined
in §1.2. Let U denote the nx2n matrix whose column-vectors are
"li"-. U2n ar)d Fthemx2m matrix whose column-vectors are vu---, vzm-
We have then
SU = VM,'
so that
where bars denote complex conjugates. Now suppose that A± = Az,
n = m, Di = Dt, в^ = #2,and U = V. Since the matrix if) is inver-
tible, we, see -that the rational representation M is equivalent to the direct
sum of' the analytic representation S and its complex conjugate S.
3." 3. Dual abelian varieties. Let (C"/A в) be an analytic rep-
representation of an abelian variety A. We shall now define the dual
of Cn\D. Let x = (xu) and у = (yu) be two vectors in Cn with the
components х„ and yv. Put
C) <х,у> = Т>(х„у„+х&„).
Then, we see that < x, у ) is a non-degenerate symmetric .R-bilinear
form on Cn with values in R; hence, О is considered as the dual
vector space over R of itself, with respect to this inner product <«•, y).
We have furthermore
D) < л/^Ъс, у > = <*, - V^l
Denote by D* the set of all vectors у е О such that < x, у > is an

[3.3] ANALYTIC THEORY OF ABELIAN VARIETIES 23
integer for every x e D; then D* is a discrete subgroup of Cn,so that
CnjD* is a complex torus of dimension n, which we call the dual of
O/Z). We can show that O/Z)* has a structure of abelian variety,
and consider Cn\D* as an analytic representation of a Picard variety A*
of A in the following manner. Let T be the set of all homomorphisms
of D into the group of complex numbers of absolute value 1. We see
easily that the mapping
О э у _, exp [Ъс VM < у, >]
giv.es an isomorphism of Cn/D* onto T; so we identify Twith Cn/D*
by this isomorphism. Let X be a divisor of -4. Take a normalized
theta function / on Cn(D such that (/) = в{Х). Suppose that X is
algebraically equivalent to 0; we have then E{X) = 0. Since / is
normalized, we can easily verify that / satisfies the formula
f(x+d) = (i{d)f(x) for deD,
where ft is an element of T; it can be seen that ft is determined only
by X. Let ф(Х) denote the point on CnjD* corresponding to ft. Then
we can establish an isomophism в* of A* onto Cn\D* by the relation
0*(C\(X)) = ф(Х).
This implies, as we have said, that CnjD* is considered as an analytic
representation of A*; we call {CnjD*, в*) the dual of (Сп/Д в).
Now we shall consider the mapping <pr of A into A*, defined by D)
of § 1.3. Let Y be a divisor on A and / a normalized theta function
on CnID such that (f) = в(F). Put E = E{ Y); then / satisfies the for-
formula A) of § 3.1 with the form Я given by B). Put, for every и е С",
Фи(х) = ДхУУ(х-и)ехр[2ж <J^\H(u, x)\.
Let t be a point on A corresponding to и by в. We observe that
E) @u)=0(Yt-Y),
F) 0u(x+d) = Фи(х)ехр[2ж V^lE(u, d)} for d <= D.
On the other hand, we obtain an .R-linear mapping Gc of Cn into itself
by the relation
<®(и), vy = E(u, v).

24 PRELIMINARIES ON ABELIAN VARIETIES [I]
By the properties (Rl-3) of Riemann forms and the relation D), we
see that <£ is C-linear and maps D into D*, so that S gives a homo-
morphism of Cn/D into CnjD*. Then the relations E) and F) show
that <£(u) represents the point ф{ Yt — Y) on Cn/D*; namely, we have
In other words, S is the analytic representation of <py with respect to
0 and в*.
Let {tti,---, u2n} be a basis of D over iT; we can find In elements
иi* of Cn such that
< Hi, uj* > = 5y,
where the <5y denote Kronecker's delta; the Uj* form a basis of D*
over iT. Let Eo = (et/) be the matrix of degree 2и which represents
the form E with respect to the basis {щ}; this means
2я 2я
where a and 6 denote respectively the vectors of R2n with the com-
components ai and bi. We have then
this shoXvs that £0 is the rational representation of pr with respect
to the bases {щ} and {u{*}.
Let N be a positive integer and u, v two vectors in Cn such that
Лгм g D, JVb eD; let s and t be the points on A corresponding to и
and v by #. The functions / and Фи being defined for the divisor Y
as above, put
g{x) = 0u(Nx), h(x) = Фи(х)».
We observe that g and h are considered as functions on O/D and
satisfy
(A) = e(N(Y,- Y)\ h(Nx) =
g(x+v) = g(x)exp[2n V=lE(u, Nv)}.
Using the notation of § 1.4, we have

[3. 3] ANALYTIC THEORY OF ABELIAN VARIETIES 25
G) eY,n{U s) = ехр[2тг V^iNE(u, v)].
It follows that, for ever}' rational prime /, Eo coincides with the /-adic
representation of F, defined in §1.4, for a suitable choice of Z-adic
coordinate-system.
Let Ai be another abelian variety defined over С and (Cm/A, Bi)
an analytic representation of At; let At* be a Pjcard variety of Ax and
(O/A*, tfi*) the dual of (Cm/Du Oj). We shall now consider the trans-
transpose *Л of a homomorphism ^ of ^4 into A- Take bases {щ} of Z),
{u<*} of D*, {vt} of А, Ы*} of A*, over Z, such that
< ut, us* > = d(j, < w<, Vj* > = 5O-.
The homomorphism X has an analytic representation Л which is a
linear mapping of C" into Cm and a rational representation M = (с4Д
for which we have
We can obtain a C-linear mapping 'A of Cm into Cn by the relation
< x, lA{y) > = < A(x), у >
for ж G Cn, у е Cro. We shall now show that 'A is the analytic rep-
representation of 4. with respect to в* and в{*, and 'M is the rational
representation of lX with respect to {vt} and {и<*}. То prove this,
take a divisor ЛГ on Аг which is algebraically equivalent to 0 and a
normalized theta function / on Cm/Di such that if) = 6i(X); as is seen
above, / satisfies
d,) = Лу)ехр[2* v^T< du y* >] for & e A.
where y* is an element of Cm which represents Cl(-X"). Put g =f°A;
we have then,
(g) = вB-КХ)),
g(x+d) = g(x)exp[2;r V^lid, 'A(y*) >] for d e A
These relations show that 'Л(у*) is a point on Cn corresponding to the
point СЛ(С\(Х)) = СЦЛ-^Х)) on A*. Thus we have proved that 'A is
the analytic representation of CX with respect to в* and 0ц*; it follows

26 PRELIMINARIES ON ABELIAN VARIETIES [I]
that 'M is the rational representation of 'X with respect to {»<*} and
{".*}• ■
A, A*, (CnID, (?) and (C/D*, в*) being as above, let F be a non-
degenerate divisor on A, and E(x, y) the alternating form defined by
F. Let 1 be an element of <j*fo(A); let E and Л be respectively the
analytic representations of <pv and 1 with respect to в and 0*. We
have then
in other words, pr 'Лрг is the adjoint of 1 with respect to the form
E{x,y). The involution 1 —» ^' = py'Лру of wsfo(A) is thus described
by means of the alternating form.
4. FIELDS £}F MODULI AND KUMMER VARIETIES.
4. 1. Polarization of a" variety. Let F be a complete variety,
non-singular in Co-dimension 1, defined over a field k, and X & divisor
on V which is rational «over k. We denote by L(X; k) the set of
functions^/ on V, defined over k, such that (f) > —X, where (/) de-
denotes tne divisor of the function /. If k' is an extension of k, we
have Lj(X\ k') — L(X; k)®kkf, so that the dimension of the vector
space L(X; k) over k (which is always finite) is independent of k; we
denote this dimension by l(X). Let {/o,---,/r} be a basis of L(X; k)
over k and x a generic point of Foverfe. Consider (fo(x),---,fr(x)) as
a point of the projective space P* of dimension r and denote by U
the locus of ifo(x),---,fr(x)) over k. Then we obtain a rational mapping
Ф of V onto U defined by
Ф(х) = (fo(x),-Jr(x))
with respect to k. We say that the divisor X (or the complete linear
system defined Ъу X) is ample if the mapping Ф is birational and
biregular; this definition does not depend upon the choice of k and
ifth
V, X being as above, we denote by "€{X) the set of all divisors
X' on V for which there exist two positive integers m, m' such that

[4.2] FIELDS OF MODULI AND KUMMER VARIETIES 27
mX is algebraically equivalent to m'X'. The set &(Х) is called a
polarization of F if it contains an ample divisor. A variety V is said
to be polarizable if there exists a polarization of V. We understand
by a polarized variety a couple (F, ^) formed by a variety F and a
polarization ^ of F; every divisor X in ^ is called a £oZar divisor
of (F, ^). & being a field of definition for V, we say that (F, ^)
or ^ is defined over k if ^ contains a divisor which is rational
over k. If F has a structure of abelian variety, we call (F, ^) with
this structure a polarized abelian variety; in this case, (F, <£) is said
to be defined over k only when the structure of abelian variety is also
defined over k. Let a be an isomorphism of k into a field k'; (F, ^)
being defined over k, we denote by 'to' the polarization 'e'(.Y') of V',
where X is a divisor in Ч§ which is rational over k.
Proposition 10. If V is defined over k, every polarized variety
(F, ^) is defined over a finite algebraic extension of k.
Proof. Take a divisor X in 4o. We can find a specialization
X' of X over k, which is rational over a finite algebraic extension of
k. Then, X' is algebraically equivalent to X, so that X' is contained
in 4o. This proves the proposition.
Proposition 11. If V is defined over k and polarizable, then V has
a polarization defined over k.
Proof. By Proposition 10, every polarization contains a divisor
X which is rational over a finite algebraic extension k! of k. Put
Y = pm 2] X', where the sum is taken over all the isomorphisms a of
k' into the algebraic closure of k, and p denotes the characteristic or 1
according as the characteristic is a prime or 0. Then, for a suitable
m, Y is rational over k; and we can easily see that F determines a
polarization of V; our proposition is thereby proved.
We shall now confine ourselves to abelian varieties. A theory for
a more general case can be found in Matsusaka [28].
4. 2. Fields of moduli of polarized abelian varieties. The
following two propositions are due to Weil [56].
Proposition 12. Every abelian variety is polarizable.

28 PRELIMINARIES ON ABELIAN VARIETIES [I]
Proposition 13. Let X be a divisor on an abelian variety. If there
exists an integer n > 0 such that nX is ample, X is non-degenerate. Con-
Conversely, if X is a positive non-degenerate divisor, there exists an integer
и0 > 0 such that nX is ample for n ^ и0-
Let {A, 'rg) and (A', 4g') be two polarized abelian varieties, of
the same dimension. A homomorphism (resp. an isomorphism) X of
A onto A' is called a homomorphism (resp. an isomorphism) of (A, 'To)
onto {A', 4d') if there exists a divisor X' in 4g' such that X~\X') is
contained in 'rg; if that is so, for every Y in 4o', X~\Y) is contained
in ^.
We now give an important theorem which is a base of the de-
definition of field of moduli.
Theorem 2, Let A be an abelian variety and 4g a polarization of
A. Then, there exists a field k0 with the following property:
(M) k and в heing respectively a field of definition for (А, "го) con-
containing kg, and an isomorphism of k into a field, (А, Ч/д) is isomorphic to
(A', &*) if and only if a is the identity on k0.
', »
A proof is given in Shimura [36]. For our later use, we need
only the case where A is defined over an algebraic number field of
finite degree. In this case the field k0 is easily given by Galois theory.
Before snowing this, we give an easy consequence of Theorem 2.
Proposition 14. If the characteristic of the universal domain is 0,
the field k0 with the property (M) of Theorem 2 is uniquely determined
by (A, 4g) and is contained in every field of definition for (A, 4g).
Proof. Let k0 and kg' be two fields with the property (M). Take
a field of definition for (A, 'rg) containing k0 and k0'. Then for every
isomorphism a of k into a field, a is the identity on k0 if and only if
<t is so on k0'; this implies k0 = k0'. Let k^ be a field of definition
for {А, 'ё') and г an isomorphism of &„&! into a field such that r is
the identity on kj,. Then we have {А, *ё) = (A', '€T), so that by the
property (M), т is the identity on k0; this, proves k0 С kv
Now let us consider the case where A is defined over an algebraic

[4. 2] FIELDS OF MODULI AND RUMMER VARIETIES 29
number field k,. of finite degree. By Proposition 10, (A, 4^) is defined
over a finite algebraic extension h! of ki. Take a Galois extension k"
of Q containing k' and call G the Galois group of k" over g. Let
H be the subgroup of G composed of the elements a e G such that
(A', jt>') is isomorphic to (A, 4g\ and k0 the subfield of k" corre-
corresponding to H. Then, it is easy to see that k0 has the property (M).
We call the field k0 with the property (M), which is uniquely
determined by (A, 'tg) if the characteristic is 0, as we have seen in
Proposition 14, the field of moduli of (A, 4g). Obviously, two polarized
abelian varieties, isomorphic to each other, have the same field of
moduli. We can define the field of moduli also in case of positive
characteristics; for details we refer to [28], [36].
If the characteristic is 0, (A, 4o) determines a point z on the
Siegel's space of degree n, where и is the dimension of A. It is
plausible that the field of moduli of (A, *&) is generated over Q by
the values of certain Siegel's modular (or paramodular) functions
(cf. Siegel [38]) at the point z; this is at least true for " generic"
polarized abelian varieties (A, 4g) (cf. [30] exposes 18-20).
Proposition 15. Let ^ be a polarization of an abelian variety.
Then, there exists a divisor Y in 4g such that every divisor in 'To is
algebraically equivalent to a multiple mY with a positive integer m.
Proof. Consider, for each X e 4g, the homomorphism <px of A
into its Picard variety A*. By Proposition 13, every X in Ч§ is non-
degenerate, so that we have v{yx) > 0. Since the v((pz) are integers,
there exists a divisor Y in 4o such that v{(pr) = Шт{ъ>(<рх)\Х е &}.
We shall prove that Y has the property of our proposition. For every
X in '€, by our definition, there exist two positive integers a and b
such that aX is algebraically equivalent to bY; we have then a<px = b<py.
We can find two integers q and r such that b = aq+r, 0<[r<a.
Assume that r > 0; then putting Z = X—qY, <paZ = a<pz = a(pz—aq<py
= r<pr = <ртг- This implies that aZ is algebraically equivalent to r Y,
and hence Z is contained in 4g. By the relations a<pz = r<pr and r < a,
we get v(<pz) < v(<pr); this is a contradiction; so r must be 0. We
have then b = aq, and hence <pz = q<pY- It follows that X is algebra-

30 PRELIMINARIES ON ABELIAN VARIETIES [I]
ically equivalent to qY; this completes the proof.
We call a divisor Y with the property of Proposition 15 a basic
polar divisor of 'rS.
4. 3. Quotient of an abelian variety. We want to prove
Proposition 16. Let A be an abelian variety and G a finite group
of automorphisms of A; let k be a field of definition for A and the ele-
elements of G. Then, there exist a projective variety W and a rational
mapping F of A onto W, both defined over k, satisfying the following
conditions:
(Kl) F is everywhere defined on A;
(K2) F{u) = F(v) if and only if there exists an element jeG such
that и = r(v);
(КЗ) if F' is a rational mapping of A into a variety W satisfying
F' — F'-y for every у e G, then there exists a rational mapping Ф of W
into W such thaf F' = Ф-F and Ф is defined at F{a) whenever F' is
defined at a. f ,
Pkoof. Mere we borrow an idea from Serre [32], where quotient
of a variety, whjch is not, necessarily abelian, is treated in a little dif-
different forrn. We assume that A is a projective variety. This is pos-
possible by virtue of Propositions 11 and 12. For every point a of A, we
can find, a homogeneous polynomial ЩХ) with coefficients in k, other
than the constants, such that H{y{a)) i= 0 for every у е G. Put
B'= {xEi
Then, В is an open set of A, in the sense of Zariski-topology, which
is biregularly equivalent over k to an affine variety. By our choice of
H, В contains the given point a; moreover, we have ?{B) = В for
everj' ;- e G We obtain in this way a finite open covering {Bi} of
A such that each Bt is biregularly equivalent over k to an. affine variety
Vi and r(Bt) = Bi for every у е G. Let k' be an arbitrary extension
of k, x a generic point of A over k', and £t the point on F,- cor-
corresponding to x for each i. Now we fix our attention to one of the
Vt, say F2. Since A is non-singular, so is F2; hence &'[£i] is inte-
integrally closed. As we have y{Bi) = Bu we can define the operation of

[4. 3] FIELDS OF MODULI AND KUMMER VARIETIES 31
the elements of G on the variety F2. Every element;- of G determines
an automorphism of k'(x) = £'(£i)- We denote by W the image of
м е k'(x) by this automorphism. Moreover, since ;- operates on Ft
and &'[£i] is integrally closed, this automorphism induces an automor-
automorphism of A'[£i]. Denote by K, K', R, R' respectively the set of G-
invariant elements of k(x), k'(x), A[£J, A'[£i]. Then, it is easy to see
that К and K' are respectively the quotient fields of R and R', and
we have R' — R^)tk'. Now let (gu) be the coordinates of f1( and ijr
the coefficients of the polynomials Pi(X) = ]~[ (X— £ur). Then we see
т
that Щ,] z> R z> k[y] and A[£t] is integral over k[rj]. As A[^] is finitely
generated as k[y]-mod\i\e, the submodule i? is also finitely generated.
Hence there exists a finite set of elements (d) such that R = k[d];
we have then R' = A'[Ci]- Since &'[&] is integrally closed and
R' — К' П £'[&]> we see that A'[Ci] is integrally closed^ We obtain in
the same manner, for each i, a finite set (&) such that
К = *(«, AKO = /С П A[f«], A'Kd = X' П A'[fi].
Let a be a point of Bt and a the point on V% corresponding to a.
We have obviously
{x-+a; *'] = [?,-»*; A'],
where [м-^w; A'] denotes the specialization-ring of v in k'(u) (cf.
§ 9.2). Let [} be a specialization of Ci over £j —> a ref. A'. We shall
now prove
A) ЛГ'П [&-»«; A'] = [d-*j8; A'].
First we observe that for every j- e G,
B) K' n [fi - «; A'] = *' П [f, - r(«); *']• <
In fact, if и is an element of К' П [fi —»ar; A'], we have an expression
и = /(£i)/g(£i), where / and g are polynomials with coefficients in A'
such that g(a) =jfc 0. As и is contained in K', we have ы = f(£ir)lg(£ir).
Since the £t/ are contained in А[^], there exist polynomials hi with
coefficients in k such that flii' = /г;(^!). Put
Then we have ы = />(£i)/?(£i) and ?(r~'(a)) = g(«0 ¥= 0. Hence м is con-

32 PRELIMINARIES ON ABELIAN VARIETIES [Г]
tained in K' n [£i —> 7~l{a) ; k']; so we have
Я7 П [6 -» *; А'] с К' П [f, - 7-4"); П
The inverse inclusion is similarly proved; so we obtain the equality
B). Now и being an element of К' Л [f i —>«; &'], let rf be a speciali-
specialization of м over Ci —> /S ref. &'. Extend this specialization to a speciali-
specialization
(fi,Ci, «)-(«'. ft <*) ref. k'.
As &'[fi] is integral over £'[&], «' must be finite. Consider the poly-
polynomial
M(T) =
T 1
We observe that the coefficients of M(T) are contained in A[Ci], so
that the - specializations
(fi, CO -»(«, |S) ref...*', Й,С1)-*(«',Я ref.ft'
lead to the equality
where ;-(а^ and- y{a')i denote respectively the /-th coordinates of the
points y(ct) and y(a'). This shows that there exists an element yoiG such
that ?-(«).*= «'■ By the relation B), и is contained in K' n [fi —» a'; A'];
so its specialization <tf must be finite. This implies that every element
of К' П [£i -» or; k'} is integral over [Ci —> /8; fc'J. On the other hand,
as A'fCi] is integrally closed, [Ci —♦ /9; A'] is integrally closed; this
proves the equality A). Now let Wi be the locus of Qi over k. Con-
Consider a specialization
(Ci,-,W —(A,-,ftO ref. A'.
Extend this to a specialization
(ж, £i,-, £л, Ci.-, Сл) -»(a. «I.--, «a, ft,--. /8*) ref. k'.
Since ^'[ft] is integral over k'[Qt] and A'fft] э &'[C»], we see that, for
every i, at is finite if and only if fit is finite. As {St} is a covering
of Д at least one of the at is finite; and for such an i, we have

[4. 3] FIELDS OF MODULI AND KUMMER VARIETIES 33
[x-*e;ft'] = [fi-*e«;*n,
К' П [ft -» at; k'] = [Qt -> fit;, k'],
and hence
C) X'n[x-*e;*/] = Ki-jSi;ft'].
Therefore the specialization-ring [Ct —► /Si; &] does not depend on the
choice of i, so far as fit is finite. It follows that the affine varieties Wt
determine an abstract variety TV if we regard the & as corresponding
generic points with respect to k. Denote by z the point on TV whose
representatives are the Qu and by F the rational mapping of A onto
W defined by F(x) = z with respect to k. The equality C) is then
written in the form
D) Kf П [x - a; k'] = [z — F(a); *'].
By our construction, .F is everywhere defined on A-and F°y = F for
every ;- e G. We shall now prove that (W, F) satisfies the condition
(КЗ). (W, F') being as in (КЗ), let k' be a field of definition for W
and F', containing k; let x be a generic point on ^4 over k'. Put
F(x) = z, F'(x) = z'. Since F'°r = F' for every rsG, *'(*') is con-
contained in k'(z). Hence we obtain a rational mapping Ф of TV into И7'
defined by Ф(г) = г' with respect to k'; we have then F' = 0°F. Now
suppose that F' is defined at a point a e A. Then we have
[*-.e;ft']:>[*'->.P(e);*'].
By the equality D), we have
[z^F(a);k']Z>[z'^F'(a);k'];
this proves that Ф is defined at F{a). Therefore (W, F) satisfies (КЗ).
We have thus constructed an abstract variety TV and a rational map-
mapping F, both denned over k, satisfying the conditions (Kl), (КЗ) and
the " if" part of (K2). It remains to prove the " only if" part of
(K2) and realize W as a projective variety. For this purpose, take a
generic point x on A over k and consider the Chow point у of the
O-dimensional cycle 2 (т(хУ) on -^- Let TVt be the locus of у over k
and TV0 a projective normalization of Wx with respect to k. Then we
obtain a rational mapping /, of A onto Wt defined by fi(x) = у with

34 PRELIMINARIES ON ABELIAN VARIETIES Щ
respect to k and a birational mapping / of Wo onto Wx defined by
the normalization. Put /o =/~1°/i, t=fo(x). We have obviously
k(i) = К = k(z). Hence there exists a birational mapping W of W onto
Wo, defined over k, such that W{z) — t. Let (a, b) be a specialization
of (x, t) over k. As Wo is a normalization of PFt with respect to k,
[t —> b; A] is integrally closed. Using this fact, we can prove the
equality
Kr\ [x->a;k] = [t-»A;A],
in the same way as in the proof of A). Hence we have
It follows that IVO and W are biregularly equivalent over k. Further-
Furthermore, it is easy to see that (Wufi) and (W0,f0) satisfy (K2). This com-
completes our proof.
The couple (W, F) is uniquely determined by the conditions
(Kl-3) up to biregular ,biratipnal mappings. We call (W, F) a quotient
of A by G, defined over k.
Remark 1. Notations and assumptions being as in (КЗ), let k' be
a field of definition for W and F', containing k. Then the rational
mapping Ф is defined over k'. This is included in the above proof.
4. 4. Kummer varieties. By Weil [54], Matsusaka [28] we
know
Proposition 17. Every polarized abelian variety has only a finite
number of automorphisms.
Here we reproduce the proof of [54]. ^ being a polarization of an
abelian variety A, take a divisor Xin *& and consider the involution
«—»«' = (px'1 la<px of <j/0(A). For every automorphism a of (A, 'to),
there exist two positive integers m гид. т' such that
m la(px<x — m'<px
on account of the relation G) of § 1.3. Taking the degree of both sides,
we get m = m', so that aa' = 1д. Thus we obtain tr(aa') = 2n for
every automorphism a of (A, &), where n is the dimension of A.

[4. 4] FIELDS OF MODULI AND KUMMER VARIETIES 35
Since tr(aa') is a positive non-degenerate quadratic form and wtf(A)
is finite]}' generated over Z, only a finite number of such a can exist.
(A, 'rg) being a polarized abelian variety, let G be the group of
automorphisms of (A, ^f); Proposition 17 asserts that G is finite.
By a Kummer variety of (A, 4f), we understand a quotient of A by G.
Theorem 3. Let {A, &) be a polarized abelian variety and k0 the
field of moduli of (A, <g). Then there exists a Kummer variety (W, F)
of (A, &) satisfying the following conditions:
. (N1) W is defined over kn;
(N2) F is defined over every field of definition for (A, "rg) con-
containing k0;
(N3) k being a field of definition for (A, So) containing k0, if a is
an isomorphism of k into a field and if rj is an isomorphism of (A, <&)
onto {A", &'), then we have F = F»°)?.
We note that if {A, So) is isomorphic to (A', Sg'), then a is the
identity on k0 by virtue of the property (M) of Theorem 2, so that
W = W by the property (N1).
We shall call a Kummer variety (W, F) of (A, *€) satisfying the
conditions (N1-3) a normalized Kummer variety of (A, '&), which is
uniquely determined for {A, &) up to biregular birational mappings de-
defined over k0, if A is defined over a separably generated extension of k0.
We give a proof of Theorem 3 only in case where (Д '&) is
defined over an algebraic number field; the same method is applicable
to the general case; as for this, see Remark 2 at the end of the proof.
Let G be the group of automorphisms of {A, '&) and k a field
of definition for (A, *&) and the elements of G; we assume that k
is a finite Galois extension of Q; this is possible by our restriction.
Then, as is seen above, there exists a Kummer variety (TV, F) defined
over k. Let © denote the Galois group of k over k0. For every
ff e ®, by virtue of the property (M) of k0, there exists an isomorphism
a. of (A, &) onto {A', <ig'). We see that G' = {fir ^ G> is the
group of automorphisms of (A', ¥>') and (W', F') is a Kummer
variety of (A', 'if'). Then F°°a, is a rational mapping of A onto
W' which is everywhere defined on A; and for every у е G, we have

36 PRELIMINARIES ON ABELIAN VARIETIES [Q
(F'oa,)oy = F°°a,; in fact, as a. is an isomorphism of (A, 'if) onto
(A', 'if'), there exists an element f e G° such that a,°-[ = f°ac, so
that F"°a,<>Y = F'«y'«a. — F'°a,. Hence, by (КЗ), there exists a ra-
rational mapping p. of W onto И* such that F°°a. = /S,,°.F; and /5„ is
everywhere defined on W. We obtain similarly a rational mapping
/S/ of W' onto Й7, everywhere defined on W', such that j5/»F' = Foa,.
It is easy to see that ft, is birational and biregular, and B/ = /S,.
The birational mapping /S. is uniquely determined by <r and does not
depend upon the choice of «„; in fact, for any other isomorphism a/
of (A, 'if) on (A't 'if'), there exists an element 7- e G such that
or/ = «„07-; it follows that F"°a/ = F«»a,;'this shows the uniqueness
of p.. We can easily verify that f}, is defined over k. Now a and г
being two elements of ©, we have /S.'-i5"' = F"<-a.r and /Sr°F= Fr»ar
so that {icr°Pr°F =.F"°a.r°ar. By the uniqueness, we have #,'<>& = £„.
Put /r>, = /Sr°/S, for every о- е @, г е ®. Then we observe that the
relations
hold for-teyery p, а, т ^ ®. Hence, applying the results of Weil [55]
to the present case, we obtain a variety Wo and a birational biregular
mapping <p of Wo onto Й7 such that: i) Wo is defined over k0; ii) <p is
defined over A; iii)/r>» = (p'^ifp')'1 for every <r, г e ©, Put Fo = (p~l°F;
then (Й7О> ^o) is clearly a Kummer variety pf (A, ^") and Fo is de-
defined over k. We shall now show that (Wo, Fo) satisfies the conditions
(N1-3). (N1) is just the property i). Let k' be a field of definition
for {A, '&); then k' contains Ao by virtue of Proposition 14. Let a
be an isomorphism of kk' onto a field which fixes every element of
kn; denote by the same letter a the element of © induced by a.
Then, for every isomorphism a, of {A, *to) onto (A', 'rS'), we have
Fo'oa, = (yy^F'ea,, = {y)-1°p,°F= 9J-!.F= Fo;
This shows that Fo satisfies (N3). In particular, if a is the iden-
identity on k', we have {A', 'if') = (A, &), so that we can take a. = 1A;
we have hence iV = Fo for every isomorphism a of A&' over A'" It
follows that Fo is defined over k'; so Fo satisfies (N2).

[4.4] FIELDS OF MODULI AND KUMMER VARIETIES 37
Remark 2. We can prove Theorem 3 without any condition for
the fields of definition, using also the theory of [55]. In order to
perform this, it is necessary to avoid inseparable field-extensions,
since [55] deals with only separable or regular extensions. This is
certainly possible by virtue of Proposition 4 of [36].
Remark 3. We can take'the variety W in Theorem 3 to be a
projective variety; this is proved by means of Proposition 16 and [55].

CHAPTER II. ABELIAN VARIETIES WITH •
COMPLEX MULTIPLICATION.
5. STRUCTURE OF ^fo(A)»
5. 1. Let 31 be a simple algebra over Q and 3 the center of 31;
put [91:3] = У8, lS-Q] = d. Then, 91 has d inequivalent irreducible
representations in the algebraic closure of Q, which are all of degree
/. We call a representation 5 of 3i in an extension of Q a reduced
representation of 31 if 5 is equivalent to the direct sum of those d ir-
irreducible representations. S being a reduced representation of 31, the
characteristic polynomial of S(a) has rational coefficients for every
a e 91. Put
N(a) = det S(a), Tr(a) = tr S(a)
for абЛ. We call N(a) and Tr(a) the reduced norm and the reduced
trace of a; these are independent of the choice of 5.
1. Let 31 be a simple algebra over Q and S a representation
of 3i in/an extension of Q. Suppose that for every a e 9t, the charac-
characteristic polynomial of S(a) has rational coefficients. Then S is equiva-
equivalent to the sum of a multiple of a reduced representation of 5ft and a
^-representation.
Proof. Let the Su for 1 5J i ^ d, denote the inequivalent irredu-
irreducible representations of 9t in the algebraic closure L of Q. Then 5
is equivalent to the direct sum of representations rritSi and a 0-rep-
resentation, where the mt denote the multiplicities. Let the 04, for
1 ^ 1: :£ d, be all the isomorphisms of the center 3 °f 91 into L. Then
we observe, after reordering, that St(a) is the diagonal matrix «"«1/
1) Some of the results of §§ 5-6 have their source in Lefschetz [27]. The
reader will also find a more general structure-theory of ^jnf^A) in Albert [1, 2].
[38]

[5.1] STRUCTURE OF ^f^A) 39
for a e 3. f°r every i. Hence, for every a e 3> the characteristic
polynomial of S(a) is of the form Xh\\ (X—a'i)m\ Our assumption
i
implies that the ntt are the same; this proves our lemma.
Proposition 1. Let A be an abelian variety of dimension n and @
a commutative semi-simple subalgebra of wtfo(A). Then we have
If [©:£?]= 2n, then the commutor of © in ^(fo(A) coincides with ©.
■ Proof. Let the Ki denote the simple components of ©; then, as
<S is commutative, the Kt are fields. Put [Kt: Q] = dt. Let Si be a
reduced representation of Kt. Take a prime I other than the charac-
characteristic of the fields of definition for A, and consider an /-adic rep-
representation Mi of cj*fo(A). By Lemma 1, the restriction of Mi to Kt is
equivalent to the direct sum of a multiple mtSi of 5£ and a 0-repre-
sentation. Considering Mi on ©, we see that Mt is equivalent to the
direct sum of the mtSi and a O-representation. As Mi is faithful, every
mt must be positive. Hence we have 2я 2; 2 от^ ^ 2 <^i = [®: £?]•
This proves the first assertion. Now suppose that 2n = [©: Q]. Let
©' be the commutor of © in ud<>{A). We can find a matrix P with
coefficients in the algebraic closure of Qi such that PMiffiP'1 is a
diagonal matrix for every fee. As we have [@: Q] = 2я, there exists
an element a of © such that the diagonal elements of PMi{a)P~l are
distinct. For every t] e ©', PMi{rj)P-* commutes with PM^P'1, so
that PMi(rj)P~i is a diagonal matrix for every rj e ©'. This shows that
©' is a commutative semi-simple algebra. Then, applying to ©' what
we have just proved fcr S, we get [S': Q] ^2n; so we must have
@ = ©'; this completes the proof.
Proposition 2. Let A be an abelian variety of dimension n; let 91
be a simple subalgebra of wtfo(A) and 3 the center of 5ft; and put
[Ю:3]=Л 13 •■Qi = d.
Suppose that 31 contains the identity element of ^o(A). Then fd divides
2n; and putting 2n = fdm, we have, for every a e 31,

40 COMPLEX MULTIPLICATION [II]
v(a) = N(a)», tr(a) = m Tr(a),
wfiere N(a) and Tr(a) denote the reduced norm and trace of a e SR.
Proof. Let S be a reduced representation of 8t. Take a prime 7
other than the characteristic and an /-adic representation Mi of <jaf 0(/4).
By Lemma 1, the restriction of Mi to 31 is equivalent to a multiple
пг£ of 5; so we have 2и = fdm; and the characteristic polynomial of
Mi(a) is the wz-th power of that of S(a). This proves the proposition.
Proposition 3. Let A be an abelian variety of dimension n. If
tjdb(A) contains a field F of degree In over Q, then A is isogenous to a
product Bx---xB with a simple abelian variety В; the commuter of F in
<jtfo{-A) coincides with F; and, for every a e F, we have
v(a) = NF/Q(a), tr(a) = TrF/Q(a).
Proof. By the results of n°55-6 of Weil [46], there exist simple
abelian varietje$ Alt---,AS such that A is isogenous to the product
(A1x--xA1)x--x(Atx---xA,)
and the At are not isogenous to each other. Let hi be the number
of the factor At occuring in the product, for each i. Then, denoting
by Sli^the total matric ring of degree hi over ^fo(Ai), ^„(AiX--- xAt)
is identified with 314, and и^ъ{А) is identified with the direct sum of
the ЭD. Let е4 denote the identity element of 5П4 for each i. Fet is
not {0} for at least one of the eb say ev Then, Fet is isomorphic to
F, since F is a field. We see that Fet is a • semi-simple commutative
subalgebra of «^г/0(^1 X • • X -^i) ; hence, by Proposition 1, we have
[F: Q] g2/i,-dim(^1). By the assumption [F: Q] = 2n, we must have
s— 1 and A is isogenous to ^1x---X.41. This proves the first asser-
assertion of our proposition. The second assertion follows from Proposition
1 and the last from Proposition 2.
Proposition 4. The notations А, В and F being as in Proposition
3, let m be the dimension of В and h the number of the factor В in the
product Bx--xB which is isogenous to A. Let К be the center of <jtfo(B).
Then, К is a subfieM of F; and, if we put [K: Q] =/, МУ-В): Щ = g\
we have In = fgh, 2m = fg.

[5.1] STRUCTURE OF *j*fD(A) 41
Proof. We first note that ^fo(B) is a division algebra, since В is
simple. wf<^A) is identified with the total matric ring of degree h
over uxf^B); hence К is the center of <jdu{A) and is contained in F
by virtue of Proposition 3. We observe that <j>fo(A) is a central simple
algebra over К and [^„(A): K] = g1}?. By a well-known theorem of
ring theory, [F: K] = Injf divides gh ; and if we put gh — [F: K]q, the
commuter of F in ^n{A) is an algebra of degree cf over F. By
Proposition 3, we must have q = 1, so that [F: K] = gh = 2njf and
hence 2m = fg.
Lemma 2. .Let X" 6e an algebraic number field of finite degree; let
p be an automorphism of К such that p* = 1 and Ko the subfield of К
consisting of all the elements of К fixed by p. Suppose that
TrK,Q(№) > 0
for every element f ф 0 in K. Then Ko is a totally real field. If p is
not the identity on К, К is a totally imaginary field, and, for every iso-
isomorphism т of К into С, £'г is the complex conjugate of £*.
Proof. We have Tnco/g(£2) > 0 for every £ ^ 0 in Ko. Let гь--, тт
denote all the isomorphisms of Ko into C. Assume that rt is not
real; then the complex conjugate of t1 coincides with one of the iso-
isomorphisms tv ■•,*•„, say r2. We see that Ko*> is dense in C, so that
there exists an element rt in Ko such that Re({7f)'i) < — 1. By the
approximation theorem, for any small positive number e, we can find
an element f in Ko such that
|f'i-7'i| = \?r2-rf*\ < e, |£'i| < e B < i ^ m).
If we take a sufficiently small e, we have
><) < -2.
This is a contradiction; hence Ko must be totally real. Now suppose
that p is not the identity. Then we have [K: Ko] = 2, so that there
exists an element Q of К such that К = K0(Q, C2 e /Co, С = — С-
For every element a ^ 0 of .Ko. we have

42 COMPLEX MULTIPLICATION [И]
Again by the approximation theorem, we can find, for each i, an ele-
element a of Ko such that
\ач\ > 1к\ач\ <е 0>f)
for a sufficiently small positive number e; we have then (a2t?L < 0;
this shows that — (C2)'* is positive. Hence — C2 is totally positive, so
that К = K0(C) is totally imaginary; and for every isomorphism r of
К into C, Cr is a purely imaginary number. Therefore, we have, for
every a, p e Ko,
where z denotes the complex conjugate of г; this completes our proof.
Proposition 5. Let В be a simple abelian variety and К the center
of u>fo(B)- Then К is a totally real number field or a totally imaginary
quadratic extension of a totally real number field.
Proof. In § 1.3, we have noted that ^0(В) has an involution £ —> £'
with the property tr(ff') > 0 for every £ j= 0 in ^fu(B). As К is the
center of oasfo(-B), the involution maps .KTonto itself; and by Proposition
2, we have tr(?) = mTrit/g(£) for £ e X", for a suitable positive integer
m. He^ice we have Trjc/g(££') > 0 for every q =j= 0 in if. Our prop-
proposition is then an immediate consequence of Lemma 2.
P*roposition 6. Notations being as in Proposition 4, suppose that the
characteristic of the fields of definition for A is zero. Then, we have
g = 1 and ^fo(B) = K.
Proof. If the characteristic is 0, we can consider a rational rep-
representation of _л^0(В), defined in § 3.2, which is of degree 2m, m being
the dimension of B. As ^fn{B) is a division algebra, the degree of
any rational representation of wtfo(B) is divisible by [^0(В): Q] =fg2.
Hence, by the equality 2m = fg, we must have g = 1 ; this implies
uSo(B) = K.
5. 2. CM-type. Let 3? be an algebra over Q, with an identity
element 1. We understand by an abelian variety of type (91) a couple
(A,, i) formed by an abelian variety A and an isomorphism с of 31

[5.2] " — STRUCTURE OF ^fo(A) 43
into ^0(А) such that i{\) — 1A- When there is no fear of confusion,
we write (A, t) simply by A and identify an element a of 31 with i{a).
Let F be an algebraic number field, {A, c) an abelian variety of
type (F) and я the dimension of A. By Proposition 2, [F: Q] divides 2я.
We shall now investigate the structure of (A, e) for which [F: Q] = 2n
holds, in case where the characteristic is 0. If the characteristic is 0,
A is isomorphic to a complex torus, and we obtain a rational represen-
representation M and an analytic representation S of ^0(A), with respect to
an analytic coordinate-system; M is of degree 2n, and S is of degree
п.; М is equivalent to the direct sum of 5 and the complex conjugate
S of S (cf. §3.2). Let <plt—, (ргп be all the isomorphisms of F into C.
Then, by Lemma 1, the representation M restricted to F is equivalent
to the direct sum of the <pi; hence 5 is equivalent to the direct sum
of a half of 2n isomorphisms <pi, say <pi,---, <pn. Then 5 is equivalent
to the direct sum of y>n+i>"'> <Ргп, which is equal, as a whole, to pi,---, <рп~
Therefore, we observe that there are no two isomorphisms among
<Pi,--><Pn which are complex conjugate of each other. Moreover, we
see that F must be totally imaginary, {pi,---, <pn) being thus determined*
we say that (A, c) is of type {F; {<pi,---, <Pn})- Recalling that S is equiv-
equivalent to the representation of ^jd^A) by invariant differential forms,
we can find я invariant differential forms o>lt---,a>n of degree 1 on A
such that, for every a e F,
Bda)<Oi = afitOi A g i ^ и).
Conversely, if there exist such ши {А, с) is of type (F; {<pi}); and the
Юг form a basis of ®0(^)- We shall often use these facts afterwards.
Now let us consider the center К of ^я£п(А), which is also the
center of <_я#й(В), where Б is a simple abelian variety determined as
in Proposition 3. Proposition 5 asserts that К is totally real or a
totally imaginary quadratic extension of a totally real field. By Prop-
Propositions 4 and 6, we have [K: Q] = 2-dim(B); so we can apply to
В and К what we have proved for A and F; then we see that К
must be totally imaginary. Let S' be an analytic representation of
u>tfo(B)- Then, as A is isogenous to the product of h copies of B,
the restriction of S to К is equivalent to h times of Sr. Hence the

44 COMPLEX MULTIPLICATION [ГЦ
restriction of <pu-■■, tpn to К yields exactly h times of//2 isomorphisms
<J>i,-■•,</>/p. of К into C, where/= [iff: 2]; an(i S' is equivalent to the
direct sum of the фу; there are no two isomorphisms among <pj which
are complex conjugate of each other.
In general, for an algebraic number field F of degree 2n and n
distinct isomorphisms <рг of F into C, we say that (F; {<pi,-",<Pn}) is
a CM-type if there exists an abelian variety of dimension n of type
(F; {<pi}). The above discussion gives us a necessary condition for a
CM-type.
Theorem 1. In order that (F; {wt}) is a CM-type, it is necessary'
and sufficient that F contains two subfields К and Ko satisfying the follow-
following conditions.
(CM1) Ko is totally real and К is a totally imaginary quadratic
extension of Kq.
(CM2) There are no two isomorphisms among the у>4 which are com-
complex conjugate of each other pn K.
The sufficiency will Be proved in the following section (Theorem 3).
t
b3 CONSTRUCTION OF ABELIAN VARIETIES
WITH COMPLEX MULTIPLICATION.
6. 1. Analytic structure of an abelian variety of type (F; {u>t}).
{A, 0 being of type (F; {<р(}), put
r = rn^f(A) n t(F)];
then r is a subring of F which is finitely generated over Z and F = Qz;
hence r is a free 2-module of rank 2n, where n = dim A. Take an
analytic coordinate-system (Cn/D, в) of A and denote by 5 the analytic
representation of c_&fo(A) with respect to в. We see that S(c(a)) is non-
singular for every a i= 0 in F. As we have S(e(a))D с D for every a
in r, D is considered as an r-module. Choose a vector x0 ^= 0 in D
and put D' = S(:(t))xo. Then the mapping a —»■ S(c(a))x0 is an r-isomor-
phism of r onto D'. Hence D' is of rank 2n, so that there exists a
positive integer g such that gD С D'. Fixing such a number g, we

[6.1] CONSTRUCTION OF ABELIAN VARIETIES 45
obtain an r-isomorphism x —»fi of D into r by means of the relation
gx = S(c(/i))x0.
Let m denote the image of this isomorphism; m is then an ideal of
т. Put X! — g'^Xt,; we have then
D = S(t{m))xv
We observe that S(c{a))D с D if and only if am с m; hence r consists
of all the elements a e F such that am с m, namely т is the " order "
of the module m. Now, for a suitable choice of coordinate-system,
S(e(a)) is the diagonal matrix with the diagonal elements or",---, a?».
Let (bu---,bn) be the components of the vector xt with respect to this
coordinate-system. Then D is the set of all vectors with the com-
components (ftnblt...,/tr*bn) for fi e m. Assume that we have b{ = 0 for
some i; then D is contained in a proper subspace of C"; this is a
contradiction, since D is a discrete subgroup of C" of rank 2n. There-
Therefore, we must have bt ф 0 for every i. Change the coordinate-system
(Zi,--,zn) for (b1~lz1,---,bn~1Zn); we see that S(c(a)) is expressed again
by the same diagonal matrix as before with respect to this new system;
and D is the set of all vectors with the components (ftri,---,/jr*) for
ji e= m. We have thus proved
Theorem 2. Let F be an algebraic number field of degree In and the
<Pi, for 1 <: i: ^ п, я distinct isomorphisms of F into C; denote by u(a)
the vector in Cn with the components (ar*V--, a?») for a e. F, by D(m)
the set of all vectors u(a) for a in a free Z-module m in F of rank 2n,
and by S(a) the diagonal matrix with the diagonal elements an,--ta?n.
If (A, i) is an abelian variety of type (F; {<pi}), then there exist a module
m and an isomorphism в of A onto C"/Z)(m) 63; which c(a) corresponds to
the linear transformation of Cn given by S(a) for every a e F; and if
we denote by r the set of all elements a in F such that am. с m, we have
с(т) = C(F) п Л{А).
Corollary. Any two abelian varieties of the same CM-type are
isogenous to each other.
Proof. If m, m' are two free Z-submodules of F of rank 2n, then

46 COMPLEX MULTIPLICATION [11]
there exists a positive integer g such that gm с m'; we have then
gD(m) с -D(m'). Hence x —► gx gives a homomorphism of CnjD(m) onto
CnjD{m'); this proves the assertion.
Remark. The homomorphism x-+gx commutes obviously with
the operation of F.
6. 2. Construction. We shall now prove that the existence of
the fields К and Ko satisfying (CM 1,2) is sufficient for (F; {р4}) to
be a CM-type.
Theorem 3. The notations F, <pi, D(m), S(a) being the same as in
Theorem 2, suppose that F contains two subfields K, Ko satisfying the
conditions (CM 1,2) of Theorem 1. Then, for every free Z-submodule m
of F of rank In, CnjD(m) is isomorphic to an abelian variety A, and, for
every ceF, the linear transformation of Cn given by S(a) corresponds to
an element of wtfo{'A); if we denote this element by :(a), (A, c) is of type
(F; {<pi}). Morebver, if F does not coincide with K, A is not simple.
Proof. Take a basis {cci,---,ain} of m over Z; then 2n vectors
"(аД-•-, u(ain) form a basis of -D(trt) over Z. In order that D(m) is
discrete jn Cn, it is sufficient that the u(cti) are linearly independent
over R,\ this is equivalent to that the matrix of degree 2я with the
columns (а?1,---, «iTn, a%^,---, a^n), for l^ig 2я, is non-singular. The
latter is the case, since the at form a basis of .F over Q and 2n iso-
isomorphisms <piy ipi give all the isomorphisms of F into C. Hence
CnjD{m) is a complex torus. We shall now prove that C"/D(m) has a
structure of abelian variety. It is sufficient to show this for a certain
m, because, for every two free Z-submodules m and m' of F, -D(m) and
D(m') are commensurable to each other. Put [Ko: Q] — m, [F': K] = h;
we have then n = mh, [K: Q] = 2m. Let n be a free Z-submodule
of К of rank 2m and (j-i,-, д} a basis of F over K; put
А-, = {u{fi) I p e v-Ti}.
Then D(m) is the sum of Au---,Ah, and each At has 2m generators

[6.2] CONSTRUCTION OF ABELIAN VARIETIES 47
which are linearly independent over R. Denote by Vi the subspace
of C" generated over R by these 2m vectors. Then it is easy to see
that CnjD(m) is isomorphic to the direct product of the Vi\At as real
analytic manifold. We shall now show that the Vi are complex
vector subspaces of C". Let ф1,---,фк be the distinct isomorphisms
of К into С induced by the y>4. By virtue of the conditions (CM1.2),
we see that k = m and the <pi are all the isomorphisms of jF into С
inducing the cf>j on K. Denote by v(fi), for (SeK, the vector of Cm
whose components are /S*1,---, /3<*т, and by A the set of all vectors v{§)
for £ e ii. Then, applying to A what we have proved above, we see
that A is a discrete subgroup of Cm and CmjA is a complex torus. If
{&,•■•, ргт} is a basis of n over Z, 2m vectors w(/St) give a basis of Cm
over R. The linear mapping x—> V— \x of Cm onto itself, regarded
as an i?-linear mapping, determines a matrix (ctt) of degree 2wi with
real coefficients with respect to the basis
A) V=l /ЗЛ = S с«0Л A ^ j ^ от, Utg 2m).
j=i
Now we see that the vectors (TinPtn,..., 7ir*Ptr»), for 1 ^ t gj 2m, form
a basis of Fj over /?. As every $о4 induces one of the ф] on i?, we
find, multiplying A) by jft,
B) V-l г^'^ч = S с„гг'ф,ч A ^ г" ^ п, 1 ^ t ^ 2m).
This implies V— 1 Fa С Fj for every Л, so that Vx is a complex sub-
space of C" of dimension m; and it can be easily verified that C"/Z)(m)
is isomorphic, as complex manifold, to the direct product of the complex
tori VifAi. Moreover, if we correspond w(/S() to u(ji(ii), we obtain an
/?-isomorphism уг of Cm onto Vi; we see easily 7]i{A) = Ai- By the
relations A) and B), we have V—1 r)i = ^ V—1, so that rjx gives a
complex analytic isomorphism of Ст/Л onto Vi\Ai. It follows that
CB/£)(m) is complex analytically isomorphic to the direct product of
h copies of CmjA. Hence, if we show that CmjA has a structure of
abelian variety, CnjD(m) has also a structure of abelian variety. We
prove this in constructing a non-degenerate Riemann form on СШ\А.
By the condition (CM1), there exists an element C, in К such that

48 COMPLEX MULTIPLICATION [II]
К = K0(Q) and CJ e Ko; as К is totally imaginary, — C2 must be totally
positive. We can take £ in such a way that
Im(C^) > 0 A &j ^ m).
In fact, if this is not so, we choose an element a of K, such that
c?Am(£fj) > 0 for 1 ^j^m, and adopt aC in place of Q. Now, г,
и> being two vectors of Cm with the components («i,---, гто) and
(k>i,---, гото), we define an /?-bilinear form E(z, го) on Cm by
£■(«, го) =
We see easily £B, го) = —E(w, z), and
E(z, V-l w) = - V-l Z Q*]{zflOi+SflBj).
}=1
Hence E(z, V—Л го) is я symmetric form and is positive non-degenerate
since the £«V are purely imaginary and we have Im(Q?j) > 0. Denoting
by p the automorphism of K^ over Ko, other than the identity, we
have С = — С and £'*i = f^t for every £ in К (cf. Lemma 2 of § 5. 1
and its proof). Ёу means of these relations, we have, for every a, $
in K, '
i ~ ■ E{v{a), vtf)) = TrK/Q&ap).
We can find a positive integer g such that all elements of g£mv" are
algebraic integers; then the values of gE(z, го) on Jxd are integers.
Thus we obtain a non-degenerate Riemann form gE(z, w) on CmjA.
This proves that CnjD(m) has a structure of abelian variety. The rest
of our theorem is almost obvious.
Notations being as above, denote by T(£) for £ e К the diagonal
matrix with the diagonal elements f*1,---, £<*»>. Then, by virtue of the
relation £кч = $?i, we have
E(z, Ttfyzo) = E(T(e>)z,v>)
for every £ e K. Thus we have proved the first part of the follow-
following theorem.
Theorem 4. Let Ko be a totally real field of degree m, К a totally
imaginary quadratic extension of Ko and p the automorphism of К over

[6.2] CONSTRUCTION OF ABELIAN VARIETIES 49
Ko other than the identity; let (K; {^t}) be a CM-type and it a free
Z-submodule of К of rank 2m. Denote by w(/S) for /S e К the vector of
Cm with the components fl*1,---, fitm, by T(j3) the diagonal matrix with the
diagonal elements ^3*V"»jS*»> and by D(n) the set of all vectors w(/S) for
/Sen. Let Q be a number of К such that —C2 is a totally positive ele-
element ofK0 and Im(Ci) > 0 for every i. Put, for two vectors z — (zi,--, zm)
and w = (»!,-■•, zvm) of Cm,
m
E(z, w) = 2 Q^Zi'uii—ZiWi).
«=1
Then, for a suitable positive integer g, the form gE is a non-degenerate
Riemann form on CmjD{v); and we have
C) E(
for every t ift K. Conversely, every non-degenerate Riemann form on
Cm/D(n) satisfying C) is obtained from an element Q of К in this man-
manner. If CmjD{VL) is simple, every Riemann form on CmID(n), other than
0, is non-degenerate and satisfies the relation C).
Proof. We have only to prove the second and the last assertions ;
we first prove the last. As is remarked in § 1.3 and § 3.3, for every
non-degenerate Riemann form E(z, w), we obtain an involution Л —> Л'
of j/,(Cm/Z)(n)) by the relation
E(z, Aw) = E(A'z, w).
If Cm/D(n) is simple, every Riemann form on it, other than 0, is
non-degenerate, and ^0{CmjD(u)) coincides with T(K) by virtue of
Proposition 6; so the involution corresponds to an automorphism г
of K; we have namely Щ)' = T(p); and we have Тгк,е(££г) > О for
every $ =± 0 in K. By Lemma 2, if we denote by Kt the subfield of
К consisting of the elements fixed by r, X is totally real and
[K: Kx] = 2. As Ko and Kt are totally real, we must have Ko = Kx,
and hence т = p; so E satisfies C). Now we prove the second asser-
assertion. If E is a Riemann form, the mapping $ -* E(v($), v(l)) is a Q-
linear mapping of К into Q, so that there exists an element Q of К
such that E(v(£), w(l)) = Ttk/q(Z£) for every $ e K. Suppose that E

50 COMPLEX MULTIPLICATION [II]
satisfies C). We have then
Since E is alternating, we have
this implies С = —С- Hence —С2 = CC' is contained in Ко and
K = Ko(C). As К is totally imaginary, — £2 must be totally positive.
By the same argument as in the proof of Lemma 2, we find £'** = £**
for every £ e /sT, so that
Since the vectors «(£) for. f eif form a dense subset of Cm, we have
i E{z, w) = £ &t(z<Wi—z<ub)
on CmxCm. The inequality Imft?*) > 0 follows from the fact that
E(z, V—1 г) = —2 V^l 2 C**[^i|2 is a positive form. This completes
our proof;'
We^can give another expression for the form E on D(n)xD(n).
Let {ух, Yh.) be a basis of К over Ко'. Put
then we have 7' = rj, so that 51 is an element of Ко- For au a2,
/S2 e Ко, we have
Hence we obtain
6. 3. Picard variety. Notations being as in Theorem 4, let A
be an abelian variety isomorphic to CmjD{n); we fix an isomorphism
в of A onto Cm/D(n) and denote by c(a) for a e К the element of
.jafo(^) corresponding to Г(а); then (A, c) is of type (К; {ф,}). We

[6.3] CONSTRUCTION OF ABELIAN VARIETIES 51
shall now consider a Picard variety A* of A. Recall the form < z, w >
on Cm introduced in § 3.3. We have
m
< z, w > = 2 (z,w,+z.u\)
v=l
for any two vectors z, to with the coordinates zp го„ so that for every
D) < t<«, v(V) > =
Let (CmjD*, в*) be the dual of (O/D(n), 0), defined in § 3.3, which
is" an analytic representation of A*; D* is the set of vectors z such
that < z, a> > e Z for every w e £>(n); we see then easily that every
vector in D* is of the form «(£) for f e£ Hence if we denote by
n* the set of elements feK such that
we find D* = Z>(n*) on account of D). Thus the dual of 0/.0(п) is
given by CmID(n*). By the relation D), we see
< T{a)v^\ v{V) > = < t<«, 7VM?) >.
Since the vectors u(f) for £ e i£ form a dense subset in Cm, we have
< Г(а)г, w> = <z, T(a')zv >
for every a e K. This shows that T(a') is the analytic representation
of the element *с(а) of ^sf9(A*) with respect to в*. Put, for a e K,
E) <*(«) = '<«')•
Then we see that c* is an isomorphism of К into ол/о(-^*); and c*{a)
is represented by Г(а). It follows that (A*, c*) is of type (K; {(/>,})■
Now let E(z, to) be the Riemann form on Cm/D(n) obtained from an
element С of K, as in Therem 4, and X a divisor on A corresponding
to E. Since we have
E(z, го) = < T{Qz, w >,
the homomorphism <pz of A into A* corresponds to the mapping of
CmID(n) onto O/Z)(n*) given by T(Q. We see easily that
<pxi(a) = ^*(a)?>jr

52 COMPLEX MULTIPLICATION [Щ
for every a e K.
7. TRANSFORMATIONS.AND MULTIPLICATIONS.-
7. 1. Definitions. Let К be an algebra over Q with an identity
element 1. We shall understand by a lattice in SR a free Z'-submodule
of !R of rank [3? : £?J. We call a subring о of SR an or<f«- in SR if it is a
lattice in SR and contains the identity element of SR. Let a be a lattice
in SR; let or (resp. o{) be the set of all ellements a of Ж such that
aa с a (resp. aa с a). Then, or and ог are orders in 9?. We call or
(resp. ог) the right (resp. left) order of a.
9? being as above, let (A, c) be an' abelian variety of type (SR).
Recall that i is an isomorphism of SR into ол/о(^) such that i(l) is the
identity \A of cj*fo(^)- Put
t = г^(А) Л
V
It is easy to See that r is an order in SR. r is called the order of
{A, c). We say that ('Д г) is defined over a field A if A is a field of
definition for A and every element of i(z). Let {A', c') be another
abelian variety of type (SR). A homomorphism (resp. an isomorphism)
/ of A into A' is called a homomorphism (resp. an isomorphism) of (Д г)
into (Л', /), or an ^-homomorphism (resp. ^.-isomorphism) of Л into
^', 4f it satisfies
&(«) = :'(a)X
for every «e3l. An endomorphism or an automorphism of (A, c) is
similarly defined.
Let {A, t) and {A', c') be abelian varieties of type C1); let i be
the order of {A, c) and a a lattice in SR contained in i. A homomor-
homomorphism /. of (A, c) onto (A', c') is called an ^-multiplication of (A, c)
onto (A', c') if there exist a field k of definition for (Д с), {А', с') and
/, and a generic point x of A over A:, such that k(hc) is the composite
of all the fields k(c(a)x) for a e a. We note that if Я is an a-multipli-
cation, then, for any field of definition kx for {A, c), (A', i') and A,
and for any generic point у of A over ku ki(ty) is the composite of
all the fields ki(e(a)y) for a e a. It is easy to see that every a-multi-

[7.1] TRANSFORMATIONS AND MULTIPLICATIONS 53
plication is an ra-multiplication. {A', c') is called an a-transform of
(A, i) if there exists an a-multiplication Я of (A, e) onto {A1, c'). We
call also the system (A1, c'; ?.) an a-transform of (A, c). By our def-
definition, every a-transform of (A, c) is of the same dimension as {A, c)
and every a-multiplication is an isogeny.
Proposition 7. Let SR be an algebra over Q with, an identity ele-
element and (A, c) an abelian variety of type (fR);'let r be tlie order of {A, c)
and a a lattice in 91 contained in r. Then, there exists an a-transform
(A', c'; A) of (А, с); {А1, с'; Я) is uniquely determined by (A, c) and a
up to an Si-isomorphism; and the order of {A', i') contains the right order
of a. Moreover, if k is a field of definition for {A, c), we can find {A', c')
and ). so that they are defined over k.
Proof. The uniqueness follows immediately from our definition.
Let A be a field of definition for {A, c) and x a generic point of A
over k. Take a basis {av> <*<г} of a over Z. Let A' denote the locus
of c(a1)xx---Xe(ad)x over k in the product of d copies of A. Then,
A' is an abelian variety with the origin Ox---xO, where 0 denotes
the origin of A. Define a rational mapping Л of A onto A' by
with respect to k; then A is a homomorphism of A onto A'. We see
easily that k{lx) is the composite of the fields k(c(a)x) for a S a, since
{аи--,ай} is a basis of a over Z. Let r' be the right order of a. Let
,9 be an element of r'. Put fa = аф; then the fa are contained in a.
We can find a positive integer g such that g/3 e I. Let у be a point
of A such that gy = x; we have then
c{fa)xx -xc{fa.)x = Z(i(gP)y),
so that the point c{fa)xx---Xc(fa<)x is contained in A'. Since the ele-
elements fa are contained in a, the field k{c{fa)x,---, c(fat)x) is contained in
k(Ax). Hence we can define a rational mapping pi of A' into itself
by /jBx) = e(fa)xx ••• X ({faipc, with respect to k. ц is an endomorphism
of A', since it maps the origin onto the origin. Denote this endo-
endomorphism p. by c'(fa). We have then

54 COMPLEX MULTIPLICATION \Щ
It follows from this relation that /3 —> /(/3) is an isomorphism of r' into
wsf(A'); and c'{\) is the identity element of <_sd(A')\ so (A',_cr) is an
abelian variety of type {Ж) and the order of {A', t') contains r'. If a
is contained in г П r', we have c'{a)hc = h{a)x by our construction.
This completes our proof.
Notations being as in Proposition 7, we denote by g(a, A), or
simply by g(a), the set of points t on A such that c{a)t = 0 for every
a e a.
Proposition 8. The notations SR, {A, c), r, a being as in Proposition
7, let (A', t'; Я) be an a-transform of (A,1 c) and k a field of definition for
{А, с), {А1, с') and Л. Then, for every point t on A, the field k(ti) is the
composite of all the fields k(c(a)t)for a e a; and the kernel of Я is g(a, A).
Proof. J3y the uniqueness, it is sufficient to prove our conclusion
for the a-trar)sforn> (A', ic'; Я) constructed in the proof of Proposition
7; but this is easily seen by the relation Xt = ^a^tx-'-Xcia^t.
(A, c) and r being as above, let a be a regular element of 91 contain-
contained in r. Define an isomorphism c' of 91 into <л!?й(А) by с'(у) = c^oera'1).
Th&n, we see easily that c(a) is an ra-multiplication of (A, c) onto
(А; с'). If 9? is commutative, we have of course {A, c) = {A, i').
Proposition 9. The notations Ш, (A, c), r, a being as in Proposi-
Proposition 7, let Xi be the right order of a, and Ь a lattice in St contained in
Vi. Let (At, c^; ).) be an a-transform of {A, c) and (A2, c2; p) a Ъ-trans-
form of (At, fi). Then, [A2, сг\ fj2) is an ab-transform of (A, c).
Proof. piX is obviously an 3I-homomorphism of A onto A2. Let
k be a field of definition for {A, c), (Au it), (A2, cz), Я, р; take a generic
point x of A over k and put у = he. Then k(fty) is the composite
of the fields fc(fi(j8)y) for /3 e 6. We can find a positive integer m
such that «let and a point z of A such that mz = x. We have
then c0)y = fo(mp)z, and, by Proposition 8, /г(Яс(т[1)г) is the compo-
composite of the fields k(c(a)c(mj$)z) for a G a. As we have ab с arx с a, a/3
is contained in r, so that we have c(a)t(mfi)z = c(aj3)x for every a e a,

[7.2] TRANSFORMATIONS AND MULTIPLICATIONS 55
/3 e b. Hence k(pihc) is the composite of the fields k{c{afi)x) for a e a
and ^ e Б. Since ah. is generated by the elements aft, this proves that
pd is an ab-multiplication.
7. 2. From now on, we assume that SR ij a simple algebra over
Q2) and denote by N(£) the reduced norm of f E3i (cf. §5.1). We
first recall the ideal-theory in SR; for details we refer to Deuring [7].
An order о in SR is said to be maximal if there is no order containing
о other than itself. Let a be a lattice in SR and о an order in SR. We
call a a right (resp. left) o-ideal if we have ao с a (resp. оа с a), a
is said to be normal if both its right and left orders are maximal; it
is known that if one of the left and right orders of a is maximal, then
a is normal, a and Ь being normal lattices in SR, the product аЪ is said
to be proper if the right order of a coincides with the left order of b.
The set of all normal lattices in SR form a groupoid ® with respect
to the operation of proper product; the maximal orders are the units
of ® and the inverse of a is given by
a = {f |f e SR, a$a с a}.
or and Oi being the right and left orders of a lattice a in SR, if we have
а с or, then a С ог, and conversely; we call such a integral. If a is
integral and normal, the numbers of elements in the factor modules
or/a and otja are the same; we denote this number by N^a). Let 3
be the center of SR; and put
A) [» = 3]=Л [3:g]=rf.
Then, we can prove that N^a) is the /-th power of a positive integer;
we put N(a) = N^ayif. We can define N(a) in a natural manner for
every normal lattice in St which is not necessarily integral; and if the
product ab is proper, we have
N(ab) = N(a)N(b).
2) For our principal aim in Chap IV, it is sufficient to consider the case
where 5ft is an algebraic number field and 2(dim^) = [31: Q]; so the reader who
is interested only in this case may dispense with the trouble of considering the
general case.

56 COMPLEX MULTIPLICATION [ГЦ
If £ is a regular element in SR, we have, for every maximal order o,
N(oS) = Що) = \W)\-
Now consider an abelian variety (A, i) of type (SR). Let n be the
dimension of A and /, d be as in A). Then, by Proposition 2, fd
divides 2и; putting 2и = mfd, we call m the index of (А, с). (А, с) is
said to be principal if the order of (Д c) is. maximal. By Proposition
7, if (Д i) is principal, then, for every integral left o-ideal a, an a-
transform of {A, c) is also principal. In the following treatment, {A, c),
(A', c') etc. will denote abelian varieties of type Ci) which are assumed
to be principal.
Proposition 10. Let о be the order of {A, c) and a. an integral left
o-ideal; let {Ax, cr; ?.a) be an a-transform of {A, c). Then we have
where m is- ihe index of (A, c).
Proof. Let o' be the right order of a. Then we can find an in-
integral left o'-ideal Ь such that ah is a principal ideal oy and (N(b), v{Xa))
= 1 (cf. [7] VI, Satz 27). Let (Ait cz; JLb) be a Ь-transform of (A, «0;
theft (A2, сг; ^a) is an o/--transform of (A, c) by virtue of Proposition
9. , On the other hand, putting c'{a) = c(rcq-'1) for a e 5R, we see that
(Д c'; c{f)) is an o^-transform of (A, c). Hence there exists an isomor-
isomorphism t) of {As., cs) onto (A, t') such that г]к^)<а — c(f). It follows that
<h>Q-a) = Mr))-
By Proposition 2, we have Mr)) = N(r)m = N(a)mN(b)m, so that
B)
Since v(Za) is prime to N(b), и(ла) must divide AT(a)m. Applying this
result to b, we see that ufo) divides N(b)m. Therefore the equality B)
shows v{).a) = N{a)m.
Proposition 11. Let о be the order of {A, c); let a and Ъ be integral
left o-ideals. Let {Au ^ ; Xa) and (Ait c2; %ь) be respectively an a-trans-
a-transform and a Ъ-transform of (A, c). Then the following three conditions are
equivalent to each other.

[7.2] TRANSFORMATIONS AND MULTIPLICATIONS 57
1) а з Ь.
2) There exist a field of definition k for {A, c), (Alt it), (A2, c2),
?.a, Ль and a generic point x of A over k such that k()-ax) z> k(?.f,x).
3) There exists a hotnomorphism (x of (Au cy) onto (A2, c2) such that
P?-a = h-
Proof. It is easy to see that 1) =£> 2) <£==?> 3). Suppose that qJi6;
and put с = a+b. We have then c^a, so that N(c) < N(a). Hence,
if Ac is a c-multiplication of (A, :), we have k(^cx) $ k(?.ax) by virtue
of Proposition 10. On the other hand, as we have с = а+Ь, the field
k(?.cx) is the composite of k{Xax) and k(?.tfc); so we must have
Щах) J> Щ$х). This proves 2) => 1).
Proposition 12. Notations being as in Proposition 11, let g be a
positive integer such that garlb is integral. Then, there exists a (ga"^)-
multiplication /j of {Au ct) onto (A2, c2) such that рйа = gXb.
Proof. Put с = ga^b. Let (As, сг; Лс) be a c-transform of (Au ct).
Then, as both (A2, сг) and (Аг, f8) are gb-transforms of (A, c), we obtain
an isomorphism -q of {At, ct) onto {A2, c2) such that y]ic^a — 8^-b- We
see easily that t]Xc is a c-multiplication; this proves our proposition.
Now we impose the following condition on our abelian varieties
of type (SR).
(C) If 3 denotes the center of SR, the commuter of c(g,) in ^й{А) is
contained in c(fR).
This is trivially satisfied if ;(iR) = u*fo(A). If the index of (A, c)
is 1, (A, c) satisfies (C). In fact, the degrees/ and d being as in A),
SR contains a subfield g such that [g:3] =/. If {A, c) is of index 1,
we have 2 dim(^4) = [%: Q]. By Proposition 3, ^й{А) must be simple;
and if we denote by St the center of ^й{А), cffi) contains St. It
follows that гC) contains 5t As we have [^0(A): г©)] = (г®): Й] and
[Я : 5] = [g: 3]. we get [лГ0(А): с(Щ = КЗ): Я]. Let 8 be the com-
mutor of гC) in ^Bifo(A); then S contains c(fR); and by a property
of central simple algebra we have [ъл#9(А):&] = [c(S): Я]. This shows
S = <9t). Hence (A, i) satisfies (C).
We note that if (A, c) satisfies (C), every a-transform of {A, c), for

58 COMPLEX MULTIPLICATION [П]
an integral lattice a, satisfies (C).
Proposition 13. Suppose that (A, e) satisfies (C). Let о be the
order of {A, c) and a an integral left o-ideal; let {Ax, i\; Za) be an a-trans-
form of {A, c). Denote by c* and ct* the restrictions of с and cx to the
center 3 of Ш. Then, every homomorphism of {A, c*) onto (Au ij*) is a
t-muhiplication of {A, t)for an integral left o-ideal с Moreover, the set
of all homomorphisms of {A, i*) into {Au ct*) coincides with ^„•г(а~1).
Proof. Take a positive integer g such that gar1 is integral and
put li = ga~1. By Proposition 12, there exists a Ь-multiplication Xf, of
(Ai, ii) onto (A, c) such that ?^a = g\A; we fix 6 and %. Let pi be
a homomorphism of {A, (*) into (Ai, fi*). Then, fdf, is an endomor-
phism of (Au f,*). On account of (C), fdf, must be of the form ^G-)
for an element ? e o1( where ot denotes the left order of 6. Applying
Proposition 11 to the ideals Б and Oj^+b, we see that 7- e b. Now
suppose that jli is an isogeny; then у must be a regular element of 91.
Put с = b"Y;'Iet Dг, '2l' Л:) be a c-transform of {A, c). By the same
argument as in the proof of Proposition 10, we obtain an isomorphism
■q of A2 onto Ai such that rj^b = ct(T) — A^b- As Af, is an isogeny, we
have 7}ZC — ft. It follows that p is a c-multiplication ; this proves the first
assertion. Now let ф denote the module of homomorphisms of (A, c*)
intoi (Ah cj*). We have proved above §->ij с tj(b). Let /3 be an element
of i. If k is a field of definition for {A, t), (Alt c,) and Ль, and x is a
generic point of At over k, we have k(Xf,x) 3 A;(^(/5)x). Hence there exists
a homomorphism /. of A into At such that 2./.% = *i(/3). We see easily
that 1 commutes with the operation of 3. This shows ф-^ = ^F).
A'lultiplying by la this relation, we get §-g\A = ci(b)Xa — g-^-a'K0-'1),
so that § = P.a--'(a); this completes our proof.
Proposition 14. Notations and assumptions being as in Proposition
13, let Ъ be an integral left a~ideal and (A2, c2; Л&) be a b-transform of
(A, e); denote by c2* the restriction of c2 to Q. Then, (Au ct*) is isomor-
phic to (A2, с-,*) if and only if there exists a regular element 7- of 31
such that a = 6?-.
Proof. Let §t denote, for * = 1, 2, the set of all homomorphisms

[7.4] TRANSFORMATIONS AND MULTIPLICATIONS 59
of {A, c*) into (A{, d*). The £i are considered as right o-modules.
If (Ai, ct*) is isomorphic to (Ait сг*), §! must be o-isomorphic to ф2.
By Proposition 13, this amounts to saying that a is isomorphic to
b as right o-modules; this implies that a and Ь are isomorphic as left
o-modules. This proves the "only if" part of our proposition. Con-
Conversely, suppose that there exists a regular element у in SR such that
ay = b. Take a positive integer g such that gf is contained in the right
order of a. Then, by the same argument as in the proof of Proposi-
Proposition 10, we can find an isomorphism у of Av onto A2 such that
Vc.i(gf)^a — Sh- I* is easY to see that i] commutes with the operation
of 3. This proves the "if" part.
7. 3. Now we consider the case where SR is an algebraic number
field F. In this case the condition (C) is reduced to the following
form.
(C) The commuter of c{F) in <jxfa{A) is c(F) itself.
This is satisfied if e{F) = <j>f<>(A) or if (A, c) is of index 1.
Assuming the condition (C) to be satisfied, we give the following-
definition: с being an ideal-class of F, we call (A', c') a c-transform
of (A, c) if {A', c') is an a-transform of {A, c) for some integral ideal
a in с; if that is so, for every integral ideal с in с, (А', с') is a c-trans-
form of {A, e). Let с and d be ideal-classes of F; let (At, tb) and
{Ad, cd) be respectively a c-transform and a <f-transform of {A, t).
Then, by Proposition 12, {Ad, cd) is a c~V-transform of {Ac, ce).
Furthermore, by Proposition 14, {Ac, cc) and {Ad, id) are isomorphic if
and only if с — d.
7. 4. Let us consider the case of characteristic 0. {F; {p<}) being
a CM-type, let {A, c) be an abelian variety of type {F; {<pt})\ by our
definition, {A, c) is of index 1. By Theorem 2, {A, c) is represented
by a complex torus CnjD{a) for a free Z-submodule a of rank 2n in
F, notations being as in that theorem. We observe that {A, c) is
principal if and only if a is an ideal (not necessarily integral) of F. It
is easy to see that, for every ideal-class с of F, a c-transform of {A, c)
is also of type {F; {<pi})-

60 COMPLEX MULTIPLICATION [И]
Proposition 15. Notations being as in Theorem 2, let a and b be
two ideals of F; let (Alt a) and (A2, i2) be abelian varieties of type
(F; {<pt}), respectively represented by the complex tori CnjD{a) and
Cn[D(b). If у is an element of а~'Б other than 0, the diagonal matrix
S{y) with the diagonal elements y*i,-", y*n represents а {yb'^aymultiplica-
tion of (Au tt) into (A2, c2); conversely, every homomorphism of (Au ^)
onto {A2, t%) corresponds to some S(y) such that у е о'*Ь.
Proof. If у е a~% we have S(y)D(a) с D(b), so that S(y) gives a
homomorphism of CnjD{a.) into Cn/D(b). Hence S(y) represents a
homomorphism Л of At into A2. Since I commutes with the operation
of F, Я is a "homomorphism of (Alt e,) into (Аг, с2). Suppose that
у i= 0. We see easily that the kernel д(Л) of 1 corresponds to D(y~1b)jD{a)
and
£>(r-ib) = {u\ueCn, S(a)u e D(a) for every a e ydb'1}.
This implies fa(X) = д(?-аЬ, А^). By Proposition 8, it follows that X
is a (?-ab)-niultiplication of {Au ct) onto (Ait сг). The last assertion
of our proposition follows from this and Proposition 13.
Proposition 16. Let (Д c) and {A', c') be two abelian varieties
which* are principal and of the same CM-type (F; {<pi}). Then (A', ?')
is a c-transform of (A, c) for an ideal-class с of F.
'This is an easy consequence of Proposition 15. Furthermore, on
account of Proposition 14, we obtain
Proposition 17. Let (F; {<pi}) be a CM-type and h the number
of ideal-classes of F. Then, there are exactly h abelian varieties of type
{F; {<pt}), which are principal and not isomorphic to each other.
7. 5. Ideal-section points. Now coming back to the case of
arbitrary characteristic, denote by о the ring of integers in the number
field F; let {A, c) be an abelian variety of type (F), which is principal.
Let a be an ideal of о and (А', с'; Ла) an а-transform of (Д c). We
observe that Vi(Xa) and ^г(?.а) depends only upon {A, c) and a, and not
on the choice of (А', с'; ?<а); so we denote them by Nt(a, A) and
N,(a, A). Let с be an ideal-class of F and (Д., ic) a c-transform of

[7.5] TRANSFORMATIONS AND MULTIPLICATIONS 61
{A, c). Then we have
Ща, А) = Ща, Ас), Ща, А) = Ща, Ac).
In fact, let (A/, cc'; Xq1) be an a-transform of (Ac, cc). Take an in-
integral ideal Ь in the class c, prime to the characteristic p; let %ь an<^
Xbr be respectively Ь-multiplications of {A, c) onto (Ac, cc) and of {A1, c')
onto (Acr, cc')- Then both ?.f,'Xa and /2а'Л& are аб-multiplications of (A, c)
onto (Ac', cc'); so there exists an automorphism j) of (A/, ccf) such
that Яъ'Ла = vXa'Xb. We have then п(Хъ')щ(Ха) = w(V>«tfb)- By our
assumption that 6 is prime to p, the degree v(ib) = v(&b') — N(b)m is
prime to p, where m denotes the index of (A, c); hence we have
Vi()-b) = i>i(JLb') = 1, so that vi(Xa) = щ(Ла'). This proves the above re-
relations.
Now let <У = {(A, c)} be a system of abelian varieties of type (F),
whose members are* transforms of each other by ideals of o. Then,
Nt(a, A) and Ns(a, A) for (А, с) е <bf does not depend on the choice
of (A, c); so we denote them by Nt(a, <У) and N,(a, j/7) or simply
by Ni(a) and Ns(a) when we fix our attention to a given system <y.
We can easily verify
Ща)Ща) = ЛГ(а)-
m denoting the index of the members of j/', and
ЩаЬ) = Ща)ЩЪ), ЩаЬ) = Ща)ЩЪ).
If Ла is an а-multiplication of (A, c), v,B.a) is the order of the kernel
д(Ла) of Xa. As we have д(Ла) = д(а, Л), this shows that Ns(a) is the
order of д(а, ^4); in particular, if a is prime to the characteristic of the
fields of definition for A, we get Ni(a) — 1, and hence д(а, A) is of
order N(a)m.
Proposition 18. Let a and Ъ be ideals of o. Then we have
д(а+Ь, A) = Q(a, А) л д(Ь, А),
д(а Г)Ъ,А) = д(а, А)+ф, А).
Moreover, if а й prime to Ь, д(аЬ, Л) й the direct sum of д(а, A) and
8(Ь, ^).

62 COMPLEX MULTIPLICATION [11]
Proof. The first equality is obvious. We see easily that
g(a Л Ъ, А) з g(a, А)+ф, А).
By an elementary theorem of group-theory, the order of g(a, A)+Q(b, A)
is equal to
[g(a, A): {0}] [g(b, A): {0}]/[g(a, А) Л g(b, Л): {0}].
As we have g(a+6, A) — g(a, Л) Л sF, ^), this number is equal to
Лг!(а)Л",(Ь)Лг,(а+Ь). On the other hand, by means of the relation
ab = (n n Ь)(а+6), we have Ща)ЩЬ)Ща+Ъ)^ = N,(a П Ь). Hence
both sides of the above inclusion-relation have the same order; this
proves the second equality. If a is prime to b, we have а Л Ь = аЪ
and а+Ъ = о, so that
В(а, А) П g(b, А) = д(о, А) = {0}.
This implies the last assertion.
Proposition 191 Let p be a prime ideal ofo. Then, Nt(p) and N,(p)
are powers of N(p). In particular, if (A, c) is of index 1, we have
Proof. We can easily verify that g(p, A) is invariant under the
operation of o; moreover, g(£, A) is considered as an (o/p)-module.
Sirfce p is a prime ideal, o/p is a finite field with N(p) elements; and
g(p, A) is a vector space over the field o/p. This proves the first as-
assertion. If (A, 0 is of index 1, we must have Ni(p)N,(p) = N(p); this
proves the second assertion.
As an example, we shall determine Nt(a) in the case of dimension
1. Suppose that [F: Q] — 2 and {A, c) is of dimension 1; then the
index of {A, i) is 1. Let A: be a field of definition for (A, c). If the
characteristic of k is 0, we have Л^(а) = 1 for every а; so there is no
problem. Suppose that k is of characteristic p ^ 0. If a is prime to
p, we have Nt(a) = 1; so we have only to consider Nt(p) for the prime
ideals p dividing p. Since F is of degree 2, there can occur three
cases:
i) (p) = PiPt, Pi Ф h,

[7.5] TRANSFORMATIONS AND MULTIPLICATIONS 63
ii) (P) = P,
Hi) (p) = p2,
where plt p2 and p denote prime ideals of F. By Proposition 7 of
§ 2, we have Nt((p)) = vt(plA) = P or p2. Hence, in the cases ii) and
iii), we must have Ni(p) > 1; so by Proposition 19, we have Nt(p) = N(p),
Vi(plA) = p2. By the same reason, in the case i), we can not have
M(Pi) = Щр2) = 1. Assume that M(Pi) = Щр2) = p. Let ^ and ).г
be respectively a p!-multiplication and a p2-multiplication of {A, c).
Then, as A is of dimension 1, if л; is a generic point of A over k, the
fields k(?.ix) and k{X2x) must be contained in k{xp). On the other hand,
as we have о = p!+p2, the field k{x) is the composite of kfax) and
At(/2x); we have thus arrived at a contradiction. Hence we must have
Ni(pa) = 1 for one of pt and p2, say p%. Then, we have Nt(p2) = p and
hence Vi(plA) = p. This completes the analysis of the case i).
Returning to the general case, we call a point t of g(a, A) a proper
a-section point on A if c{a)t = 0 implies a e a.
Proposition 20. {A, c) being principal and of index 1, let m бе ая
iVfea/ of о and t a proper va-section point of A. Then, for every point t'
of в(т), there exists an element a of о such that t' = c(a)t; and the map-
mapping a —» i(a)t gives an o-isomorphism of o/m onto g(m). The point c(a)t
is a proper m-section point on A if and only if a is prime to m.
Proof. It is easy to see that the mapping a —*■ c(a)t is an o-homo-
morphism of о into g(nt); as t is a proper nt-section point, the kernel
of this homomorphism is in. Hence the module o/m is isomorphic to
a submodule of g(m). On the other hand, o/m is of order N(m) and the
order of g(m) is Af,(m), which is not greater than N(m). It follows
that the mapping a —»c(a)t is an isomorphism of o/m onto g(m); all the
assertions easily follow from this fact.
Proposition 21. (A, c) and m being as in Proposition 20, there
exists a proper m-section point on A if and only if Ni(m, A) = 1. In
particular, if m is prime to the characteristic of the fields of definition for
A, there exists a proper m-section point.
Proof. By Proposition 20, if there exists a proper m-section point,

64 COMPLEX MULTIPLICATION [П]
then the order of g(m) is equal to N(m), so that iVj(m) = 1. Conversely,
suppose that Ai<(m) = 1. Let m = piei—prer be the factorization of in
into prime ideals pu. By Proposition 18, g(m) is the direct sum of_ the
g(p«eu). We have clearly Ni(pu) = 1, and hence g(|vO is of order N(puf).
Therefore, we can find a point tu in g(pue») which is not contained in
g(pB<:u-1). Let a be an element of о such that c(a)tu = 0. Put ao+pueu
= puf; then we see that tu is contained in g(pu-O, so that eu =/; this
implies that a is contained in pueu. Hence tu is a proper pKE«-section
point. Put t — tx-\ \-tr. Then we can easily verify that г is a proper
m-section point. This proves our proposition.
Proposition 22. {A, c) being principal and of index 1, let Jj be a
finite subgroup of A, such that e{o)'<) с t). Then, there exists an ideal n
of о such that Ij = g(a, A).
Proof. Let {tu---,t,,,} be a system of generatois of I) over г(о).
Let au, for each' м, denote the set of elements a e о such that г(а)ги = О.
Then, au is an ideal of ff and ги is a proper au-section point. Hence,
by Proposition 20, we have g(au) = г(о)г„. Putting a = at П ■ • ■ П ад, we
obtain, by Proposition 18,
This proves our proposition.
Proposition 23. Let (A, c) and (A' c') be abelian varieties of type
(F), which are principal and of index 1. Let Я be a homomorphism of
{A, c) onto {A', e') such that vt(Z) = 1. Then, 2. is ana-multiplication of
(A, c) onto {A', c') for an ideal a of a.
Proof. Put i) = §{?.) and apply the argument of the proof of Prop-
Proposition 22 to this case. We obtain then g(i) = g(a, A); moreover by
Proposition 21, we have Ni(au) = 1 for each u, so that JVi(a) = 1. Hence,
if (Al3 c1; Xa) is an a-transform of {A, c), we have ^i(^i) = 1. The
equality д(Л)= д(а, A) shows that ?■. and ?.a have the same kernel. It
follows from this and the relation vt(?.) — vt{X.a) = 1 that there exists an
isomorphism tj of A' onto A± such that )?<? = Xa. Since both X and Xa
are o-homomorphisms, -q is an o-isomorphism. This proves that
{A', c'; 2.) is an a-transform of (A, c).

[7.6] TRANSFORMATIONS AND MULTIPLICATIONS 65
Proposition 24. Let m be an ideal of о and с an ideal-class of F.
Assuming {A, c) to • be principal and of index 1, let {A', c') be a c-trans-
form of {A, c) and t a proper m-section point on A. Then, for every t'
in s(m> A'), there exist an ideal a in с and an ^-multiplication JLa of
{A, c) onto {A1, c') such that t' — ?-at; and 2.at is a proper (a, 1^)"%-
section point of A'. In particular, ).at is a proper m-section point of A'
if and only if a й prime to m; and ?.at = 0 if and only if а с m.
Proof. Let a be an ideal in с and ?.a an a-multiplication of (A, e)
onto {A', c'). We first prove that 2.at is a proper (a, nt)nt-section
point on A'. Take an integral ideal Ь in the class c'1, prime to m,
and a b-multiplication >Jg of (A1, c') onto {A, c). Then there exists an
element у of о such that 2.^ — c{f) and ab = ro. If we have c'(fiJ.at = 0
for an element fie o, we have c{fif)t — 0, so that ^em, namely
juab'Cm. As b is prime to tit, we get pi cm; this shows that
ft e (a, nt)~ 1m. Conversely, if we have ft e (a, m)~1m, then pa. с m and
hence fiy <= ftab с m, so that ?^c'(fi)?^at = 0; this shows that c'(fi)Xat
e g(b, A'). On the other hand, as t is contained in g(m, A), we have
clearly c'(fi)Xat e g(m, A'). Since b is prime to m, the intersection of
g(b, A') and g(nt, A') must be {0}; it follows that e'(jx)Xat — 0. We
have thus proved that c'(fi)Xat = 0 if and only if ft is contained in
(a, in))!!, namely, ?.at is a proper (a, m)~1m-section point. In particu-
particular, ~/.at is a proper m-section point if and only if a is prime to m;
and ?.at = 0 if and only if а с ш. Now fix an integral ideal с in c,
prime to m, and a c-multiplication ?.c of (A, c) onto (A', /). Then Xct
is a proper m-section point; hence, for every t' in g(m, A'), there ex-
exists, by Proposition 20, an element a of о such that t' ~ c'(a)Xct. This
proves the first assertion of our proposition, since c'{a)/.c is an ac-mul-
tiplication of {A, c) onto {A', c'). The rest of the proposition is already
proved.
7. 6. Let (A, c) be an abelian variety of type (F), defined over a
field k, and a an isomorphism of k onto a field k°. Let t be the order
of (A, c). Put, for every a e r, c"{a) = c(a)' (cf. § 1.5); then c° is uniquely
extended to an isomorphism of F into <jtfa{A°) which we denote also
by c. We obtain thus an abelian variety (Ac, c) of type (F), defined

66 COMPLEX MULTIPLICATION [П]
over k'; r is the order of (A', c°). Let a be an ideal of r and (Alt cy; X)
an a-transform of (A, i), defined over k. Then, we see easily that
(Af, tf; X') is an a-transform of {A', c°). If {A, t) is principal, so is
{A", c'); and if г is a proper a-section point on A, rational over k, then
V is a proper a-section point on A'. Now suppose that k is of char-
characteristic p ^= 0. Then, for every power q = pf with / > 0, we obtain
an abelian variety (A*, fl) of type (F); let ж be the 5-th power homo-
morphism of A onto A*, defined in § 1.5. We see easily that
nc{a) = fl{a)n for every a e F, so that л- is a homomorphism of (A, c)
onto (A?, fl). In particular, if {A, c) is defined over a finite field with q
elements, the 5-th power endomorphism ж of A is an endomorphism
of {A, c); hence, if further (A, c) satisfies the condition (C) of § 7.3,
there exists an element 7- of r such that ж = c{f).
• .8. THE DUAL OF A CM-TYPE.
rt
The purpose of this Section is to investigate the algebraic struc-
structure of a CM-type. !For the sake of simplicity, we assume that the
fields appearing in thij section are all containd in the field С of com-
complex numbers.
j
8. 1. Group-theoretic characterization of CM-types. We
begiri with
Lemma 3. Let L be a Galois extension of Q, G the Galois group
of L over Q and p the element of G such that fp is the complex conju-
conjugate of f for every feL Let К and Ka be two subfields of L such
that [K: Ko] = 2, and H, Ho be respectively the subgroups .of G corre-
corresponding to K, K9. Then, the following two conditions are equivalent.
i) Ko is totally real and К is totally imaginary.
ii) Ho = H U Нора'1 for every a e G.
If these conditions are satisfied, we have pHr = Нтр = оро~1Нт = Нтара'1
for every a e G, r e G.
Pkoof. If a is an element of G, the subfields Ko', K" of L cor-
correspond to the subgroups o~iHaa, a~xHa of G. If Ko is totally real

[8.1] THE DUAL OF A CM-TYPE 67
and К is totally imaginary, then K9' is a real field and K" is an ima-
imaginary field, so that p fixes every element of Ko' and does not fix some
element of K'. Hence we have p e a~1Hoa and p £ a~1Ha, so that
ара'1 e Ho and ара'1 ф H. By the assumption [K: Ko] = 2, we have
[H0:H]-2; it follows that Ho = H \J Hapa'1. Conversely, if the
relation Ha = HU Hapa'1 holds for every a e G, we can easily see,
following up the above argument in the opposite direction, that K9 is
totally real and К is totally imaginary. Now suppose that the con-
conditions i) and ii) are satisfied. Then, we see that both Hapa'1 and
apa'lH are equal to the set Ho—H for every a e G. We have hence
pH = Hp = apa'1H = Hapa'1.
It follows that pHa = Hap. Let p. be an element of G. Then, jK> is
totally imaginary and K^ is totally real; and the subgroup [л^Н/л of
G corresponds to Kf. Therefore," applying the formula pHa = Hap to
[x'T-Hp., we have p{p.~1H{i)a = (fi~1Hft)ap. Transform this by the inner
automorphism y—^-frfpT1 and put г = [лор'1- We have then ppfx~xHz
= Hr/tpft'1. This completes the proof.
Proposition 25. Let F be an extension of Q of degree 2n, {<pu- ••, <pn)
a set of n distinct isomorphisms of F into C. Let L be a Galois extension
of Q containing F, and G the Galois group of L over Q. Denote by p
the element of G such that $' is the complex conjugate of £ for every
$ G L, and by S the set of all the elements of G inducing some <pi on F.
Then, (F; {<pt}) is a CM-type if and only if we have
A) G = S U Sapa'\ Sapa'1 = apa^S
for every a e G.
Proof. If (F; {<рг}) is a CM-type, then, by Theorem 1 of § 5, F
has two subfields К and Ko satisfying the conditions (CM1, 2) in that
theorem. L, G, p, S being as in our proposition, let H be the sub-
subgroup of G corresponding to F. Let {фи---, фт) be the set of all
distinct isomorphisms of К into С obtained by restricting <pi to K.
Then there are no two members of {fij} which are complex conjugate
of each other; and it is easy to see that {<pj} gives the set of all dis-
distinct isomorphisms of Ko into C. It follows that [K^: Q] = m. We

68 COMPLEX MULTIPLICATION \Щ
observe also that {<pt} is the set of all the isomorphisms of F into С
inducing some ф) on K. Hence S is the set of all the elements of
G inducing some ф^ on K. Take, for every j, an element r,-, of G
inducing ф] on K. Then we have S — U Hzj. Every element of G
coincides with ry or zjp on K. Hence we have
G = U (Яг,- U Htjp) = S и Sp.
j
By Lemma 3, we have pHxj = H-jp = apa~1HrJ = Hrjapa'1; it follows
that pS = Sp = apa~'S — Sapa'1. Thus we have proved the " only
if" part of our proposition. Conversely, suppose that the relation A)
holds for every a e G. Put p' = ара'1. As the number of the ele-
elements in 5 is the half of the order 'of G, the sets S and p'S = Sp'
have no common element and p'S = G—S. Put
H' = {r\reG,rS = S}.
We have tben, p'H'pS = p'H'p'S - p'H'Sp' = p'Sp' = p'p'S = S, so
that we have p'H'p ф Н', р'Н'р' с Н'. It follows from this that
p'H' = H'p' = H'p., Since S does not coincide with pS, the element
p is not contained in H'. Put now
Ho' = ffu H'p.
We, -can easily verify that Ha' is a subgroup of G and [До': H'] = 2;
ai)d we have Щ = H' U H'apa'1 for every a e G. Hence, if we de-
denote by K' and i£o' the subfields of L corresponding to H' and H9',
respectively, K' is totally imaginary and K9' is totally real by virtue
of Lemma 3. If у is an element of G leaving invariant every element
of F, then every element of jrS coincides with one of the p» on F, so
that we have j-S <z S and hence у e H'; this implies F z> K'. As we
have H'S = S, S is expressed as a join of cosets: S =. U H7^,,. We
see that {//„} gives the half of the isomorphisms of K' into C; and
H'fjap does not coincide with any H'fip, since S and 5p have no com-
common element; in other words, for any a and /5, fia does not coincide
with the complex conjugate of /tf on K'; namely, the system
{Kr, Ko', {<pi}} satisfies the conditions (CM1,2) of Theorem 1. Hence
(F; {<pi}) is a CM-type. This completes the proof.
8. 2. Primitive CM-types. We call a CM-type primitive if the

[8.2] THE DUAL OF A CM-TYPE 69
abelian varieties of that type are simple; recall that any two abelian
varieties of the same CM-type are isogenous to each other (Corollary
of Theorem 2). The following proposition is a criterion for the primi-
tiveness of a CM-type.
Proposition 26. (F; {p<}) being a CM-type, let L, G, p, S, be as
in Proposition 25 and H^ the subgroup of G corresponding to F. Put
Then, (F; {<pi}) is primitive if and only if Hy = H'.
Proof. Using the same notations as in the proof of Proposition
25, we see that F contains the field K' corresponding to H'. If
(F; {pi}) is primitive, we must have F = K.' by virtue of Theorem 3
of §6, so that Hi = H'. If (F; {<pt}) is not primitive, F contains two
subfields К and Ko, satisfying the conditions (CM1,2) of Theorem 1,
such that F ^ К; this follows from the discussion in §5.2. Denoting
by H the subgroup of G corresponding to K, there exist elements ту
of G such that S — U Нт}-, as is seen in the proof of Proposition 25.
We have then HS = S, and hence H с Н'. By the relation F^K,
we have H 5 Hi, so that H,. ^= Н'; this completes the proof.
We shall now give another criterion with no use of Galois group.
If (K; {pi}) is a primitive CM-type, Theorem 3 of § 6 shows that К
is a totally imaginary quadratic extension of a totally real field Ke.
We have seen in the proof of that theorem that there exists an element
С of К such that К = K9(Q, —C2 is totally positive and Im(CK) > 0
for every i. Conversely, suppose a totally real field Ko and a totally
positive element -q of K9 to be given; let С be a number such that
— C2 = f}\ and put К = Ka(Q. Let {©;} be the set of all the isomor-
isomorphisms <p of К into С such that Im(C>) > 0. Then it can be easily seen
that (K; {<pi}) is a CM-type. We shall denote this CM-type by K0((Q).
We have K0((Q) = K0((C)) if and only if £/£' is a totally positive ele-
element of K9.
Proposition 27. Ka((Q) is primitive if and only if the following two
conditions are satisfied:

70 COMPLEX MULTIPLICATION [И]
ii) for any conjugate С of С other than С itself, over Q, C'/C « not
totally positive.
Proof. First we note that С 1С" is totally real for any two, con-
conjugates C' and C" of £ over Q, since £' and C" are purely imaginary num-
numbers. Put F = К = KB(Q); let {fi} be the set of all the isomorphisms
со of F into С such that Im(C) > 0. Define L, G, p, 5 for (F; {<pt})
as in Proposition 25 and denote by H the subgroup of G corresponding
to K\ put H' — {f\f e. G, yS = 5). Then we see easily that
S = {<? \ о e G, Im(C) > 0}. If у e IF, we have ya e S for every
<t S 5, so that Im^') > 0 for every a e S. As Cr°/C° is real, we see
£>/£' > 0. If a £ 5, then <r e 5p, 74т e r«5p = «5^. and hence ya §Ё 5,
so that Im(Cr') < 0, Im(C) < 0. Thus we have £>/£' > 0 for every
a e G, namely C/C is totally positive. Conversely, if C/C is totally
positive for an element у of G, ImfC") has the same sign as Iir^C") for
every a e G; it follows that yS = S. Therefore, H' is the set of all
the elements jsG such that C/C is totally positive. By Proposition 26,
^o(@) = (-KV {ipi}) Is primitive if and only if H' = H. Our proposition
is an immediate consequence of these facts.
8. 3. Now we cfefine the dual of a CM-type.
PftOPOSiTiON 28. (F; {(o4}), L, G, p, S, being the same as in Propo-
Proposition 26, put
S* = {o-i\o e5}, H* = {y\y e G, rS* = »S*}.
Zy«f K* be the subfield of L corresponding to H* and {<£,} the set of all
the isomorphisms of K* into С obtained from the elements of S*. Then,
(K*; {<jij}) is a primitive CM-type and we have
K* = Q(E £*|e e F).
i
(K*; {<pj}) is determined only by (F; {<pi}) and independent of the choice
of L.
Proof. Let a be an element of G. Then, by the relation A) of
Proposition 25, we have G = S* \J S*apa~1, S*apa~1 = opo'^S*. We see
easily that 5* is the set of all the elements of G inducing on K* some

[8. 3] THE DUAL OF A CM-TYPE 71
<bj and {<pj} is the half of all the isomorphisms of K* into C. Hence, by
Proposition 25, (K*\ {</>]}) is a CM-type, and is primitive by virtue of
Proposition 26. Let у be an element of H*. We have then j-~1S* = S*,
so that Sy = S; hence {<ptf,--, <pny} coincides with {<pi,---, <pn} as a
whole. We have therefore B £ e«)r = £ £w f°r every £ e Л Converse-
i i
ly, suppose that an element у of G fixes 2 f« for every f eF. Then,
r*
we have for every integer a,
By an elementary theorem of algebra, we see that {£*»V> fp»r} coin-
coincides with {f*1,--•,£»»} as a whole; this shows that {<pi,---,u>%} coincides
with {<PiC---,<PnY} on F as a whole; in other words we have 5У = S,
so that ?• e Д*. Thus we have proved that H* is the set of all the
elements of G leaving invariant every element of <9(S £Pi|£ e -^)'.
this implies K* = Q(J2 f"'|f e ^)- The last assertion of our proposition
follows easily from the definition.
We call the CM-type (K*; {<pj}) of the above proposition the dual
of(F; {<pi}). By the proposition, the dual of every CM-type is primi-
primitive. If a CM-type (F; {<pi}) is primitive, then (F; {<pi}) coincides
with the dual of its dual; this follows immediately from Propositions
26 and 28. For every type (F; {<pi}), we can find two subfields К and
Ko of F satisfying the conditions (CM1,2) of Theorem 1. Let {&■} be
the set of distinct isomorphisms of К into С induced by the coj. Then
it is easy to see that (К; {%Л) is a CM-type; and we observe that
(F; {<pi}) and (K; {yj}) have the same dual; this is also an immediate
consequence of the definition.
Proposition 29. Let (F; {<pi}) be a CM-type and (К*; {ф}}) the
dual of (F; {{£>,}); denote by p the automorphism of С which correponds to
£ £ С its complex conjugate. Let a be an element of K*; put fi = П <*?*■
i
Then, fi is contained in F; and we have /S/S' = Nk*iQ(jx). Let further a
be an ideal of K*; put Б = Г] a<*/. Then, Б is an ideal of F; and we
i
have ББ' = Nk*/q(o.).

72 COMPLEX MULTIPLICATION [И]
Proof. Define L, G, S as before. Let H be the subgroup of
G corresponding to F. Then, by the definition of S, we have HS = S,
so that S*H = S*, if we define S* as in Proposition 28. It follows
that, for every a e H, {<pjo} coincides with {<pj} as a whole. Hence
we have p> = f] a*s° = ]\a*i = /S for every a e H; so jS is contained in
F. As we have G = S* U «S1*^ {^/, ^до} gives the set of all distinct
isomorphisms of K* into С; this proves the relation (I(I* = Nk*/q(o:).
Now, as for the assertions concerning ideals, it is sufficient to prove
it in case where a is an integral ideal. Take a non-zero element a of
K*, divisible by a. We can find an element у of K* divisible by a
such that fa is prime to Nk*/q(o:). Put {) = \] a*J and S = Ц T*1-
i j
Then, by what we have already proved, ft and 8 are contained in F.
Put b= IJa<*/; it is clear that both fi and S are divisible by b. As
У
ya'1 is prime to NK*/Q(a), we see that 8b'1 is prime to /S. It follows
that Ь is the greatest common divisor of /S and o; this proves that Ь
is an ideal of F. The relation &Б' = Nk*/q(P) is proved by the same
argument as for flfjr =
8. 4. Examples. We shall now give some examples of CM-type.
A) First we consider a CM-type (F; {<pi}) such that F is a Galois
extension of Q. We can take F itself as L of Proposition 25; the
subgroup of the Galois group G corresponding to F is then the
subgroup consisting only of the identity; and we have 5= {<pw,<pn},
S* = (уГ1)■■■> фп'1}, where we consider the <pi as elements of G. Hence,
if F is abelian over Q, the subgroup H* defined in Proposition 28
coincides with the subgroup H' = {y\y e G, j-S = S). If moreover
(F; {{£>;}) is primitive, H' must be the identity-subgroup, so that the
field corresponding to H* is F. Thus we get the following result: if
F is abelian over Q and if (F; {<pi}) is primitive, the dual of (F; {<pi})
is (F; {(pf1))- In the classical case, F is an imaginary quadratic field ;
so the dual is the same as itself.
Now we give an example of a primitive CM-type (F; {<pi}) where
F is a cyclotomic field. Let p be an odd prime and С = e2lri/J>. The
automorphisms of Q(Q are given by С —► С" for the integers a such that
1 ^a^p — 1; Put я = (р—1)/2 and denote by y>( the automorphism

[8.4] THE DUAL OF A CM-TYPE 73
C—>C4 for i= 1,••-,«■ Then we see that there are no two automor-
automorphisms among (pt which are complex conjugate of each other; and
Q@ is totally imaginary quadratic extension of the totally real field
<Э(С+С-1)- Hence (Q(Q; {<pi}) is a CM-type. Let j be an automor-
automorphism of £>(C) such that {r^i,-■ ■, r<P") — {<Pi,---, <Pn}; let a be an in-
integer such that С = С°- Then, {l-a,---,n-a mod (p)} coincides with
{1,■■■,n mod (p)} as a whole, so that we have
l-a-\ b«-asl-| )-я mod (p).
As 1-| Ья = (p2—1)/8 is relatively prime to p, we have a = 1 mod (/>),
so that 7- is the identity. Therefore, our CM-type (Q(Q', {?>«}) is
primitive.
There can exist many CM-types with the same field F. Consider
for example F= Q(Q with p = 13. We normalize 5 = {<pi,---,<Pn} taking
the identity as <pv Then, we obtain 32 CM-types (F; {<pt}). By an
easy calculation, we see that two of them are non-primitive and the
remaining primitive 30 CM-types are divided into 5 families in the
following way: each family consists of 6 types; and any two types be-
belong to the same family if and only if they are transformed onto each
other by an automorphism of F. We can similarly treat the case
where F is cyclic over Q.
B) Let Ко be a real quadratic field and £ a number such that
—£* is a totally positive number of K<,. Then we obtain a CM-type
Ko((£)). Put К = Ko($). For the sake of simplicity, suppose Im(f) > 0;
this amounts to assuming that 5 contains the identity. If К is a
Galois extension of Q, the Galois group G is an abelian group of
degree 4, so that G is cyclic or the product of two cyclic groups of
order 2.
a) The case where G is the product of two cyclic groups of or-
order 2. Denoting by p the element of G such that a' is the complex
conjugate of a for every a e K, the subgroup of G corresponding to
Ka is {1, p). Put »S = {1, a}. Then we have G = {1, a, p, op) ; and
the elements ;• such that yS = S form the subgroup {1, a). Hence
K0((E)) is not primitive. Let K* be the subfield of К corresponding
to the subgroup {1, a). Then we see that K* is an imaginary quad-

74 COMPLEX MULTIPLICATION [II]
ratic field, К is the composite of Ko and K*, and the dual of
is (K*, {1}).
b) The case where G is cyclic, p and S = {1, a} being as'above,
we have G = {1, a, <r2 = p, <r8}. We see that there is no element у other
than the identity such that yS = S, so that the CM-type Ko(($))
= (K\ {1, a}) is primitive. The dual is \K; {1, «Г1» by the result
of A). We can easily verify that Ko(($')) = (K; {1, a}).
c) The case where К is not Galois over Q (The case of
Hecke [20]). Put Ko((£)) = (K; {1, jo}). As /<ч, is a real quadratic
field, there exists a positive integer d such that Ko= Q(V d); and
— c2 is expressed in the form — f2 = лг+j' Vrf, where x and у are ra-
rational numbers; as — £2 is totally positive, we have x+y Vd > 0,
я—у V d > 0, and $= Vx+yVdi, $ * — V x—y7~d i. We see that
four elements ±£, ±£<? are the conjugates of f over Q. As .K" is not
Galois over Q, f r is not contained in Ki so that K^r) ^= Ko(£). Put
d' = хг—у*Л; then, we have </' > 0. On the other hand, we have
d' = C^»"J; hence jd' is not contained in KQ, since we have Ko(£»)
?fc ifo(f). Therefore, 6( V5') is a real quadratic field different from
Q{ Vd) = Ko. Put 'now L = <3(f, f'); then we see that L is Galois
ove,r Q, L = Щ Vd') and hence [Z.: Q] = 8. The Galois group of L
over J? consists of eight automorphisms which map the couple (£, £«■)
ojfto (±f, ±fr), (+f, +f»), (±f<=, ±f), (±fp, +f). Denote by a and г
respectively the elements of G which send (?, £») onto (£*", —f) and
(f", f). Then we see that a2 = p, a* = r2 = 1, r<r = ff*r and G is ge-
generated by a and r. We can easily verify that the subgroup of G cor-
corresponding to К = g(f) is {1, стг} and 5 = {1, a, r, err} ; moreover, we
have {\,a-} = fr^eG, ^= £}, which proves that (Я"; {1, <p}) is prim-
primitive. If we define H* as in Proposition 28, then we have H* = {1, r} ;
and the subfield of L corresponding to H* is Q(£+£*). As we have
5* = Я* U H*<7r, the dual of (g(f), {1, jo}) is
latter is also written as Q( Vd7)
8. 5. Fields of definition for (A, c).
Proposition 30. Let (F; {ipi}) be a CM-type and (К*; {фЛ) its
dual; let (A, e) be an abelian variety of type (F; {<pt}) and k a field of

[8.5J THE DUAL OF A CM-TYPE 75
definition for A. Then, if every element of t{F) Л <л#{А) is defined over
k, we have k 3 K*; conversely, if k гэ К* and if (F; {<pi}) is primitive,
every element of ^f{A) is defined over k.
Proof. Let 5 be a representation of ^0(A) by invariant differen-
differential forms on A. Then, for every £ e F, £",•••> £p» are the characteristic
roots of £(*(£)). Suppose that every element of c(F) Л cjd(A) is de-
defined over k. Then we can find a basis of invariant differential forms
on A with respect to which S(i(£)) has coefficients in k for every
£ e: F(cf. §2.8); hence the trace of £(/(£)) is contained in k, namely,
we have £ ff' e * f°r every feF. This proves k Э K*. Conversely,
i
suppose that k гэ К* and (F; {$»<}) is primitive, namely A is simple.
Then, by Proposition 6 of § 5.1, <j£u(A) coincides with t(F). Let a be
an isomorphism of С into itself which is the identity on k; we have
then A' = A. Our proposition is proved if we show that X' = X for
every Л e <_stf(A). First we see that /?—»/?" gives an automorphism of
<jf<,(A). As we have u>#o(A) = c(F), there exists an automorphism г
of F such that г(£т) = c($)' for every ?eF. As is remarked in §5.2,
we can find я invariant differential forms <i>lt---, (on on A such that for
every f e F,
B) Sd$)o>i = £па>г (l^i^ n).
We have then dc($)°a>i° = pi'mf A <: i ^ я); this shows that S(c(£)')
has the characteristic roots £«',•••, £*"•• On the other hand, S(c(£')) has
the characteristic roots £r?>1,---, £**«; so, {£"',■••, f5"""} coincides with
{?r5>V", £r(""} as a whole. Since we have assumed k D .К*, <т fixes every
element of K*; hence, for every feF, we have 2 £ p'° = (S£w)*
= Zlf9"'; so by the same argument as in the proof of Proposition 28,
we see that {£«',-■■,£("»1'} coincides with {f*,--, £Pl>} as a whole. Con-
Consequently, {£«V> £*"} coincides with {£*Р1,---, £rp»} as a whole; this im-
implies that {<pi,--,<P"} coincides with {т<ри-■ ■, т<р„} as a whole. Now
define L, G, 5 as in Proposition 25; and let Hi denote the subgroup
of G corresponding to F. Take an element of G inducing г on F,.
and denote it again by r. Then as г gives an automorphism of F,
we have тНг — H^. Take for every i an element 74 of G inducing

76 COMPLEX MULTIPLICATION [II]
*Pi on F. We have then S = U Htft. The above result shows that
{.HjrTv.-ffitT»} coincides with {Htfu---, Htfn} as a whole. We'have
therefore,
S= V HlTi = U H^t = L) ri?1?-i = tS.
i i i
By our assumption that (F; {<pi}) is primitive, г must be contained in
Hi by virtue of Proposition 26. This proves that г is the identity on
F; so we have *(£)• — c(g) for every £ e F; this completes the proof.
Proposition 31. Let (F; {p(}) be a CM-type and (К*;{ф})) its
dual; let (A, c) be an abelian variety of type (F; {<pi}) and k a field of
definition for (A, c). Let a be an isomorphism of k into C. Then, a fixes
■every element of K* if and only if there exists a homomorphism of (A, c)
onto (А', с'У
Proof. We note that k must contain K* by virtue of Proposition
30. Extend a to an isomorphism of С into itself and denote it again
by a. Take я invariant differential forms а>4 on A for which the re-
relation B) holds for every f e F. Then, the а>4" form a basis of in-
invariant differential forms on A'; and we have 8c'(£)a>i° = £п°Ш{°. This
shotvs that {A", c) is of type (F; {<pta}). Suppose that there exists a
hoifiomorphism X of {A, c) onto (A', c"). By Theorem 1 of § 2, <5/ is
an isomorphism of ^0(A") onto Ъ0{А); putting со/ = (д2.)~1ш(, we see
easily dc'(£)o>i' = f"<w/ for every £ e F. Hence (A', c°) is of type
(F; {{Si}). It follows that {<pi<j} coincides with {<pt} as a whole; so we
have, for every feF, B £«)' = £ £n° — 2 f"; namely, <r leaves in-
invariant every element of K*. This proves the " if " part of the prop-
proposition. Converselj', suppose that a leaves invariant even' element
of K*; then, by the same argument as in the proof of Proposition 28,
we see that {tpto} coincides with {<pi} as a whole; so (A", i°) is of the
same type (F; {щ}) as (Д t). By Corollary of Theorem 2 of § 6 and
Remark below it, we can obtain a homomorphism of (A, c) onto (A", c°).
This completes our proof.

CHAPTER III. REDUCTION OF
CONSTANT FIELDS.
The aim of §§ 9-12 is to complement the theory of reduction
modulo p of algebraic varieties, given in [33], with a particular interest
in abelian varieties, toward the later use. We will first recall definitions
and results from the general theory with a slight modification, and
then proceed to the main subject. For omitted proofs in § 9, we refer
to [33].
9. REDUCTION OF VARIETIES AND CYCLES.
9. 1. Places. Let К be a field. We denote by Ka the join
К U {oo} of the set К and one additional element oo; and we define,
besides the operation in K, the operation in Km as follows:
a + oo = oo, a/oo = 0 for a e K,
a-oa = a/0 = oo for a e Кос, a ^= 0.
Let k and k' be two fields; we call a mapping <p of ka, onto k'a,
a place of k if it satisfies
<p(a+b) = o{a)+<p{b), <p{ab) = w(a)<p(b);
k' is called the residue field of <p. Put
D = {x\x <E k, <р(х)т± oo},
p = {x I x e k, <p(x) = 0}.
Then d is a valuation ring of k, p is the maximal ideal of о and ojp is
isomorphic to k'; so we denote the place also by p and <p{x) by p(x). It
there is no fear of .confusion, we denote p(x) by x; similarly, the res-
residue field is denoted by p(k) or k. We call о the ring of p-integers.
If F{X) is a polynomial with coefficients in o, P{X) or F9(X) denotes
the polynomial whose coefficients are the images of the corresponding
coefficients of F by p. The following lemma is fundamental and well-
known. A proof is given in Weil [49].
[77]

76 COMPLEX MULTIPLICATION [II]
<pi on F. We have then S = U Нф. The above result shows that
i
{Н1т-г1,--,Н1туп} coincides with {H^w^Hifn} as a whole. We'have
therefore,
S= U Hl7i = U Hizri = U *Нф = rS.
i i <
By our assumption that (F; {<Pi}) is primitive, г must be contained in
Ht by virtue of Proposition 26. This proves that г is the identity on
F; so we have c(£)' = c($) for every f eF; this completes the proof.
Proposition 31. Let (F; {<р(}) be a CM-type and (K*;{<pj}) its
■dual; let {A, c) be an abelian variety of type (F; {<pi}) and k a field of
■definition for (A, c). Let a be an isomorphism of k into C. Then, a fixes
every element of K* if and only if there exists a homomorphism of (A, c)
onto (A', c): ,
<
Proof. We note that k must contain K* by virtue of Proposition
30. Extend a to an isomorphism of С into itself and denote it again
by a. Take я invariant differential forms a>i on A for which the re-
relation B) holds for every f e F. Then, the а>(" form a basis of in-
invariant differential forms on A'; and we have 8e'($)a)t" = zri°a>ic. This
shoivs that (A", c°) is of type (F; {<pi<?}). Suppose that there exists a
hofnomorphism Л of (A, c) onto (A', t°). By Theorem 1 of § 2, <5/ is
an isomorphism of £)(,(A°) onto ®o(-^) > putting ац' = (oP.)~1mi, we see
easily di'(£)o>i' = Hno>i for every £ e F. Hence (A", c°) is of type
(F; {wt}). It follows that {<pio} coincides with {<pi} as a whole; so we
have, for every ?eF, (£ £«■')" = £ £n° = T, £"; namely, a leaves in-
invariant every element of K*. This proves the " if " part of the prop-
proposition. Conversely, suppose that a leaves invariant every element
of K* ; then, by the same argument as in the proof of Proposition 28,
we see that {<pio} coincides with {<p(} as a whole; so {A', c') is of the
same type (F; {<pi}) as (A, c). By Corollary of Theorem 2 of § 6 and
Remark below it, we can obtain a homomorphism of (A, i) onto {A", c").
This completes our proof.

CHAPTER III. REDUCTION OF
CONSTANT FIELDS.
The aim of §§ 9-12 is to complement the theory of reduction
modulo p of algebraic varieties, given in [33], with a particular interest
in abelian varieties, toward the later use. We will first recall definitions
and results from the general theory with a slight modification, and
then proceed to the main subject. For omitted proofs in § 9, we refer
to [33].
9. REDUCTION OF VARIETIES AND CYCLES.
9. 1. Places. Let К be a field. We denote by Km the join
К U {oo} of the set К and one additional element oo; and we define,
besides the operation in K, the operation in Km as follows:
a + oa = oo, a[<x> = 0 for a e K,
a • oo = a/0 = oo for a e Кос, a ^= 0.
Let k and k' be two fields; we call a mapping <p of km onto k'm
a place of k if it satisfies
<p(a+b) = u>(a)+<p(b), <p(ab) = <р{а)ф);
k' is called the residue field of w. Put
d = {x\x e k, <р(х)фао],
p - {x [ x e k, w{x) = 0}.
Then d is a valuation ring of k, p is the maximal ideal of о and o/p is
isomorphic to k'; so we denote the place also by p and <p(x) by p(x). If
there is no fear of confusion, we denote p(x) by x; similarly, the res-
residue field is denoted by p(k) or k. We call о the ring of p-integers.
If F(X) is a polynomial with coefficients in o, P{X) or F9(X) denotes
the polynomial whose coefficients are the images of the corresponding
coefficients of F by p. The following lemma is fundamental and well-
known. A proof is given in Weil [49].
[77]

78 REDUCTION OF CONSTANT FIELDS [III]
Lemma 1. Let S be a subring of a field ft, containing the identity
of k. Then, every homomorphism of S into a field k', which maps the
identity of k onto the identity of k', is extended to a place of k whose
residue field is contained in the algebraic closure of k'.
We call a place p of a field k trivial if p is the identity mapping
of k onto itself. We say that a place pt of a field ki is an extension of
a place p of a field ft if fti 3 k and pi(a) = p(a) for every a e ft.
9. 2. Specializations over a place. For the sake of simplicity,
we fix two universal domains К and К with the same or different
characteristics and deal only with the places defined on a subfield of К
taking values in Кж; this restriction is kept until the end of §11.
We denote by P", Pn, Sn, Sn respectively the projective spaces and
affine spaces, of dimension я, denned with respect to К and K. Let
(x) = '(xlt--,xn) be a set of я elements in Koo and (£) = (£,-••,£») a set
of я elements in Йоо. We say that (£) is a specialization of (x) over
a place p of a field K, and write
if there exists an extension p' of p such that p'{xi) = ft for every i.
When p is trivial, we write (x) —»(£) ref. &. If the xt are contained in
И», then (x) has the only specialization over p, which we denote by
P(x) or (x). We can easily verify
(x) - (V) ref. ft, (*') -» (f) ref. p . =ф (*) -» (f) ref. J>,
(,r) - (?) ref. p, (c) -» (f') ref. k ^> (x) - (f 0 ref. p.
If we have (ж) —»(с) ref. p and (y) is a set of elements in Kco, then
there exists a set of elements (jy) in Koo such that (x, y) —»(£, 7) ref. p.
This is an immediate consequence of Lemma 1.
We call a set of elements in Kco or Kco finite if it does not con-
contain 00; so, if (x) is finite, (x) is considered as a point of S", and the
same for Sn. Let (£) be a specialization of (x) over a place p of a field
ft. Then, if (£) is finite, so is (x). Assuming (?) to be finite, we de-
denote by [(л:) —» (£); p] (or [(ж) —» (£); ft], if p is trivial,) the set of elements
of the form a(x)[b(x) in k(x), where a and b are polynomials in o[-X"]

[9.3] REDUCTION OF VARIETIES AND CYCLES 79
such that Щ) i= О, о being the ring of p-integers. It is easy to see that
[(x) —> (£); p] is a local ring having k(x) as its quotient field.
9. 3. Reduction of affine varieties. We call a place p discrete
if the ring of p-integers is a discrete valuation ring of rank 1. The
theory in [33] concerns the reduction of varieties or cycles with
respect to a discrete place. It is not difficult to extend the theory to
non-discrete places, as is indicated in the appendix of [36], by means
of the following lemma, which is a restatement of Proposition 26 of
[33].
Lemma 2. Let p be a discrete place of a field k and (f) a speciali-
specialization of (x) over p. Then there exists a discrete place pt of k(x), which
is an extension of p, such 'that p^x) = (£)•
Corollary. Let p be a place (not necessarily discrete) and (£) a
specialization of (x) over p. Then there exist a field кг containing (x)
and a discrete place pi of kx such that pi{x) = (?).
Proof. Let k0 be the prime field contained in k and p0 the restric-
restriction of p to ke; then we see that p0 is discrete and (f) is a specialization
of (x) over p0. Hence by Lemma 2, there exists a discrete place pi of
ko(x) such that р^я) = (£).
In the present treatment, however, we restrict ourselves to discrete
places, in order to avoid the complication arising from the generaliza-
generalization ; therefore, from now on, a place or an extension of a place means
a discrete one, except when the contrary is specifically stated.
Let k be a field, p a place of k, and о the ring of p-integers. Un-
Until the end of § 11, we use these notations alwa}'s in this sense. Now
let V be an algebraic set in the affine space S", defined over k. De-
Denote by 21 the set of polynomials F(X) in o[X] such that F(x) = 0
for even' (x) e V; then Ж is an ideal of o[X]. We denote by p(V) or
V the set of points (£) in S" such that P($) = 0 for all F e SI. p{V)
is clearly an algebraic set defined over k. We can prove
p(F) = {(a) | (a) e Sn, (a) -* (a) ref. p for some (a) <E V).
We call p{V) the reduction of V modulo p. In general, 2 being an

80 REDUCTION OF CONSTANT FIELDS [III]
algebro-geometric object defined with respect to k, we call the algebro-
geometric object obtained by considering £ modulo p (for which a pre-
precise definition will be given in each case) the reduction of X modulo p,
and denote it by p(Z) or S. Let p' be an extension of p; then we can
easily verify p(V) = p'(V). We have also р(Ж) = p'(X) for every algebro-
geometric object I, whose reduction modulo p is to be defined later.
The following relations can be easily verified:
p(V UW) = p(V) и P(W), p(V n W) с p(V) П P(W),
p(VxW) = p(V)xp(W).
It may happen that P is an empty set even if V is not empty.
If the components of V are all of the same dimension r, and if P is
not empty, then the components of P are all of dimension r.
We shall now define the reduction of cycles in Sn modulo p. We
begin with the cycles of dimension 0. Let Y be a cycle of dimension
■ i *
0 in Sn, rational over k; let Y = 2 tniat be its reduced expression,
*=i
where the mt are rational integers and the a( are points of Sn. Con-
Consider a specialization
Л ' (аь■••, a,)->(«!,-•■, a,) ref. p
and remove those a* having oo as one of their coordinates. Then the
sum Yj'miai of the remaining щ gives a cycle in S"; and it can be
proved that this cycle is uniquely determined by Y and p, and does
not depend upon the choice of the above specialization. We call the
cycle the reduction of Y modulo p and denote it by p(Y) or Y.
Now consider a cycle Z in Sn, of dimension r, rational over k.
Let tij, for 1 <; i^r, 0 ^j g n, be г(я+1) independent variables, over
k, and TiJt for l^i^r, Ogy^n, be r(n + l) independent variables
over k. Let L and Z denote the linear varieties defined respectively
by
У=1

[9.3] REDUCTION OF VARIETIES AND CYCLES 81
Then the intersection-product Z-L is defined and is a cycle of dimen-
dimension 0 in Sn, rational over k(U/). We see that the specialization
(Uj) —>(Uj) ref. p gives a discrete extension p' of p in k(ttj); so we can
consider p'(Z-L). Now, it can be proved that there exists a cycle Z
of dimension r in Sn, rational over k, such that 2-L = p'(Z-L); and
such a cycle Z is uniquely determined by Z and p. We call 2Г the
reduction of Z modulo p and denote it by p(Z). If Z is positive, the com-
components of Z are the components of the reduction of the support of Z.
Let V be an algebraic variety of dimension r in S", defined over
k and a a point of V. We say that a is simple on F if there exist
n—r potynomials F1{X),---,Fn~T{X) in o[X], vanishing on V, such that
mnk(dPildXj(a)) = n-r.
Let XX be a sub variety of F; we say that U. is simple on V if U has
a point which is simple on V. If л: is a generic point of V over k
and if a point £ of P is simple on V, then [* —* £; p] is a regular local
ring, so that it is integrally closed.3' Let 35 be a component of P.
Then, 93 is simple on V if and only if 2? is of multiplicity 1 in the
cycle p(V). Suppose that a component SB of P" is simple on V; then,
a point £ of 35 is simple on F if and only if $ is simple on 35; if x
is generic on V over & and £ is generic on 93 over the algebraic closure
of k, then [x—*$; p] is a discrete valuation ring of rank 1 and every
prime element of о is a prime element of [x —* $; p].
Let V and И7 be two affine varieties, defined over k, and T a
birational correspondence between Fand W, defined over k. Let xXy
be a generic point of T over k and £ a point of F. We say that T
is regular at £ if the coordinates of у are all contained in [x —> £; p];
if that is so, there exists one and only one point у on W such that
£ X)? e !T. Let £' be an extension of £>; then T is regular at a point
£ of F with respect to p if and only if T is so with respect to p'.
3) The proof in [33] of this fact is based on Proposition 17 of that paper,
which the author considered as a simple translation of Proposition 19 of Weil
[44] Chap. V. It is not easy, however, to prove the former proposition by the
same argument as in the proof of the latter. A complete proof is given in
Appendix of [24].

82 REDUCTION OF CONSTANT FIELDS [III]
£ X j? being a point in T, we say that £ and 17 are regularly correspond-
corresponding points by T if T is regular at £ and at y.
9. 4. Reduction of abstract varieties. Let [Va; Fa\ 2>„] be
an abstract variety, defined over k; let, for each a, %a be an algebraic
subset in Va, other than Va, defined over k, containing F.. We call
the system
[Va; F«; g.; 7>e]
a р-юялеф; if the following condition is satisfied:
(V) If $ is a point in Va—%a and у is a point in ¥?—%? such that
f X17 is in "Tfa, then f and -q are regularly corresponding points by T?a.
V= [Va ; F. ; ga; 7)„] being a p-variety, let f£/a№ ; F.w П £/a(i) ; R^}
be a subvariety of [ V,; Fe; 7^], defined over й. Then the system
• [Uaii); Faii) П иящ; ??«(л П С/«<л ; Rpi]
defines a p-variety, whfch we call a subvariety of the p-variety V. Let
И^= \WT; <?r; ©r; 5*jr] be another p-variety and [F.X PFr; H.T; U>t,.T\
the produpt-variet}'t of the abstract varieties [Va\ F,; Тра] and
[WT; GT; SSr]. Put §„ = (g.x ЙРГГ) U (P«X®r). Then the system
[ V.icWr; Har; ф„.; Uft,<,r] defines a p-varierj' which will be called the
product variety of the p-varieties V and W. Every projective variety
denned over k defines a uniquely determined p-variety with empty %„.
Let V = [ V« ; Fa ; %, ; Tfa\ be a p-variety and the xa corresponding
generic points of the Va over k by the Tpa; and let (£1(■•■,!;,.) be a
specialization of (Xi,---, жд) over p. By a full set of representatives at-
attached to P~, we understand the set ($ai,---,£ax) of all the £„ which are
finite and not in ga. We say that V is ^-complete if no full set of
representatives attached to V~ is empty. The following facts are easily
proved: i) the underlying abstract variety of a p-complete p-variety
is complete; ii) every subvariety of a p-complete p-variety is p-com-
p-complete ; iii) the product of p-complete p-varieties is p-complete; iv) every
projective variety defined over k defines a p-complete p-variety.
V being as above, let SB = [SB;; ©;; ©„;] be an abstract variety
defined over an extension ^ of k. We say that SB is a variety in V

[9.4] REDUCTION OF VARIETIES AND CYCLES 83
if the following conditions are satisfied : i) there exists a full set of
representatives (£„(!),■■•, £«d)) attached to F such that, for every Л, Шг
is the locus of £„<,!) over %i\ ii) ®i = SB* П г?«(л '. i") ®/.i is the variety
in !Г„(,)вН) with the projection SBj on F,,^) and SB,, on F,,;,). We
call ЭВл the representatives of SB in Fe(i). For every variety SB. in Fo
which is not contained in g,,, there exists one and only one variety
SB in F such that SB. is a representative of SB in V*. By a point in
F we understand a O-dimensional variety in F. It is obvious that all
the representatives of a point in F form a full set of representatives
attached to F and conversely. Let л: be a point in V and f a point
in F. We say that f is a specialization of x over p if x and £ have
representatives д;в and f. in some F« such that $„ is a specialization
of xa over }); since [xa —> £„; p] is independent of a, we denote it by
[x —> £;})]. A variety SB in V is called simple on F if a representative
SBa of SB is simple on Va.
Let V =[Va\Fa;%a;Tfa] be a p-variety of dimension n. V is
called p-simple if there exists one and only one variety 35 of dimen-
dimension я in Г and if every representative SB,, of SB in F. is of multi-
multiplicity 1 in the cycle \)(Va). If that is so, SB is an abstract variety
defined over k. We call the abstract variety SB the reduction of V
modulo p and denote it by p(V).
V and SB being as above, we see that every subvariety of SB is a
variety in F and conversely; and SB is simple on SB if and only if SB
is simple on V. Let W be an algebraic subset of F, rational over k.
We denote by p(W) or W the set of points of SB which are speciali-
specializations of points in W. Then p(W) is an algebraic subset of SB,
rational over k. Let X = 2 niiAi be a cycle on V, rational over k,
i
where the A( are subvarieties of V. Let k' be a field of definition
for the Ai containing k and p' an extension of p in k'. Take, for each
i, a representative Aia of At in Va and put p'(-<4i«) = E ^®л« where
the 93;„ are varieties in Fe. Put 2C{ = 2' ^©y, where 93^ is the variety
in SB having S5y, as the representative in Va and the sum is taken
over all S3/ which is simple on SB. Then the cycle 2 ™№t on S3 is
rational over %. and determined only by X and £. We denote it by

84 REDUCTION OF CONSTANT FIELDS [III]
p(X) or X. The reduction of cycles preserves the operation on cycles:
Proposition 1. A) Let Vbe a p-simple p-variety; let X and Y be
positive cycles on V, rational over k such that the intersection-products
X- Y and p(X)-p(Y) are defined. Then we have
B) Let V and W be p-simple p-varieties, X a cycle on V and Y
a cycle on W, both rational over k. Then Vx W is p-simple and we
have
) = p(X)xp(Y).
C) Let V and W be p-simple p-varieties; denote by 35 and SB the
reduction of V and W modulo p, respectively. Suppose that W is p-com-
plete and SB has no multiple point. If X is a cycle on Vx W, rational
over k, the<h we have
; ' ' p(prr(X)) =
10. REDUCTION OF RATIONAL MAPPINGS
AND DIFFERENTIAL FORMS.
i 10.1. Reduction of rational mappings. Let V and W be two
p-varieties and / a rational mapping of V into W defined over k. Let
? be a point in V and x a generic point of V over k; put f(x) = y.
We say that/ is defined at £ if there exists a point -q on W such that
[y — i); p] с [x -> f; p];
this definition is independent of the choice of x and any extension of
p. It is easy to see that r? is uniquely determined by / and f. We
shall write /(£) = jj.
V, W and / being as above, suppose that V and W are p-simple.
Let T be the graph of /, x xy a generic point of T over k and £ a
generic point of P' over k. Since [x —> £; p] is a valuation ring, / is
defined at £ whenever W is p-complete. Suppose that / is defined at
some point in P"; then/ is defined at £; put /(£) = -q. Define a ra-
rational mapping / of V~ into W by /(£) = -q with respect to %.. We

[10.2] RATIONAL MAPPINGS AND DIFFERENTIAL FORMS 85
call / the reduction, of f modulo p. We see easily that if / is defined
at a e V, f is also defined at a and /(a) = /(a). Denote by £ the
graph of /. Then we have clearly t Z) %; and % is the only com-
component of T whose projection on V is V.
Proposition 2. Let V and W be two p-simple p-varieties, f a ra-
rational mapping of V into W, defined over k. Suppose that f is every-
everywhere defined on V. Let f be the reduction of f modulo p; denote by
T and % the graphs of f and /, respectively. If % is simple on Vx W,
we' have p(T) = 2, where T and % are considered as cycles.
Proof. The notations x, y, $, rj being as above, let, a x /S a point
of T. As / is defined at a, we have fi = flat) = /(a), so that a X /S is
a point of X. Hence £ is the only component of T. Put p(T) = m%,
where m is a positive integer. If W is p-complete and p{W) has no
multiple point, we have m = \ applying C) of Proposition 1. We can
prove m — 1, without such assumptions on W, applying Theorem 12
of [33] to a representative of T.
Let U, V and W be p-simple ^-varieties; let / and g be rational
mappings of U into V and of V into W, respectively. Suppose that
/ is defined at a point £ on О and g is defined at /(f). Then it is
easy to see that the rational mapping g°f is defined at £ and
g'№) = S(f(?))- We see easily g(f(£)) = g(№) = §(f@)- This shows
10.2. Reduction of functions. Let V be a ^-simple variety and/
a generalized function on V, defined over k, namely, a rational mapping
of V into the projective space P1 of dimension 1. Then / gives a
rational mapping of P' into P1. We say that / is p-finite if / is not
the constant oo.
Proposition 3. Let V be a p-simple p-variety, and f a function on
V, defined over k. If f is p-finite and f is not the constant 0, we have
This is a restatement of Theorem 20 of [33].
Now denote by k* the completion of k with respect to p and by

86 REDUCTION OF CONSTANT FIELDS [III]
o*, p* the closure of о, p in k*; let fi denote the normalized ex-
exponential valuation of k*; namely, fi is the mapping of k* into Z\J {00}
such that
fi(xy) = fKx)+fi(y), Кх+У) ^ Min{ft(x),
o*={a|ae k*, fz(d) ^ 0},
and fi{a) = 1 for every prime element a of 0*.
Lemma 3. Let ^Ж be a vector space over k* of dimension n. Let
Л be a mapping of <Ж into R U {00} such that, for every f,ge *Ж and
every a e k*,
^ Min{Z(f),
iii) Л(аЛ = ft(a)+*(f).
Put ^Го ^ {f\fe W, -Kf) > 0}, Л = {/|/e UL, W2 !}■ Then
^Жи is a ffee o*-module of rank n; and ^Ж^^1 is a vector space of
dimension n over" the residue field p(k).
Proof. It is easy to see that 0^0 forms an o*-module, ^Ж =
and ,Ль\^Л' is considered as a vector space over k of dimension g n.
Ltet /i,-'",/m be elements of r^0 giving a basis of сЛ^'оД-^' over k;
Now we shall show that ^Жй — o*ft-{ \-o*fm. Take and fix a prime
Element ж of o*. Let g be an element of <^V Then there exist m
elements aoi of 0* such that g— 2 aotf( e ^f. Put g — 2 aatf( + 7tgi;
then g! is contained in ^#0. Applying the same argument to gu we find
m elements au of 0* and an element g2 of <^<f 0 such that ^ = £ anfi+ng2.
Repeating this procedure, we obtain m sequences {a,i} of elements of 0*
and a sequence {#„} of elements of c^o such that 5r« = 2
i
00
Since ft* is complete, the series 2 e»i»f"> for each 1, has a meaning
and defines an element bt of o*. Then, it can be easily verified
that g = 2 btfi. This proves that <^f0 = "*/H ho*/m. As we have
c^ = к*^Же, we must have w = я; this proves our lemma.
Let V be a p-simple p-variety and X a divisor of F, rational over
k. Consider the set L{X; k) of the functions / on V, defined over k.

[10.3] RA TIONAL MAPPINGS AND DIFFERENTIAL FORMS 87
such that (/) > —X. Denote by L<,{X; k) the set of p-finite elements
in L(X; k). Suppose that L(X; k) is of a finite dimension я over k.
We shall now prove that L0(X; k) is a free o-module of rank я.
The notations k*, o* and p* being as above, we can choose our uni-
universal domain IT so that it contains k* as subfield. Let k(V) and k*(V)
denote the fields of functions on V defined over k and over k*, re-
respectively. Then, k(V) and k* is linearly disjoint over k; so we can
define a tensor product ^Ж of L(X; k) and k* over k as a submodule
of k*(V). Let j; be a generic point of V over k* and £ a generic
point of V over p*(k*). Then [x —» £; p*] is a discrete valuation ring
and every prime element of о gives a prime element of that valuation
ring. Therefore, considering the isomorphism between k*(x) and k*{ V),
we obtain an exponential valuation Л of k*(V), which satisfies the con-
conditions, of .Lemma 3.. Hence, by that lemma, if we_ denote.by ^Жо
the set of p*-finite elements in ^^, ^^^ has a basis {gi,---,gn} over o*.
Let {hlt---,hn} be a basis of L(X;k) over k; then the gt are expressed
in the form gi = £ Cijhj with cy in k*. As A: is dense in k*, there
exist elements dij in & such that the matrix (rfy) (cy) is congruent
with the unit matrix modulo p*. Put /i = 2 rf^/г^. Then we see that
{/i,••-,/»} gives a basis of *Жъ over o*, and hence a basis of L0(X; k)
over o. Thus we have proved that Le(X; k) is a free o-module of гап/г
п. By Proposition 3, we have (/4) > — p(X). As the /4 are linearly
independent over k, we obtain the inequality
A) l(X) ^ l(p(X)).
10. 3. Local parameters at a point of R Let V be a p-simple
p-variety of dimension r; let £ be a simple point on V ; then f is
simple on V. We call a set of r functions <pu •••,yr in k{V) a system
of local parameters for F at £, defined over ft, if the following con-
conditions are satisfied:
(Lo 1) k{V) is separably algebraic over k((pu---,<pr).
The ^ are all defined and finite at £.
For every f in k(V), defined and finite at f, df/dpt is defined
and finite at £ for every i. (For the notation djd<fit see § 2. 2.)

88 REDUCTION OF CONSTANT FIELDS [III]
Proposition 4. V and £ being as above,, let x be a generic point of
V over k and Va, xa, f „ be representatives of V, x, f; and let Sn be the am-
ambient space for Va. Then, r elements $»!,•••, <pr in k[V) form a system of
local parameters at $ if and only if the following conditions are satisfied.
(IV 1) The <pt are defined and finite at £.
(Lo'2) There exists a set of n polynomials Fi(X1,---, Xn, Tlt-~, Tr)
in o[Xb—, Xn, Ti,—, Tr] such that Ft(x., t) = 0 for 1 £ i ^ я and
where U = <pt(x), n = ^i(f) for lgi'^r.
Proof. We first prove the "if" part. Let (xal,---,x*n) be the
coordinates of л;« and fi the function on V denned by fi(x) = х„{, with
respect to h- By Corollary of Theorem 1 of Weil [44J Chap. I, (Lo' 2)
implies that k(x) is separably algebraic over k(t); so k{V) = k(fu---,fn)
is separably algebraic oVer к((р^---, <pr). Differentiating the equations
Fn(xa, t) = 0, we have
=h -
As, we have det(dPa/dXj(£a, r)) ^= 0, we observe that the functions
dfijd<pj are all defined and finite at f. If g is an element of k(V) de-
defined and finite at f, we get a representation g(x) = G{xa)lH{xa), where
G{X) and H(X) are polynomials in o[X] such that Й($.) =f= 0. We
have then
/\ о /y\ f-ffv \
OWj
for ] <^j^n. This shows that the functions dg/d<pj are all defined
and finite at £. The " if " part is thereby proved. Conversely, suppose
that the (pt satisfy the conditions (Lo 1-3). As $„ is simple on Va, there
exist n—r polynomials A,{X) in o[X] such that Ay(xa) = 0 and
D) rank( .^(f»)) = n—r.
\ oJi.i /

[10.3] RATIONAL MAPPINGS AND DIFFERENTIAL FORMS 89
From the relation A,(xa) = 0, it follows that
By (Lo 2), each <pt has an expression ^(л;) = Bt(xa)/Ct(xa), where Bi
and Ci are polynomials in o[X] such that d(£.) ^= 0; then, differen-
differentiating Вл(ха)—(ph(x)Ch(xa) = 0 and substituting £ for x, we have
1=1
where fty = 0 or 1 according as hj=j or /г = j. Put
Fh{X, T) = Sfc(J0- ГлСл(^) (ISigr.,
Fr+>,(X, Г) = Л(-Х) Ag^ я-г).
We have then F((xa, t) = 0 for 1 ;S i ^ я, and, by the relations D), E),
F),
This completes the proof.
We have to show the existence of a system of local parameters.
Let the notations be the same as in Proposition 4. As fo is simple
on Va, there exist n—r polynomials Gt(X) in o[X] such that Gt(xa) = 0
and
We can find r linear forms Ht(X) — j cj/X/ A g i ^ r) with Ci;- in o,
>=i
such that
n
Let coj, for each i, be the function on V defined by <ft(x) — 2 djX<.j
У=1
with respect to k, where (xal,---, xan) denotes the coordinates of xa. Put

90 REDUCTION OF CONSTANT FIELDS [III]
Fh(X, T)=Th-Z CKfXj A S Л S r).
Fr+i(X, T) = Gt{X) A £ i S n-r).
Then it can be easily verified that the set {<pi,---,(pr} satisfies the
conditions (IV 1-2) of the above proposition for these F<; so the щ
are local parameters at f.
Proposition 5. Let V be а p-simple p-variety, £ a simple point on
V and {<pi,---, <pr} a system of local parameters for V at f, defined over
k. Then {&!,■■-,<рт} is a system of local parameters for V' at £. More-
Moreover, if f is an element of k{V), defined and finite at f, then f and the
dfjdwi are all p-finite and we have, for every i,
G) 1; = ll.
dipt d<pi
Proof. ■ The first assertion follows from Proposition 4. Let / be
an element jof k(V), defined and finite at f. Then by (Lo 3), the
df\dtpj are defined' and* finite at f. It follows that / and the df/3<pi
are all defined and finite at a generic point i) of V' over k; this
implies that/and th&dfjdipi are all £>-finite. Considering the equations
B) ajid C) in the proof of Proposition 4 modulo the maximal ideal
of the valuation ring [x —»• у; р], we obtain the equality G).
•' 10.4. Reduction of differential forms. V being as in § 10.3,
let a> be a differential form on V, defined over k. We say that a> is
p-finite if it has an expression
@
where the /(i) and the gt are p-finite elements of k(V). We saj' that
o) is finite at a simple point f of P" if the fw and the gt in the above
expression can be taken in such a way that they are all defined and
finite at $. со is p-finite if and only if it is finite at a generic point
on V over k. Let {<fi,---,<pr} be a sj'Stem of local parameters at $.
We have then
where

[10.4] RATIONAL MAPPINGS AND DIFFERENTIAL FORMS 91
e( } .') denoting the sign of the permutation ( .* .*). By (Lo 3), a»
is finite at £ if and only if the hu) are defined and finite at f. Sup-
Supposing that the fw and the gt are p-finite, we have, by Proposition 5,
(О V)
so that
(8) fwdgil--dgf,= 2 Ti^dipj^-dipj,.
As the h(j) are determined only by ft» and <pi,--% <pr, the relation (8)
shows that the differential form £ fn)dgu---d^it on P" is determined
only by a» and is independent of the choice of the /(,) and the g4.
We denote the differential form by p((o) or <5 and call it the reduction
of ей modulo p. We can easily verify the following facts.
i) If <o and си' are p-finite, then (o+a>' and ш-ш' are p-finite and
we have p(w+w') = p(w)+p(a>'), р(ш-а>') = р{и))-р{ш').
ii) If w is p-finite, then dw is p-finite and we have
dp(oi) = p(dw).
in) If ш is a differential form other than 0, there exists an element
a of k such that aai is p-finite and р(аш) f= 0.
The last assertion is a special case of the following proposition.
Proposition 6. Let V be a p-simple p-variety; let ^Ж be a vector
space over k of differential forms on V of degree s, defined over k, and
e^fo the set of p-finite elements in ^Ж. If *Л is of a finite dimension m over
k, then ьЖь is a free o-module of rank m, and the set {p{w)\w e ол£о}
is a vector space of dimension m over k.
Proof. Let {<р%,—,фг} be a system of local parameters for V at
a generic point of t? over k. Denote by Z the exponential valuation of

92 REDUCTION OF CONSTANT FIELDS [Ш]
the field k(V) introduced in §10.2. For every differential form
put
X(w) = Mm (*(hw)}.
W
Then X satisfies the conditions i-iii) of Lemma 3, and a> is ^-finite if
and only if X{a>) ^ 0. So our proposition is proved by the same argu-
argument as in the last part of § 10. 2.
Proposition 7. Let V and W be p-simple p-varieties, T a rational
mapping of V into W and ш a differential form on W, defined over k.
Suppose that there exists a point f on V' such that T is defined at f and
<o is finite at T(£). Then <w° T is defined and finite at f; and we have
p(w°T) = p(a>j°T.
This follows easily from our definition.
Proposition 8. Let V be a p-simple p-variety of dimension r and
<o a differential form op. V defined over k. Suppose <o to be p-finite and
p(w) i= 0- Then we have
4 .
■wherk (y) denotes the divisor of a differential form ij. If ш is of degree
r, then we have
{For the definition of the divisor of a differential form, we refer to
Nakai [28].)
Proof. Let SI be a simple subvariety of P of dimension r—1;
let {<Pi,---,<Pr} be a system of local parameters on V at some point of
SI. Then, w has an expression
w = ^ fwd(Pii---d<pit,
ii-=:■••-a,
"where the fw are p-finite elements of k(V); we have then
р(ш) = 2 f'wdfiu---dpir

[10.4] RATIONAL MAPPINGS AND DIFFERENTIAL FORMS 93
Take an extension k! of k and an extension p' of p in k' such that
the components of the divisors (/<«) are defined over k' and 81 is de-
defined over p'(£')• Let А1г—, At be the components of (/(t)) such that
Я is a component of p'(A,). By Proposition 4, we see that the ps
are local parameters for V along A, for every v. Denote by v,(a) the
multiplicity of A, in the divisor (or) of a function or a differential form
a, and similarly by v the multiplicity of 21; denote further by ft(A, SI)
the multiplicity of SI in the cycle p'(A). Then, by Proposition 3, we
have, for every (i),
This proves (p(a>)) > p((<*>)). If <u is of degree r, w is written in the
form a» = fdtpi—dtpr, so that f>(a>) = fdip^'-dipr- We have then
«(/) - s к^, адл = s a<4, ад«);
this implies {p(a>)) = })((<»)).
Proposition 9. LeZ F ie a p-simple p-variety; suppose that V is
p-complete and P has no multiple point. Let ш be a differential form on
V of the first kind, defined over k. If ш is p-finite, then p(a>) is of the
first kind.
Proof. We first note that V has no multiple point. Now by
Proposition 5 of Koizumi [23], a differential form tj on a complete
non-singular variety is of the first kind if and onty if (tj) > 0. Our
proposition follows from this and from Proposition 8.
U being a complete non-singular variety, denote by h,(U) the
number of linearly independent differential forms on U of degree s,
of the first kind. Then, by Propositions 6 and 9, we get:
Proposition 10. Let V be a p-simple p-variety. Suppose that V is
p-complete and & has no multiple point. Then we have, for every s,
h,(V) £h(P).
We conclude this section by a simple application to curves:
Proposition 11. Let С be a p-simple curve. Suppose that С is p-

94 REDUCTION OF CONSTANT FIELDS [III]
complete and С has no multiple point. Then С and С have the same
genus.
Proof. Take a differential form w on С such that <o is £>-fmite
and }>(u>) ,fc 0. This is possible by Proposition 6. Let g and § be the
genera of С and C, respectively. Then, by Proposition 8, we have
2g-2 = deg(o>) = deg(K<*>)) = 2g-2,
and hence g — g.
11. REDUCTION OF ABELIAN VARIETIES.
11.1. Let A be an abelian variety defined over k. Denote by /
the rational mapping of Ax A into A defined by
f(x,y) = x+y.
and by g the rational mapping of A into itself defined by
g(x) = -x.
Suppose that a structure of p-variety is defined on A. We say that
A has яо defect for p (with respect to this structure) if the following
conditions (Al-3) are satisfied.
(Д1) A is p-simple and p-complete.
(A2) / is everywhere defined on Ax A.
(A3) g is everywhere defined on A.
Under these conditions, A becomes, in a natural way, an abelian
variety defined over k. As is remarked in § 10.1, / and § are every-
everywhere denned on Ax A and on A, and we have /(?, 7) =/(£, 7),
gtc) = £(f) for csi,?ei. Put f+т) = /(£, 7). Then it can be
easily verified that Л is a group variety with respect to this law of
composition, defined over ft, and g(f) gives — f. As A is p-complete,
A is a complete variety, so that Л is an abelian variety; if 0 denotes
the origin of A, then p@) is the origin of A. We call the abelian
variety A the reduction of the abelian variety A modulo p.
Proposition 12. Let A and В be two abelian varieties having no

[11. 1] REDUCTION OF ABELIAN VARIETIES 95
defect for p ; denote by <^'{A, B;k) the set of all homomorphisms of A
into B, defined over k, and by J£f(A, В; k) the set of all homomorphisms
of A into B, defined over k. Then, for every X e Jtf{A, В; k), the re-
reduction /, of the rational mapping X modulo p is an element of J&{A, B\k);
and the graph of X is Иге reduction of the graph of л modulo p, in the
sense of reduction of cycles. The correspondence Л —»2 defines an isomor-
plnsm of the additive group J^{A, В; k) into J£f{A, B;k). If A = B,
this isomorphism is a ring-isomorphism. If A and В have the same di-
dimension, vie have v(X) = v(l) for every 1 e ^JC^{A, В; k).
■ Proof. We shall first show that every ;. e J(f{A, В; k) is every-
everywhere defined on A. Let £ be a point of A and rj a generic point of
A over k($). Then, f+tj is generic on A over k, so that 2. is defined
at у and s+4- Take two independent generic points x, у of A over
k and define a rational mapping h of AxA into В by
then h is defined at £ x^ since 2. is defined at у and at £+9. Put
С = A(£, if). We have clearly h(x, y) = 2(x). Hence we have
Since the ring in the left hand side is containd in k(x), we have
W*)-C;|>] с [*-£;*>]
by virtue of Proposition 7 of [33]; this shows that Л is defined at £.
Thus /. is everywhere defined on A. Then, by Proposition 2, the
graph Г-i of / is the reduction of the graph Гг of к modulo p, both
considered as cycles. It is easy to see that 1 is a homomorphism
of A into В and 2. —> 1 gives an additive mapping of <J£f(A, В; k) into
cJC^iA, В; k). If / is not 0, we have dim^hc) ^ 1 for a generic point
x of A over k. Let D be the locus of X(x) over k. As В is f>-com-
plete, p(D) is not empty, so that there exists a specialization у of X(x)
over p such that dimj^) 2: 1. As A is p-complete, there exists a point
£ such that x x Л(ж) —» £ x i) ref. jj; we have then J(£) = у; this shows
that / is not 0. It follows that 1 —> 1 is an isomorphism. Now assume
that A and S have the same dimension. By the definition of vB),

96 REDUCTION OF CONSTANT FIELDS [III]
we have ргд(Г^) = v(Z)B, pri(/*i) = v@)B. As we have p(B) = Д
p(Fi) = Гi, we get, by C) of Proposition 1, v(l) = v{X). It is clear that,
when A = B, ?.—>% gives a ring-isomorphism.
Proposition 13. Notations being as in Proposition 12, let 2. be an
element of J(f(A, В; k). Suppose that 1 is an isogeny of A onto В and
every element of the kernel д(Я) of X is rational over k. Then the reduc-
reduction of points modulo p defines a homomorphism of д(Л) onto the kernel
of 1. If v{B) = 1, this homomorphisms is an isomorphism.
Proof. Denote by Fx and Fi the graphs of ?. and X respectively.
We have then
prA[/V(^x0)] = 2 att, ргЛЛ.(ЛхО)] = 2 атт,
where the sums are taken over all elements t e д(Л), г е д(/Е) with
certain multiplicities at, ar, respectively. By the relation p(Ft) — Fi
and by Proposition 1, we obtain
' p{prAlFi-(Ax0)]} =ргл[/>(Лх0)].
This shows that the reduction modulo p gives a surjective mapping
of q(P.) onto gC). It as clear that this mapping is a homomorphism.
Hence the order of д(Л) is not less than the order of gB), so that
vt(Z) jk '"«(/I). If щA) = 1, we get, by the relation v(X) = v(I), the equality
V'Q)- This proves the last assertion.
Proposition 14. Notations being as in Proposition 12, let I be a prime
other than the characteristic of k. Then, we can choose l-adic coordinate-
systems of Si(A), Si(B), %i{A), Qi{B) in such a way that:
i) for every 2 e Jtf(A, B; k), we have Mt(Z) = МД);
ii) for every divisor X on A, rational over k, we have Ei{X) — Ei(X).
Proof. Let k' be an extension of k over which every point of
Qi(A) and Qz(-B) are rational. Take an extension p' of p in k'; p' may
not be discrete. We will now consider the reduction of the points in
8i(A) and 6i(B) modulo p'. Since every point of &i(A) and Qi(B) is
rational over a finite extension of k, the reduction modulo p' of a
point of Qi(A) and Qi(B) is in substance the same as the reduction
modulo a discrete place. Then, we see by Proposition 13 that the

[11.2] REDUCTION OF ABELIAN VARIETIES 97
reduction modulo p' gives an isomorphism of Qi(A) onto 9i(A) and an
isomorphism of Si(B) onto сц(Ё). Hence we can choose Z-adic coordi-
coordinate-systems of 8i(A), Qi(A), Qi(B), Qi(£) in such a way that a point x
in Qi(A), or Qi(B), has the same Z-adic coordinates as p'(x). Then, we
get obviously, with respect to these systems, Mt(Z) = MiQ) for every
;. e J£f(A, В; k). Let k" be the algebraic closure of k' and Ui the
set of roots of unity in k" whose orders are powers of /. Take an
extension p" of p' in k"; then p"(Ui) is the set of roots of unity in
\>"{k") whose orders are powers of /. Choose isomorphisms of Ui
and p"(Ui) onto QijZi in such a way that for every Q e Ui, £ and p"(Q
have the same image in Qi/Zi. Then we can easily verify, following
step by step the definition of the matrix Ei(X), the relation Ei(X) = Et(X)
for every divisor X on A, rational over k.
11.2.. k and.p being as before, let F be an algebraic number
field; and let (A, c) be an abelian variety of type {F), defined over k.
Put r = Г1 [unf(A) П i(F)]. If A has no defect for p, then, for every
/ e c(x), we obtain, by reduction modulo p, an element Z of ^(A).
Put t(fi) = Aji) for every [i e r. Then с is an isomorphism of r into
^(A) such that ?(l) = li; we can extend this isomorphism to an
isomorphism of F into ^0{A), which we denote again by I. Thus
we obtain an abelian variety (A, i) of type (F), defined over k. We
call (A, f) the reduction of (A, c) modulo p. (Д t) is clearly of the
same index as (A, c). If {A, t) is principal, so is (Д i).
Proposition 15. Let {A, c) be an abelian variety of type (F), defined
over k, ivhich is principal. Let a be an integral ideal of F and {Аъ tt; X)
an a-transform of (A, c), defined over k. Suppose that A and Ax have
no defect for p. Then, (Alt «i; 1) is an a-transform of (Д I).
Proof. It is clear that 1 commutes with the operation of F. Let
.г" be a generic point of A over k and f a generic point of A over k.
For even' a e a, we have k(c(a)x) С k(Ax), so that we obtain a rational
mapping fi of At into A, defined over k, such that fx{)x) = c(a)x; /x is
clearly a homomorphism. We have then p%£ = c(a)£, and hence
k(t(a)£) С k(X£). Therefore, if lt is an a-multiplication of A, defined
over k, we have kB£) z> kQ.^). On the other hand, by Proposition 10

98 REDUCTION OF CONSTANT FIELDS [III]
of §7.2 and Proposition 12 of § 11.1, we have
[Щ) •• km = <ъ = «w - wwro = «ft) = iko: ад,?)],
where m denotes the index of (A, c). Hence £E?) = k(Xi$). This shows
that J is an a-multiplication.
Proposition 16. (Д 0 Ьегя^ as in Proposition 15, /e£ a be an integral
ideal of F. Suppose that A has no defect for p and every point of g(a, A)
is rational over k. Then, the reduction modulo p defines a Iwmomorphism
of g(a, A) onto g(a, A). Moreover, if a к prime to the characteristic of
k, tliis homomorphism is an isomorphism.
Proof. We can find an integral ideal b, prime to a and the char-
characteristic p of k, such that ab is a principal ideal G-). By Proposition
18 of §7.5, we have
) = 8@, А)+ф, A), g((r), A) = g(a, Л)+8(Ь, А).
Take an extension k' oft k such that every point of 9((f), A) is rational
over k', and an extension p' of p in k'. Then, the reduction modulo
p' gives honiomorphisms of д((^), А), д(а, А), д(Ь, A) respectively into
$)((;•), Л), g(a, Л), д(Ь, »Л). By Proposition 13, q((t), A) is mapped onto
й((?-)^Л); so g(a, ^4) must be mapped onto g(a, Л). If a is prime to p,
g(n, (A) and g(a, A) are of the same order N(a)m, where w is the index
of £A, c) ; hence the reduction modulo p gives an isomorphism of
g(a, A) onto g(a, Л).
11.3. Consider now the case where both k and k are of charac-
characteristic 0. Let k be a subfield of С and p a place of k taking values
in C. Let (F; {<fi}) be a CM-type and (Д f) an abelian variety of
type (F; {cpt}), defined over k. Suppose that A has no defect for p,
k contains U Ffi and for every £ e U -F1", p(f) = f. Under these as-
assumptions, we shall prove that the reduction (Д t) of (A, t) modulo
p is of type (F; {<pi}). By the definition of CM-type, there exist
invariant differential forms aib---,ain on A such that
d(c(a))o>i = a*4Di A^!^ я)
for every a e r, where я is the dimension of A and r is the order of
(A, c). On account of the results of § 2, we may assume that the cot

[12.1] THE THEORY " FOR ALMOST ALL p " 99
are defined over a finite algebraic extension k' of k. Take an exten-
extension p' of p in k' and consider the reduction modulo p'. By the as-
assertion iii) of § 10.4, which is a particular case of Proposition 6,
we may assume that the o>i are p'-finite and р'(ац) =*f 0 for every i.
Then we get, by Proposition 7,
A) 8{с(а))р'{Ш1) = p'ianyp'im,) (l^ig я).
By our assumption, we have p'{op<) = a*'; so the relation A) shows
that (Д t) is of type (F;
12. THE THEORY " FOR ALMOST ALL p ".
12.1. Preliminary lemmas. The notations k, о, р being as be-
before, let V be a variety of dimension r in the affine space Sn, defined
over k. Let (x) be a generic point of V over k and the ty, for 0 jj г ^ r,
n
1 ^; g я, be (r+l)n independent variables over k(x). Put yi — S UjXj
for Ogi^r. Then we have kit, x) = k(t,y). As <уй,---,ут) is of
dimension r over k(t), there exists an irreducible polynomial F(T, Y)
in k[T, Y] such that F(t, y) = 0. Substituting 2 ТцХ] for Y{, we
obtain from F(T, Y) a polynomial G{T, X) in (Гу) and (Xj). We
write G(T, X) as a polynomial in (T1^) with coefficients in k[X] and
denote by the Н„(Х) for l^itgJ those coefficients. We call the set
{Ha(X)} a k-basic system for F. We can take F in such a way that
all its coefficients are contained in о and some coefficient is a p-unit.
F being taken as this, we call {Ha(X}} a p-basic system for V.
Lemma 4. Let V be a variety of dimension r in Sn, defined over
k, and {Ha{X) 11 g a g s) a p-basic system for V. Then a point (£) of
Sn is contained in V if and only if #«(£) = 0 for Igags.
Proof. Let (x), {t), (y), F, G be as above; and let (?) be a spe-
n
cialization of (x) over p. As we have F(Uj, 2 UjXj) — 0 and the tij are
.7 = 1
independent over k(x), we have Ha(x) — 0 for 1 ^ a ^ s, so that
Й«(с) = 0 for 1 ^ a ^ s. This proves the " only if" part. Converse-

100 REDUCTION OF CONSTANT FIELDS [III]
ly, let (?) be a point of Sn such that #"„(£) = 0 for Iga^s. Let
the r^ for 0 ^ i:^ r, 1 gy ^ n be (r+l)n« independent variables over
£(?). Put щ - Z *ij£j for Ogi^r. We have then
Let W be the locus of (r) over £(«) in 5r+1. The specialization (t) -* (r)
ref. p gives an extension p' of f> in &(£). By Theorem 21 of [33], we
have
p'(W) = {(Q | @ 6 5-«, /(r№ 0) = 0}.
Hence (jj) is contained in p'(W); so (w) is a specialization of (y) over
t>'. Let (f) be a specialization of (x) such that
By Proposition 16 of [33], (f) is finite, so that it is a point of' Sn;
and we have 2 ry£'* = m = 2 ^yf,• for O^i'^r. Assume that
f> =5*= f j for one of the j, say 1; we have then
As j(f') is a specialization of (ж) over )j, we have dimjt(f') ^ r. Hence
we' have
dimsw)(r) ^ dimjm(f, r«0" > 1)) ^ г+(г+1)(я-1) < (г+1)я.
This contradicts the assumption that the Ti}- are (r+l)n independent
variables over £(£). Therefore we have (f) = (f). Hence (f) is a point
of J7; this completes the proof.
We will now study the reduction of algebraic varieties modulo
infinitely many p. Let k be a field and 2 a set of discrete places
of k. In the rest of this section, we use k and 2 always in this sense.
We call a subset a of 2 an орея sei of 2 if there exist a finite num-
number of elements аи---,аг, other than 0, in k such that
<j= {PlpeZ, P(ai) Ф 0,-, Jj(or) =*= 0}.
For any set of elements {ab---, ar} in k, none of which is 0, the set

[12.2] THE THEORY "FOR ALMOST ALL p" 101
of p in 2 such that the a4 are p-units, is an open set. In fact, we
have
о = {P | P e 2> P(ei) ^ 0, fla,) ^ °.-. Wer) ¥= 0, Кат») ?= 0}.
We say that a proposition P(p) concerned with p in 2 holds for almost
all p if .P(p) holds for all p in an open set of 2-
Lemma 5. k and 2 being as above, let k' be a finitely generated
extension of k, of dimension s over k. Let 2' be a set of discrete places of
k' satisfying the following conditions:
• i) there exists a set of elements (tu- ■•,£«) in k' such that we have
dimr№,G>'(*i),-, »>'(*,)) = s for every p' in 2';
ii) for every p in 2, there exists an extension of p in 2'-
Let a' be an open set of 2' and о the subset of 2 consisting of all p in
2 such that p has at'least one extension p' in 2' Then a contains an
open set of 2-
Proof. By our definition, there exists a set of elements (yi,---,yr)
in k' such that y( j= 0, and if р'(у,) ^ О,---, p'(yr) j= 0 and if p' e 2'.
then p' e a'. By our assumption, tu--,ts are independent variables
over k, and k' is algebraic over k(t). Let 2 Ьь Y" = 0 be an irreducible
equation for yf1 over k(t), for each i. We may assume that the Ьы
are polynomials in k[tlt---,t,]. Let a0 be the set of all p in 2 such
that all non-zero coefficients of the polynomials bu are p-units. Then
a0 is an open set of 2 • ^et f be a place in a0; by our assumption,
there exists in 2' an extension p' of p, and p'{t-i),---,P'{ts) are inde-
independent variables over p{k), so that the bi, are all ^'-integral; moreover,
any one of the bit>, other than 0, is a jj'-unit. It follows that the yf1
are all (/-integral, so that p'(yi) ^ 0 for every i; namely, p' is contained
in a'. Hence ae is contained in a; this proves our lemma.
12.2. Now, it is easy to verify that the results of [33] § 6 are
extended to the present case. Namely, Proposition 29, Lemma 3,
Proposition 30, Theorem 26 in that section are all true when we use
the terms " for almost all p " in the sense explained above. In parti-
particular, we have

102 REDUCTION OF CONSTANT FIELDS [III]
Proposition 17. Let V be a variety in Sn defined over k. Then,
for almost all p, p(V) is not empty and V is p-simple.
We have to show that p(V) is not empty for almost all p, as it
was not explicitly proved in [33]. Take a point (alt~-,an) in V such
that the at are contained in a finite algebraic extension k' of k. Let
2' be the set of all extensions in k' of all p in 2 and a' be the set
off)' in 2' such that the a4 are p'-integral. Let a be the set of all p in
2 having an extension p' in o-'. Then by Lemma 5, a is an open
set of 2- It is easy to see that p(V) is not empty for every p in o.
12. 3. We shall now give several properties preserved for almost
all p in the process of reduction mo'dulo p.
Proposition 18. LetF1(X),---,Fr(X) be r polynomials in k[Xu--,Xn]
and U the algebraic set in Sn given by
Then, we have
for^ almost all p.
) Proof. We may consider only those p for which the coefficients of
rfie F4 are all ^-integral. It is easy to see
so we will now prove that the inverse inclusion holds for almost all
p. Suppose that U is not empty. Let Ult---, Us be the components
of U and k' be a finite algebraic extension of k such that the Ui are
all defined over k'. Let 2' be the set of all extensions in k' of all
{) in 2. Let {H^a{X); 1 ^ a ^ h) be a &'-basic system for Uu for
each i. Then, by the definition of basic system, {Hwa} is a p'-basic
system for U( for almost all p' in 2'-
НЯг....г(Х) =
put Hx(X) = 1, if U is empty. Then, by Lemma 4, we see easily
P'(U) = {@ | Я.,....^) = 0 for every («)}

[12; 3] THE THEORY " FOR ALMOST ALL p" 103
for almost all p' in 2'. By Hilbert's theorem, there exists a positive
integer p such that Hai...at{X)' = £ Q(вИ(Х)^(А'), where the £>(eL
are polynomials in fc'[.X]- The coefficients of Qwi are all (/-integral
For almost all Jj' in 2У- F°r those p', we have
and hence
p'(U) з.{@ I ^p-(f) = 0 A g x g r)}.
By virtue of Lemma 5, this proves our proposition.
Proposition 19. Let U and V be two algebraic sets in Sn, defined
over k. Then, we have p(U П V) = p(U) П P(V) for almost all p.
Proof. Let a and Ъ be the ideals of k[Xu---, Xn] given by
a = {F(X).\ F(jc).= 0 for every (x) e U),
Ъ = {G(X) | G(x) = 0 for every (x) e V}.
Let {F1(X),-,F,(X)} and {G^X),-, GS{X)} be bases for a and 6,
respectively. Then we have
U = {(x)\F((x) =0(Utg r)}, V = {(x)\Gf(x) = 0(l^ii)},
U П F = {(xMFiO) = 0, G/a:) = 0 A ^ f ^ r, 1 gj g j)}.
Therefore, our proposition is an immediate consequence of Proposition
18.
Proposition 20. Let U be an algebraic set, defined over k, in
5"+m = SnxSm and V the projection of U on the first factor Sn. Then,
p( V) is the projection of p( U) on the first factor of p{S"™) = p(Sn) X P(Sm),
for almost all p.
Proof. It is sufficient to prove our proposition in case where U
is a variety defined over k, since the general case is easily reduced to
this particular case by means of Lemma 5. Assuming U to be a
variety defined over k, let (x, y) be a generic point of U over k, with
the projection (x) on Sn and (y) on Sm; and let s be the dimension
of (у) = (Уи"-,Ут) over k(x). If s is not 0, we may assume that ylt---,ys
are independent variables over k(x) and (y) is algebraic over к(х,у^--,Уш).

104 REDUCTION OF CONSTANT FIELDS [III]
The locus of (x,yu---,ys) over k is the variety VxS'. As we have
p(VxS*) = p(V)xp(S'), the projection of p(VxS') on the first factor
of p(Sn) Xp(Ss) is p(V). Therefore, our proposition is proved if
we show that the projection of p(U) on the factor p(Sn) x P(S') of
p(S")Xp(S')xp(Sm-') is p(VxSs) for almost all p. Hence it is sufficient
to prove our proposition in case where (y) is algebraic over k(x).
Suppose that this is so; let ty, for l^igr, 1 ^/ <; n, be rn independent
variables over k(x), where r is the dimension of (x) = (xt,-•-,«„) over k.
n
Put Zi = 2 *y*y for 1 s£ i s£ r; then (г, г) is г(я + 1) independent variables
over & and (x) is algebraic over k(t, z). By Proposition 17, p(V) is a
variety defined over p(k) for every p ' in an open set и of J. Let
Tijp and Qip, for 1 g i ^ r, 1 jjy g n, be г(я+1) independent variables
over p(A). Take and fix, for each p in a, a point fp in the intersection
of p(F) and the generic linear variety denned by
, S гур^-£р = 0 (lg.g r).
.7 = 1
Then, fp is a generic point of p(V) over p(&). We see that (fp, Гр, Ср)
is a specialization of (x, <, z) over Jj. Let 2' be the set of all ex-
extensions p' of p in k{x, t, z, y) such that
■; (x, t, z) -^ (fp, Гр, Ср) ref. p'.
Let o-' be the set of all p' in 2' such that the y( are all (/-integral,
and let a0 be the set of all p in a such that p has at least one exten-
extension p' in <t'. Then, by Lemma 5, a0 contains an open set of 2 ■ Let
% denote the projection of p(U) on the first factor p(Sn) ofp(Sn)xp(Sm).
For every p in a0, we can find a point зур in pE"m) such that
(x>y)~~~>(£p> 7p)ref-P- Hence 3Sp contains fp; this implies 93p Z) p(F).
On the other hand, if ap is a generic point of a component of 35F
over the algebraic closure of p(k), there exists a point (ap, /3p) in K(t7).
As (ap, j8p) is a specialization of {x, y) over f), the point ap is contained
in p(V); so we have 93p С p(V). Thus we have proved 93p = p(V)
for every p in <70. This proves our proposition.
Proposition 21. Let V be a variety in Sn, defined over k, andf a
rational mapping of V into Sm, defined over k. Let F be an algebraic

[12. 3] THE THEORY " FOR ALMOST ALL p " 105
set contained in V, defined over k, different from V. Suppose that f is
defined at every point in V—F. Then, for almost all p, f is defined at
every point in p(V)—p(F).
Proof. It is sufficient to prove the proposition in case where
m = 1. Let x be a generic point of V; then we have an expression
f(x) — Q(x)IP(x), where P and Q are potynomials in k[Xu--,Xn]. Let
the Wlf denote the components of the algebraic set {a\a e V, P(a) = 0}
which are not contained in F. Then, r being the dimension of V,
we have dim WJf^r—1. Denote by kx the algebraic closure of k;
an'd let xtf be a generic point of Wlf over ft,. As xu is not con-
contained in F, we have expressions f{x) = Qif(x)tP1?(x) where Q1? and
P,/) are polynomials in k[X] such that Plf(x1?) Ф 0. Let the W2r be
the components of the algebraic set
{a | a e V,P(a) = 0, Pw(a) = 0 for everj' ft,
which are not contained in F. We have then dim W2r gr-Z After
repeating (at most r times) this procedure, we obtain a set of polyno-
polynomials Pij{X) in k[X] such that
F 3 V Л {a | Рц{а) = 0 for every i and j) ;
and for each (i, j), we have f(x) = Qij(x)IPij(x), where Оц is a poly-
polynomial in k[X]. By Propositions 18 and 19, we have
P(F) э p(V) П {f| Pyp(f) = 0 for every ," and /}
for every p in an open set a. If p is in a, we see that, for every
point i) in f>(F)—p(F), there exists a polynomial P,j such that PypO?) ¥= 0,
namely, / is defined at rj. Our proposition is thereby proved.
Proposition 22. Let V be a variety in 5", defined over k, and F
an algebraic set contained in V, defined over k. Suppose that every
point in V—F is simple on V. Then, for almost all p, every point in
p(V)—p(F) is simple on V.
Proof. Let {Gt(,X),--,G,(X)} be a basis for the ideal a of k[X]
given by
a = {G(X) | G(x) = 0 for every x e V}.
Let the H,{X) denote the determinants of degree n—r belonging to

106 REDUCTION OF CONSTANT FIELDS [III]
the matrix (dGi/dX/), where r is the dimension of V. Our assumption
implies
Fzd {x\ Gi(x) = 0, H»(x) = 0 for every i and every v).
By Proposition 18, we have, for almost all p,
p(F) 3 {f I G,p(?) = 0, Я„„(?) = 0 for every i and every v).
Hence, if у is a point in p(V)—p(F), there exists a polynomial .Hi,
such that H,v{r;) -/= 0, namely, we have rai\k(dGif,ldXj(yy) 2: n—r. This
proves our proposition.
Proposition 23. Le< V =[Va; Fa; Tfa] be an abstract variety de-
defined over k. Then, for almost all p, the system [Va; Fa; p(Fa); Tfa]
defines a p-variety. If V is complete, the p-variety is p-complete for
almost all p. Moreover, let H be an algebraic set in V such that every
point in V—H is simple on V. Then, for almost all p, every point in
p{V)—p{H) is simple on V.
Proof. Let Bfa be the set of points in Va such that the projection
from Tfa to Va is regular at x if and only if x is not contained in
B?a. Then, Bfa is an algebraic set in Va defined over k. This fact
is well-known and is proved b}' the same argument as in the proof
of proposition 21. By Proposition 21, for almost all p, Tfa is regular
at every point in p(Va)—p(Bfa) for even' (or, /3). By the definition of
abstract variety, we have
Ти П KB^xV,) U (F.xB.,)] с 7V, n {(F.x Vt) U (V.xF,)].
By Proposition 19, for almost all p,
PBV) П ШВ»)хр(У,)) U (p(Vo)xp(Baf))]
с р(Г,„) П MF.)xp(V,)) U (p(Va)xp(Ff))].
This shows that the system [ Va; F*; p(F«); T?a] is a p-variety for
almost all p. Let the (x«u--, ха„а), for 1 ^ a ^ h, be corresponding
generic points of the Va by T?a. Let U(ai)} be a set of integers which
are equal to 1 or — 1. Let W, denote the locus of
over k, where we omit xai which are equal to 0. Let {Ga,(X)} be a

[12. 3] THE THEORY " FOR ALMOST ALL p " 107
basis of the ideal
{G(X) | G{X) e k[X], G(x) = 0 for every x e Fa},
for each a. Then, we have, for almost all p,
p(Fa) = {? | G«p(£) = 0 for even' v),
for every a. Suppose that V is complete. Then, we have, for every e,
j. txt i, s , .1 1—s(or) nr
ф = W, П Uuii) X • • • X (мм)' -z Mai = 0 tor every а, г,
and С,(м„4)| I r—- = 0 for every a, v\.
By Propositions 18 and 19, we have, for almost all p,
-4 f^ = 0 for every a, i,
and СяпК«)[\ 2(<И) = ° for every a' v
for every £. This shows that F is p-complete for almost all p. The
algebraic set i? has an expression H = U-ff'"', where, for each a, i?<«' is
the join of the components of H having representatives in Va. Let
Я, be the join of the representatives in Va for the components of
if1. Then, Ha is an algebraic set defined over k. By our assump-
assumption, every point in Va—(Ha U Fa) is simple on V«. Then, by Pro-
Proposition 22, every point in p(V«)—(p(H«) U P(Fa)) is simple on Va for
almost all p. This proves the last assertion of our proposition.
Let V = [ Va; Fa; Tfa] be an abstract variety defined over k.
Then, by Proposition 23, the system [Fo; Fa; p(Fa); Tfa] defines a
p-variety for almost all p; and, by Proposition 17, the f>-variety is p-
simple for almost all p. Thus we obtain, for almost all p, an abstract
variety p(V) - [p(Va): p(Fa); P(T?a)], defined over p(k). Proposition
23 shows that, if V is complete, then, for almost all p, p( V) is com-
complete, and, if V has no multiple point, then, for almost all p, p{V)
has no multiple point.
Proposition 24. Let V= [V.\ FB; 7%] and W= [Wr, Gr, S*]
be two abstract varieties defined over k; and let f be a rational mapping

108 REDUCTION OF CONSTANT FIELDS [III]
of V into W and H an algebraic set in V, defined over k, such that f
is defined at every point in V—H. Then, for almost all p, f is defined
at every point in p(V)—p(H).
Proof. We may consider only those p for which [ F,; jF. ; p(Fa); Tfa]
and \WX\ Gx; p(Gi); S^] define p-varieties. Let Ha be an algebraic
set in F_ defined for H in the same manner as in the last part of the
proof of Proposition 23. Let Z be the graph of / and Z.i the repre-
representative of Z in F« X Wi; in the following we shall consider only
those pairs (a,X) for which Z has the representative in VaxWi. Let
Bal be the algebraic set in V« such that the projection from Zal to Va
is regular at a if and only if a is not contained in Bai. Let Cai be
the projection of Zal П (F.xGj) on Va. We will now prove
A) • nEeJU QcH.Ui?..
If a point x« in Fo'is not contained in H« U jF«, / is defined at the
point x having xa as its representative in Va, so that there exists a
suffix 2. such that Z0jtcontains a point xaxyt with the projection xa
on Va, yu on Wi—Gx and Zai is regular at xa. Suppose that xa is
contained in Bal U С„д; then, by the definition of Bai, we have xa e Cax-
Thij implies that there exists a point лг'^х^'д in Zal П (F^xGj)
such that лг'а —> л:„ ref. k. As Z«j is regular at xa, we must have
x'cXy'i—> xaxyi ref. A, so that we have ул е Gj; this is a contradic-
contradiction; so xa is not contained in Bai U Caj. We have thus proved the
above inclusion A). By Propositions 19, 20, 21, there exists an open
set a of Yi such that, if p e a, then we have
B) n (}>(£.,) и KCO) с К-Н.) и КЛ)
for every a, Z<,; is regular at every point in p(Va)—p(Bal) and p{Cai)
is the projection of p{Z.x) П (K^«)xK^)) on P(V«)- Now fi being in
a, let £ be a point in $>(F)—К-^О- Then there exists a representative
*„ of $ in v(F«) such that £« ^ р(Я„) U ^(F«). By B), f. is not con-
contained in p(Bai) U p(Cai) for some Я. For such a ^, p(Z«x) contains a
point c«x^ and Zal is regular at £„. As f« is not contained in р(С„г),
т)х is not contained in p(Gi); so there exists a point ^ in p{W) having

[12.4] THE THEORY "FOR ALMOST ALL p" Ю9
tji as its representative in Wi. This shows that / is defined at f and
/(£) — -q. Thus we have proved that, for every p in a, f is defined at
every point in p(V)— P(H).
Proposition 25. Let A be an abelian variety defined over k. Then,
A has no defect for almost all p.
This is an immediate consequence of Propositions 17, 23, 24.
12. 4. We shall now consider the case where k and 2 are given
as follows. Given a field k0, we take as Л a finitely generated exten-
extension of k0 and as 2 the set of discrete places p of k, taking values in
the universal domain over k, such that p(a) = a for every a e k0. Let
A be an abelian variety defined over k. By Proposition 25, there exists
a set of non-zero elements {xi,—,xr} in A such that, if p e 2 and
p(^i) ^f 0 for every i, then ^4 has no defect for p. Take elements
Xr+i,---, x, so that A = AoO*!,---, xr, av+i,--, я:,) and denote by Fthe locus
of (xlt---,xs) over ko- V may not be absolutely irreducible. As we
have Xt ^ 0 for 1 5j г ^ r, F carries a point (fli,---, a*) such that at j= 0
for 1 g » ^ r and all the at are algebraic over k0. (Cf. [44] Chap. IV,
Proposition 3). By Lemma 2, we can find a place p in 2 such that
p(xt) = a; for every i. Then, A has no defect for p; and p(A) is
defined over p(k). - Suppose that dim*^ > 0. Then, one of the Xi is
not algebraic over k0. Since the a* are algebraic over k0, we see that
dimjto)}^) < dinijfc0^. We shall use this result in the proof of the
following proposition.
Proposition 26. Let (F; {<pt}) be a CM-type and [A, c) an abelian
variety of type (F; {<Pi}). Then, tliere exists an abelian variety of type
(F; {(Pi}), isomorphic to {A, t), defined over an algebraic number field of
finite degree.
Proof. Take an abelian variety (Au ci) of type (F; {<pi}) and a
field k of definition for (Au C]). Let k0 be the composite of the
fields F*i. We may assume that Л is a finitely generated extension of
Q and k contains k0. If dim^u > 0, we obtain, by means of the above
argument, an abelian variety (A2, c2) of type (F), defined over an ex-

110 REDUCTION OF CONSTANT FIELDS [III]
tension kt of k0 such that dim*,,^ < dim*^. By the result of §11.3,
(A2, c2) is of type (F; {ipt}). Repeating this procedure, we get an
abelian variety (Ao, c0) of type (F; {tpi}), defined over an algebraic
number field. Now let (A, c) be an arbitrary abelian variety of type
(F; {<pi}). Then, by Corollary of Theorem 2 of §6 and Remark below
it, there exists a homomorphism X of (AB, cB) onto (A, i). We can find
a finite algebraic extension k0' of к„, over which (Ao, i0) is defined and
every point of g(>!) is rational. Taking kB' in place of k0, apply the above
argument to (A, c). Then we obtain from (A, i), after several times of
reduction, an abelian variety (A', c') of type (F; {ipi}), defined over a
finite algebraic extension of A</; moreover we obtain, at the same
time, a homomorphism ).' of (Ao, c0) onto (A', c') as the result of re-
reduction of the homomorphism X. We observe that Ao and g(X) never
change in the reduction process; so the kernel of X' coincides with
It follows that "{A, e) is isomorphic to (A', c'). This proves our prop-
proposition. '
13. PRIME IDEAL DECOMPOSITION OF
POWER HOMOMORPHISMS.
j
We shall now prove a fundamental relation for an abelian variety
with (£omplex multiplication, which is a generalization of Kronecker's
congruence formula for elliptic functions with singular moduli. The
relation is described as follows in terms of the reduction modulo p of
an abelian variety belonging to a given CM-type.
Let (F; {an}) be a CM-type and (К* ; {фа}) the dual of (F; {<?*})•
Let (A, c) be an abelian variety of type (F; {<p(}), defined over an
algebraic number field k of finite degree. We assume that {A, c) is
principal. By Proposition 30 of § 8. 5, we know that k contains K*.
We extend the и„ to isomorphisms of k which we denote again by
<ba. Let p be a prime ideal of K*, ?P a prime ideal of k dividing p,
and p the rational prime divisible by p. Suppose that A has no defect
for S|5 and p is unramified in F. Denote by (Д t) the reduction of
(A, i) modulo $.
Theorem 1. Notations and assumptions being as above, let згр denote

[13] ЩруТН POWER HQMOMORPHISMS 111
the N(p)-tk power homovwrphism of A onto A» and jr the N($)-th
power endomorpkism of A. Then:
(я1) UP*' " an ideal of F, and (AN&\ W' ; 7tp) is a flp**-transform
a a
of {.A, I);
(гг2) there exists an element зг0 in F such that фг0) = к; and we have
Proof. Let n be the dimension of A and о the ring of integers
in F. By the definition of CM-type, there exist и invariant differen-
differential forms он on A such that
@) dc(fi)wi = fffuoi (lgi'S n)
for every fi e o. In view of the results of § 2, we may assume that
the ац are defined over a finite algebraic extension of k. Let ki be a
Galois extension of Q, containing k and F, over which the щ are de-
defined. Let Spx be a prime ideal of ki dividing S|5. We indicate by tilde
the reduction modulo ^. By the property iii) (or Proposition 6) of
§ 10.4, taking a suitable multiple of each wt in place of o>i, if necessary,
we may assume that шг is ^-finite and 5* =f= 0 for every i. We shall
show that the Zt are linearly independent over ^(k{). Let {Pi,---,^zn}
be a basis of о over Z. Then, denoting by вг the complex conjugate of
the isomorphism <pi for each i, the determinant of the matrix
A4
is not divisible by 5p1( because /) is not ramified in F. Hence, the
matrix composed of the first я columns of A) has rank n, modulo S&;
so suitable n rows of the matrix are linearly independent modulo SfJi-
It follows from this that there exist n elements jui,---, fin in о such
that
B) det(fifi) ■£ 0 mod fpt.
n
Suppose that 2 £«*>* = 0 for some £t in the residue field tyifa). Mul-
1

112 REDUCTION OF CONSTANT FIELDS [III]
tiplying by ос(т) this relation, we get, by means of @) and B), &<пг = О
for every i. Since we have wt ^ 0, the coefficients f< must be all 0.
This proves that the <5j are linearly independent over 5Pi(£i), so that
they form a basis for the linear space of invariant differential forms
on A. Now, as is seen in §11.2, {A, t) is an abelian variety of type
(F), defined over the finite field with ЩЩ elements. Hence, there
exists an element гг0 of о such that f(jr0) = к (cf. § 7. 6). We have
obviously N(t:0) = v{tz) = JV($P)n. Let
0r0) = h'l-V
be the decomposition of the principal ideal (гг0) in prime ideals pt of
F. Extend <pcl to an automorphism of ku for each i, and denote it
by ot. Let dt be the number of i such that SPj"* divides pt. If we de-
denote by h the class number of F, pefte< is a principal ideal; put ptne' = (yt)
for each t. Then, it is easy to see that dt is the number of i such
that ftn e 5рх. Considering modulo Sp1( we have
дсХп)ЯГ= г7'п* (U«^ я).
Since the Si form a basis for the vector space 3H(Д) of invariant dif-
differential forms on A, wfe see that the linear mapping 5c(j-t) of Ъ0{А)
into itself is of rank n—dt. Furthermore, if if is a generic point of A
over k, we have
k{i(rt)x) з k(nhx) = k(xN^h),
since KOh e pthet = (ft)- By Theorem 1 of § 2. 8, we have
NQ>«*»«) = N(rt) = v{
so that
C)
Put Nfcj/jtCSPi) = 5Pr; we have then
D) N(№^ N(
where the product is taken over all i such that 5Pi°* divides pc. This
shows in particular that dt is not 0, so that every pt is divisible by
at least one of the SP^. Each $!«< divides at most one of the pt.
Hence, from the relations D) and (зг0) = Pic^--p,'> follows

[13] N(p)- ТН POWER HOMOMORPHISMS 113
We note that both sides are equal to ЩЩпг. Therefore, C) and D)
must be equalities; and every ф^ must divide exactly one of the pt.
Put Nttirf^i'*) = $>«"' for every pt divisible by $&•*; as kx is a Galois
extension of F, щ does not depend on the choice of i. We have then
N(^i) = N(pt)u', and, by the equality D), N(pt)re' - N(pt)utd'. Hence
we get ret = Utdt, so that
where the product is taken over all i such that ty^t [ pt. We obtain
therefore,
E) (noy = pt"t ■ ■ -p,". = Ntj r{ П 5Ei")-
.... - • - «=i
Now take a Frobenius substitution a for SPj over K*; a is an auto-
automorphism of &! over K* such that SJ5i" = ^ and г' s zNip) mod SJ5i. Put
■^(f) = 9- As $!» = $!, Л" has no defect for $& and the reduction
modulo Spj of (^4", «•) is identified with (A", f). By Proposition 31
of § 8. 5 and Proposition 16 (or Proposition 23) of § 7, (^4-1, c) is a c-
transform of (A, c) for an ideal-class с of F; so by Proposition 15 of
§11.2, {A\ pi) is a c-transform of (Д f). By the result of § 7. 6, гг„ is
a homomorphism of (A, c) onto (A% It); so by Proposition 13 of § 7. 2,
rrp is an a-multiplication for an ideal a of o. Put iVi/^»(SP) = p"; then,
iVE|3) = qv. Now let jra denote the q-th power homomorphism of Aia
onto Aqa for each positive integer a. Then, as (Aqa , A4*, ~a) is an
isomorphic image of (А, А?, ягр), we observe that (A«a, cia; ка) is an
a-transform of {А*а \ (i'). Hence х,---ягщ is an ^-multiplication of
A onto АК1Ф = A. On the other hand, we have
Ovtti)* = x*" = х"Ф = тех,
so that к = ггс---!Гд. Since я is a (^-multiplication, we have (.-0) = ^^
therefore, by the relation E), we obtain
F) a- = (гг„)' = iV*,/K Д ft4).

114 REDUCTION OF CONSTANT FIELDS [III]
Let G denote the Galois group of &j over Q; let H and H* be the
subgroup of G corresponding to F and K*, respectively. Then, by
the definition of dual of CM-type, denoting by S* the set of elements of
G which induce some фа on K*, we have
S* = U OiH = U Н*фа.
I a
It follows from this and F) that
G) a» = П $ir = П( П &')*• = IK^/jp^i)*- = П №")'-■
res* a peH* о <r
Hence we have a — Цр*° ; this proves the assertion (;rl) of our theorem.
As we have (xB) = 0° and pv = JV*/x*0P), we obtain (гг2); so our theorem
is completely proved.
Theorem 2. Notations and assumptions being as in Theorem 1, sup-
suppose that A is simple. Then, if \> is of absolute degree 1 and the rational
prime p divisible by p is unramified in F and in К*, A is simple.
■<
Proof. We use the same notations as in the proof of Theorem
1. We observe that, for a sufficiently large positive integer g, every
element of <j4(A) is defined over a finite field with N($y> elements.
g being taken as this, -" is contained in the center of ^j4a{A); by Prop-
Proposition ■$ of § 5. 1, we can find an element $ of F such that -' = f(f).
Let a be an element of G. If £• = f, we have aw = (a"")", and hence,
by virtue of G),
(8) П %' = П SJ$i".
reS* reS*
Denote by Z the subgroup of G consisting of the elements - such
that 5Pjr = 5{?i; then, by the assumption of our theorem, Z is contained
in H*. The relation (8) implies S*a с ZS* С H*S* = S*; so we have
S*<r = S*. As (F; {tpi}) is primitive, we must have, by Proposition
26 of § 8. 2, a e H. We have thus proved that £• = £ implies a e H;
this shows F = Q($). Therefore, every element of *sda(A) commutes
with the elements of c(F). By Proposition 3 of § 5.1, the commutor of
c(F) in сл/oiA) coincides with i(F) itself; hence we have <jdo(A) = l(F).
This proves that A is simple; in fact, if A is not simple, <j*fo(A) can
not be a field; so our theorem is proved.

CHAPTER IV. CONSTRUCTION OF
CLASS-FIELDS.
Throughout this chapter, we shall denote by « the complex con-
conjugate of a complex number a; a being an ideal of an algebraic number
field, 5 will denote the ideal consisting of the elements a for a e n.
14. POLARIZED ABELIAN VARIETIES OF
TYPE (K; {9i}).
14.1. Let (K; {<pt}) be a primitive CM-type, (К*; {ф„}) the dual
of (K; {(pi)). As is seen in §8.2, К must be a totally imaginary
quadratic extension of a totally real field KB; put n = [Ko: Q]. The
automorphism of К over Ko other than the identity is given by a —* a.
Let о denote the ring of integers in K. In §§ 14-17, the notations
(К; {<р(}), (К*; {фа}), Ko and n will be always used in this sense.
Let {A, i) be an abelian variety of type (K; {<pt}). Then, by
Proposition 6 of § 5.1, ^fo{A) = t(K).
Proposition 1. (K; {<pt}) being a primitive CM-type, let (A, :)
and (A', c') be abelian varieties of type (K; {<pi}). Then, every homo-
morphism of A into A' is a homomorpMsm of (A, c) into {A', c').
Proof. By Corollary of Theorem 2 of § 6. 1 and by Remark
below it, there exists a homomorphism ?. of {A1, c') onto {A, c). L^-t
a be a homomorphism of A into A'. Then Хц is an element of
^s4a{A). As we have ^fo{A) = i{K), there exists an element f of К
such that Xft = t(£); so we have /u = i~4c)• Hence fi commutes with
the operation of K; this proves our proposition.
14.2. We shall now consider polarizations. j3 being an element
of K, we denote by w(/9) the vector of C" with the components j8pv, ft"
and by T(fi) the diagonal matrix with the diagonal elements /S<V> ft*-
If m is a free'Z-submodule of К of rank 2n, we denote by D(m) the
set of all vectors v(fi) for ft em. Let (A, i) be an abelian variety of
[115]

114 REDUCTION OF CONSTANT FIELDS [III]
Let G denote the Galois group of fcj over Q; let H and H* be the
subgroup of G corresponding to F and K*, respectively. Then, by
the definition of dual of CM-type, denoting by S* the set of elements of
G which induce some фа on K*, we have
S* = U otH = и Н*фа.
i a
It follows from this and F) that
G) a- = П «Pi' = IK П &')*■ = IT(JVv*.!Pi)*- = П №")'•■
Hence we have a = Ц$*°; this proves the assertion (зг1) of our theorem.
As we have (жг0) = 0У and pv = JV/t/jE*0P), we obtain (гг2); so our theorem
is completely proved.
Theorem 2. Notations and assumptions being as in Theorem 1, sup-
suppose that A is simple. Then, if p is of absolute degree 1 and the rational
prime p divisible by p is unramified in F and in K*, A is simple.
Proof. We use the same notations as in the proof of Theorem
1. We observe that, for a sufficiently large positive integer g, every
element of ^j4{A) is defined over a finite field with ./VEJ5)» elements.
g being taken as this, л-» is contained in the center of <jrfo(A); by Prop-
Proposition j'-of §5. 1, we can find an element $ of F such that -» = ?(£).
Let a |be an element of G. If f» = f, we have a"r = (a">r)", and hence,
by virtue of G),
(8) П 4V = П ft". .
res* rES*
Denote by Z the subgroup of G consisting of the elements r such
that SJJj' = 5|3i; then, by the assumption of our theorem, Z is contained
in H*. The relation (8) implies S*c с ZS* С Я*5* = S*; so we have
5*a = 5*. As (F; {(pi}) is primitive, we must have, by Proposition
26 of § 8. 2, a e i£ We have thus proved that £• = £ implies <r e H;
this shows F = £)(£). Therefore, every element of <jd<>(A) commutes
with the elements of i(F). By Proposition 3 of § 5.1, the commutor of
l(F) in j/t(i) coincides with c(F) itself; hence we have <j*fo(A) = £"(F).
This proves that A is simple; in fact, if A is not simple, ^0(A) can
not be a field ; so our theorem is proved.

CHAPTER IV. CONSTRUCTION OF
CLASS-FIELDS.
Throughout this chapter, we shall denote by a the complex con-
conjugate of a complex number a; a being an ideal of an algebraic number
field, й will denote the ideal consisting of the elements a for a e n.
14. POLARIZED ABELIAN VARIETIES OF
TYPE (K; {9i}).
14.1. Let (K; {<pt}) be a primitive CM-type, (K*; {<pa}) the dual
of (K; {<Pi}). As is seen in § 8.2, К must be a totally imaginary
quadratic extension of a totally real field Ko; put n = [Ko: Q]. The
automorphism of К over Ko other than the identity is given by a —> a.
Let о denote the ring of integers in K. In §§ 14-17, the notations
(K; {<pt}), (К*; {фа}), Ko and n will be always used in this sense.
Let (A, c) be an abelian variety of type (К; {<рг}). Then, by
Proposition 6 of § 5.1, ^fo(A) = c(K).
Proposition 1. (К; {ич}) being a primitive CM-type, let (A, c)
and (A', t') be abelian varieties of type (K; {<pi}). Then, every hotiio-
morphism of A into A' is a homomorpfusm of (A, c) into (A', c').
Proof. By Corollary of Theorem 2 of § 6.1 and by Remark
below it, there exists a homomorphism ?. of {A', e') onto {A, c). Lot
,u be a homomorphism of A into A'. Then ?.ц is an element of
^snfo(A). As we have ^jtfo(A) — i(K), there exists an element f of К
such that l(i = c($); so we have fi = Jryi{£). Hence fi commutes with
the operation of K\ this proves our proposition.
14.2. We shall now consider polarizations. £ being an element
of K, we denote by v(fi) the vector of C" with the components jSfi,---, ^S1»
and by Тф) the diagonal matrix with the diagonal elements ^v, /S*».
If m is a free'^-submodule of К of rank 2n, we denote by D(m) the
set of all vectors w(/S) for /6 e m. Let (A, c) be an abelian variety of
[П5]

116 CONSTRUCTION OF CLASS-FIELDS [IV]
type (К; {<рг}). By Theorem 2 of § 6. 1, (A, c) is represented by a
complex torus C"/Z)(m) for a suitable m. Let E(u, v) be a non-degenerate
Riemann form on Cn/D(m). By Theorem 4 of § 6. 2, there exists an
element С of К such that
A) £(«(?), w(?)) = Tr*/e(Cf 9)
for every £ e K, j^eX; the element С satisfies
B) С = -C, Im(C«) > 0 A ^ i ^ я).
Conversely, any such element £ of К determines a non-degenerate
Riemann form on CnjD(m) by the relation A).
- Let A* be a Picard variety of A; put, for every a ^ K,
Then, we have seen in §6.3 that (A*, c*) is of type (K; {<pi}); and
(A*, i*) is analytically represented by the complex torus C"/D(m*),
where m* is given by
C) m* *= {/31 /3 G X; Tr*/e08m) с Z}.
Furthermore, if -Y is, a divisor on A corresponding to the Riemann
form defined by A), then the homomorphism <px of A onto A*- is
represented by the matrix T(Q. The following proposition is an easy
consequence of this fact and the relation B).
Proposition 2. (A, c) and C"ID(m) being as above, let Xand Y be
tzco non-degenerate divisors on A; and let Еъ Ег be the Riemann forms
on C"ID(m) defined by X, Y, respectively. Let £<, for i = 1,2, be the
elements of К determined by the relation
Then, СГ'Сг w a totally positive element of Ko and we have
Now put
ipx~l4>r =
then r is an order in K.
Proposition 3. Notations being as in Proposition 2, the polarized

[14.3] POLARIZED VARIETIES OF TYPE (K; {<«}) 117
abelian varieties {A, Sg{X)) and (A, Sg(Y)) are isomorphic if and only
if there exist a unit e of x and a positive rational number s such that
&:»& = see.
Proof. As we have ^s4{A) — г(т), every automorphism of A is
given by c(e) for a unit t of r. Suppose that г(е) maps Sg(X) onto
So{Y). Then, by the definition of So{X), there exist two positive
integers m and m' such that mc(e)~l(X) is algebraically equivalent to
m' Y. Consider the homomorphisms ipx and <py of A onto the Picard
variety A* of A; we have then by G) of §1.3, mcc(e)<pxc(e) = m'tpy.
Put i = mjm'. By the relations E) and F) of § 6. 3, and by Proposition
2, we have sc(ee) = tpx'^-tpr = '(СГ'Сг)- This proves the " only if" part.
The " if" part is proved by following up the above argument in the
opposite direction.
Corollary. (A, i) and x being as above, let So be a polarization of
A. Then, for every root of unity e contained in r, i(e) gives an auto-
automorphism of (A, So). Conversely, every automorphism of (A, jg) is given
by '(s) for a root of unity e contained in x.
Proof. Put ff = &(X) = W( Y) and £i = C2 in the above proposi-
proposition. We see then that if £ is a unit of r, c(e) gives an automorphism
of (A, 'if) if and only if её = 1. Since (e)p« is the complex conjugate
of £P< (cf. Lemma 2 of § 5.1), the condition её = 1 is equivalent to that
the £<"* are all of absolute value 1. It is well-known that an algebraic
integer a is a root of unity if and only if every conjugate of a over
Q is of absolute value 1. This proves the assertion of our corollary.
14.3. (K; {<pi}) being as before, by a polarized abelian variety of
type (K; {<Pi}), we shall understand a triplet JP = {A, c, W) formed
by an abelian variety (A, c) of type (K; {<pt}) and a polarization Чо of
A; JP = (A, 1, "rg) and ^ = (Alt eu Wi) are said to be isomorphic if
there exists an isomorphism of (A, c) onto (Ait ct) which sends So
onto 4fi. As we have restricted ourselves to primitive CM-type, every
isomorphism of (A, So) onto (Au So\) gives an isomorphism of JP onto
J^i on account of Proposition 1.
Now we impose one more condition on our abelian varieties

118 CONSTRUCTION OF CLASS-FIELDS [IV]
(A, c). From now on, until the end of § 16, by an abelian variety
(A, c) of type (K; {tpi}), we shall understand a principal one; namely,
we assume
where о is the ring of integers in K.
Let {A, t) be an abelian variety of type (K; {(pi})- (A*, i*),
CnjD{m) and C"/Z)(m*) being determined as above, we observe that
(A*, c*) is also principal and m, m* are ideals of K. Moreover, if we
denote by b the different of К with respect to Q, the relation C) of
§ 14. 2 shows that
m* = (bin).
Let X be a non-degenerate divisor of A, corresponding to the Riemann
form given by A) of § 14.2. The homomorphism <px of A onto A* is
represented by the matrix T(Q. Hence, by Proposition 15 of § 7.4,
<px is a (Cbmm^-multiplication of (A, i) onto (A*, c*). Put
f = Cbrnm.
We shall now prove that f is an " ideal of KQ", namely, there exists
an ideal f0 of Ko such that f = of0. Let b0 be the different of Ko with
respe/;t to Q and bj the different of К with respect to Ko ; we have then
b =:boi>i. The ideal bj is generated by the elements 6 — 6 for Jeo.
By jkie property B) of the element C, we see that £F—6) is contained
in Ko for every 9e», so that C,bi is an ideal of Ko. It is obvious
that bomm is an ideal of Ko. Hence f = (£bi)(bomm) is an ideal of Ko.
Let ^ be a polarization of A. Take a basic polar divisor Y of
(A, 'ig) (cf. § 4. 2). Then, the above argument shows that there exists
an ideal f0 of Ko such that <pr is an (ofo)-multiplication of {A, c) onto
{A*, c*). We observe that the ideal f0 is determined by <&> = (A, i, "€)
and does not depend on the choice of the basic polar divisor Y; so
we say that the polarized abelian variety ,_^ is of type (K; {tpi} ; f0).
Proposition 4. Let f0 be an ideal of Ko. Then, there exists a
polarized abelian variety of type (K; {<pt} ; s\^)for some rational number
s if and only if there exists an ideal m of К and an element С of К such
that

[14.3] POLARIZED VARIETIES OF TYPE (K; {<*}) 119
of0 = Cbmm,
B) £=-C, Im(C«)>0 (U^«),
where b denotes the different of К with respect to Q.
Proof. We have already proved the " only if " part; so we shall
now prove the " if" part. We may assume that f0 is an integral
ideal. If m and £ are given as above, we obtain an abelian variety
(A, t) of type (K; {©<}) by means of the complex torus C"/Z)(m);
and Q defines a bilinear form E(u, v) by the relation A) of § 14. 2. As
£binm = of0 is integral, the values of E on D(m)xD(m) are rational
integers; hence by Theorem 4 of § 6.2, E is a non-degenerate
Riemann form on C'ljD{m). If X is a divisor on A corresponding to
the form E, <px is an (of0)-multiplication; so {A, c, ^(X)) is of type
(K; {tpi} ; sfo) for a suitable rational number s; this completes the
proof.
(A, c) being as above, let ^ and Wz be two polarizations of A; sup-
suppose that (A, i, ^x) and (A, c, ^2) are of the same type (K; {<pt} ; f0).
Then ^i contains a divisor Yt such that <pYt is an (of0)-multiplication,
for each i. Let £t be the element of К for which the bilinear form
Trx-/g(Cif]7) defines the Riemann form defined by Yt. Then, by Prop-
Proposition 2, we have (<fr )~l4>r = '(СГ'Сг); and СГ'Сг is a totally posi-
positive element of Ko. Since both the <py are (of0)-multiplications, СГ'Сг
is a unit of Ko. Conversely, take a totally positive unit e of Ko and
put £ = e£i- Then £ satisfies the relation B); and hence there exists
a divisor X on A corresponding to the form Tr(Cf^). Then, we see
easily that (A, c, ^{X)) is of type (К; {ср{} ; f0). By Proposition 3,
(A, c, Wi) is isomorphic to (А, г, ^2) if and only if Ci^Cz is of the
form da for a unit a of K. The following proposition is a conse-
consequence of these considerations.
Proposition 5. Let U be the group of totally positive units of Ko
and Ui the subgroup of U consisting of the elements Nkik (e) for units e
of K. Then, there exist exactly [U: U{\ polarized abelian varieties
{A, i, "&) of the same type (K; {<pi} ; f0) with the same underlying abelian
variety (A, e) of type (K; {(pi}), of which no two are isomorphic.

120 CONSTRUCTION OF CLASS-FIELDS [IV]
We note that the index [U: U,] is finite.
14.4. Let (A, i, 'ig) and {Аъ clt Wi) be two polarized abelian
varieties of the same type (K; {ipt} ; f0). Put f = of0. Let X and, Y
be respectively a basic polar divisor in ^ and a basic polar divisor in
^i; then both ipx and <py are f-multiplications. Let ?> be a homomor-
phism of (A, c) onto (Au e^. By Proposition 23 of § 7. 5, X is an a-
multiplication of {A, c) onto (^4I( ct) for an ideal a of o. Put Z = 1~*-(Y).
Then y>x~lipe is an element of <ji?0{A); so there exists an element a
of К such that f(a) = (px~1<pz. We observe that a is determined only
by 2 and does not depend on the choice of X and У; we write
Let {A*, c*) and (Д*, ct*) be respectively duals of {A, c) and (Д, cj).
Now we need the following fact.
Proposition 6. Notations being as above, if ?. is an a-multiplication
of {A, c) onto (Au ii) then 4. is an a-multiplication of (A,*, ct*) onto
(A*, £*). ' ,
Proof. By Proposition 1, C2 is a homomorphism of (Ax*, ct*) onto
(A*, i*). By Proposition 13 of § 7. 2, and by Proposition 1, we have
ej£S(Af Aj)'= Xifo'1). Now consider the mapping ц —»• l(i which gives
an isomorphism of J6f{A, A{) onto Jtf(Af, A*). As we have
КНЬ = КчШ = (^i@ = Ч'1*(£), we obtain ,/(Л* ^*) = '^^((a)).
This proves our proposition. We can also prove the proposition by
means of Proposition 15 of § 7.4.
Now, since Z = ).'1{Y), we have wz = l/.oy/., and hence
As both ips and азу are f-multiplications, we see, by the above prop-
proposition, that *(/(/.)) is an ай-multiplication. On the other hand, by
Proposition 2, /(/.) is a totally positive element of Ka. We have thus
proved that f(X) is a totally positive element of Ko such that
cm = (f(X)).
Conversely, let Б be an ideal of о such that there exists a totally posi-
positive element /6 of Ko for which we have 66 = (Д). Let (Ait ct; /t) be

[14.5] POLARIZED VARIETIES OF TYPE (K; {&}) 121
a Б-transform of (A, c). By the results of § 7. 4, we can take complex
tori Cn/D(m) and CnID(b~1m) as analytic representations of (A, i) and
(^2, ft)- Let £ be the element of К corresponding to the basic polar
divisor X. We have then
D) f = Cbmm = (^C)bF-1m)(b-1m);
and we see that pC satisfies the condition B) of Proposition 4. Then,
it can be easily verified, by means of the same argument as in the
proof of that proposition, that Аг has a polarization ^2 for which
(A2, cit Wi) is of type (К; {щ} ; f0). Thus we have proved :
Proposition 7. Let (A, c, *&) be a polarized abelian variety of
type (K; {ipi} ; f0) \ let a be an ideal of о and (Au ct) be an a-transform
of (A, i). Then, At has a polarization Wi such that (Au cu Wi) ts of
type (K; {<pi}; f0) if and only if there exists a totally positive element
a of Ko such that aa — (a).
X being as above, we have J&^{A, At) = ^(a'1) by virtue of Prop-
Proposition 13 of §7. 2; we shall now prove that for every $ e a,
E) /(ад=/«с#.
Put ft = ietf). We have then i(f(jt)) = Vx'1 ■ lfi<pYfi = <рх~1 • cc($ytiprh($) =
c(l)yx-1 ■ гЬргЩ) = <£У(ЛШ?) = '(Л*Ш- This proves the relation E).
Therefore, / is considered as a Hermitian form defined on the module
14.5 Class of Hermitian forms. Before proceeding further,
we give a definition concerning a certain class of Hermitian forms on
K.. Let a be an ideal of К such that there exists a totally positive ele-
element p of Ко for which we have aa = (p). Then the form p$§ for £ e К
is a positive Hermitian form on К taking algebraic integral values on
a. Now let {ab pt} be another pair such that a^ = (pt) and pt is
totally positive. We say that {аь рг} and {a, p} are equivalent if there
exists an element ft e К for which we have ца — at and pt = pfifi, in
other words, if the module a with the form p$£ is isomorphic to a^1
with p!$i by the mapping f-^/r1^. The class determined by this
equivalence relation will be denoted by (a, p); we call it a class of

122 CONSTRUCTION OF CLASS-FIELDS [IV]
positive Hermitian forms in K. Define the multiplication of two classes
(a, p) and (ab Pl) by
(a, p) (ai, pi) - (aax, ppj.
Then the set of classes becomes a group; and the identity element is
given by (o, 1). We denote this group by
14. 6. Let J* = (Д c, <£) and ^ = (Au iu tfi) be two polarized
abelian varieties of the same type (K; {<pt} ; f0). Take a homomor-
phism ?. of (A, i) onto (Au i,); there exists an ideal a of о such that
/. is an a-multiplication. We have shown that f(X) is totally positive
and an = (/(/.)); so we obtain an element (a, f(X)) of <£(Я). By means
of the relation E), we see that the clas^ (a, /(/!)) does not depend on
the choice of X. We write
{J?VJ2*} = (a, /(*)).
Proposition 8. If S>, ^u ^2 are three polarized abelian varieties
of the same typje (K; {ipt} ; f0), we have
Proof. By our definition, there exist ideals а, Ь of 0 and an
n-multiplicatjon / of JP onto ^y and a Б-multiplication ц of J^ onto
J&i sifcb that
= (a, f(Z)), {J»*: ^,} = F,
Then ^;. is an аб-multiplication of ^> onto JPZ. Let X, У, Z be
respectively basic polar divisors of J^, J^, ^2. We have then
Hence we have {^2 : ^} = (ab, f(?~]f(,^)) ', tr"s proves the proposition.
Proposition 9. Let <_&> = (A, c, "€) and JP^ = (Ль ^i, ^1) ie fzuo
polarized abelian varieties of the same type (K; {ipi} ; f0); опй? fef 17 бе
an isomorphism of A onto At. Then, -q is an isomorphism of ^ onto
Proof. Let X and Y be basic polar divisors of 'ig and ^lt re-
respectively. Put Z = 7)~1{Y). If 7] is an isomorphism of JP onto ^b

[14.7] POLARIZED VARIETIES OF TYPE (K; {<pt}) 123
there exist two positive integers m and m' such that mZ is algebraically
equivalent to m'X, namely, m<pz = m'<px; we have then f{rj) = m'jm.
Since 9 is an o-multiplication, we have о = (f(rf)). It follows that
f{rf) — 1; this proves the " only if" part. Conversely, if f(y) = 1, we
get <px~l<pz = I a and hence <px = <pz, so that if^Y) = Z is algebraically
equivalent to X; hence rj is an isomorphism of J^> onto J^. This
completes the proof.
Proposition 10. Let <£> and J^ be two polarized abelian varieties
of the same type. Then, JP and J^ are isomorphic if and only if vie
have {J^: J?>} = (o, 1).
Proof. The " only if " part follows directly from Proposition 9; so
we prove the " if" part. If {«^: J5} = (o, 1), there exist an ideal a
of о and an a-multiplication X of A onto At such that (o, 1) = (о, /(Л)).
By our definition, we can find an element у of К such that a = (f)
and f(X) = if. Then, as i(X) is an a-multiplication of A onto itself,,
there exists an isomorphism -q of (A, i) onto (Au ct) such that 7)i(f) = Л.
By the relation E) of § 14.4, we have f(rf) = 1, so that by Proposition
9, J^ is isomorphic to J^. This completes the proof.
Corollary. Let ^>, ^>u JP2 be three polarized abelian varieties
of the same type. Then, J^j and J^2 are isomorphic if and only if we
have
This is an immediate consequence of Propositions 8 and 10.
14. 7. Let JP — (A, c, 'to) be a polarized abelian variety of type
(К; {щ} ; f0). We say that «_^ is defined over k if (A, c) is defined
over k and jf is defined over k. If that is so, by Proposition 30 of
§8.5, k must contain the field K*. Let г be an isomorphism of k
onto a field k'', which leaves invariant the elements of K*. Then we
obtain naturalty a system <_^г = (AT, r, 'gfr). By Proposition 31 of
§ 8. 5, (A% p) is of type (K; {<pt}). Let (A*, c*) be a dual of (A, c)
and Ai a field of definition for (A*, e*) containing k. We extend г to
an isomorphism of klt which we denote again by r. It is easy to see
that (A*', i*T) gives a dual of (Ar, iT). Let X be a basic polar divisor

124 CONSTRUCTION OF CLASS-FIELDS [IV]
in 'gf; we take kt so large that X is rational over kt; <px is an f-
multiplication. As is seen in § 7. 6, (о(Л-г, = (fox)r is an f-multiplication.
Therefore, JP* -{A', cr, &') is of the same type as <_^>.
Let J^ = (Ab iu ^j) be another abelian variety of the same type
as S> and 1 a homomorphism of (A, i) onto (Au ct); let У be a basic
polar divisor in ^t. We may assume that (Au cj and X is defined
over Aj and У is rational over kt; if this is not so, we take a suitable
extension of Ax instead of kv Now X being an a-multiplication, we
have cm = (f(Z)) and г(/(Л)) = ^-'-'Vl We see easily that c(f(X))
= PBrr)~1-e(^rV(i'r)^r an<i к is an a-multiplication. We have thus proved
the following proposition. '
Proposition 11. Let JP and JP± be two polarized abelian varieties
of the same type (K; {cp(} ; f0), defined over k; let т be an isomorphism
of k onto a field k', which leaves invariant the elements of K*. Then,
JPT and ^!r are of the same type (K; {cp(} ; f0) as JP and JPX; and we
have
/•15. UNRAMIFIED CLASS-FIELD OBTAINED
\ FROM THE FIELD OF MODULI.
15. 1. We shall now proceed in the theorj' of construction of class-
fields. First we introduce some notations concerning ideal-groups.
Let k be an algebraic number field of finite degree; let m be an in-
integral ideal of k. We denote by A(m) the group of ideals of k which
are prime to m, and by Pk(m) the subgroup of Ik(m) consisting of all
principal ideals (a) such that a e k, a = 1 mod m. The factor group
It(m)/Pb(m) is then the group of ideal-classes modulo m. For our
purpose in this section, it is not necessary to consider infinite primes.
If k' is a Galois extension of k, we denote by G(k'/k) the Galois group
•of k' over k.
As before, let (K; {<pi}) be a primitive CM-type, [K: Q] = 2и,
{К*; {ф„}) the dual of (K; {p<}), Ko the totally real subfield of К of
■degree n, and о the ring of integers in K. Let (A, t) be an abelian

[15.2] UNRAMIFIED CLASS-FIELD 125
variety of type (K; {<pi}) which is principal. We assume in the sequel
that (A, c) is defined over an algebraic number field. Let ^ be
a polarization of A and Y a basic polar divisor in 'ig. We may assume
that Y is algebraic over Q; in fact, if Y is not algebraic over Q, we
can find a specialization Y' of Y algebraic over Q; we see easily that
Y' is also a basic polar divisor in 'ig. By the result of § 14, there
exists an integral ideal f0 of Ko such that ipy is an (of0)-multiplication
of (A, c) onto its dual. Put «^ = {A, c, W) and f = of0. Let k be an
algebraic number field of finite degree satisfying the following condi-
conditions :
i) k is normal over K* ;
ii) A is defined over k;
iii) for every о e G(klK*), all the elements of J^~(A, A') are de-
defined over k;
iv) Y is rational over k.
Such a field k really exists, since there are only finitely many trans-
transforms A' of A over K*. As (K; {><}) is the dual of (К*; {фа}), k
contains K. By Proposition 30 of § 8. 5, every element of ^snf(A') is
defined over k. Let k0 be the field of moduli of (A, tf). Then, ko
is contained in k. Put ko* = k0K*. These notations and assumptions
will be never changed until the end of § 16.
15. 2. Our purpose is to describe the extension ko* of K* in terms
of class-field theory. Let о be an element of G(kjK*). By Proposition
11, ,JP° is of the same type as ^; so {^°: J9} has a meaning. Put
By Propositions 8 and 11, we have, for every a, ~ e G(kjK*),
[or] =
namely, a —»[a] gives a homomorphism of G(k/K*) into the group
ЩК). Let Д"Ье the kernel of this homomorphism. By Proposition 10,
we have a e H if and only if J^' is isomorphic to J^. On the other
hand, by Proposition 1 and by the definition of field of moduli, J3°°

126 CONSTRUCTION OF CLASS-FIELDS [Щ
is isomorphic to JP if and only if a leaves invariant the elements of
the field of moduli k0. Therefore, H is the set of elements of G(k/K*)
which leave invariant the elements of k0*. If follows that a —> [a] in-
induces an isomorphism of G(ka*jK*) into <£(K). As <£(K) is an abelian
group, k0* must be an abelian extension of K*.
15. 3. Now we consider reduction of {A, c) modulo prime ideals
of k. By Proposition 25 of § 12. 3, A has no defect for almost all prime
ideals of k. Here and in the following, the terms " almost all" mean
" all except a finite number of". Let m be the product of prime
ideals p of K* satisfying at least one of the following conditions.
i) There exists a prime ideal of ft dividing p, for which A has
defect.
ii) The rational prime divisible by p is ramified in k.
Let p be a prime ideal of K* which does not divide m, and Sp a
prime ideal of ^k dividing p; then, by our definition of nt, for every
o e G(k/K*), A' has, no defect for Sp. In the following treatment, we
denote by tilde the reduction modulo $. Put N(p) = q. Let a be z
Frobenius automorphism of k for Щ/р. We can identify (A*, ?«) with
the reduction of (A', c°) modulo $. Let ж be the <?-th power homo-
morpbism of A onto A>. Put
) 4 = X\P*';
* a
then, by Theorem 1 of § 13, ж is a ^-multiplication of (A, t) onto
(Aq, ci). If с denotes the ideal-class of q, (Aq, cq) is a c-transform of
(A, ij, so that, by Proposition 15 of §11.2, (A", c") is a c-transform of
{A, c). Hence there exists a q-multiplication [x of (A, c) onto (A', >.°).
Then p. is a q-multiplication of (Д I) onto (A*, I*); so by Proposition
7 of § 7.1, there exists an automorphism -q of (Aq, I*) such that ж = ур.
By Proposition 3 of §5.1, 7] must be of the form с*(е) for s e o. Put
Z = /i-i(s). We have then 2 — ж; and as £ is a unit of о, Л is a q-
multiplication. We have thus proved the existence of a q-multiplica-
q-multiplication / of (A, c) onto {A', c") whose reduction modulo $ is ж.
Put Z= ?r\Y')\ we have then Z = ж'1^"), and hence Z-q7.
Take a prime I which is not divisible by p and consider Z-adic repre-

[15.3] UNRAMIFIED CLASS-FIELD 127
sentations of the divisors Y, Z, Y, Z. By Proposition 14 of § 11.1, with
respect to suitable /-adic coordinate-systems, we obtain Et(Z) — Ei(Z);
and Ei(Y) = Ei{Y), so that Et{Z) = qEt(Y); this implies that Z is
algebraically equivalent to q Y. Hence we have (pr'^z = q ■ 1 л so that
/ being determined as in § 14. 4. We have thus arrived at an important
conclusion
We have seen above that a —> [a] induces an isomorphism of G(ko*/K*)
into <£(K); we denote this isomorphism also by [a]. Now, p being a
prime ideal of K* which is prime to m, let a(p) denote a Frobenius
automorphism of ko*/K* for p; we note that, since kB* is abelian over
K*, a(p) is uniquely determined by p. For every ideal a e IK*(m),
consider the prime ideal decomposition a = П£е"" ап^ put
a(a) = П^СР)'1"'.
P
Then, by means of the relation A), we obtain, for every a e Ik*(™),
Let Hi denote the kernel of the homomorphism a —> a(a) of 1к*Ь^)
into G(ko*/K*). Since a —► [a] is an isomorphism of G(ko*jK*) into
<£(ЛГ), fli is the set of ideals a of ideals of /к*(т) such that
(По*-, Me» = (o,l).
а
Let Ho be the subgroup of /*•((!)) consisting of ideals a such that there
exists an element p e К for which we have Па#" — (M) and N(a) — ^ju.
a
Then, we see easily that Но П 1к*(т) — Hi- Let b = (/3) be a principal
ideal in K*. Put j- = П/З^; then, by Proposition 29 of §8. 3, 7- is
a
contained in К and 77 = JV(/J). As ff > 0, we get 77 = Л^(Ь). This
shows that Ho contains PK*(A)). Therefore, according to the results
of class-field theory, kB* is the unramified class-field over K* cor-
corresponding to the ideal-group Ho. We have thus established our first
main theorem.

128 CONSTRUCTION OF CLASS-FIELDS [IV]
Main theorem 1. Let (К*; {фа}) be a primitive CM-type and
(K; {<pi}) the dual of (К*; {ф.}). Let Ho be the group of all ideals a
of K* such that there exists an element p e К for which we have
= Си), N(a) = fa
zuliere fi denotes the complex conjugate of fi. Let {A, c) be an abelian
variety of type (K; {<pi}) and <£ a polarization of A. Let kc be the field
of moduli of (A, tf). Then, Ho is an ideal-group of K*, defined modulo
A); and the composite kc* of the fields k0 and K* is the unramified
class-field over K* corresponding to tlie ideal-group Ho.
Note 1. We have to explain why we may drop the condition
that (A, c) is defined over an algebraic number field. Let (A, c) be
an abelian variety of type (K; {<Pi}) and "tg a polarization of A. By
Proposition 26 of § 12. 4, there can be found an abeliau variety (Au ct),
defined over an -algebraic number field, which is isomorphic to (A, c).
If we denote 'by ^\ the image of ^ by an isomorphism of (A, c)
onto (Au .a), the fields of moduli of (A, <£) and of (Au ^j) are the
same. Therefore we obtain the results of our theorem, applying the
above discussion to (A[, i,).
N^TE 2. ' The field k0* = k0K* depends only upon (K*\ {ф.}; and
is independent of the choice of {A, c) and 'to. A similar fact holds for
another type of abelian varieties, which is related to automorphic
functions (cf. [36] я°23).
Note 3. In the classical case where the dimension of A is equal
to 1, every abelian variety of the same type as A is isomorphic to
some conjugate A' of A over K*; this implies the so-called " irreduci-
bility of the class-equation ". We shall now consider this problem in
a general case. {A, t, <ig) being as above, let (Au et) be another abelian
variety of type (K; {(pi)). By Proposition 16 of §7.4, there exists an
ideal Ь of К such that (Au c{) is a Ь-multiplication of {A, c). By Prop-
Proposition 7, Ay has a polarization ^j such that (Au clt 'toy) is of the
same type as {A, e, 4g) if and only if there exists a totally positive element
p of 'Ka such that bb = (p). Let h' be the number of ideal-classes of
К whose members 6 have this property. Then there exist exactly h!

[15.4] UNRAMIFIED CLASS-FIELD 129
distinct abelian varieties (Au it) of type (K; {<pt}) on which we can
find a polarization ^ such that (Alt clt &i) is of the given type
(K; {wt} ; f0). On the other hand, by Proposition 5, there exist [U: E7i]
polarized abelian varieties, of which no two are isomorphic, having the
same (Au t]) as the underlying abelian variety, U and t/t being as in
that proposition. Put [U: E7J = d. Thus we observe that there exist
exactly dh' polarized abelian varieties of type (K; {<pt} ; f0), not iso-
isomorphic to each other. We have shown that for every a e 1к*(т)>
(A'ia\ c'ia)) is a (na#")-transform of (A, c). Let h0' be the number of
ideal-classes of К containing ideals of the form Па*" f°r ideals a of K*.
a
Then, we see that there exist exactly /г/ conjugates A' of A over K*,
of which no two are isomorphic; /г0' is clearly a divisor of h!. If A'W
is isomorphic to A, Y[o^B is a principal ideal (jt) of K; then NtyQifiy1
is a totally positive unit of Ko. Let Ua be the group of units e in Ko
of the form e = N(a)(/ii/l)~l, where a is an ideal of K* such that Ha*°
is a principal ideal (fi) of K. Then we have U n Uo 13 Ut; put
rfo = [t/o: C7i]. We see that there exist exactly dah0' conjugates J?>°
of JP — {A, e, ^) over K*, of which no two are isomorphic; and the
number djia' is a divisor of dh'. If we have doko' = dh', the analogy
of the irreducibility of class-equation holds for {A, t, <g). It is not
easy, however, to see in which case the equality djia' = dh' holds".
15.4. Examples. 1) We shall first consider the " classical case "
where я = 1. In this case, К is an imaginary quadratic field and
(K; m) = {К*; ф). The ideal-group Ho coincides with PK(A)). Hence
k0K is the absolute class-field over K. As A is of dimension 1, A
has only one polarization ^. It is easy to see that k0 is generated
over Q by the value of the classical modular function j(z) (cf. [35] n" 7).
2) Let I be an odd prime and С the plane algebraic curve defined
by the equation y2 = 1 — xl. Then the genus g of С is equal to (/—1)/2;
and <uv = x"dxjy for u = 0,1, ••-,#—1 give a basis for the differential
forms of the first kind. If С denotes a primitive Z-th root of unity,
then (x, у) —> (Слг, у) gives a birational correspondence of С onto itself,
which is represented with respect to the basis {<dv} by the diagonal
matrix having CiC2,---, С as the diagonal elements. Let Co be a com-

130 CONSTRUCTION OF CLASS-FIELDS [IV]
plete non-singular curve, denned over Q, birationally equivalent to С
over Q; we can find a Jacobian variety J of Co, defined over Q (cf.
Chow [5], Weil [51], [52]). Now denote by c@ the endomorphism' of
J corresponding to the above birational correspondence (x, y) —> (Слг, у)
of С. Then we can easily verify that £ —♦ КО is extended to an iso-
isomorphism с of Q(Q into zjgfoCJ)- Let <p, denote the automorphism
С -»С" of б(С) for v = 1, 2,-.., g. Then (/, t) is of type (Q(Q ; {<?„}). As
is seen in § 8. 4, (Q(Q', {<p*}) is a primitive CM-type and its dual is
given by (Q@; {<р,~'}). Therefore, the abelian variety J must be
simple; and we have ojj/oC/) = Q(Q- Since ^ [C] is the ring of integers
in Q(Q and t@ e ЛЦ), we have cj/C/) = <Z[Q), so that G, c) is
principal. Hence we can apply our theorem to this case; so, for every
polarization ^ of/, the field of moduli k0 of (J, *io) generates a class-
field &o@ over QXO, which, corresponds to the ideal-group Нц. Since
J is defined dy«r Q, we can find a polarization *& of J, denned over
Q; then k0 mUst coipcidej with Q. Therefore in this case, the class-
field k0K* is not a proper extension of K*. We can conclude from
this the following interesting fact. By class-field theory, HB must
coincide with the whole'ideal-group IK((Y}). Hence, putting o, = p,,
we se^ that, for every ideal a of Q(Q, there exists an element и of К
such .that ]7a#" — C") ar)d N(a) = ^. This is a particular case of
" Stiq&elberger's relation " (cf. [39], [6]); our result gives a proof of
this relation. Finally, we note that this example shows the invalidity
of " irreducibility of class-equation " in a general case.
3) Let us consider the example B. c) of § 8. 4. Notations being
as there, put ф = от. We have then Па#° = аа!* = NLiK(a) for every
ideal a of K*. Now assume that the class-number h of К is an odd
number. Let H denote the subgroup of /к(A))/^к(A)) consisting of
the classes which contain NlIkC&) f°r an ideal 21 of L. Then H is
of index :£ [L: K] — 2; as h is odd, the index must be 1. Hence,
every ideal-class of К contains an ideal of the form NLjK{%) for some
ideal 81 of L. Let H' be the subgroup of Ik(Q))IPk((\)) consisting of
the classes whose members b have the property ЬЬ = (f) for a totally
positive element £ of Ко. As the order h' of H' is odd, for every
с e H', there exists an element c' of H' such that с = с'%. Take an

[16.1J THE CLASS-FIELDS 131
ideal 2t of L such that ЩК(Щ e c'. Put Ь = 8ОД'И*Й*'; then Ь is an
ideal of K*. Since Nz.Ik(%) is contained in c', by the definition of H',
there exists a totally positive element f of KB such that we have (c) =
NLiK(W) NLiz(%). We can easily verify that
NLlK(b) = iVL/*(SI)^).
This shows that the class с contains Nlik^)- It follows from this that
h' = %. Since Ко is a real quadratic field, the group of units in
Kq is the direct product of {1, —1} and the free cyclic group
{еп|я e Z} generated by a unit e, which we call a fundamental unit of
Ko. Assume that NKoiq(e) = — 1. Then the group of totally positive
units in Ko is the free cyclic group generated by e2; and, since
NxiK0(e) = £2, we have d = d0 = 1. Therefore, if the class-number h
of К is odd and if the norm Nkoiq(s) of a fundamental unit e of .Ко
is — 1, we have <й' = <foAo'> so that the analogy of the irreducibility of
class-equation holds in this case. This result is due to Hecke [20].
16. THE CLASS-FIELDS GENERATED BY
IDEAL-SECTION POINTS.
16.1. Notations and assumptions being as in the preceding sec-
section, let (V, F) be a normalized Kummer variety of (A, *ig) (cf § 4. 4).
Let b be an integral ideal of К and t a proper Б-section point of A;
then we say that a system (^>, t) = {A, c, "g", t) is of type (К; {(рг} ; fo; b).
We shall denote (K; {ipi} ; f0; b) briefly by Й(Ь). Our purpose is to
show that the field ko*(F(t)) is a class-field over K* and to determine
the corresponding ideal-group of K*.
Let (^, tj) = (Alt cu &,., t,} be another system of type Я(Ь). By
Proposition 24 of § 7. 5, there exists a homomorphism X oi A onto .^
such that M = ty; and Л is an a-multiplication for an ideal a of if
which is prime to b. Now consider the function /(Л) defined in § 14. 4;
f(X) is a totally positive element of Ko, and /(-k(£)) =/(%)££ for every
£ e a. Let <?0 be another homomorphism of A onto y4x such that
Xat = b. As we have J^{A, AJ = M11'1), there exists an element
p. eo such that ?.a = fa{p). Then, >io is a (//^-multiplication and

132 CONSTRUCTION OF-CLASS-FIELDS ' [IV]
/(Л) = /Mfifi- Moreover, we see that X—).a = fa(\— ft) is a A— fi)a-
multiplication. Hence, as (/.—20)t=0, we have (l-/a)acb, by virtue
of Proposition 24 of § 7. 5. Since a is prime to b, we get /u = 1 mod b.
Thus we are led to the following definition.
Б being as above an integral ideal of K, consider a pair {a, p)
formed by an ideal a of К which is prime to Ъ and a totally positive
element p of Ko such that
aa = (p).
(ai> pi} being another pair satisfying these conditions, we say that
{a, p) and {ab pj) are equivalent modulo b, if there exists an element
ft e К for which we have
at = /ла, p! = pfifl, [t = 1 mod b.
The class determined by this equivalence relation will be denoted by
(a, p)b. Define the multiplication of two classes (a, p\ and (ab />iN by
)' (a. P)b(ai, Pl)b = C^i. PPi)b-
Then, it can be easily, verified that the classes (a, p)^ form a group by
this law of multiplication; we denote this group by &{K; b); the
identity element of ЩК; b) is given by (o, l)j.
Now consider the pair {a,f(X)} determined by {^>, t) and (^>lt tj;
by the above considerations, the class (a, f{X))b does not depend on the
choice of к; so we write
{(«^i, h): {J>, t)} = (a, p)b.
If furthermore (^2, i2) >s of type ЙF), we have
*i) : (Л «i)
This is proved in a straightforward way, using the proof of Proposi-
Proposition 8.
An isomorphism rj of JP onto ^j is called an isomorphism of
t) onto (<£>!, ti) if we have y]t = ^.
Proposition 12. (J*, t) and (^>lt ti) being of type Я(Ь), we have
^i): («-^, t)} = (o, l)j */ and only if there exists an isomorphism
of (JP, t) onto (JPU tl).

[16.2] THE CLASS-FIELDS 133
Proof. The " if" part is an easy consequence of our definition
and Proposition 9. Suppose that {(<_^i, *i): {^>, t)} = (o, 1N. Take an
a-multiplication X of A onto Ax such that Xt = tlt By our definition,
there exists an element ft of К such that a = (ji), f(X) = ftp, and
p = 1 mod 6. Then, we can easily verify that XcijT1) is an isomorphism
of (<&>, t) onto (J*i, h). This proves the " only if" part.
Proposition 13. {^>, t) and (JPX, tx) being of type Щ), let k be
afield of definition for JP and <^>x, over which both t and tx are rational.
Let a be an isomorphism of k into a field k', which leaves invariant the
elements of K*. Then, (^-, f) and {^S, V) are of type Щ); and
we have
', t')} = {(^, tx): (JP, t)}.
This is an easy consequence of Proposition 11 and the above def-
definition.
16.2. Now we fix a system («^, t) = {A, c, <£, t) of type Щ).
Let (V, F) be a normalized Kummer variety of (A, '€). Let k be an
algebraic number field of finite degree satisfying the conditions i-iv)
of § IS. 1 and the following condition v).
v) t is rational over k.
Then, for every a e G(kjK*), we obtain, by Proposition 13, an element
>, t)} of &(K; 6); put
We can prove, in the same way as for [a], that a —* [a]t gives a homo-
morphism of G(kjK*) into ЩК; b). Let § be the kernel of this
homomorphism. If a e ф, there exists, by Proposition 12, an isomor-
isomorphism 7] of J* or&oJP° such that T]t = f. By the definition of field
of moduli, a leaves invariant the elements of ke*. Since V is defined
over ko, we have V' = V; moreover, by the property (N3) of Theorem
3 of §4.4, we have F= F'oy. Hence we have
F(t)' = F'{f) = F'{vt) = F(t).
This shows that a leaves invariant the elements of &o*(.F(«)). Con-

134 CONSTRUCTION OF CLASS-FIELDS [IV]
versely, suppose that an element a of G(k/K*) fixes the elements of
ko*(F(i)). Then; by the result of §15. 2, there exists an isomorphism
e of JP onto <^>°; and again by the property (N3), we have F = i7°°e.
Hence, we get F(t) = F(t)° = F(e'H"). By the property (K2) of Prop-
Proposition 16 of § 4. 3, there exists an automorphism ft of JP such that
e~H* = fit. Then е/л is an isomorphism of JP onto JP' satisfying
(efi)t = t'; so, by Proposition 12, we have [a]t — (o, 1)&, and hence
ст е £. We have thus proved that ko*(F(t)) is the subfield of k cor-
corresponding to ф; in other words, a -* [a]t induces an isomorphism of
G(ko*(F(t))/K*) into £(K; Б); we shall also denote this isomorphism
by the same notation [a]t. Since &(K; b) is an abelian group, ko*(F(t))
is an abelian extension of K*.
16. 3. in being denned for the field &.as in §15. 3, let p be a prime
ideal of K* which does not divide m, and ?p a prime ideal of k divid-
dividing p. Put NCp) = q. Let и be a Frobenius automorphism of kJK*
for 5p/p. Consider now the reduction modulo % By the result of
§ 15.3, there exists a (rib^-multiplication X of A onto ^4" such
a
that the reduction 1 of ?. modulo 5p is the q-th power homomorphism
jt of Л onto Л»; and we have f(X) — q. We see easily
A) J ■ P = i* = ra = 7.1 = (Tt).
Assume that b is prime to iV(b); then, by Proposition 16 of §11.2,
the reduction modulo Sp gives an isomorphism of g(b, A') onto gF, A*).
Therefore, A) implies f = Xt, observing that both f and )A are con-
contained in gF, A'). Using this homomorphism 2, we obtain
B) M«
Put « = mN(b). For every prime ideal b e /jr*(n), let ct()j) denote the
Frobenius automorphism of ko*(F(t))IK* for b; here recall that ko*(F(t))
is an abelian extension of K*. For every ideal a = Пьс(р> 'n /sr*(n)» put
P
Ф) = П^)е(р)-
p
Then, we get, by means of B), for every a e /к*(п),

[16.3] THE CLASS-FIELDS 135
Let i?! be the kernel of the homomorphism a —> a(a) of /г»(п) into
G(ko*(F(t))/K*). Since a -»\o\t is an isomorphism of G(ko*(F(t))IK*)
into (£(K; Б), jHi consists of the ideals a e ijr*(n) such that
Now let b be the smallest positive integer divisible by b; and let H(b)
be the subgroup of 1к*((Ь)) consisting of the ideals a for which there
exists an element ft of К such that ПаЛг — 0")> N(a) = fijX and fi = l mod b.
a
Then we see easily that H(b) П /x*(«) = Hi- Let f be an element of
K* such that £ = 1 mod F). Put 7- = П^"; then by Proposition 29 of
а
§ 8. 3, t" e i^, 77 = AT((f)); moreover, it is obvious that 7- = 1 mod b.
This shows that Рк*{(Ь)) is contained in Я(Ь). Hence, by class-field
theory, we observe that ko*(F{t)) is a class-field over K* corresponding
to the ideal-group H(b). We have thus arrived at the following con-
conclusion.
Main theorem 2. Notations and assumptions being as in Main
theorem 1, let (V, F) be a normalized Kumnter variety of (A, 'if). Let
Ъ be an integral ideal of К and b the smallest positive integer divisible by
b. Let H(b) be the group of all ideals a of K*, prime to (b), such that
there exists an element ft e К for which we have
J7a#« = (ft), N(a) = ftft, ^sl mod b.
a
Let t be a proper b-section point of A. Then, H(b) is an ideal-group
of K* defined modulo (b); and ko*(F(t)) is the class-field over K* corre-
corresponding to the ideal-group H(b).
The reason why we may dispense with the condition that A is
defined over an algebraic number field, is the same as in Main theorem 1.
We now consider the case of dimension 1; it is then clear that
H(b) coincides with PK(b) Г) ЫФ)) \ it follows that ko*(F(t)) is the class-
field over К corresponding to the ideal-group Рк(Ь). We can also
easily verify that the normalized Kummer variety (V, F) is explicitly
given by Weierstrass' /o-function, or more precisely, by Weber's r-func-
tion ; then the above theorem implies the main theorem of the classical

136 CONSTRUCTION OF CLASS-FIELDS [IV]
theory of complex multiplication, which asserts that the abelian ex-
extensions of an imaginary quadratic field are generated by the values of
certain elliptic or elliptic modular functions for singular moduli.
The classical theory of complex multiplication contains much more;
for details, the reader is referred to Kronecker [25], Weber [34], Takagi
[40], Hasse [17], Deuring [11]; further references will be found in these
works.
17. THE CASE OF NON-PRINCIPAL ORDERS.
17.1. Let (K; {<pt}) be as before a primitive CM-type and
(K*; {<pa}) the dual of (K; {<pi}). We shall now consider an abelian
variety (A, c) of type (K; {<pi}) which is not necessarily principal. In
view of Proposition 26 of § 12.4, we assume that (A, c) is defined over
an algebraic number field. Put
then, r is ah;'order,in K± We denote by о the ring of all integers in
K. Let e be the sum of all o-ideals contained in r; then e itself is
an o-ideal contained in r. We call с the conductor of the order r.
e being the conductor of r, let (Alt cy; ft) be an e-transform of (A, i).
Then, by Proposition 7 of § 7. 1, we have ^(o) = <_?d(A^), so that
(Ai,\ ij) is principal; moreover, we may assume (Alt tj) and p. to be
defihed over an algebraic number field. As (Alt ct) is also of type
(K; {q>i}), we can apply ro (Alt c{) the theory developed in §§15-16.
Let e be the smallest positive integer divisible by e. By Proposition
8 of § 7. 1, for every point t of g(js), we have et = 0. Hence there
exists a homomorphism / of At onto A such that lp = e-lA. Let 4g
be a polarization of A and X be a divisor in £\ Put ^, = ^g{)r\X)).
Put
JP = (A, c, <£), J2\ = (A, iu tf,).
Then, it is easy to see that /u is a homomorphism of JP onto ^x and
/. is a homomorphism of <JPX onto JP. Let k0 and ki be respectively
the fields of moduli of {A, <€) and (Alt Wi); let (V, F) and (Fb Ft)
be respectively normalized Kummer varieties of (A, 4f) and (Au ^{j.
Put kB* - k0K*, ki* = k^K*. Since (A, &) and {Ax, ^) are defined

[17. 2] THE CASE OF NON-PRINCIPAL ORDERS 131
over an algebraic number field, k0 and ky are algebraic number fields
of finite degree.
Proposition 14. Notations being as above, if a is an automorphism
of the algebraic closure of Q, which is the identity on k0*, then there
exists an isomorphism of JP onto JP°. Moreover, for every isomorphism
7) of JP onto JP°, there exists an isomorphism e of J^t onto t_^V such
that 7]°Z = >»£.
Proof. By Proposition 31 of §8.5, (A', c) is of type (К; {<рг}).
Hence the first assertion follows directly from the definition of field
of moduli and Proposition 1. By Proposition 8 of § 7.1, g(^) = g(e, A).
It follows that Q(fj.c) — g(e, A"). Therefore, if -q is an isomorphism of
^> onto J?>°, we observe that the kernel of (f'-q coincides with the
kernel of ft. Hence there exists an isomorphism e of A± onto Af such
that e°fi = /n°°y. As we have X°fx = еЛл, we obtain Я°°е = зу=-?. Using
this relation, we can easily verify that £ is an isomorphism of ^± onto
^i'. This proves the second assertion.
Corollary. Notations being as above, k0* contains ki*.
Proof. If a is an automorphism of the algebraic closure of Q,
leaving invariant the elements of k0*, then, by the above Proposition,
(Au 'if1) is isomorphic to (AL't tff); so by the definition of field of
moduli, a is the identity on k,*. This implies the inclusion k0* n &i*.
17.2. Let и be a point on A of finite order. Our purpose is to
characterize the extension ka*(F(u)) of K*. The homomorphisms X
and fi being as in § 17. 1, we can find a point v on Ax such that
A) u = ?,v;
v is also of finite order. Let r be a positive multiple of e such that
rv = 0. Let t be a proper (ro)-section point on At; put i) = д(Л). We
have then
so there exists a submodule m of о such that о n m z> rv and
B) Ъ = е1(тI.

138 CONSTRUCTION OF CLASS-FIELDS [IV]
Proposition 15. Notations being as above, t is the order of m,
namely, r = {a \ a e K, am С m}.
Proof. If a is an element of r, we have ^(afy = с(а)й) = 0, ^so
that ii(d)\) С I), This shows that ant с m. Conversely suppose that
arm С nt. We have then fi(ar)l) С 1) and hence fai(a)i) = 0; namely, the
kernel of /.^(or) contains the kernel of Л. Hence there exists an en-
domorphism j of A such that yi. = JLti(a); we see easily that у = c(a).
This shows that e(a) is contained in urf(A); so a is contained in r.
Proposition 16. Notations being as above, the field k^F^t)) con-
contains ko*(F(u)).
Proof. Let a be an automorphism'of the algebraic closure of Q,
which is the identity on k^F^t)). By the result of § 16. 2, there exists
an isomorphism в of JPX onto ^>£ such that it = f. We have hence,
in view of B),.
< I)" = ei'(™)t' = Cl°(m)et = ec^rCjt = e\).
It follows that the kernel of Л°°е coincides with f); so there exists an
isomorphism j? of A onto A' such that yl — >°e. It can be easily
verified that rt is an isomorphism of «^ onto t^'; so a leaves invariant
the eleihents of k0. Since we have rv — 0, by Proposition 20 of § 7. 5,
there exists an element /3 of о such that v = ci(P)t; so we have
C) *' V* = Cl'(P)st = «!($* = SV.
On the other hand, by the property (N3) of normalized Kummer
variety (cf. Theorem 3 of §4.4), we have F"'r = F. By means of this
relation and C), we obtain
F(u)" = F'(X-v") = F'(/.°ev) = F'(t]1v) = F{?.v) = F{u);
namely, a leaves invariant the point F(u). It follows that a is the
identit5' on ko*(F(u)). This proves our proposition.
17. 3. Notations being as above, let k be an algebraic number field
of finite degree such that ^>, JPU ?., /u are defined over k and t is
rational over k. In § 16. 2, we have established an isomorphism

[17.3] THE CASE OF NON-PRINCIPAL ORDERS 139
of GiktfF^fylK*) into ЦК; (г)). We shall now determine the sub-
subgroup of G(k1*(F1(t))/K*) corresponding to ko*(F(u)), by means of this
isomorphism. Let a be an element of G{kl*{F1(t))jK*), which is the
identity on ko*(F(u)); we extend a to an isomorphism of k, which we
denote again by a. By Proposition 14, there exists an isomorphism
rjo of JP onto <^>"; we have F°°T}a = F by virtue of the property of
normalized Kummer variety, so that F(u) = F(u)° — F'(W) = F(tjo~1w).
Hence, by the property (K2) of Proposition 16 of §4.3, there exists
an automorphism у of JP such that yu = f]u~lu°. Put -q = 1707-. Then
37 is also an isomorphism of JP onto t^") for which we have
D) u" — г/и.
By Proposition 14, there exists an isomorphism £ of jPi onto <&f
such that
E) vX = X*t;
so ij = ii(rn)t denoting as above the kernel of X, we observe that ei) is-
the kernel of Xm. It follows that
F) b' = el).
On the other hand, there exist an ideal a of 0, prime to (r), and an
a-multiplication Aa of At onto A{ such that
G) lat = t-;
we have [a]t = (а,/(Ла))(г>. By Proposition 9, we get /(e) = 1. Since
е~Ч„ is an endomorphism of Au there exists an element or of о such
that Za = «i(ar); It follows that а = (a) and /(^а) =/(е)аа — аа. We
have, in view of B), F) and G),
and hence
This implies та С m, since m 3 ro. Therefore, by Proposition 15, a
is contained in t. Moreover, 0 being an element of 0 such that
v = ii(fi)t, we have
(8) V = fi'(j8)V
By A), D), E), (8), we have

140 CONSTRUCTION OF CLASS-FIELDS [ГУ]
rju= u' — X'V = X'scx{a)v = rjXcii^v = yc(a)?.v = r]c(a)u.
As 7] is an isomorphism, this shows [1—c(a)]u = 0. Now let to denote
the set of all the elements £ of r such that c(^)u = Q. Then, to is an ideal
of r; and we obtain a = 1 mod №. We have thus proved that if a
is the identity on ko*(F(u)), there exists an element a of r, which is
prime to (r), such that
(9) a = 1 mod to,
A0) [a]t = ((a), aS)m.
Conversely, о being an element of G{ki*(Fx(t))IK*), suppose that
there exists an element a of r, prime to (r), satisfying the relations (9)
and A0). Then, by the definition of [a]t, there exists a 6-multiplica-
tion ?.f) of Ax onto Ax' such that ty = f and
We can find an element p of К such that /j = 1 mod (r), (a) = /ob
and aa = f(h)'pp- ^Щ Za =? 2f,ci(p)- Then, we observe that 2a is not
only an element of J&Ct{Ax, Ax'), but also a true homomorphism of
Ax onto Л,*, because a is contained in o. We see easily f(ia) = aa;
as p = 1 mod (r) and >Jjif = /», we have
(И) ■>. ■ 4* = *-,
and h^nce, using the fact that v is contained in cx(o)t,
A2) Лаг> = v.
As both ^(a) and ^a are (^-multiplications of Alt there exists an iso-
isomorphism £ of Ax onto y^i* such that
A3) 4 = Ul(a).
We have then /(e) = 1, so that by Proposition 9, £ is an isomorphism
of ^x onto *&x- Since a is cotained in r, we have arm С m b}' virtue
of Proposition 15, and hence, by B), A1), A3),
Ij' = :i'(m)p = ^'(m)^ = ix'(yc^ui[a)t ~ щ(хааI С «i(m)* = £^.
As £ is an isomorphism, ef) and 1)" have the same order; so we must
have et) = ly. This shows that the kernel of >°e coincides with the
kernel i) of L Hence there exists an isomorphism у of A onto A'
such that

[17.3] THE CASE OF NON-PRINCIPAL ORDERS 141
(И) ;.'.<■ = v°x.
We can easily verify that -q is an isomorphism of JP onto JP'. It
follows that a is the identity on ke*. Furthermore, as a — 1 mod to,
we have
A5) <«)ii = «i;
so, by A), A2), A3), A4), A5), we get
u° = Z'v° = ?.'lav = ?.'eCi{a)v = TjXtjJiap = tjc(o)Xv = r)c{a)u = rju.
Hence we have F(u)' = F'(uc) = F'(t]u) = F(u); this shows that a leaves
invariant the point F(u); consequently, a is the identity on ka*(F(u)).
Thus we have shown that an element a of Gfk^F^i^lK*) is the
identity on ko*(F(u)) if and only if there exists an element a of r,
prime to (r), satisfying the relations (9) and A0). In § 16. 3, we have
proved that for every a e 1к*(п),
з
where n is a suitable integral ideal of K*, which is a multiple of (r),
and a(a) is the element of Gik^CF^ty/K*) defined as in that section.
Now we introduce the following notation, r', e' and to' being respec-
respectively an order in K, the conductor of r' and an ideal of r', we write,
for an element a of K,
a = 1 mod (r'; to'),
if there exist two elements /3 and у of r', both prime to e', such that
«■ = jS/r. P = 7 = 1 m°d to'.
Now come back to the case of the above r, e, to. Let i be the smallest
positive integer contained in e Г) to; then r is a multiple of s. Let
H(v; Id) be the group of all ideals a of K*, prime to (s), such that
ther exists an element a of К for which we have
ЦсЛ = (a), N(a) = aa, a = 1 mod (r; to).
i
Then we see easily that H(r; to) contains PK*((s)). The above consid-
considerations show that, for every а е /г*(п), а(а) is the identity on ka*{F(u))
if and only if a e H(x; to) n 1к*(п). Therefore, kB*{F(u)) is the class-
field over K* corresponding to the ideal-group H(x; to). We have thus
proved:

142 CONSTRUCTION OF CLASS-FIELDS fIVjf
Main theorem 3. Let (K*; {cpj}) be a primitive CM-type and
(K; {<pi}) the dual of (K*; {<pj}). Let r be an order in К and e the
conductor of t. Let {A, t) be an abelian variety of type (K; {p*}) such
that c(x) = ^jxf(A), and и a point on A of finite order; and let to he the
ideal of r defined by
to = {£ | £ e r, <£)ii = 0}.
Im 'if be a polarization of A, (V, F) a normalized Kummer variety of
(A, &), and k0 the field of moduli of {A, <£). Let s be the smallest
positive integer contained in г П to. Let H(t; Id) be the group of all
ideals a of K*, prime to (s), such that there exists an element a of К for
which zee have
JJa*j — (a), N(a) — aa, a == 1 mod (r; to).
i
Then, H(x;to) is an ideal-group of K* defined modulo (s); and the
composite of the fields ku{F(u)) and K* is the class-field over K* corre-
corresponding to the ideal-group H(x ; to).
Main theorems 1 and 2 a"re of course included in the above theorem
as particular cases.
t
18. T/IE ZETA-FUNCTIONS OF ABELIAN VARIETIES
; WITH COMPLEX MULTIPLICATION.
Our object of this section is to determine the zeta-function of any
abelian variety of the given CM-type, defined oyer an algebraic number
field. We begin by recalling some definitions.
18.1. L-functions with Grossen-characters. Let k be an
algebraic number field of finite degree; let tn be an integral ideal of
k. A homomorphic mapping -/ of the ideal-group i*(tn) into the group
of complex numbers with the absolute value 1, is called a Grossen-
character of k, if there exist rational integers /, and real numbers f,
with the following property: for every principal ideal a = (a) such
that a = 1 mod m,
Z(a) =

[18.2] ZETA-FUNCTIONS OF ABELIAN VARIETIES 143
where a runs over all the isomorphisms of k into С which are not com-
complex conjugate of each other; we say that x 's defined modulo nt and call
in a denning ideal of %. Two Grossen-characters are called equivalent
if they coincide whenever they are both denned; among the defining
ideals of all the characters which are equivalent to a given one there
is one which divides all the others; it is called the conductor of those
characters. A Grossen-character is called primitive if it is defined
modulo its conductor.
Now, after Hecke [22], we define the L-function with the Grossen-
character ■/ by
us, x) = z'xwm)" = n'ti-^om
where the sum and the product are respectively extended over all in-
integral ideals in /*(т) and all prime ideals in /*(т), т being a defining
ideal of x- The function L(s, %) is holomorphic on the whole .f-plane,
unless the character x 1& the constant 1. If we put 2(a) = x(a)> then
X is a Grossen-character with the same conductor as x- Assuming x
to be primitive, if we form a product $(s, x) of L(s, x) and a suitable
/■-factor, and similarly £(s, %) f°r L(s, f), then there exists a constant
W(x) such that
This is called the functional equation for L(s, x)- For details, the reader
is referred to [22].
18. 2. The zeta-function of a variety defined over a finite
field. Xow let к be a finite field with q elements. Let V be a com-
complete non-singular algebraic variety defined over к. Denote by Nm the
number of points on V rational over the extension of к of degree
m. The zeta-function Z(u; V/к) of V with respect to к is defined by
(cf. Weil [47])
4~\ogZ{u; VJk) = 2 Nmu™~K
аи m=i
If V is an abelian variety, Z{u; V\k) is easily obtained by means of the
characteristic roots of the q-th power endomorphism:
Proposition 17. Let A be an abelian variety of dimension n, defined

144 CONSTRUCTION OF CLASS-FIELDS [IV]
over a finite field к with q elements; and let xlt---, Ягп be the diaracteristic
roots of the q-ih pozoer endomorphism of A. Put
() П( v«)
@
where the product is extended over all the combinations {*!,•••, t't) о/ £
fetters taken from {1,---,2n}. Then the zeta-function of A with respect
to к is given by
As a simple consequence, we obtain:
Corollary. A and к being as in the above proposition, let A' be
an abelian variety, defined over к, such that there exists an isogeny of A
onto A', defined over к. Then the zeta-functions of A and A' with respect
to ic are the same;
Weil [47] conjectured that, for any V, Z(u; Vjk) is a rational
function, satisfying a functional equation, with the zeros of absolute
value qKl2 for h «= 1, 3,--, In—1 and the poles of absolute value qh' for
h' = 1, 2,j--, n. .Proposition 17 and Weil's result which asserts that
\xi\ = q4%, show that this conjecture is assured for abelian varieties.
18.'3. The zeta-function of a variety defined over an alge-
algebraic number field. Let k be an algebraic number field of finite
degree. Let V be a complete non-singular variety defined over k.
Consider the reduction V(p) of V modulo a prime ideal p of k. Then,
by Proposition 23 of § 12. 3, we obtain, for all except a finite number
of p, a complete non-singular variety V(p) defined over the residue
field k{p). Put, for such p,
C(s ;V;p) = Z(N(py; V(p)/k(p)).
Now the zeta-function Cfa; Vjk) of V with respect to k is defined by
p
where the product is extended over all prime ideals p of k, for which

[18.4] ZETA-FUNCTIONS OF ABELIAN VARIETIES 145
V(p) is a complete non-singular variety. Hasse conjectured, in case
where V is of dimension one, that £(s; V/k) is meromorphic on the
whole 5-plane and satisfies a functional equation of the usual type.
This conjecture was generalized by Weil for varieties of any dimen-
dimension. We know at present few cases where this conjecture is true;
the following is a list of those known cases:
1) algebraic curves Y' = rXf+8 (Weil [50]);
2) elliptic curves with complex multiplication (Deuring [12]);
3) abelian varieties with complex multiplication (Taniyama [42]);
4) models of certain automorphic function fields (Eichler [13],
Shimura [35], [37]).
The results of the case 3) include as particular cases those of 1) and
2) so far as only the conjecture concerns. We shall now give a treat-
treatment for the case 3) in a little weaker form than [42].
18. 4. Let (F; {q>t}) be a CM-type, which is not necessarily prim-
primitive ; put [F: Q] = 2n. Let (A, c) be an abelian variety of type
(F; {<pi}), defined over an algebraic number field k; A is of dimension
n. Put
r = r4^f(A) n
Then, r is an order in F. If t is not the ring о of all integers in F,
consider the conductor e of r. By Proposition 7 of § 7.1, we can find
an e-transform (Au ct; X) of (A, c) defined over k, for which we have
A) Ф) = ^f(A) П
By Corollary of Proposition 17, if both A and Ax have no defect for
a prime ideal p of k, the functions C(s; A; p) and £(s; A^\ p) coin-
coincide. Hence, C(s, A/k) differs from C(s, AJk) only by a finite number
of factors in the Euler products. Therefore, if such factors are left
out of consideration, it suffices to deal with abelian varieties satisfying
A); so we assume henceforth that гЧ^л/(А) Л c(F)] is the ring о of
all integers in F.
Let Ь be an integral ideal of F; take a proper Б-section point t
on A. Let k' be a Galois extension of k, over which t is rational.
For every a e G(k'/k), f is also a proper Б-section point on A"; so

146 CONSTRUCTION OF CLASS-FIELDS [IV]
by Proposition 20 of § 7. 5, there exists an element ft, of o, prime to
b, such that c(fi,)t = f; fi, is uniquely determined modulo Ь by <7.
Denote by fj[a] the class of [i, modulo 6. Then, we have, for
а, т <= G(k'/k),
t" = (t(fJL,)ty = t(pL.)t< = c{{l.fU)t,
so that the mapping a —+ ft[<x] gives a homomorphism of G(k'/k) into
the multiplicative group (o/b)* of the residue class ring o/b. We see
easily that the kernel of this homomorphism corresponds to the sub-
field k(t) of k'. Hence a —* tt[a] induces an isomorphism of G(k(t)/k)
into (o/b)*; as (o/b)* is commutative, this implies that k(t) is an abelian
extension of k. ,
Let m be the product of all prime ideals of k for which A has
defect. 6 being as above, let b be the smallest positive integer divis-
divisible by b. Let.}) be a prime ideal of k which does not divide Ant,
and Sp a prime "ideal of k(t), dividing p. Put N(p) = q. Consider the
reduction of A modulo 5K; ,we denote by tilde the reductions modulo
5K. Let op be a Frobenius substitution of k(t)/k for p. As Л is de-
defined over the residue field of k modulo p, we obtain the g-th power
endomorphism it of A. By Theorem 1 of § 13, there exists an element
izp of о such that Г(ягр) = л. We see easily that c(xv)t and t'r have the
same reduction modulo 5K. As 5K is prime to b, by Proposition 16 of
§ 11. 2, ,the reduction modulo 5K gives an isomorphism of gF, A) onto
gF, A). Hence we have
CGtp)t = fp.
This implies that ap is uniquely determined by p. It follows that if
p is prime to bin, p is unramified in the extension k(f). We see further-
furthermore, that ft[<tp] is the class of жр modulo Б. Let d be the discrimi-
discriminant of F with respect to Q. For every ideal a = [jf<:('') of I^bdm), put
P P
Then, the above considerations show that the correspondence aa —* xa
gives an isomorphism of G(k(t)/k) into the group (o/Ъ)*; in particular,
<ra is the identity if and only if жа = 1 mod 6.

[18.4] ZETA-FUNCTIONS OF ABELIAN VARIETIES 147
Now let (K*; {<}>«}) be the dual of (F; {<pt}). Then, by Theorem
1 of § 13, we have
and hence
For every element £ of A, put с* = П-Wifc/Wf)**; then, by Proposition
a
29 of § 8. 3, £* is contained in F, and we have, for every isomorphism
т of F into C, £**!*? = N*/g(£), so that
Let Яр'1',---, л-р'2"' be the characteristic roots of c(xp). Then as is re-
remarked in § 18. 2, by virtue of Weil'a result, we have, for every i,
|irp««J = V7W).
By Proposition 14 of § 11.1, the ггр(<» are the characteristic roots of
с(яр). For any isomorphism т of F into C, np~ coincides with one of
the y-p"», so that
It follows that, for every a e Ik(bdm),
From these considerations, we conclude that, for every principal
ideal а = (£) of Ik{bdm), and for every isomorphism - of F into C, we
have |(яа/?*)г| = 1; as we have (яга) = (f*), ^0/f* is an algebraic in-
integer, so that ;га/£* is a root of unity. We have thus proved that for
even' principal ideal а = (f) of Ik(bdm), there exists a root of unity £
in F for which we have
Now, by class-field theory, the abelian extension k{t) of A is the
class-field over k corresponding to an ideal-group H of k. Let f be
the conductor of H, and / the smallest positive integer contained in
f П Ь П (d). If an element £ of k satisfies £ = 1 mod /m, then the

148 CONSTRUCTION OF CLASS-FIELDS [IV]
ideal a = (f) is contained in H, and hence aa is the identity, so that
ла = \ mod b. On the other hand, from ? = 1 mod (/) follows the
relation £ * = 1 mod (f). We have therefore
B) £ = ,Ta/f* = lmod6,
if a = (£), f = 1 mod /m. The discussion as far as here is valid for
any integral ideal 6 of k. We take now 6 in such a way that, for'
every root of unity e in F, other than 1, e—1 is not divisible by b.
Then, the relation B) implies ra/f* = 1 ; namely, we have
C) *„
if a = (f), £ = 1 mod /m. Put, for ever}' a
Then, x(a) *s а character of /«(/m); and the relation C) shows that
X{a) is a Grossen-character of k defined modulo /m.
Let {?!,•••, Г('„} be the set of all isomorphisms of .F into C. Put
for a e /tC/m), *
F) &—i,(a) - ХфУ-хф)-
Then, the xt and the £«,•••*, are Grossen-characters of k defined modulo
/in. Ob account of Proposition 17, we obtain the following result.
Main theorem 4. Let (F; {<р(}) be a CM-type; put In — [F: Q].
Let {A, c) be an abelian variety of dimension n of type (F; {<pi}), defined
over an algebraic number field. Let k be an algebraic numbe field over
which A and the elements of c(F) П *j*f(A) are defined. Let m be the
product of all prime ideals of k for which A has defect. Then there exist
a positive integer f and Grossen-characters #,--ч, of k defined modulo
/m by means of which the zeta-function of A with respect to k is expressed
in the form
Z(s,Alk) = R(s)n nL(s-~y.i
1=0 (i) \ /
•where R(s) is a product of rational functions of N(p)~s for a finite number
of prime ideals p of k, and L(s, xw) is the L-function with the character

[18.5] ZETA-FUNCTIONS OF ABEUAN VARIETIES 149
Xm ; the product is extended over all combinations {h,---,it} taken from
{l,---,2n}. The characters xm <*re defined by the relations D), E), F).
As a consequence of this result, we observe that Q(s, Ajk) is a
meromorphic function satisfying a functional equation of the usual type.
The above theorem has an insufficiency in that it says nothing
about the coductors of the characters; this is due to that we have
only dealt with " almost all " prime ideals in our treatment. Deuring
[12] has considered, in case of dimension 1, "all" prime ideals, and
obtained a highly precise result. Hasse [19] has treated this problem
in case of the curves Xl+Yl — 1. The theory of Taniyama [42] gives,
in the case of dimension > 1, a more general and precise result than
ours, in which one have some informations about the conductors of
the characters.
- 18.5. -The attentive reader -will find - from the above proof of
main theorem 4 that the zeta-function of A is closely connected with
the abelian extensions generated by the points of finite order on A.
We shall now consider the connection in a more general case.
Let A be an abelian variety of dimension n, defined over an
algebraic number field k of finite degree. For every rational prime /,
denote by ui(A) the set of points on A whose orders are powers of /,
and by kil) the extension of k generated by the coordinates of the
points t in Qi(A). It is easy to see that kw is a Galois extension of
k*K Let G(kil)/k) denote the Galois group of £(!> over k. Since every
element of G(kmIk) induces an automorphism of Qi(A), we obtain,
choosing an Z-adic coordinate-svstem of %i(A), a representation of
G(ka)jk) by matrices of degree 2n with coefficients in Zi, which we
call an Z-adic representation of G(k^/k) and denote by 5№г. It is
clear that this representation is faithful. Let p be a prime ideal of k.
Take a prime divisor 5K in kll) which divides p and a Frobenius auto-
morpism a of km for 5}5/{>. Since 5№г is faithful, if we know the matrix
5№г(сг) for every p, we get much informations on the arithmetic of the
extension ka)[k.
4) On account of Mordell-Weil's theorem, £'!) is of infinite degree over k.

150 CONSTRUCTION OF CLASS-FIELDS [IV]
Suppose that A has no defect for p and / is prime to p. By the
proof of Proposition 14 of §11.1, we can choose 7-adic coordinate-
systems of Qi(A) and Qi(A(p)) in such a way that everypoint t of Si(A)
and its reduction <($) have the same /-adic coordinates. Let xp be the
N{p)-th power endomorphism of A(p). Then, we have, for every
t e 6i(A),
This together with the definition of 2Kj yields
G) ЗВД = Af,(«p).
This shows that a is uniquely determined by 5}5. It follows that p is
unramified in kw. Denote by /pj the' characteristic polynomial of
3Ki(<r). Then, fpi is also the characteristic polynomial of згр; so we
obtain the following result.
Proposition. 18. Notations being as above, suppose that A has no
defect for p. Then for every rational prime I which is prime to p, the
prime ideal p is unramified in kw ; and the characteristic polynomial fPti
has coefficients in Z. Moreover, for any two primes I and I' prime to
p, we have fp>l = fp,v. '
It rnay be said therefore that every prime ideal p of k for which A
has no, defect behaves similarly in the distinct extensions kil> of k for
(I, P) = 1.
On the other hand, the zeta-function Cfc. A(p)jk(p)) is determined
by the characteristic roots of яр. Hence the above equality G) shows
that the zeta-function ^(s, Ajk) dominates an essential part of the
arithmetical structure of the extensions k<-l) of k. In case where A
belongs to a CM-rype, the extensions k{l) are abelian over k; and this
is why C(s, A/k) is described by means of abelian characters. A more
precise and deep analysis in this case has been given in Taniyama [42].
If the abelian variety A has no or few complex multiplications, the
extensions ka) of k are not necessarily abelian. Denote by k(l; m) the
extension of k generated by the coordinates of the points on A of order
lm. It is easy to see that k(l; 2m) is abelian over k(l; m); hence k(l; m)
is a solvable extension of k(l; 1) if m ^ 1. However, it may happen

[18.5] ZETA-FUNCTIONS OF ABELIAN VARIETIES 151
that k(l; 1) is not a solvable extension of k. Therefore the determi-
determination of the zeta-function of an abelian variety in a general case gives
us some important knowledge of non-abelian or non-solvable ex-
extensions of algebraic number fields. As for the zeta-function of an
abelian variety having no or few multiplications, at present, only the
case of models of certain automorphic function-fields is known ([13],
[35], [37]). We end this monograph, in which we have treated ex-
exclusively abelian varieties with sufficiently many complex multiplica-
multiplications, by emphasizing the importance of zeta-functions of abelian
varieties with no or few multiplications.

BIBLIOGRAPHY
[ 1 ] A. A. Albert, On the construction of Riemann matrices, I, II, Ann. Math.,
35 A934), 1-28, 36 A935), 376-394.
[ 2 ] A. A. Albert, A solution of the principal problem in the theory of Riemann
matrices, Ann. Math., 35 A934), 500-515.
[ 3 ] I. Barsotti, Abelian varieties over fields of positive characteristic, Rendiconti
del circolo di Palermo, 5 A956), 1-25.
[ 4 ] O. Blumenthal, Uber Modulfunktionen von mehreren Veranderlichen, I, II,
Math. Ann., 56 A903), 509-548, 58 A904), 497-527.
[ 5 ] W. L. Chow, The Jacobian variety of an algebraic curve, Amer. J. Math.,
76 A954), 453-476.
[ 6 ] H. Davenpor^ und H. Hasse, Die Nullstellen der Kongruenzzetafunktionen
im gewissen zyklischen Fallen," Journ. Reine Angew, Math., 172 A935), 151-182.
[ 7 ] M. Deuring, Algebren,' Ergebnisse der Math., Berlin, 1935.
[ 8 ] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionen-
korper, Abh. Math. Sem. >Univ. Hamburg, 14 A941), 197-272.
[ 9 ] M. Peurinjj, Reduktion algebraischer Funktionenkorper nach Primdivisoren
des Konstantenkorpers, Math. Zeitschr., 47 A942) 643-654.
[10] My Deuring, Algebraische Begriindung der komplexen Multiplikation, Abh.
Math!* Sem. Univ. Hamburg, 16 A949), 32-47.
[11] M. Deuring, Die Struktur der elliptischen Funktionenkorper und Klassen-
korper der imaginaren quadratischen Zahlkorper, Math. Ann., 124 A952) 393-426.
[12] M. Deuring, Die Zetafunktion einer algebraischen Kurve vom Geschlechte
Eins, I, II, III, IV, Nachr. Akad. Wiss. Gottingen, (UBS) S5-94, A955) 13-42,
A956) 37-76, A957) 55-80.
{13] M. Eichler, Quaternare quadratische Formen und die Riemannsche Ver-
mutung fiir die Kongruenzzetafunktion, Arch. Math., 5 A954), 355-366.
[14] M. Eichler, Der Hilbertsche Klassenkorper eines imaginerquadratischen
Zahlkorpers, Math. Zeitschr., 64 A956), 229-242.
[15] R. Fricke, Lehrbuch der Algebra III, Braunschweig, 1928.
[16] R. Fueter, Vorlesungen iiber die singularen Moduln und die komplexe Multi-
Multiplikation der elliptischen Funktionen, I, II, 1924, 1927.
[17] H. Hasse, Neue Begriindung der komplexen Multiplikation, I, II, Journ.
Reine Angew. Math., 157 A927), 115-139, 165 A931), 64-88.
[18] H. Hasse, Abstrakte Begriindung der komplexen Multiplikation und Rieman-
Riemannsche Vermutung in Funktionenkorpern, Abh. Math. Sem. Univ. Hamburg,
[152]

BIBLIOGRAPHY 153
10 A934), 325-348.
[19] H. Hasse, Zetafunktion und L-Funktionen zu einem arithmetischen Funk-
tionenkorper vom Fermatschen Typus, Abh. Deutscher Akad. Wiss., 1955.
[20] E. Hecke, Hohere Modulfunktionen und ihre Anwendung auf die Zahlen-
theorie, Math. Ann., 71 A912), 1-37.
[21] E. Hecke, Ober die Konstruktion relativ-Abelscher Zahlkorper durch
Modulfunktionen von zwei Variablen, Math. Ann., 74 A913), 465-510.
[22] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur
Verteilung der Primzahlen, I, II, Math. Zeitschr., 1 A918), 357-376, б A920),
11-51.
[23] S. Koizumi, On the differential forms of the first kind on algebraic varie-
varieties, Journ. Math. Soc. Japan, 1 A949), 273-280.
[24] S. Koizumi and G. Shimura, On specializations of abelian varieties, Scienti-
Scientific Papers of the College of General Education, University of Tokyo, 9 A959),
187-211.
[25] L. Kronecker, Zur Theorie der elliptischen Funktionen, 1883-1889, Werke IV.
[26] S. Lang, Abelian varieties, Interscience Tracts, New York, 1959.
[27] S. Lefschetz, On certain numerical invariants of algebraic varieties with
application to abelian varieties, Trans. Amer. Math. Soc, 22 A921) 327-482.
[28] T. Matsusaka, Polarized varieties, fields of moduli and generalized Kummer
varieties of polarized abelian varieties, Amer. J. Math., 80 A958), 45-82.
[29] Y. Nakai, On the divisors of differential forms on algebraic varieties, Journ.
Math. Soc. Japan, 5 A953), 184-199.
[30] Seminaire H. Cartan, E.N.S., 1957/1958, Fonctions automorphes.
[31] J.-P. Serre, Quelques proprietes des varietes abeliennes en caracteristique
p. Amer. J. Math., 80 A958), 715-739.
[32] J.-P. Serre, Groupes algebriques et corps de classes, Hermann, Paris, 1959.
[33] G. Shimura, Reduction of algebraic varieties with respect to a discrete
valuation of the basic field, Amer. J. Math., 77 A955), 134-176.
[34] G. Shimura, On complex multiplications, Proceedings of the International
Symposium on Algebraic Number Theory, Tokyo-Nikko, 1955, 23-30.
[35] G. Shimura, Correspondences modulaires et les fonctions £ de courbes
algebriques, Journ. Math. Soc. Japan, 10 A958), 1-28.
[36] G. Shimura, On the theory of automorphic functions, Ann. Math., 70 A959),
101-144.
[37] G. Shimura, Fonctions automorphes et correspondences modulaires, Pro-
Proceedings of the International Congress of Mathematicians, 1958, 330-338.
[38] C. L. Siegel, Einfuhrung in die Theorie der Modulfunktionen n-ten Grades,
Math. Ann., 116 A939), 617-657.
[39] L. Stickelberger, Ober eine Verallgemeinerung der Kreisteilung, Math.
Ann., 37 A890), 321-367.
[40] T. Takagi, Ober eine Theorie des relativ-Abelschen Zahlkorpers, Journ.
Coll. Science, Tokyo, 41 A920), 1-132.

154 BIBLIOGRAPHY
[41] Y. Taniyama, Jacobian varieties and number fields, Proceedings of the
International Symposium on Algebraic Number Theory, Tokyo-Nikko, 1955,
31-45.
[42] Y. Taniyama, L-functions of number fields and zeta functions of abelian
varieties, Journ. Math. Soc. Japan, 9 A957), 330-366.
[43] H. Weber, Lchrbuch der Algebra, III, Braunschweig, 2 Auflage, 1908.
[44] A. Weil, Foundations of algebraic geometry, New York, 1946.
[45] A. Weil, Sur les courbes algebriques et les varietes qui s'en deduisent,
Hermann, Paris, 1948.
[46] A. Weil, VarictiSs abeliennes et courbes algebriques, Hermann, Paris, 1948.
[47] A. Weil, Number of solutions of equations in finite fields, Bull. Amer. Math.
Soc, 55 A949), 497-508.
[48] A. Weil, Number-theory and algebraic geometry, Proceedings of the Inter-
International Congress of Mathematicians, 1950, 90-100.
[49] A. Weil, Arithmetic on algebraic varieties, Ann. Math., 53 A951), 412-444.
[50] A. Weil, Jacobi sums as " Grossencharactere", Trans. Amer. Math. Soc,
73 A952), 487-495.
[51] A. Weil, On' ajgebraic groups of transformations, Amer. J. Math., 77 A955),
355-391. , )
[52] A. Weil, On1 :i!gebraic gfbups and homogeneous spaces, Amer. J. Math.,
77 (J955), 493-512.
[53] A. Weil, On a certain type of characters of the idfele-class group of an
algebraic number-field, Proceedings of the International Symposium on Al-
Algebraic Number Theory, Tokyo-Nikko, 1955, 1-7.
[54] A. Weil, On the theory of complex multiplication, ibid., 9-22.
[55] Aj Weil, The field of definition of a variety, Amer. J. Math., 78 A956).
509-54.
[56] A. Weil, On the projective embedding of abelian varieties, Algebraic geome-
geometry and topology, a symposium in honor of S. Lefschetz, Princeton, 1957.
[57] A. Weil, Introduction a 1'etude des varietes kahleriennes, Hermann Paris,
1958.

TABLE OF NOTATIONS
), <^fo(A) l KM)) 69
<j*f(A; k), Л0(А; k) 17 (K; {<pt}; f0) 118
(Д c) 42 (K; {wi}; f0; b) 131
(A, t, <€) 117 Щ) 131
{A, c; X) 53 Us, x) 143
(a-, p) 121 l(X) 26
(a, ^ 132 L(X; k) 26
'a 3 A'1 for an isogeny ^ 2
(О/Д в) 21 Af,( ) 2
ЩК) 122 mod(r; in) 141
&(K;b) 132 N( ) 38
3 N(a), Ща) 55
26 Ща), Ща) 61
13 Ч ), vt( ), vt( ) 1
k) 5 J* = (A,e,V) 117
; k) 5 {J*r:J»} 122
) 9, 17 (^, t) = (A, c, V, t) 131
ea{u,X) 4 {(J*u h):(J»,t)} 132
E{X) 20, 21 Pt(nt) • 124
(F; Ы) 43 XV) 80
№ 120 <px 3
fl(a), 9(a, A) 54 [a] 125
Bi(A) 2 [a]t 133
6(X),S(m,A) 1 tr( ) 2
ЛГ{А, В), JCT^A, В) 1 Tr( _) 38
JCT{A, B; k) 94 (V, <£) 27
/*(nt) 124 (x)-+(£) ref. p (or k) 78
K* 71 [(xHf); p], [(*)-»(«; k) 78
Kco 77 Z(u; Vjk), C(*; VIk) 144
k(V) 5
155]

INDEX
Abelian variety of type >Type
almost all (for — p) 100, 126
Alternating form
— defined by a divisor 21
— defined by a theta-function
20
Ample (divisor, linear system) 26
a-multiplication 52
Analytic
— coordinate system of an
abelian variety 21
— representation of ^JC^0(A, B)
21
a-section point 63
a-transform 53
Basic polar divisor 30
Basic system for a variety 99
Characteristic polynomial, roots
of an element of <jtfo(A) 2
Class of positive Hermitian forms
121, 132
CM-type 44
Complex torus 19
Conductor
— of a Grossen-character
143, 149
— of an order 136
Defect for p 94, 108
Differential form 5
— of the first kind 8
Discrete place 79
Dual
— of a CM-type 71
— of an abelian variety 23
Equivalent
({a, P)) 121,
(Grossen-character)
Extension of a place 78
Field of moduli 29,
Finite (differential form)
— along (or at) a subvariety
— at a point of V"
Functional equation 143,144,
Grossen-character
Hasse's conjecture
Homomorphism
— of an abelian variety
- of (Д <g)
— of (Д t)
Index of (A, c)
Invariant differential form 9,
132
143
, 79
125
6
90
145
142
145
1
28
52
56
, 12
Involution of <jxfo(A) determined
by a divisor
3
Irreducibility of the class-equa-
class-equation 128,
Isogenous, isogeny
Isomorphism
— of an abelian variety
- of {A, <g)
- of (A, t)
— of J» = {A, i, <£)
Jacobian variety 17,
A-basic system for a variety
Kummer variety
/-adic coordinate system
/-adic representation
— of a divisor
- of J6T&A, B)
130
1
1
28
52
117
130
99
35
2
4
2
[157]

158
INDEX
Lattice in an algebra 52
Left invariant differential form 9
L-function with Grossen-char-
acter 143
Local parameters 7, 87
Modular function
elliptic — v, 129, 135
Hilbert — v
Siegel — 29
Multiplication (a-) . 52
Non-degenerate divisor 3, 28
N(p)-th power endomorphism,
homomorphism 110
Normal lattice in an algebra 55
Normalized
— Kummer variety 35
— theta-function 20
Open set of discrete places, 100
Order
— in an algebra . 52
— of (Д e) , 52
p-basic system for a variety 99
^-complete 82
— differential form
— rational mapping
p-simple
).i-variety
Picard variety
Place of a field
Polar divisor
Polarizable, polarization
Polarized abelian variety
— of type (K; {9i})
— of type (K; {ot}; f0)
Polarized variety
Point in V~
Primitive
— CM-type 68,
90
85
83
82
3, 51
77
27
27
27
117
118
27
83
72, 73
— Grossen-character 143
Principal ((A, e)) 56
Proper a-section point 63
q-th power endomorphism,
homomorphism 4, 5, 66
Quotient of an abelian variety by
a group of automorphisms 34
Rational representation of
JCT^A, В) 22
Reduced norm, — trace, — rep-
representation of a simple algebra
38
Reduction modulo \>
— of a cycle 80, 81, 83
— of a differential form 91
— of a function 85
— of a homomorphism 94
— of a rational mapping 85
— of an abelian variety 94
— of an abstract variety 83
— of an affine variety 79
Regular at a point of V' (bira-
tional mapping) 81
Regularly corresponding points
82
Representation
analytic — of ЛГ9(А, B) 21
Z-adic — of a divisor h
/-adic — of J^0{A, B) 2
rational — of Л/й{А, В) 22
— of *jzfu{A; k) by invariant
differential forms 17
Representative in V 83
Residue field of a place 77
Riemann form 19
— defined by a divisor 21
— defined by a theta-function
20

INDEX
159
Riemann hypothesis for con-
congruence zeta-functions v, 5
Ring of ^-integers 77
Section point (a-) 63
Simple (a point or a subvariety
in 9 — on V) 81, 83
Specialization over a place 78
— of a point 83
Specialization ring 31, 78
Stickelberger's relation 130
Subvariety of a ^-variety 82
System of local parameters 7, 87
Theta-function 20
Transform (a-) 52
Transpose of an element of
Ж<кА, В) З
Trivial place 78
Type
— (F), C1) 42, 43
— (F; Vlt:;9n) or (F; {wt})
43
— (K; {<*}; fо; b) or «F) 131
Variety in P 82
Weber's r-function 135
Weierstrass' p-function 135
Weil's conjecture
— on Z(u; V{k) 144
— on COr; V\k) 145
Zeta-function of a variety
— defined over a finite field
143
— defined over an algebraic
number field 144