Lecture Notes in Mathematics A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and 6. Eckmann, Zurich
Series: Forschungsinstitut f i r Mathematik, ETH, Zurich . Adviser: K. Chandrasekharar
Goro Shimura Princeton University, Princeton, New Jersey
Automorphic Functions and Number Theory
Springer-Verlag Berlin Heidelberg New York
Preface
These notes a r e based on lectures which I gave at the
Forschungsinstitut f a r Mathematik, Eidgenossische Technische
Hochschule, Ziirich in July 1967. I have attempted to make a
shor t comprehensible account of the latest resul ts in the field,
together with an exposition of the mater ia l of an elementary nature.
No detailed proofs a r e given, but there i s an indication of basic ideas
involved. Occasionally even the definition of fundamental concepts
may be somewhat vague. I hope that this procedure will not bother
the reader . Some references a r e collected in the final section in
order to overcome these shortcomings. The reader will be able to
find in them a more complete presentation of the resul ts given here,
with the exception of some resul ts of §lo, which I intend to discuss
in detail in a future publication.
It is my pleasure to express my thanks to Professors K.
Chandrasekharan and B. Eckmann for their interest in this work,
and for inviting m e to publish it in the Springer Lecture Notes in
Mathematics. I wish a lso acknowledge the support of the
Eidgenossische Technische Hochschule, Institute for Advanced Study,
and the National Science Foundation (NSF-GP 7444, 5803) during the
summer and fall of 1967.
Princeton, January 1968 G. Shimura
All rights re\crued. No part uf this book m q be translated or reproduced in any form without wrincn permission from Springer Veriag. 0 by Springer-Vdag Berlin. Hddelberg 1%8
.I,ibmty of Congress Catalog G r d Number 68-2>132. Printed in Germany. Title No. 7374
Contents Notation
Introduction 1
Automorphic functions on the upper half plane, especially modular
functions
Elliptic curves and the fundamental theorems of the classical
theory of complex multiplication
Relation between the points of finite order on an elliptic curve
and the modular functions of higher level
Abelian var ie t ies and Siege1 modular functions
The endomorphism ring of an abelian variety; the field of moduli
of a n abelian variety with many complex multiplications
The c l a s s -field-theoretical characterization of K' ( (z)).
A further method of constructing c lass fields
The Hasse zeta function of an algebraic curve
Infinite Galois extensions with l -adic representations
Fur ther generalization and concluding remarks
Bibliography
We denote by 2, Q, R and C respectively the ring of rational
integers, the rational number field, the r ea l number field and the
complex number field. F o r an associative ring Y with identity ele- X
ment, Y denotes the group of invertible elements in Y, M (Y) the n
r ing of al l matr ices of size n with entries in Y, and GLn(Y) the X group of invertible elements in M (Y), i. e. , Mn(Y) . The identity n
element of M (Y) i s denoted by 1 and the transpose of a n element n t n '
A of Mn(Y) by A a s usual. When Y i s commutative, SLn(Y)
denotes the group of a l l elements of M (Y) of determinant 1. For a n
typographical reason, the quotient of a space S by a group G will be
denoted by S/ G, even if G acts on the left of S. If F is a field
and x i s a point in an affine (resp. a projective) space, then F(x)
means the field generated over F by the coordinates (resp. the
quotients of the homogeneous coordinates) of x. If K i s a Galois
extension of F, G(K/ F) stands for the Galois group of K over F.
1. Introduction
Our starting point i s the following theorem which was stated by
Kronecker and proved by Weber:
Theorem 1. Every finite abelian extension of Q contained & 2ni/ m
a cyclotomic field Q(5) with an m-th root of unity 3 = e - for - some positive integer m.
As i s immediately observed, 5 i s the special value of Qe expo- 2 niz
nential function e a t z = l/ m. One can naturally ask the following
question:
2 niz Find analytic functions which play a role analogous to e -
for a given algebraic number field. -- Such a question was ra ised by Kronecker and la ter taken up by
Hilbert a s the lzth of his famous mathematical problems. F o r an
imaginary quadratic field K, this was settled by the works of Kronecker
himself, Weber, Takagi, and Hasse. It turns out that the maximal abelian
extension of K i s generated over K by the special values of certain
elliptic functions and elliptic modular functions. A pr imary purpose of
these lectures is to indicate briefly how this resul t can be generalized
for the number fields of higher degree, making thereby an introduction
to the theory of automorphic functions and abelian variet ies. I will a h o
include some resul ts concerning the zeta function of an algebraic curve
in the sense of Hasse and Weil, since this subject i s closely connected
with the above question. Fur the r , it should be pointed out that the auto-
morphic functions a r e meaningful a s a means of generating not only
abelian but a lso non-abelian algebraic extensions of a number field.
Some ideas in this direction will be explained in the l a s t par t of the
lec tures .
2. Automorphic functions on the upper half plane,
especially modular functions
Let 5 denote the complex upper half plane:
b We l e t every element a = d) of GL2(R), with det ( a ) > 0, ac t
(2.1) a (z) = (az + b ) / (cz t d)
It i s well known that the group of analytic automorphisms of ff i s
isomorphic to S L (R)/ {tl ). Let r be a discrete subgroup of 2 2
SL2(R). Then the quotient $ 1 ~ has a structure of Riemann surface
such that the natura l projection $ + $/I? i s holomorphic. If
$ /r is compact, one can simply define an automorphic function on
f respect t_o l? to be a meromorphic function on -$ invariant
under the elements of I? . Such a function may be regarded a s a
merornorphic function on the Riemann surface $11. in an obvious way,
and vice versa . We shall la ter discuss special values of automorphic
functions with respect to r for an arithmetically defined l? with
compact #/r . But we f i r s t consider the most classical group l? =
S L (2). Since 2 $/I' i s not compact in this case , one has to impose
a certain condition on automorphic functions. It i s well known that
every point of f can be transformed by an element of T = SL (2) 2
into the region
No two distinct inner points of F can be transformed to each other
by an element of T . Now $/I? can be compactified by adjoining a 2riz point a t infinity. By taking e a s a local parameter around this
point, we see that $1 r becomes a compact Riemann surface of
genus 0. Thus we define an automorphic function with respect to l?
to be a meromorphic function on this Riemann surface, considered a s
a function on . In other words, le t f be a r- invariant meromor- /B 1 phic function on f. F o r y = 6 , we have y (z) = z t 1. Since
0 27rinz f (y (2)) = f (z), we can express f (z) in the form f (z) = Z c e n=-w n
with c c C. Now an automorphic function with respect to r is an n
f such that c = O for a l l n < n for some n , i. e . , meromorphic n
2riz in the local parameter q = e a t q = 0 . Such a function is usually
called a modular function of level one. Since $ 1 is of genus 0,
all modular functions of level one form a rational function field over C.
As a generator of this field, one can choose a function j such that
Obviously the function j can be characterized by (2.2) and the
property of being a generator of the field of a l l modular functions
of level one.
Now le t K b e an imaginary quadratic field, and a a f r a c -
tional ideal in K. Take a bas is {wl, w 2 ) of 8t over Z. Since K
is imaginary, wl/ w2 i s not real . Therefore one may assume that
wI/ Y2 ' $ . by exchanging w and u2 if necessary. In this setting. 1 we have:
Theorem 2. T h e maximal unramified abelian extension of K
can be generated 9 j (w / w2) over K. -- 1
This is the f i r s t main theorem of the classical theory of com-
plex multiplication. To construct ramified abelian extensions of K ,
one needs modular functions of higher level (see below) or elliptic
functions with pe r iods w w2 . Even Th. 2 can fully be understood
with the knowledge of elliptic functions o r elliptic curves, though such
a r e not explicitly involved in the statement. Therefore, our next task
is to recal l some elementary facts on this subject. But before that,
i t will be worth discussing a few elementary facts about the fractional
l inear t ransformat ions and discontinuous groups.
Every a = % :) r GL (C) acts on the Riemann sphere C U{m) 2 - 1
by the rule (2.1). With a suitable element fj of GL2(C), P = Sac
has one of the following two normalized forms:
(i) P(z) = z t A,
(ii) P(z) = K z ,
with constants h and K . This can be shown, for example, by taking
the Jordan form of a . In the f i r s t case , we call a parabolic; in
the second case, a is called elliptic, hyperbolic, or loxodromic,
according a s 1 K I = 1, K real , o r otherwise. In this classification,
we exclude the identity transformation, which is represented by the
scalar matrices.
If a s GL2 (R) and det (a) > 0, a maps -f onto itself, and a
i s
elliptic if a has exactly one fixed point in $ 8
hyperbolic if a has two fixed points in R U {m),
parabolic if a has only one fixed point in R U {m).
No transformation in GL (R) with positive determinant is loxodromic. 2
If we put
then i t can easily be verified that SO (R) i s the se t of al l elements 0.l 2
SL2(R) which leave the point i fixed. Therefore the map
\
gives a diffeomorphism of the quotient SL2 (R)/ Slb2 (R) onto # . It i s a fruitful idea to regard $. a s such a quotient. But I shall not
pursue this view point, from which one can actually s t a r t investi-
gation in various directions.
We see easily that a differential form
- 2 Y d x A d y (z = x t iy)
on is invariant under SL2 (R). Therefore we can introduce an
invariant measure on # by means of this form.
To speak of an automorphic function for a r with non-compact
1 , we have to introduce the notion of cusp. Let J? be a discrete
subgroup of SL2(R). We call a point s of RU{m) a = o f J? if
there exists a parabolic element y of I7 leaving s fixed. Let
Then one can find an element p of SL 2 (R) so that p(w) = s , and -1
PTsp i s generated by (1O i) and possibly by -I2 . Then we define
an automorphic function on #. w x respect 2 r to be a meromorphic
function on -# satisfying the following two conditions:
(i) f ( y ( z ) ) = f ( z ) for all y e r .
(ii) Lf s g a cusp of I? a n d p k a s above, then f(p(z)) k a 2riz
meromorphic function in q = e iz _n neighborhood o_f q = 0.
(Here note that if f satisfies (i), then f(p(z)) is invariant under
z H z + 1, hence f(p(z)) i s always meromorphic a t least in the
domain O < ( q[ < r for some r > 0 . ) Let 9 be the join of # and all the cusps of I? . Then J?
acts on 9 . One can define a structure of Riemann surface on
fj*/r by taking 2 z ] a s a local parameter around the
point s . (Actually the proof of the fact that */I? is a Hausdorff -% space i s not difficult, but non-trivial. ) Then an automorphic function
with respect to r , defined above, is nothing else than a meromorphic
function on the Riemann surface PIT, regarded a s a function on
-f . The above discussion about SL ( Z ) i s a special case of these 2
facts. Now the following facts a r e known:
Proposition 1. j'/r is compact if and only if d / r has - - a finite measure with respect to the above invariant measure. --
Proposition 2. Suppose that /r has a finite measure.
- $ Then I is compact if and only if I' has no parabolic element.
of r - finite -
As for elliptic elements, the following proposition holds :
Proposition 3. k t r & a point of -$ fixed 9 an e c element
. - Let = a a z = z . Then Tz 2 a cyclic group of - order. - 6
Such a point z is called an elliptic point of I?, and the order of rz. {+12}/ {+12} is called the o rder of the point z (with respect
to I?). Two elliptic points or cusps a r e called equivalent if they a re
transformed to each other by elements of I?. If -f/r is of finite
measure , there a r e only a finite number of inequivalent elliptic
points and cusps, and the following formula holds:
Here g i s the genus of p / r ; h i s the number of inequivalent
cusps; CZ i s the sum extended over al l inequivalent elliptic points;
e is the order of z. F o r r = SL2(Z), one has g = 0, h = 1, eZ =
2 o r 3 according a s z = cl or z = (-1 + c 3 ) / 2.
F o r every positive integer N, se t
I'm) = { a c S ~ ~ ( z ) 1 a r l2 mod N. M ~ ( z ) ) .
An automorphic function with respect to I 'm) i s called a modular
function of level N. ---
3. Elliptic curves and the fundamental theorems
of the c lass ica l theory of complex multiplication
Let L be a lat t ice in the complex plane, i. e . , a f ree
Z - submodule of C of rank 2 which i s discrete. Then C / L i s a
compact Riemann surface of genus one. An elliptic function with
periods in L i s a meromorphic function on C invariant under the
translation u u + w for every w t L. Define complex numbers g2, g3 and meromorphic functions P(u) and 8. (u) on C by
where C denotes the sum extended over a l lnon-ze ro w in L.
Then it is well-known that
(3. 3) The field of a l l elliptic functions with periods in L coincides -
with C ( 9 , ), the field generated b~ P and $g E r C. - - Now le t E be the algebraic curve defined by
Here we consider E a s the se t of al l points (x. y) satisfying (3.4) '
with x, y in C, together with a point (w, a!. Then the map
gives a holomorphic isomorphism of C / L onto E in the sense of
complex manifold. It is a lso known that any elliptic curve (i. e . an
algebraic curve of genus one) defined over C is isomorphic to a
curve of this type, and hence to a complex torus.
Take a bas is {ul, w ) of L over 2. We may assume that 2
w / u f , Then one can easily show that 1 2
defines a one-to-one correspondence between $11. & al l the iso-
morphism-classes of elliptic curves. Fur thermore we have an
important relation
One should note that the right hand side can be obtained purely a l -
gebraically f rom the defining equation (3. 4) for E, while the lef t
i s defined analytically. This coincidence of an algebraic object with
an analytic object has a deep meaning, though we know, from (3. l ) ,
that g2 and g depend analytically on wl and w2 . We call the 3
number expressed by (3 .5) the invariant o_f E. Two elliptic curves
have the s a m e invariant if and only if they a r e isomorphic.
Let us now observe that any holomorphic endomorphism of E =
C/ L is obtained by u H Xu with a complex number X satisfying
A. L C L. Let End(E) denote the ring of all such endomorphisms. It
can easily be proved that End(E) i s isomorphic to Z unless
Q(w / w ) i s an imaginary quadratic field. Assume that Q(w / w ) is 21 2 1 2 imaginary quadratic, and put K = Q(w / ). Then End(E) i s iso-
1 ' 2 morphic to a subring of the ring 0' of all algebraic integers in K ,
which generates K. In this case we say that E has complex multi-
plications. In particular, if L = Zwl + Zw2 i s an ideal in K, End(E)
is isomorphic to Q . Put jo = j(wl/ %). For a given L (or wl, w2),
one can find the equation (3. 4) s o that g2 and g3 a r e contained in
Q(j ). Moreover j is an algebraic number if E has complex multi- 0
plications.
Now write E a s E(&) if L = DL for an ideal 6t in K.
We choose the equation for E(8L ) so that g2, g3 c Q(jo). Suppose
we could somehow prove that K(j ) i s an abelian extension of K.
(Anyway this i s not the most difficult point of the theory. ) Take a
pr ime ideal J7/ in K unramifiedin K ( j ), a n d l e t a = [T , KGo)/K]
(= the Frobenius automorphism of KG ) over K for 2 ). Then g2
and g3 a r e meaningful. Therefore we can define an elliptic curve
E ( @ L ) O by
Then one has a fundamental relation:
If we denote by j (a ) the invariant of E(& ), then (3.6) i s equi-
valent to
F r o m the relation (3.6) or (3.7), one can easily derive Th. 2
and also the reciprocity law in the extension KGo) of K. Here I do
not go into detail of the proof of (3. 6), but would like to call the reader1 s
attention to the following point: Although no elliptic curves appear in
Th. 2 , they conceal themselves in it through the above (3.6) and the
following facts.
(3.8) The quotient $/I? 2 in o n e - 5 - 0 2 correspondence with all - the isomorphism classes of elliptic curves.
(3. 9) j(w 1 w ) is the invariant of an elliptic curve E isomorphic 1 2 --
to C / (Zwl + Zw2). - (3.10) 2 Q(ul/ uZ) 2 imaginary quadratic, End@) i s non-trivial.
Let us now consider the question of generalizing Theorems 1 and
2 to the fields of higher degree. We observe that there a r e three
objects:
(A) elliptic curve,
(B) modular function,
(C) imaginary quadratic field.
Among many possible ideas, one can take the most naive one, namely
ask whether there exist generalizations of (A), (B), (C) whose inter-
relation i s s imi lar to that of the original ones, as described in (3.8-10).
The answer is affirmative but not unique. It may be said that the
world of mathematics is built with a great harmony but not always
in the form which \r*e expect before unveiling it. This certainly
applies to our present question. I shall , however, f i r s t present a
comparatively simple answer which consists of the following three
objects:
(A1 ) abelian variety,
(B' ) Siege1 modular function,
(C1 ) totally imaginary quadratic extension of a totally r e a l
algebraic number field.
At leas t this will include the above resul t concerning elliptic curves
a s a special case. A different type of theory, which I feel rather un-
expected, and which also generalizes Th. 2, will be discussed later.
4. Relation between the points of finite order on
an elliptic curve and the modular functions of
higher level.
Before talking about abelian varieties, le t us discuss the topic
given a s the title of this section. Any hasty reader may skip this
section, and come back afterward.
F ix a positive integer N. Observe that any point t on E
such that Nt = 0 can be expressed a s
with integers a , b. Now, for each ordered pair (a, b) of integers
such that (a, b) ? (0, 0) mod (N), we can define a meromorphic N
function f (z) on # by ab
where z = w / w and L = Zwl t Zw2 . This i s possible because the 1 2
right hand side depends only on z = w / w Then 1 2 '
N N f a b ( z ) = f z ( a , b ) = _ ( c , d ) m o d ( N )
cd or (a, b ) ( - C , -d) mod (N).
By a simple calculation, we can show that, for every a r SL2(Z),
N N Therefore, f (2) = f (a (2)) for a l l (a, b) if and only if a belongs
ab ab
to I? (N). {iJL}. It follows that j and the f:b , for a l l (a, b ) ,
generate the field of a l l modular functions of level N. Roughly speak-
ing, the modular functions of level N can be obtained from the in-
variant of elliptic curves and points of o r d e r N on the curves. Now
we have the following resul t which is an analogue of Th. 1 for an
imaginary quadratic field.
Theorem 3. Let K be a s above, and @, an ideal in K. Take
@L L, and let 6 = Zw1 t Zw2 wifh wl/ w2 6 6 . Suppose that
g Z ( 6t )g3(8L) # 0. Then the maximal abelian extension f K i s N
generated over K & j (wl/ w2) and the fab (wl/ w2) for al l N, a , b ,
with a fixed . 8L . --- N
It should be observed here that fab(wl/ w2) i s a special value
of an elliptic function and a special value of a modular function of
level N a s well. This coincidence will not necessari ly be retained
in one of our later generalizations.
We note that g (a ) = 0 or g3 (a ) = 0 according a s K = 2 a ( ( - 1 t a)/ 2) or K = Q( m). In these special cases , we can still
h T
obtain the same type of resul t by modifying the definition of fLY ab
suitably. N
The function field C ( i , f ), with a fixed N, is a Galois ex- ab
tension of C(j) whose Galois group is isomorphic to r ( l ) / r ( N ) . {&12};
the lat ter group is isomorphic to SL (Z/ NZ)/ (21 ). Since our purpose 2 2
is to construct number fields by special values of functions, i t i s
meaningful to take Q, instead of C, a s the basic field. Then we
obtain:
N Theorem 4. Q(j, fab) i s a Galois extension o_f Q(j) whose
Galois group is isomorphic to GL2(Z/ NZ)/ {f12}, and the following
statements hold. (i) F o r every a E GL2(Z/ NZ), the action of a an element
of the Galois group is given by --- fib I-+ fN with (c d) = (a b ) a . cd -
N (ii) If y E SL ( Z ) , the action of y mod (N) Q(j, f ) is 2
N ab -
o b t a i n e d h 'f'(z)rj y(y(z)) for Y E Q(j, fab).
N (iii) Q(j, f ) contains 5 = e 2nil N, and
ab - a r GL2 (Z/ NZ)
sends 5 2 6 de t (a )
We shall la ter extend this theorem to the field of automorphic
functions with respect to a more general type of group.
5. Abelian varieties and Siege1 modular functions
A non-singular projective variety of dimension n , defined over
C, is called an abelian variety if i t i s , a s a complex manifold, iso-
morphic to a complex torus of dimension n. An elliptic curve is
nothing but an abelian variety of dimension one. We know that any
one dimensional complex torus defines an elliptic curve, but such
is not t rue in the higher dimensional case. To explain the necessary
condition, le t L be a lattice in the n-dimensional complex vector
n space cn , i. e . , a discrete f ree Z-submodule of rank 2n in C . n
Then the complex torus C / L has a structure of projective variety,
and hence becomes an abelian variety, if and only if there exists an
R-valued R-bilinear form G(x, y) on Cn with the following
properties:
We call such 6 a Riemann form on Cn/ L. Take a basis n
{vl, . . . , v ) of L over Z , and regard the elements of C a s 2n
column vectors. Then we obtain a matr ix
of nX2n type, which may be called a p e r i o d matr ix for Cn/ L. De-
fine a m a t r i x P = (p..) of size 2n by p.. = f*(vi , v.). Thenthe U 1J J
above a r e equivalent to the following (Ri-3):
i ) 'P = -P;
(R;) p.. Z ; 1J
5 ) 52p-l . = 0, & - ~ 1 a ~ - I . t~ --- i s a positive definite - hermitian matrix.
The matr ix P (or its inverse) i s called a principal matr ix of 52 .
Assuming these conditions, le t A be a projective variety iso- n
morphic to C / L. Shifting the law of addition of cn/ L to A , we
can define a structure of commutative group on A. Then the map
A X A 3 ( x , y ) C , x + y E A
can be expressed rationally by the coordinates of x and y. This is
classically known a s the addition theorem of abelian functions. '
In general, a projective variety A, defined over any field of
any characterist ic, is called an abelian variety, i f there exist rational
mappings f : A X A -+ A and g: A --t A which define a group s t ruc-
ture on A by f(x, y) = x + y, g(x) = -x. Additive notation is used since any such
group s t ructure on a projective variety can be shown to be commuta-
tive. It should be observed that such a variety defined over C ,
being a connected compact commutative complex Lie group, must be
isomorphic to a complex torus.
If n = 1, there is a single universal family of elliptic curves
parametrized by the point of # . If n > I, however, there a r e
infinitely many families of abelian varieties depending on the elementary
divisors of P , a s shown in the Supplement below. But we shall f i r s t
fix our attention to one particular family by considering only abelian
varieties for which P = J . where n
Under this assumption, le t q and w2 be the square matrices of size
n composed of the f i r s t and the l a s t n columns of respectively. -1
One can show that w i s invertible. Pu t z = w u1 . If we change the n
coordinate system of C by w2 , we may assume that 52 i s of the
form
Now it can be. shown (see Supplement below) that z is symmetric and
Im(z) i s positive definite. We denote by $-- the s e t of a l l such z
of degree n. Thus every abelian variety, under the assumption that
P has the form (5. l ) , corresponds to a point of , though z is #n
not unique for a given abelian variety. Moreover, every point of
corresponds to such an abelian variety. Obviously
As an analogue of SL (R), we introduce a group 2
F o r every U = [: 1 1 6 Sp(n, R) with a , b, c , d in M n (R), we
define the action of U on
Put
When n > I, we can define an automorphic function with respect to
Sp(n, Z ) to be a meromorphic function on invariant under
Sp(n, Z). Fortunately, if n > 1, i t i s not necessary to impose any
further condition like that we needed in the case n = 1. Such a
function i s us'ually called a Siege1 modular function (of degree n
and level one).
Pu t r = Sp(n, Z). Now one can ask whether the quotient fillr i s in one-to-one correspondence with all the isomorphism classes of
abelian varieties of type (5.1). This is almost so but not quite. To
get an exact answer, we define 2n rea l coordinate functions x (u), n n
1 . . . , x (u) (u E C ) by u = xi(u)vi , and consider a cohomology
2n c lass c on A represented by a differential form
of degree 2. Such a c is called a polarization of A, and the s t ruc-
ture (A, c ) formed by A and i t s polarization c i s called a polarized
abelian variety. T&n f n represents a11 the isomorphism classes
of polarized abelian var ie t ies of type (5. l ) , the isomorphism being de- - fined in a natural way.
Our next question i s about the existence of some functions s imi lar
to j and the analogue of (3.5). F i r s t one should note that there exists
a Zar iski open subset V o_f 5 projective var ie ty V* and a holomorphic -- mapping of fn into V which induces a biregular isomorphism of
$n/r Onto V. This was proved by W. L. Baily using the Satake
compatification of I . We call such a couple (V, y ) a model
fo r Sn/r . This is-not sufficient for our purpose. In fact , in the
case n = 1, the function ( a j + p) / (y j + 6) plays a ro le of ? f o r
any [: GL2(C). Of course one can not replace j by such a
function in Th. 2. Fu r the rmore , we would like to have an analogue of
(3.5). Therefore a further refinement i s necessary , and can be given
a s follows:
Theorem 5. There exists a couple (V, ) with the following
properties:
(i) (V, y ) i s a model for f n f r .
(ii) V is.defined over Q.
(iji) k t (A, c ) a polarized abelian variety with a P of type
(5. l ) , defined over a subfield k f C , a n d a & isomorphism of k
into C. Let z and z' be points on - f n corresponding t_o (A, c)
and (A, c ) ~ respectively. Then the coordinates of the point (z) - belong to k, a n d ~ ( z ) ~ = Y)(zl ).
In (iii), we of course consider A a s a projective variety de-
fined by some homogeneous equations. Now one can prove that the
cohomology c lass c i s represented by a divisor on A (i. e. an (n - 1)-
dimensional algebraic subset of A). If the defining equations for A
and such a divisor have coefficients in a field k, we say that (A, c)
i s defined over k. If o i s a s in (iii), the transforms of the equations
by o define an abelian variety together with a divisor, which turns out
to be a polarized abelian variety of type (5. l ) , which we write a s (A, cIa . We can actually prove a stronger statement than (iii), which is roughly
a s follows:
(iv) If (A' , c 1 ) 2 specialization of (A, c ) o v e r Q, z'
corresponds 2 (A' , c ' ), then ((A1 , c 1 ), (zl )) specialization
of ((A, c ) , cf(z)) over Q. - F o r details we refer the reader to the paper [ZO] in $12.
Thus plays a role s imi lar to j. It i s analytic on fn , and
a t the same t ime, it i s a rational expression of the coefficients of defin-
ing equations for (A, c ) , a s explained in (iii). F r o m (i) it follows that
the coordinates of y ( z ) generate the whole field of Siege1 modular
functions of degree n. The couple (V, y ) can be characterized by
these properties (i, i i , iii). Namely, if (V' , cj?') i s another couple
with the same properties, there exists a biregular isomorphism f of
V onto V' defined over Q such that y ' = f 0 (o . Moreover, from
(iii), we see that the field Q( 'f'(z)) has an invariant meaning for the
isomorphism class of (A, c) . We cal l i t the field of moduli of (A, c ) .
Actually we can prove a l l these things without assuming P = J . n
F o r each choice of P (or ra ther for a choice of elementary divisors
of P ) , one obtains a group r (see Supplement below) acting on P
and a couple (Vp , yp) with the properties (i, ii, iii) modified %n
suitably. Fur ther , by considering the points of finite order on the
abelian varieties, one can generate the field of automorphic functions
with respect to congruence subgroups of Sp(n, Z) ; one then obtains
a theorem analogous to Th. 4.
The next thing to do is the investigation of special members of
our family of abelian var ie t ies , analogous to elliptic curves with com-
plex multiplications. This will be done in s6.
Supplement t_o s5. To discuss the families of abelian varieties
of a more general type, for which P is not necessari ly of the form
(5. l ) , f i r s t we reca l l a well known
Lemma. L A P b_e invertible alternating matr ix of size 2n
with entries in 2. Then there exists an element U of GL (Z) --- 2n that -
- where the e a r e positive integers satisfying eitl = 0 mod (ei). -- i -
Therefore, to discuss 52 satisfying (Ri-3), we may assume r 0 - e l
t h a t P = L e o l with e a s in the above lemma. Let Y p be
the space of a l l such 51, and le t
In particular G = Sp(n, R) if e = 1 . If 52 c Yp and U r G P n P ' t
then a U r Yp . Obviously BJ B = P, hence BG B-I = Sp(n, R). n P
Now write = (v v l ) with two elements v and v' of Mn(C). Then,
f rom (R;), we see easily that
-1 t- e l . - ve . v ' ) is positive definite.
The las t fact implies that v and v1 a r e invertible. F r o m these
relations i t follows that evl -'v is symmetric and has a positive
definite imaginary par t , i. e. , ev' - l v r . n
If z $n and U =[: t] r Sp(n, R) , then (z ln)U r Y Jn'
hence by the above result , (z ln)U = A(w ln) with A r M (C) and n - 1
w t $ . Then one obtains w = (az t b)(cz t d) . This shows that n the action of U on -f can actually be defined. Since the action
n
of U-I can be defined, U gives a holomorphic automorphism of
Now s e t
It can eas i ly be seen that rp is a discrete subgroup of Sp(n,R). Then
f n / F p represents a l l the isomorphism classes of polarized abelian
var ie t ies with principal ma t r ix P.
The notion of polarization can a lso be defined in the case of
positive characterist ic. Given an abelian variety A defined over a
field of any character is t ic , and given a divisor X on A, let L be
the l inear space of al l rat ional functions on A whose poles a r e con-
tained in X (even with multiplicities). Take a basis {fo, fl, . . . , fN)
of L over k, and consider the map
A 3 x (fo(x), . . . , f N ( x ) ) ~ projective N-space.
We cal l X ample if this i s a biregular embedding of A into the
projective space. Now a polarization of A i s a se t of divisors
on A satisfying the following conditions:
(1) contains an ample divisor.
(2) _II X X' belong to , then there a r e two positive
integers m m t such that mX & m ' X t a r e algebraically
equivalent. .
(3) 5 maximal se t satisfying the above two conditions.
In general , two divisors X and Y a r e called algebraically
equivalent, if there exist a divisor W and i ts specializations W 1
and W2 over an algebraically closed field such that X - Y = W - W 1 2 -
If the universal domain is C, then the algebraic equivalence of di-
v isors coincides with the homological equivalence. Moreover, if a
divisor X represents the cohomology c lass c obtained f rom a
Riemann form, then 3X i s ample. Every abelian variety, defined
over a field of any character is t ic , has a polarization, since i t always
has an ample divisor.
Now a polarized abelian variety i s a couple (A, X ) formed by
an abelian variety A and a polarization of A. This definition
i s equivalent to the previous one, if the universal domain is C. An
isomorphism of A of A' i s called an isomorphism of (A, x ) to
(A' , xt ) if it sends to X I . F o r a given (A, x), we can
prove that there exists a field k with the following properties: 0
(i) If (A, x ) i s defined over k, then k i s contained & k. - 0 - (ii) F o r an isomorphism o of k into the universal domain,
(nu. EO) 5 isomorphic f_o (A, X ) if and only if o i s the identity -- mappins on k . -
If the universal domain is C, k i s uniquely determined by the
isomorphism class of (A, x), and is called the field of moduli of
(A, XI. This of course coincides with Q( Y, (2)) in the special case
P = J n .
6. The endomorphism-ring of an abelian variety;
the field of moduli of an abelian variety
with many complex multiplications
F o r an abelian variety A, we denote by End(A) the ring of
a l l holomorphic endomorphisms of A. If A i s isomorphic to a
complex torus Cn/ L , every endomorphism of A corresponds to
an element T of M (C), regarded a s a C-linear transformation n
on Cn , satisfying T (L) C L. Therefore End(A) is a f r ee Z-module
of finite rank. Let End (A) = End(A) 8 =Q, and W = Q. L. Then W is Q n
a vector space over Q of dimension 2n, which spans C over R, and
End (A) i s isomorphic to the ring Q
F o r each element of End (A), consider the corresponding element T Q
of Mn(C). Then we get a faithful representation of End (A) by complex Q
mat r i ces of s ize n , which we cal l the complex representation of End (A). Q On the other hand, with respect to a bas i s of W over Q (for example,
{vl,. . . , v ) considered in s5), we obtain a representation of End (A) 2n Q
by rational ma t r i ces of degree 2n, which we cal l the rational r ep re -
sentation of End (A). As an easy exercise of l inear algebra, one can Q
prove :
Lemma., The rational representation f End (A) equivalent Q
to the d i rec t sum of the complex representation of End (A) and i t s ------ Q complex conjugate.
Let k be a field of definition fo r A and the elements of End(A),
and le t D be the vector space of al l l inear invariant differential forms
on A, defined over k. If zl, . . . , z a r e the complex coordinate n
n functions in C , then dz l, . . . , dzn a r e considered a s invariant dif-
ferential forms on A, and one has
Now every A E End(A) acts on D a s usual; denote the action by A*. *
Then X H < can be extended to an anti-isomorphism of End (A) Q into the ring of linear transformations in D. F r o m the relation (6. l ) ,
we obtain
(6. 2) This anti-isomorphism equivalent to the transpose of the -- complex representation o_f End (A).
Q
Let be a Riemann fo rm on cn/ L. For every T E End (A), Q one can define an element T~ of EndQ(A) by
Here we identify an element of End (A) with the corresponding element Q
of Mn(C) Then p $ a positive involutinn o_f EndQ(A). In general ,
an involution of an associative algebra S over Q (or R) i s , by
definition, a one-to-one map p of S onto S such that
P Such a p i s called positive if Tr(xx ) > 0 for 0 f x r S, where
T r denotes the t r ace of a regular representation of S over Q.
If an algebra S over Q or R has a positive involution p,
then S has no nilpotent ideal other than (0). In fact, if x, f 0,
belongs to a nilpotent ideal, then T r (xy) = 0 for every y c S, but
this i s a contradiction, since ~r (xxP) > 0. It follows that S i s semi-
simple. If e is the identity element of a simple component of s ,
then eeP f 0, hence e P = e. It follows that p is stable on each
simple component of S. Thus the classification of S and p can be
reduced to the case of simple algebras.
If S i s an algebra over Q with a positive involution p, we
can extend p to a positive involution of S 8 *R . In particular ,
consider the case where S is an algebraic number field, and use the
let ter K instead of S. Pu t
Then [K : F] = 1 or 2. By the general principle we just mentioned,
p is extended to a positive involution of the tensor product K BQR
which is a direct sum of copies of R or C. F r o m this fact i t is easy
to see that the direct factors of F 8 R a r e all rea l , i. e. , F is Q
totally real . Fur ther , if [K : F] = 2, the direct factors of K @ R Q
a r e all C, i. e . , K is totally imaginary.
Conversely, le t F be a totally r ea l algebraic number field, K
a totally imaginary quadratic extension of F , and p the non-trivial
automorphism of K over F. Then p is a positive involution of K.
We fix such F , K, p, and consider a triple (A, c , 8 ) formed
by a polarized abelian variety (A, c) and an isomorphism 8 of K
into End (A) such that the map 8 (a) H 8 (aP) is exactly the r e s t r i c - Q tion of the involution of End (A) obtained a s above. (Note that End (A)
Q a may be larger than 8 (K).) We assume also that 8 (1) i s the identity
of End (A). Take cn/ L and W as above. Then W may be regarded Q a s a vector space over K , by means of the action of 8 (K). Let m
be the dimension of W over K, and g = [F : Q]. Then we have ob-
viously
(6.4) n = gm.
Now res t r i c t the complex representation of End (A) to 8 (K). Then Q
we obtain a representation of K by complex matrices of size n.
In this situation, we say that (A, c , 8 ) of type (K, 9). Since K is
a field, 9 is equivalent to the direct sum of n isomorphisms of K
into C. By our choice of K, there a r e exactly 2g isomorphisms of
K into C, which can be written as
with a suitable choice of g isomorphisms rl, . . . , 7 among them. g
Let r and sV be the multiplicity of 5 and p rV in 9 , r e -
spectively, o r symbolically, put
u Note that aPu is the complex conjugate of a for every a r K and
every isomorphism a of K into C. F r o m the above lemma it follows
that ~ t = ~ ( r ~ + s V ) (rV + p7 ) i s equivalent to a rational representation
of K. Therefore we have
Since i s of degree n and n = mg, we have
(6.7) r v + s V = m (v = I , ..., g ) .
In particular, if m = 1 (and hence n = g) , either r o r s
is 0. Exchanging r V and pry if necessary, we may assume that
i . e . , ip - z ; = ~ T ~ . F o r a given K and 7
1'"" 7g , the existence of (A, c , 8 ) of
type (K, $1, with 3 - c ~ , ~ T ~ , can be shown a s follows. For
every s E K, le t u(s) denote the element of cg with components
71 7
s , . . . , s . Take any f ree 2-submodule 6t of K of rank 2g.
Put
It can easily be shown that L is a lattice in cg , so that c g / L is
a complex torus. Take an element 5 of K so that
Define an R-valued alternating fo rm E ( x , y) on cg by
where xv(resp. y is the f h component of x (resp. y). and t
i s a positive integer. F o r a suitable choice of t , we see easily that
6 becomes a Riemann form on c g / L, hence cg/ L is isomorphic
to an abelian variety A. F rom 6 we obtain a polarization c of
A. For every a r K, the diagonal matr ix with diagonal elements
7~ 7
a , . . . , a defines an element of Endo(A), which we write B(a). .-
In particular, if aOt C a, the matrix sends L into L, hence
8 ( a ) r End(A). Thus we obtain (A, c , 8 ) of type (K, 3). Actually
one can prove that any (A, c , 8 ) of type (K, 9 ) is constructed in
this way. If 8t i s a fractional ideal in K , and 0 denotes the
ring of algebraic integers in K, then 8 (0) C End(A). If n = 1, our
(A, c , 8 ) i s nothing but an elliptic curve isomorphic to C/Bt
(provided that r1 is the identity map of K).
Now taking a period matr ix for A, we obtain a point z of $n
a s in $5. Here we as sume that (A, c ) i s such that P = J , and n
8 ( U ) C End(A). Let (V, y ) be a couple a s in Theorem 5. Let
K ( y (2)) be the field generated over K by the coordinates of the
point y(z). One may naturally ask a question:
Is K( 40 (z)) the maximal unramified abelian e x t e n s i o n f K? - This i s s o if n = 1, a s a s se r t ed by Theorem 2. But if n > 1,
this i s not necessar i ly true. To construct the maximal unramified
abelian extension of K, we shall la ter discuss a function which i s
ra ther different f rom . However, even though (P is not a
function with the expected property, y(z) has st i l l an interesting
number theoretical property, which is roughly described as follows:
Theorem 6. Let K' be the field generated over Q 5 - a " for a a c K. Then K1 ( (2)) & unramified abelian
extension of K1 . I t can be shown that K' is a lso a totally imaginary quadratic
extension of a totally r ea l algebraic number field. Obviously
K' = K if n = 1. However, both cases K = K1 and K f K' can
happen if n > 1. Even the degrees of K and Kt over Q may be
different. The field K1 ( y ( z ) ) i s not necessari ly the maximal un-
ramified abelian extension of K1 . Then how big i s Kt ('-f'(z))? We
shall answer this question in the following section.
7. The class-field-theoretical characterization
of K' ( y(z ) )
Let us f i r s t recal l the fundamental theorems of c lass field
theory. On this topic, I shall give an exposition which is somewhat
out of mode, since such will be most convenient to describe the
field K1 ( y(z ) ) .
Let F be an algebraic number field of finite degree, Z an
integral ideal in F , and .jL a (formal) product of r ea l archimedean
pr imes of F. F o r an element a of F , we write
if there exist two algebraic integers b and c in F such that
a = b / c , b =_ c =_ 1 mod $ , and b , c a r e positive for every
archimedean pr ime involved in &. We denote by I (F , Z ) the
group of a l l fractional ideals in F prime to T , and by P ( F , tg)
the subgroup of I (F , 2') consisting of a l l principal ideals (a) such V
that a 1 mod* Z U.
Let M be a finite abelian extension of F. For every pr ime
ideal in F unramified in M, the Frobenius automorphism
[g , M/ F ] i s meaningful. Let .? be the relative discriminant of
M over F. Then we can define [BL , M/ F] for every eL e I (F , 3) s o that
i s a homomorphism of I (F , 2) into the Galois group G(M/ F) of
M over F. We have now
Theorem 7. The map (7.1) $ surjective, and i ts kernel con-
tains P ( F , 36) for some G. - Therefore , if Y i s the kernel , G(M/ F) i s isomorphic to
I (F , J ) / Y. Moreover, a pr ime ideal J in F is fully decomposed
in M if and only if f c Y. The converse of Theorem 7 is given by
Theorem 8. For every group Y ' of ideals in F such that
fok some 7 a s &, there exists a unique abelian extension M of F such that Y t n I (F , $3) i s the kernel of the map
where /Q i s the relat ive discriminant of M over F.
One can actually show that Y t C I(F, &). We call this M
the c lass field over F corresponding to Y t . ---- Coming back to K, r l , . . . , 7 and K' of $6 , l e t us take the
g smal les t Galois extension S of Q containing K, and denote by G
the Galois group of S over Q. Let H be the subgroup of G c o r r e s -
ponding to K. Extend each rV to an element of G, and denote it
again by r v . Put T = Hrv , and
H' = { y c G I Ty = T I .
Then f rom our definition of K' (see Th. 6) , we observe that K t -1 -1 i s the subfield of S corresponding to H I . Since H t T = T , we
can find elements o l , . . . , oh of G s o that
Counting the number of elements, we see that [K' : Q] = 2h. More-
over, for every element a (resp. ideal ? ) in K ' ,
i s an element (resp. ideal) in K. This follows easily f rom (7. 2).
Now le t I' be the group of al l ideals /e in K1 such that
with an element f3 of K. I t can easily be seen that I t contains
P ( K t , (1)). Now Th. 6 i s refined in the following way: -
Theorem 6 ' . K t ( 9 (2)) i s exactly the class field over K 1 ---- correspondingt_o I t .
Fur thermore , we have an analogue of the relation (3.6). To
describe i t , let us denote by A ( & ) the abelian variety isomorphic
to cg/ L with L defined by (6.8) for a n ideal in K. Take a
field k of definition for A ( a ) containing K' ( y ( z ) ) . Let u be
an isomorphism of k into C such that u = [T , K1 ( y (z) ) / K t ] on
K' ( (z)) for a pr ime ideal 8 in K' . Then we have
u A (7.3) A ( a )a 5 isomorphic fo A(O(,~-'), where % = .
Further we can obtain ramified abelian extensions of K' by means
of the points of finite o rde r on A.
Let us briefly indicate how Th. 6 and (7. 3) can be proved.
F i r s t le t us derive Th. 6' f rom (7.3). Let A = A ( 6 t ) and k be a s
above, and T an isomorphism of k into C. To simplify the ma t t e r ,
let us assume that (iii) of Th. 5 i s t rue for the present A even if we
disregard the polarization; namely assume
7 . (7.4) A isomorphic to A ~f and only if y ( z ) = ~ ( 2 ) ' .
(This is t rue if g = 1, but not necessar i ly s o if g > 1. ) Now we observe
that A((%) and A ( c ) a r e isomorphic if and only if 8t and t be-
long to the same ideal c lass . Therefore, the notation being a s in (7. 3 ) ,
we see that A(UL )* i s isomorphic to A( ) i f and only if b i s a
principal ideal in K. Combining this fact with (7.4), we conclude
that a prime ideal in K ' decomposes completely in K' ( (p(z))
if and only if fl:=l ih i s a principal ideal in K. This i s almost
the desired resul t , but not quite. We could not obtain the condition
about N ( -$? ), since we disregarded the polarization. A careful analysis
of polarization leads to Th. 6' .
To prove (7.3), we have to introduce the notion of reduction of
a n algebraic variety modulo a prime ideal. Let V be a variety in
a projective space, defined over an algebraic number field k. Let 7 be a pr ime ideal in k, and k( ) the residue field modulo
;P - We consider the se t f of al l homogeneous polynomials vanishing on
V, whose coefficients a r e 7 -integers. F o r each f r ,f , we
consider f mod , which is a polynomial with coefficients in k ( P ).
Then we define the reduction of V modulo , denoted by V[ 1, to
be the s e t of al l common zeros of the polynomials f mod f? for all
f c 8 . If V i s an abelian variety defined over k, then one can show
that V[ 7 ] i s an abelian variety defined over k( - ) for all except a
finite number of . F o r such a P P , reduction mod of
every element of End(V) is well defined, and gives an element of
End (V[F I) . We apply these facts to the above A ( m ). It i s not difficult to
obtain A( 8t ) defined over an algebraic number field k. We assume
that k contains K' ('fJ (2)) and is Galois over K' . By the principle
(6. 2), we can find n l inearly independent l inear invariant differential
fo rms w , . . . , w on A, rational over k, s o that g
( a € 0 ; v = I , . . . , g).
Let us assume, fo r the sake of simplicity, that K is normal over
Q, K = K' , and the c lass number of K is one, though Th. 6' be-
comes somewhat tr ivial under the las t condition. By (7. 2) , we can
- 1 put a = 7
A A ' Let 2 be a prime ideal in K of absolute degree one,
u and le t b = m:=l 7 A . Take a p r ime ideal f? in k which divides
2 , and consider reduction modulo . Indicate the reduced objects
by putting tildes. F r o m (7.5) we obtain
.- if % = (b) with an integer b in K. Let x be a generic point of A
over k. Then the relation (7.6) shows that every derivation of %(x) -CV
vanishes on k(O (b)x), hence
where p is the rational prime divisible by 2 . Since
we obtain
On the other hand, if a is the-Frobenius substitution for P over K , then AO mod p can be identified with xp . Therefore
(7. 7) shows that, if A = A( a) ,
Then it i s not difficult to lift the isomorphism to that of A ( m ) O to
~ ( b - l m ) , hence (7.3).
8. A further method of constructing class fields
As I mentioned in §3, there a r e some other ways of generalizing
Theorem 2. F o r example this can be done by considering special
values of automorphic functions with respect to a discrete subgroup of
SL (R) obtained from a quaternion algebra. 2
A quaternion algebra over a field F is , by definition, an algebra
B over F such that B 8 F is isomorphic to M @) , where F F 2
denotes the algebraic closure of F. F o r our purpose we take F to
be a totally r ea l algebraic number field of finite degree. Then one
can prove that
where D i s the division ring of rea l Hamilton quaternions, g =
[F : Q], and r is an integer such that 0 <_ r <_ g. We assume that - - r > 0, and regard B a s a subset of BR . How many such B do
there exist? I shall answer this question afterward.
- 1 7 8 A ( ( ~ L ) O mod 'f2 2 isomorphic f_o A(b OL) mod 7 .
For a e B, le t a 1""' r a be the projections of a to the
t f i r s t r factors M2(R). Wedeno teby B t h e s e t o f a l l a in B
such that det(a ) > 0 for v = 1, . . . , r . Then every element a of t
V
B acts on the product -$r of r copies of the upper half plane ,
the action of each a on -$ being defined a s in §2. If we denote $
V t
by NB, ( a ) the reduced norm of a to F, then B i s the s e t of
a l l a such that N ( a ) i s totally positive. B / F
Observe that B i s of dimension 4g over Q. By an order in
B , we understand a subring O of B, containing Z, which is a
f r e e Z-module of rank 4g. An order i s called maximal, if i t i s not
contained properly in another order . There a r e infinitely many maxi-
m a l o rde r s in B. We fix a maximal order 0 in B, and put
Fur the r , for every integral ideal 1: in F, put
t Then r and r ( T ), a s subgroups of B , ac t on 5' . o n e can
show that / r ( ) i s compact if and only if B has no zero-
divisor other than 0. F o r example, if B = M2(Q), we can se t
(?' = M2(Z), hence r = SL2(Z) , and %/I? i s not compact.
Now le t .us assume r = 1. Then $/r (Z ) i s compact unless
B = M2 (Q). Therefore, a s was discussed in S2, the quotient $ . / I ? ( )
(or i ts compactification when B = M2(Q)) i s a compact Riemann
surface, and an automorphic function with respect to ( 2. ) i s a
meromorphic function on 8 invariant under r ( t ) (satisfying
an additional condition when B = M2 (Q)).
em ark. In S2 we considered only discrete subgroups of SL (R). 2
The action of an element a of r ( ) on 5 i s that of al, i. e . ,
the projection of a to the f i r s t factor M2(R) of BR . The element
a l may not be contained in S L (R). But this does not produce any 2 difficulty, since we only have to consider
in place of r ( Z ).
The group of the above type was f i r s t introduced by Poincar6
about 80 years ago in the case F = Q, and la ter Fr icke considered
the general case . They discussed ternary quadratic forms instead of
quaternion algebras.
We have to define "special pointsn on % relative to , analogous to w / W of Th. 2, where we shall examine the values of 1 2 automorphic functions. F o r this purpose, we notice:
em ma. Let M & a totally imaginary quadratic extension of F -- - which i s isomorphic to? quadratic subfield of B over F. Then the -- following assert ions hold.
(l) If f i s _s F-l inear isomorphism o_f M into B, then t
f (M) - (0) contained 2 B , and every element of f(M) - F
has exactly one fixed point on # which is common to al l elements - of f(M) - F. -
(2) If 'ICM denotes the r ing of in tegers M, then there - exists an F- l inear isomorphism f o_f M into B such that f ( % M) C 0 . --
The f i r s t asser t ion is quite easy to prove, but the second needs
a somewhat deep fact of ari thmetic of quaternion algebras.
We a r e going to take the fixed point of f (M) - F a s our special
point" . One can of course a sk a question: What kind of M can be
embedded in B? Leaving this question aside for a while, we a r e now
ready to s ta te the main result :
Theorem 9. There exists a couple (V, y ) formed by &projective
non-singular curve V and a holomorphic mapping o_f $. into V - with the following properties:
(i) g ives a biregular isomorphism of $ / r ( T ) into V.
( 9 surjective unless B = M (Q).) 2 (ii) V i s defined over the c lass field k over F corresponding
a0 is the product of &l archimedean p r imes to P(F, Z C ~ ) , where " - of F. (For notation, s ee § 7 . ) -
(iii) L e t M and f be a s in (2) of the above Lemma, a& z the
fixed point of the elements of f (M) - F f . Then t& composite - of k ( (2)) and M i s exactly the c lass field over M corresponding - to P ( K Z ). -
Thus and z correspond to j and w / w of Th. 2. We 1 2
ca l l such a couple (V, y ) a canonical model for $ I T ( 7 ). If
(V, 'p ) and (V' . (PI ) a r e two canonical models for the same $ 1 ~ ( 7 ), then we can show the existence of a biregular isomorphism 5 of V onto
Vf , rational over k, such that = 6 0 . In this sense , (V, (i7 )
i s uniquely determined fo r $/I?( Z ). It may be worth noting that the
maximal abelian extension of M can thus be generated, over the maxi-
ma l abelian extension of F , by special values of some specific automorphic
functions.
To answer the questions about B and M, let Fk
denote the
completion of F with respect to an archimedean or a non-archimedean
pr ime 2 of F. Put B = B @ F . Let PB b e t h e s e t o f a l l ? 2 F '?
such that B i s a division algebra. Then we have the following a s -
sert ions: 8
(8. 2) PB & a finite se t with an even number of primes.
(8.3) F o r any finite se t P with an even number of archimedean
or non-archimedean primes of F , there exists a quaternion algebra -- --- B over F , unique up to F-linear isomorphism, such that P = P -- B '
(8.4) _A quadratic extension M f F can be F-l inearly embedded
in B if and only if M @ - i s a field for every FF2 - --- P P J 3 .
These resul ts a r e special cases of Hassesf s theorems on simple
algebras over algebraic number fields. Observe that g - r factors of
(8.1) correspond to the archimedean pr imes of P B '
The reciprocity law for the extension M. k( Cp (2)) over M can
be described explicitly in t e rms of the special points (z). For
simplicity, le t us consider only the case where the c lass number of
F in the narrow sense i s one, i. e . , P (F , Go) = I@, (1)). F o r every
prime ideal J2 in F, l e t 16 be the ring of ?-integers in F 9 I '
and le t = % 0 . Then is a maximal order in B Let J B 2 t 8'
U( ) denote the group of a l l elements a of B such that a is a
uni tof 8 f o r a l l 8 3 dividing .Z , and le t U ( t ) be the subgroup
of U( ) consisting of a l l a such that a Z% I mod ru for a l l 3 2
dividing r . I t can easily be shown that U( t ) / U ( T ) i s isomorphic
to (0lfC7)~ (see Notation). F o r every a c U( t y, put
Now we have
Theorem 10. There exists a sys tem of biregular isomorphisms --- R ( a ) of V to vUb) , defined for each a r U ( f ) and rational over
7 - k, with the following properties:
(i) R ( ~ ) o @ ) R(p) = R(ap) .
(ii) R ( a ) = R ( p ) if a - l p r Uo(E) .
(iii) R(y )[ (z)] = 'p(y (2)) fo r every z r $. if y r ro). (iv) Let M, f , z be a s in (iii) of Th. 9 (still under the
condition f ( WM) C 0). Suppose that f &normalized & the sense
defined below. Let 3 be an ideal in M p r i m e to Z. , and le t --
(Such an a ~ not unique. ) With such an element a, one has
Here we say that f i s normalized if
(0 f a r M).
If we define ? by ?(a) = f 6) for a E M, then we see that ei ther f -
or f i s normalized.
It should be observed that (iv) of Th. 10 is a generalization of
(3. 7) . Fur the r , if B = M2(Q), (9 = M (Z) , and r = (N) with a 2 positive integer N, the function field of V is exactly the field
N Q(i, f ) considered in 93. Therefore the f i r s t three properties of
ab R ( a ) in Th. 10 may be regarded a s a generalization of Th. 4.
-1 Example. Let us consider the case F = Q(3 + 5 ) with 5 =
e2ni1 for d = 7, 9, or 11. By (8 .3 ) . there exists a unique quaternion
algebra B over F , for which P consists of al l but one archimedean B pr imes of F , the exception being the archimedean prime of F cor re - sponding to the identity map of F. The present F has the c lass num-
be r one in the narrow sense. Moreover, a l l the maximal orders in
B a r e conjugate to each other under the inner automorphisms of B
defined by the elements of B+ . Take a maximal order 7 3 in B , and
Then there exis ts an element a f U ( Z ) such that f ( 8 ) O = ag. . ----
define I? = r(l) a s above. Then one can prove that $ 1 is of
genus 0, and r modulo i t s center i s generated by three elements
y 2 , y3 , yd of order 2, 3, d, respectively, satisfying y y y = 1. 2 3 d
(If d = 7 , the measure given by (3.4) is 11 42, which is the smallest
value of (3.4) for a l l possible l?. ) These three elements have unique
fixed points on $ , one for each, which we denote by w2. w3 , wd . Then there exists a unique meromorphic function on $. such
that C ( y ) is the field of a l l automorphic functions on 5 with
respect to J?, and (i) (w2) = 1, f' (w3) = 0, (wd) = UJ . If we denote
by V the complex projective line, then gives a biregular isomor-
phism of $/r onto V. Now we can prove that this (V, p ) $32
canonical model for #/l7 . By (8.4), for every totally imaginary
quadratic extension M of F , there exists a F-linear isomorphism
f of M into B such that f ( % M) C c. Let {zl, . . . , z } be a se t 9
of representatives for the r-equivalence c lasses of the fixed points of
f(M) - F for a l l such f. Then q is exactly the c lass number of M,
and f rom (iii) of Th. 9 and (iv) of Th. 10, we obtain:
(8.5) The values ?(al), . . . , ?(Z ) fo rm a complete s e t of 4 -- c o n j u g a t e s f ?(z ) over F, and for each i , M( y(z i ) ) i s the maxi- 1 ma1 unramified abelian extension of M.
Thus has a strong resemblance to the classical modular
function j.
Unfortunately, the proofs of Theorems 9 and 10 a r e long and
very complicated. Therefore I have to content myself with a rough
sketch of the main ideas. We take a totally imaginary quadratic
extension K of F and consider (A, c , 8 ) of type (K, 9) in the
sense of 56 with a representation 3 of K such that
where T,, and P a r e a s in 56. Then i t can be shown that the (A, c , 8 )
of this type a r e parameterized by the point on g r , and there is a dis-
continuous group I? acting on $r such that $r/l" is inone-to-one
correspondence with a l l the isomorphism classes of such (A, c. 8) .
Taking K suitably, we can identify J?' with the above I? (in the case
r = 1). F o r this family of (A, c , 8 ) , we can find a model (V' , )
of #/rf with the properties analogous to those of Th. 5. If M and
z a r e a s in (iii) of Th. 9, the corresponding (A, c , 8 ) is such that
End (A) contains an isomorphic image of K 8 M. The coordinates Q F
of (z) generate an abelian extension of the nature described in Th. 6 ' .
This couple (V1 , ) is the f i rs t approximation to the desired (V, y l ) .
F r o m infinitely many such (V' , ), depending on the choice of K , we
can construct a canonical model (V, y). If B = M (Q), there is a family of elliptic curves, for which the 2
value of j is the modulus. F o r the basic field F of higher degree,
there is no such standard family of abelianvarieties, though infinitely
many families of abelian varieties can be loosely associated with a
given $ 1 . It is an open question whether there exists any family
of geometric s t ructures , other than the above (A, c , 8 ), of which our
canonical model (V, y ) i s a natural variety of moduli.
9. The Hasse zeta function of an algebraic curve
Let V be a projective non-singular curve of genus h defined
over an algebraic number field k. For every prime ideal 'J2 in k ,
le t k ( 2 ) denote the residue field modulo 2 . Considering the
equations for V modulo & , we obtain a curve V [ p ] over k ( & ' (see §7). It can be shown that V[ ] is non-singular, and of genus h for
almost a l l J2 . (We call such 2 good. ) Therefore one can speak
of the zeta function of V[f ] over k ( p ) which i s of the form
1 Z (u) = Z (u)/ [(I - u ) O - N ( ~ ) U ) ] ,
B 8 1
where u i s an indeterminate, and Z (u) i s a polynomial of degree 2h. Y The Hasse zeta function of V over k, denoted by Z(s; V/ k) with com-
plex variable s , i s now defined by
the product being taken over a l l "good" 8 . Qt is important to con-
s ider a l so "badf1 & , which we shall not discuss here. ) Now one
can make the following
Conjecture. Z (s; V / k) can be continued to the whole s -plane
and satisfies 5 functional equation. - The purpose of this section is to verify this conjecture for
the curve which is a canonical model for /r ( 7: ) in the sense of f §8. For the sake of simplicity, we shall consider here only the case
where T = 0) and the class number of F in the narrow sense is
one, i. e. , every ideal in F i s a principal ideal generated by a
totally positive element. In this case , for every right 0 -ideal 8t ,
there exists an element a in B' such that = a 8. . Let us introduce cusp forms and Hecke operators with respect
b to the group I? = r p ) . For every f = 6 d) c GL2 (R) with det (f) > 0,
Put
Let m be a positive integer. By a cusp form of weight m w&h respect
to r, we understand a holomorphic function f (2) on - $, satisfying
the following two conditions:
(i) f (y(z)) j (y , z)m = f ( z ) for a l l y c r . (ii) s a cusp of I-', a n d p , q a r e a s in $2, then
f (P (z))j (P, 4 holomorphic & q = 2niz, and vanishes a t q = 0. --- The latter condition is necessary only when B = M (a), since
2 I? has no cusps otherwise. All such functions form a vector space of
finite dimension over C, which we write S (r). If m = 2, the map m
gives an isomorphism of S ( r ) onto the space of differential forms of 2
the f i r s t kind on $D, hence the dimension of S (r) i s equal to 2
the genus of f / r .
t F o r every a E B , we note that the double coset F a r can
be decomposed into a finite number of one sided cosets:
r a r = u r Z l a i r
Then for f c S (I?), we define ( T a r ) f by m m
-1 -1 m ( raTlmf = z d 1 f ( a i (z) ) j (a i , 2) .
(see (8.2-4)), and the second over the remaining primes in F. More-
over, D (s) can be holomorphically continued to the whole s -plane, m
and satisfies a functional equation:
In this way we obtain a l inear transformation ( r a r ) on S ( r ) , m m which i s of course independent of the choice of the representatives
a F o r every integral ideal 8t in F , le t T(BL), denote the i '
(disjoint).
of a l l distinct ( rar ) , such that a E 0 and (N (a ) ) = 61. . B/ F
Then we define a Dirichlet se r i e s D (s) , whose coefficients a r e linear m endomorphisms of S (I?), by
m
where (9t runs over a l l integral ideals in F. It can be shown that
D (s) converges for sufficiently large Re (s) and has an Euler product: m
where the f i r s t product is taken over all the prime ideals 2 in P B
where P; means the s e t of non-archimedean 2 in P The last B' few stand for the usual gamma function.
Now we have
Theorem 11. Let (V, y ) &a canonical model for f/r . TZ, for almost al l prime ideals 3 & F , the zeta function o_f V z d 2 , --- over the residue field mod 2 , coincides with -
2 det[l - T ( 7 I 2 u + N(J?)u I / [(I - u)(l - N(Z)u) ]
1 Z (u) is the B --
(i. e . , ,., Euler 2 -factor of D (s) y& m = 2 and N(?)- ' = u). m - Thus Z (s; V/ F) coincides y& det(D (s)) u p to a finite number of - 2 2 -factors.
The proof of this theorem is roughly as follows. For every t a E B . define a subset X( I?a r ) of V X V by
We s e e that X(l?al?) is an algebraic correspondence, which depends
on ly on F a r and not on the choice of a . We observe that T ( 7 I2 = ( r a r ) , with any element a of 0 n Bt such that f = (NBI (a) ) .
F o r such an a , we write X ( I ' a r ) a s X 2 - Let M, f , and z be a s in (iii) of Th. 9 and (iv) of Th. 10.
T a k e M s o that 2 decomposes into two prime ideals 7 and
in M. We can find an element P of 6 0 B t s o that f ( 7'
=PW . T h e n 2 = NMl ( 7 = (NB1 (PI), hence (rPrI2 = T (f )2 by the
above r e m a r k . Therefore ~ ( ~ - ' ( z ) ) X y ( z ) c X? . Now le t
T = [q , M ( l (z))/ MI. By (iv) of Th. 10, we have ~ ( 2 ) ' = ~ ( ~ - ' ( z ) ) ,
hence , putting y = y(z ) , we obtain yT X y c X Consider now 2 . reduction modulo a p r ime divisor of a suitably large field, which
h
7 divides 2 , and denote the reduced object by tilde. Then y = y
hence
'V rV
L e t rr denote the Frobenius correspondence on V X V, i. e . , N
the locus of x X x N ( f ) for x c V. The above discussion shows that hl
X, has infinitely many points in common with t m , the transpose
t of . Since trr i s irreducible, we have mcT8 . F r o m our
definition of X 2 '
i t can easily be seen that t~ = X P 8 '
Therefore
+ . Again f rom the definition of X . we see that N N 8 N
X . ( x X V ) consists of N ( Y ) t l p o i n t s f o r a g e n e r i c x on V. Y
Therefore the equality
should hold..
Let A denote the diagonal of V X V. Then
The relations (9.1) and (9. 2) show that any symmetric function of
and tm can be obtained f rom a correspondence of V by reduction
modulo 8 . In part icular, for every positive integer m , we have
for a polynomial P , which i s determined by m
d -- 2 * du log (1 - XU + yu ) = x Pm(x, y)um-l
m =I
with indeterminates x, y, and u.
Let I[Y] denote the number of fixed points of a correspondence
Y , i. e . , the intersection number of Y with A . If Z (u) denotes
the zeta function of V mod 2 , then 2
hl
But if Y i s a correspondence on V X V, we have I[Y] = 1[Y], and
by the Lefschetz fixed point theorem,
i where H (V) denotes the usual r ea l i-th cohomology group of V. For
Y = X r a r ) , it i s easy to see that
1 tr(Y 1 H (V)) = 2 . ~ e [ t r ( l?ar)2],
since S (I?) is isomorphic to the space of differential forms of the 2
f i r s t kind on V = $ 1 ~ . Also i t is obvious that 2
tr(Y ( HOW)) = tr(Y ( H (V)) = the number of right cosets in r a r .
One can further show that t r (l?al?)2 is real . Therefore we have
In view of (9. 3), we obtain
d - 1 du log 1 (u) = (det[l - T(& )2u t N ( j )u2]-'} ,
2 du log
10. Infinite Galois extensions with l -adic representations.
So far we have been interested only in the construction of abelian
extensions. Now we a r e going to show that the above canonical model
f o r $/TI f') can be employed to obtain meaningfd non-abelian ex-
tensions of a number field, with some pleasant features.
Let us call (V, f', R(a) ) of Th. 10 a canonical system of level .t. F o r every integral ideal 8L in F, we fix a canonical system of level
& , and denote i t by (Va , t, RoL (a)) . (We a r e still assuming
that the class number of F i s one in the narrow sense , though our dis-
cussion can actually be done without this assumption. ) Let k a denote
the c lass f ie ldover F corresponding to P ( F . a&), where 4x0 is
the product of all archimedean primes of F (see $7). As is stated in
Th. 9, 10, Va and Ra ( a ) a r e defined over ka . If eL C d , we
can obtain a rational map
defined over kw , such that
hence our theorem.
When 6L = b , this means the uniqueness of canonical system of
level a. In the general case , the map (10.1) defines a Galois cover-
ing. If we consider the curves over the universal domain C, then
the Galois group i s isomorphic to r ( i$ )/r ( cz )E% , where E %
denotes the group of a l l units e of F such that e =- 1 mod 1 . If
we take kOL a s the field of definition, we obtain:
Proposition 1. Let y1 be an arbi t rary point of Vm a& y =
Tb, &(yt ). (y' may be generic o r algebraic. ) Then k o, (y' ) depends
only on y, and i s a finite Galois extension o_f k (y). For every a. 7 c G(k., (yl ) / k (Y)) , there exists an element P of U ( a , ) n uo( b )
7 such that y1 = Rot,(P)(yl ), & T = o on k, . --
# so that L(z) = y' , and Proposition 2. Take a point z of l e t rZ = { y I- ( 2) 1 y (z) = z ) . Then the following statements hold: -
(i) r ( e e ) . r z = {Y c r ( b ) I R,(Y)(Y') = y t ) .
(ii) F r o m the correspondence 7 H P described in Prop. 1, one obtains an isomorphism o_f G(ka(y1 )/ k (y)) into --
(iii) If y is generic on - Vg k % ' - this isomorphism i s
surjective, a n d rZ = E b .
Let us now fix an integral ideal 2: in F and an a rb i t r a ry
point y on Vy : Take a point y on Va for each 6LC Z s o a that T1; a ( ~ a ) = y% i f 01 C 1, and yr = J . (For example, choose a point Z
on f s o t h a t f'*(z)=y, andpu t y = G ( z ) . ) Let 6t Y denote the composite of the ka (ya ) for all a. By Prop. 1,
4 is a Galois extension of k r (y). Our purpose i s to investigate Y
the Galois group of & over k (y) and the behavior of the Y
Frobenius automorphisms (when y is algebraic) with respect to
certain representations of the Galois group.
F o r every prime ideal f in F, le t F1 . f . BI , and Oy
be the localizations a s defined in g8. F o r m the product lt of the X
groups 0 (see Notation) for a l l 1 , with the usual product top- 1
ology. ~ e ' t at denote the subgroup of a consisting of the elements
(ul ) with u t c 0; such that u = 1 mod f D for every . I - t F o r simplicity, we assume
(10.2) {Y r r(Z) 1 v(z) = z ) = E t
for a point z on $ such that c ( z ) = y. This is satisfied for all except a finite number of points on Vr . Under the assumption, we
a r e going to define an injection
- where E% i s the closure of EI in a. Let s r G ( & 1 k t (y)).
Y F o r every C ' 2 , we find, by Prop. 1, an element e m = 5 (a) of '7 u(a) n u o ( t ) * that YOL = Ra (ta)(y, 1. It i s not difficult to choose
the elements $ s o that OL
F o r each pr ime ideal f , the sequence { ( ( ln)}n=l , 2, converges ... to an element u of C; . We define J ( r ) to be the element of
represented by (u ). We can verify that J i s actually a t t
continuous injection. 3 depends on the choice of the sequence of points
{y }. But it i s unique up to inner automorphisms of Er . OL
Next le t us consider the case of an a rb i t r a ry algebraic point
y of Vr . Take any central simple algebra A over Q, and
consider a representation
Fur the r we have:
satisfying the following two conditions: Proposition 3. If y i s generic on Vr o ~ r k , J i s s u r - tr
jective.
Let u s now discuss the points y whose coordinates a r e algebraic
numbers; our main in teres t i s of course in this case. F i r s t we con-
s ider the point fixed by an imaginary quadratic subfield of B. Let
M, f , and z be a s in (iii) of Th. 9, and XM the ring of integers in
M. Then f ( X M ) C 0. Put
and view a s a subgroup of %. Then, f rom (iv) of Th. 10, we
obtain
Proposition 4. If y = T r ( z ) M C k (y), then the image
of J contained & (R ) / E r . - t
If M is not contained in k (y), we have to consider a l a rge r
group which contains rb;?f) fir a s a subgroup of index 2, but we shall
X X @) p is rational over Q, B and A being considered -
algebraic groups over Q. m
(2) p (a) = NF/ (a) fo r every a e with an integer m ,
independent of a.
F o r every rational prime I , let QI denote the field of I -adic
integers, and l e t B = B QQQI, AI = A @,aI . F r o m p one can I naturally obtain a representation
I Let a denote the product of the groups 0 for a l l prime
I fac tors t of I , and Eg the closure of Er in d . Put
= b . Combining J with a natural homomorphism of I m,/ B, to a:/ ET , we obtain a homomorphism
not go into details.
Now we make the following assumption:
(3) p(e) - 1 f o r a l l e c Ey . In view of (2), this is automatically satisfied if either m is even,
or the units of Er a r e a l l totally positive. Under the assumption,
observe that P I J i s meaningful, and defines a representation of x
G ( & ~ / k r (y)) into Al . Now i t i s important to investigate the be- I
- havior of the Frobenius automorphisms of .gy with respect to the
I -adic representations" p I o JI . F i r s t we notice:
Theorem 12. There exists an integral ideal p v ~ k (y) s u c h
that every pr ime ideal of k (y), p r i m e to IM , unramified & - the subfield of --- Ay which corresponds to the kernel of p 0 J I I '
Therefore, if is such a pr ime ideal of k (y), is a prime
divisor of dividing 8 , and
is a Frobenius automorphism of Y
& over k (y) for f2 , then the conjugacy c lass of p Y
determined only by J2 . In this setting, we have the following theorem,
which may be regarded a s the main resul t of this section:
Theorem 13. The roots of the principal polynomial of p 0 J (U
over QI are algebraic numbers f absolute value N( I %
- This fact leads us to a temptation of making the following conjec-
tures.
Conjecture I. T h e principal polynomial of p 0 J (U ) has rational coefficients, and i s independent of I . 1 1 2
Suppose this i s true, and le t f. denote the principal polynomial 2 of p, o J (cr . Then one can define a zeta function c(s; k y ; p) of
1 2 k Y associated with the representation p &
where the product i s extended over a l l prime ideals 8 in k z ( y )
prime to W , and n is the degree of f 7 '
Conjecture 11. 6(s; diy; P ) can be analytically continued f_o the
whole s-plane and satisfies 5 functional equation.
Fur ther , as for the nature of & one may make Y'
Conjecture 111. The image of J i s an open subgroup of 'aT/ tr , unless y = yZ(z) with a point z such that -
for some totally imaginary quadratic extension M f F and an -- F-linear isomorphism f of M into B.
It should be remarked that the above representation p is analo-
gous to the Grksen-charac te r in Hecke1 s sense, or more precisely,
to the Grijssen-character of type (A ) in the sense of Taniyama-Weil.
If y = yr(z) with a point z fixed by f(M - (0)) a s excluded in
Conjecture 111, and if M C k (y), then we can show that c(s; & ; p) Y
is a product of several zeta functions of k 7 (y) with Grossen-characters.
One may also notice that b%r/Et. is analogous to the idsle class
group of a number field modulo the connected component. Fur ther ,
if B = M (Q), 0 = M (Z), Z= NZ with a positive integer N, then 2 2 r ( 7 ) is the principal congruence subgroup of SL (Z) of level N, 2 and the choice of a point y on Vr i s almost equivalent to the choice
of an elliptic curve. Thus, if p is the identity mapping of B' = GL2(Q)
to itself. ((a, gYy, p) i s the Hasse zeta function of the curve s o r r e s -
ponding to y. A s imilar fact holds also when F = Q and B i s a divi-
sion algebra. It is an open question whether such an interpretation
exists for c(s, , P ) in the case where F is of degree > 1 . In Y
this connection, i t should be mentioned that the extension & is Y
ra ther different f rom the extension obtained f rom the points of finite
order on an abelian variety, if [F : Q] > 1.
11. Fur ther generalization and concluding remarks
We have obtained two different types of results: one is repre-
sented by Th. 5. Th. 6, and Th. 6' ; the other by Th. 9 and Th. 10.
They a r e however two special cases of a more general theorem. To
see this, le t us introduce discontinuous groups which include Sp(n, Z)
and F ( 7 ) a s special cases .
Let F, B and r be a s in S8. Since B BFT = MZ(F) for the - algebraic closure F of F, we can regard the elements of B a s - b matr ices of degree 2 with entries in F . F o r every a = d) c M ~ F ) ,
put
* I One can show that a t B if a E B, and a H a is an involution
of B in the sense of 46, which is not necessari ly positive. F o r an t :$ t *
element U = (a. .) c M (B) with a.. r B, put U = (a .. ). Then 1J n 1J 1J
t * U w U defines an involution of M (B). This involution can be
n R-linearly extended to M (B ), where B = B @ R. Define a Lie
n R R a group G by
In view of (8. l ) , we have
According to this direct sum decomposition, G can be decomposed into
a direct product:
G = G X . . . X G r X G r + l X . . . X G . 1 g
One can easily show that
where the bar means the quaternion conjugate in D. Therefore,
Gr+l,. . . , G a r e compact. Since Sp(n, R) acts on the Siege1 g
space f n , we can le t every element U of G act on the product
$, of r copies of , the action of U on the u-th factor n h
being that of the projection of U to Gv. As in 58, take a maximal
order (p in B, and put
In this way we obtain a discontinuous group acting on #nr . If B = M2(Q) and 0 = M2(Z), r is Sp(n, 2). If n = I , the
present r i s a subgroup of finite index of the group considered in
$8. The quotient gnr / T is compact in the following two cases:
(i) r < g; (ii) r = g, n = 1, and B is a division algebra. The group
r was introduced by Siegel in his paper on symplectic geometry (under
the restrict ion r = 1). Now, for this quotient kr/r , we can con-
s t ruc t a couple (V, y ) with the properties analogous to those in Th. 9,
thus unifying the above mentioned two types of results . A par t of the
resul ts of $10 can a lso be extended to such a general case.
One can investigate automorphic functions with respect to a
more general type of group. Namely one takes a semi-simple algebraic
group defined over Q and consider a Lie group gR consisting
of the points with coefficients in R. Suppose that the quotient of
by a maximal compact subgroup i s a bounded symmetric domain. Then
one can speak of (meromorphic) automorphic functions and forms on
this domain with respect to the group I? = qz formed by integral
points on 9. There a r e many interesting arithmetical problems
in this field, which I dare not enumerate here. But I should a t least
mention that almost al l important questions a r e related to automorphic
forms and zeta functions explicitly or implicitly, on which I have talked - only in $9.
12. Bibliography
Among a vast multitude of l i terature, I shall t ry to l i s t standard
reference books from the view point of accessibility and (probable)
comprehensibility, along with a few recent papers relevant to the topics
discussed in these lectures.
The reader with / standard knowledge of algebraic groups or
Lie theory may find the following volume useful:
[I] Algebraic groups and discontinuous subgroups, Proceedings
of Symposia in Pure Mathematics, vol. 9, Amer. Math. Soc. 1966.
This contains many interesting surveys of recent investigations,
most of which have abundant references. For those who a r e more
interested in the classical modular functions or modular forms, many
important papers in
[2] E. Hecke, Mathematische Werke, Gbttingen, 1959
will serve a s standard references. A more systematic and somewhat
eas ier treatment is presented in
[3] M. Eichler, Einfiihrung in die Theorie der algebraischen
Zahlen und Funktionen, B i rkh luse r , 1963 (the English revised version
is available).
At the end of each chapter of this book, there a r e plenty of
references. As a textbook on the c lass ica l theory of elliptic functions,
the following may be recommended:
[4] C. Jordan, Cours dl analyse de 1' i co le polytitchnique. Pa r i s ,
vol. 11, Ch. VII.
As fo r the theory of complex multiplication of elliptic functions,
I pick he re only two, old and (relatively) new:
[5] H. Weber, Lehrbuch der Algebra 111, 2nd ed . , 1908,
[6 ] M. Deuring, Die klassenkijrper der komplexen Multiplication.
Enzyclopadie d. math. Wiss. neue Aufl. Bd. 12, Heft loII, Stuttgart,
1958.
The fundamental mater ia l of abelian variet ies i s presented by
[7] A. Weil, Varibtes abeliennes e t courbes alg'ebriques , Hermann, P a r i s , 1948.
[a] S. Lang, Abelian var ie t ies , Interscience, New York, 1959.
The analytic theory of theta functions and abelian variet ies i s
systematically t rea ted in
[9] A.. Weil, Introduction a 1' btude des variittes kahlhriennes,
Hermann, P a r i s , 1958,
[lo] C. L. Siegel, Analytic functions of severa l complex
variables, lecture notes, Institute for Advanced Study, 1948,
reprinted with correc t ions , 1962.
The la t ter will s e rve a lso a s an introduction to the theory of
automorphic functions of severa l variables. On this topic and other
related subjects, one can not mis s
[ll] C. L. Siegel, Gesammelte Abhandlungen, 3 vol. , Springer,
1966.
Especially for Siege1 modular functions, the standard knowledge
can be obtained from
[12] H. Maass, Lectures on Siegel' s modular functions, Tata
Institute, 1954-55,
[13] SBminaire H. Cartan, 19571 58, Fonctions automorphes.
A detailed account of the resul ts discussed ,in 5 7 on the nature
of the number field K' ( (2)) etc. i s presented in
1141 G. Shimura and Y. Taniyama, Complex multiplication of
abelian variet ies and i t s applications to number theory, Publ. Math.
Soc. Japan, No. 6, Tokyo, 1961.
A recent volume
[15] A. Weil, Basic number theory, Springer, 1967
contains a modern treatment of c lass field theory, a s well a s the
s t ructure theorems of simple algebras over number fields, which
generalize (8. 2-4). The lat ter subject, in a concise style, can be
found in
[16] M. Deuring, Algebren, Ergebn, der Math., Berlin, 1935.
As for the general theory of arithmetically defined discontinu-
ous groups, I mention here only three papers:
[17] A. Bore1 and Harish-Chandra, Arithmetic subgroups of
algebraic groups, Ann. of Math. 75 (1962), 485-535.
[18] G. D. Mostow and T. Tamagawa, On the compactness of
arithmetically defined homogeneous spaces, Ann. of Math. 76 (1962),
446-463.
[19] W. L. Baily and A. Borel , Compactification of ari thmetic
quotients of bounded symmetr ic domains, Ann. of Math. 84 (19661,
442-528.
The compactness cr i ter ion, which generalizes that for #nr l r of s11, i s given in [17] and [18]. The l a s t paper 1191 proves the existence
of a Zar iski open subset of a projective variety isomorphic to a given
quotient like $ 1 in general . F o r these topics, s ee a lso the
ar t ic les in [I].
Theorems 5, 9, 10, 11 and their generalizations a r e proved in
my papers:
[20] G. Shimura, Moduli and fibre sys tems of abelian variet ies.
Ann. of Math. 83(1966), 294-338.
[21] G. Shimura, Construction of c lass fields and zeta functions
of algebraic curves , Ann. of Math. 85 (1967), 58-159.
[22] G. Shimura, Algebraic number fields and symplectic dis-
continuous groups, Ann. of Math. 86 0967), 503-592.
The l a s t section of [22] i s a weaker version of the resul ts stated
in §lo, for which a full account will be discussed in a forthcoming paper.
Some basic concepts of I -adic representations can be found in
[23] Y. Taniyama, L-functions of number fields and zeta functions
of abelian variet ies, J . Math. Soc. Japan, 9(1957), 330-366.
This verif ies a lso the Hasse conjecture for abelian variet ies
with sufficiently many complex multiplications. F o r this topic, s ee
a lso [14], and lecture notes by J. -P. Se r re to be published soon. The
connection of the Hasse zeta function of an elliptic curve with the
Diophantine problems on the curve i s discussed in a survey art icle
(of course with numerous references)
[24] J. W. S. Cassels , Diophantine equations with special
reference to elliptic curves, J. London Math. Soc., 41 Q966), 193-291.
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