Automorphic Functions and Number Theory

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 l e c t u r e s which I gave a t the Forschungsinstitut f a r Mathematik, Eidgenossische Technische Hochschule, Ziirich in July 1967.

I have attempted to make a

s h o r t comprehensible account of the l a t e s t r e s u l t s in the field, together with an exposition of the m a t e r i a l of an e l e m e n t a r y nature. No detailed proofs a r e given, but t h e r e i s a n indication of b a s i c ideas involved.

Occasionally even t h e definition of fundamental concepts

m a y b e somewhat vague. the r e a d e r .

I hope that this procedure will not bother

Some r e f e r e n c e s a r e collected in the final section in

o r d e r to overcome these shortcomings.

The r e a d e r will be able to

find in them a m o r e complete presentation of the r e s u l t s given h e r e , with the exception of s o m e r e s u l t s of §lo, which I intend to d i s c u s s in detail in a future publication. It is m y pleasure to e x p r e s s m y thanks t o P r o f e s s o r s K. Chandrasekharan and B. Eckmann for their i n t e r e s t in this w o r k , and f o r inviting m e to publish it in the Springer L e c t u r e Notes in Mathematics.

I wish a l s o acknowledge the support of the

Eidgenossische Technische Hochschule, Institute for Advanced Study, and the National Science Foundation (NSF-GP 7444, 5803) during the s u m m e r and f a l l of 1967.

Princeton, January 1968

All rights re\crued. N o 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

G. Shimura

Notation

Contents

We denote by 2 , Q, R and C respectively the ring of rational i n t e g e r s , the rational number field, the r e a l number field and the 1

Introduction Automorphic functions on the upper half plane, especially modular functions Elliptic c u r v e s and the fundamental t h e o r e m s of the c l a s s i c a l t h e o r y of complex multiplication Relation between the points of finite o r d e r on an elliptic curve

complex number field. ment,

Y

F o r a n associative ring Y with identity ele-

X

denotes the group of invertible elements in Y, M (Y) the n r i n g of a l l m a t r i c e s of s i z e n with e n t r i e s in Y, and GLn(Y) the

group of invertible elements in M (Y), i. e. , Mn(Y)X . The identity n element of M (Y) i s denoted by 1 and the t r a n s p o s e of a n element n n ' t 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. F o r a n typographical reason, the quotient of a space S by a group G will be

and the modular functions of higher level Abelian v a r i e t i e s and Siege1 modular functions The endomorphism ring of an abelian variety; the field of moduli

denoted by S/ G,

even if G a c t s on the left of S.

If F i s a field

of a n a b e l i a n v a r i e t y with many complex multiplications

and x i s a point in a n affine (resp. a projective) space, then F ( x )

The c l a s s -field-theoretical c h a r a c t e r i z a t i o n of K' (

m e a n s the field generated over F by the coordinates (resp. t h e

(z)).

A f u r t h e r method of constructing c l a s s fields

The H a s s e z e t a function of a n algebraic c u r v e Infinite Galois extensions with l -adic representations F u r t h e r generalization and concluding r e m a r k s Bibliography

quotients of the homogeneous coordinates) of x.

If K i s a Galois

extension of F, G(K/ F) stands f o r the Galois group of K o v e r F.

1.

Introduction

Our starting point i s the following t h e o r e m which was stated by Kronecker and proved by Weber: Theorem 1.

of

Q

with a n m - t h root of unity

3

E v e r y finite abelian extension

a cyclotomic field Q(5) -

contained & 2ni/ m =e for

-

s o m e positive integer m. As i s immediately observed, 2 niz a t z = l/ m .

nential function e

5

i s the special value of Qe expo-

One can naturally a s k the following

question: Find analytic functions which play a r o l e analogous to -

e

2 niz

-f o-r a given algebraic number field. Such a question was r a i s e d b y Kronecker and l a t e r taken up by Hilbert a s the lzth of h i s famous mathematical problems. imaginary quadratic field K,

F o r an

this was settled by the works of Kronecker

himself, Weber, Takagi, and Hasse.

It t u r n s out that the m a x i m a l abelian

extension of K i s generated over K by the special values of c e r t a i n elliptic functions and elliptic modular functions.

A p r i m a r y purpose of

t h e s e l e c t u r e s is to indicate briefly how this r e s u l t can be generalized f o r the number fields of higher d e g r e e , making thereby an introduction to the theory of automorphic functions and abelian v a r i e t i e s .

I will a h o

include s o m e r e s u l t s concerning the zeta function of an a l g e b r a i c curve in the s e n s e of Hasse and Weil, since this subject i s closely connected with the above question.

F u r t h e r , it should be pointed out that the auto-

morphic functions a r e meaningful a s a m e a n s of generating not only

abelian but a l s o non-abelian algebraic extensions of a number field.

and vice v e r s a .

Some ideas in this direction will b e explained in the l a s t p a r t of the

functions with r e s p e c t to

lectures.

compact

We s h a l l l a t e r d i s c u s s special values of automorphic

#/r .

S L (2). Since

2

r

f o r an arithmetically defined l? with

But we f i r s t consider the m o s t c l a s s i c a l group l? = i s not compact in this c a s e , one has to impose

$/I'

a c e r t a i n condition on automorphic functions.

f

e v e r y point of 2.

It i s well known that

c a n be t r a n s f o r m e d by an element of T = SL ( 2 ) 2

into the region

Automorphic functions on the upper half plane, e s p e c i a l l y modular functions

Let

5

denote the complex upper half plane: No two distinct inner points of F can be t r a n s f o r m e d to each other by a n element of T point a t infinity.

We l e t e v e r y e l e m e n t a =

b d) of GL2(R), with det ( a ) > 0 ,

act

point, we s e e that genus 0.

.

Now

By taking e

$1

r

$/I? 2riz

c a n be compactified by adjoining a a s a local p a r a m e t e r around this

becomes a compact Riemann s u r f a c e of

Thus we define an automorphic function with r e s p e c t to l?

t o b e a meromorphic function on this Riemann s u r f a c e , considered a s (2.1)

a ( z ) = (az

+ b ) / (cz t d)

It i s well known t h a t the group of analytic automorphisms of

ff

is

isomorphic to S L (R)/ { t l ). L e t r be a d i s c r e t e subgroup of 2 2 SL2(R). Then the quotient $ 1 ~ has a s t r u c t u r e of Riemann surface such that the n a t u r a l projection

$ /r f

$+

$/I?

i s holomorphic.

If

/B .f.

In other w o r d s , l e t f be a r - i n v a r i a n t m e r o m o r 1 phic function on For y = , we have y (z) = z t 1. Since c e 27rinz f (y ( 2 ) ) = f (z), we can e x p r e s s f (z) in the f o r m f (z) = Z 0 n=-w n a function on

6

c C. Now a n automorphic function with r e s p e c t t o r i s an n f such that c = O for a l l n < n f o r s o m e n , i. e . , meromorphic n 2riz a t q = 0 . Such a function i s usually in the local p a r a m e t e r q = e

with c

i s compact, one can simply define an automorphic function on r e s p e c t t_o l? to be a m e r o m o r p h i c function on

under the e l e m e n t s of I?

.

-$

invariant

Such a function m a y be r e g a r d e d a s a

merornorphic function on the Riemann surface

$11.

in an obvious way,

called a modular function of level one.

Since

$ 1

is of genus 0,

a l l modular functions of level one f o r m a rational function field over C. As a generator of this field, one can choose a function j such that

h a s one of the following two normalized f o r m s : (i) P(z) = z t A,

Obviously the function j c a n be c h a r a c t e r i z e d by (2.2) and the

(ii) P(z) =

K z ,

property of being a g e n e r a t o r of the field of a l l modular functions of level one. Now l e t K b e a n imaginary quadratic field, and Take a b a s i s {wl, w2)

tional ideal in K. is imaginary,

wI/

'$

Y2

.

wl/ w2 i s not r e a l .

of

8t

a

over Z.

a fracSince K

Therefore one may a s s u m e that

by exchanging w and u2 if n e c e s s a r y . 1

In this setting.

.

with constants h and

K

the Jordan f o r m of a .

In the f i r s t c a s e , we call a parabolic; in

the second c a s e , according a s

This can be shown, for example, by taking

a i s called elliptic, hyperbolic, o r loxodromic,

1K I

= 1, K r e a l , o r otherwise.

we exclude the identity transformation, which i s represented b y the scalar matrices.

we have:

If a s GL2 (R) and d e t ( a ) > 0, T h e maximal unramified abelian extension

T h e o r e m 2. c a n be --

generated

9

j (w

/ w2)

1

over

of

K

-f

$

one needs m o d u l a r functions of higher level (see below) o r elliptic

w2

.

onto itself, and a

is

hyperbolic if a h a s two fixed points in R

To c o n s t r u c t ramified abelian extensions of K ,

functions with p e r i o d s w

a maps

elliptic if a h a s exactly one fixed point in

K.

This is t h e f i r s t m a i n t h e o r e m of the c l a s s i c a l theory of c o m plex multiplication.

In this classification,

parabolic if

8

U {m),

a h a s only one fixed point in R

U {m).

No transformation in G L (R) with positive determinant i s loxodromic. 2 If we put

Even Th. 2 c a n fully be understood

with the knowledge of elliptic functions o r elliptic c u r v e s , though such a r e not explicitly involved in the statement.

T h e r e f o r e , our next t a s k

i s to r e c a l l s o m e e l e m e n t a r y f a c t s on this subject.

But before that,

i t will be worth d i s c u s s i n g a few elementary facts about the fractional l i n e a r t r a n s f o r m a t i o n s and discontinuous groups. Every a = by the rule

(2.1).

%

r GL (C) a c t s on the Riemann s p h e r e C U{m) 2 -1 With a suitable element fj of GL2(C), P = S a c ):

then i t c a n easily be verified that SO (R) i s the s e t of a l 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 r e g a r d

$.

a s such a quotient.

#

(Here note that if f s a t i s f i e s (i), then f(p(z)) i s invariant under

.

But I shall not

pursue this view point, f r o m which one can actually s t a r t investi-

z

is invariant under SL2 (R).

#

by means of this form. with non-compact

Let J? be a d i s c r e t e

We c a l l a point s of R U { m )

there e x i s t s a parabolic element y

of

2

z

a=of

I7 leaving s fixed.

.

Then J?

] a s a local p a r a m e t e r around the

(Actually the proof of the fact that

with r e s p e c t to r ,

-%

defined above, i s nothing e l s e than a meromorphic

PIT,

function on the Riemann s u r f a c e

r

and a l l the cusps of I?

*/I? i s a Hausdorff space i s not difficult, but non-trivial. ) Then an automorphic function

Therefore we can introduce an

, we have to introduce the notion of cusp.

subgroup of SL2(R).

point s .

(z = x t iy)

#

One can define a s t r u c t u r e of Riemann surface on

fj*/rby taking

To speak of a n automorphic function for a 1

b e the join of

9.

a c t s on

-2 Y dxAdy

invariant m e a s u r e on

9

Let

We s e e easily that a differential f o r m

hence f(p(z)) i s always meromorphic a t l e a s t in the

.

gation in various directions.

on

+ 1,

Hz

domain O < ( q[ < r for some r > 0 )

J? if Let

regarded a s a function on The above discussion about SL ( Z ) i s a special c a s e of these 2 facts. Now the following facts a r e known:

-f .

i s compact if and only if d/r has a finite m e a s u r e with r e s p e c t t o the above invariant m e a s u r e . --

j'/r

Proposition 1.

Proposition 2.

Then

I

Suppose that

$/r

has a finite m e a s u r e .

i s compact if and only if I' has no parabolic element.

As for elliptic elements, the following proposition holds : Then one c a n find an element p of S L2 (R) s o that p(w) = s , and PTsp

-1

i s generated by

(1O

an automorphic function on function on

-#

i)

#.

and possibly by -I2 w x respect

2 r

.

Then we define

to be a meromorphic

Lf

g

a cusp of I? a n d p k a s above, then f(p(z)) 2riz i z _n neighborhood o_f q = 0. meromorphic function in q = e s

of

r.

Let

= a

-finite order.

satisfying the following two conditions:

t r

az

&a

point of

=z .

-$

Then Tz -

fixed

9 an e

2 a cyclic

c element

group of

6

Such a point z i s called a n elliptic point of I?,

(i) f ( y ( z ) )= f ( z ) f o r a l l y e r .

(ii)

Proposition 3. k

of

ka

r z .{+12}/ {+12}

t o I?).

and the o r d e r is called the o r d e r of the point z (with r e s p e c t

Two elliptic points or cusps a r e called equivalent if they a r e

transformed to each other by elements of I?.

If

- f / r is

of finite

periods in L i s a m e r o m o r p h i c function on C invariant under the

m e a s u r e , t h e r e a r e only a finite number of inequivalent elliptic

translation u

points and c u s p s , and the following f o r m u l a holds:

u

+w

for every w

t

g2, g3 and meromorphic functions

H e r e g i s the genus of

L. Define complex numbers P(u) and

8. (u)

on C by

p / r ; h i s the number of inequivalent

c u s p s ; C Z i s the s u m extended over a l l inequivalent elliptic points; is the o r d e r of z. F o r r = S L 2 ( Z ) , one h a s g = 0, h = 1, e Z = e 2 o r 3 according a s z = c l o r z = (-1 F o r e v e r y positive integer N,

+

c 3 ) / 2.

set where C denotes the s u m extended over a l l n o n - z e r o w in L.

I'm) = { a c S ~ ~ (1 za) r l2 mod N. M ~ ( z ) ) . An automorphic function with r e s p e c t to function of l e v e l ---

I'm)

Then it is well-known that

i s called a modular

N. (3. 3) The field of a l l elliptic functions with periods in L coincides with C ( 9 , -

3.

),

the field g e n e r a t e d b~

P

and

-

$g

E

r

C.

Now l e t E be the algebraic curve defined b y

Elliptic c u r v e s and the fundamental t h e o r e m s

of the c l a s s i c a l theory of complex multiplication L e t L b e a l a t t i c e in the complex plane, i. e . , a f r e e Z - submodule of C of r a n k 2 which i s d i s c r e t e . Then C / L i s a compact Riemann s u r f a c e of genus one.

An elliptic function with

H e r e we consider E a s the s e t of a l l points '

with x, y in C ,

together with a point

(x. y) satisfying (3.4)

(w,a!. Then the m a p

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 gives a holomorphic isomorphism of C / L onto E in the sense of complex manifold.

It is a l s o known that any elliptic curve (i. e . an

algebraic curve of genus one) defined over C i s isomorphic to a

Q(w / w ) i s a n 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 iso1 ' 2 morphic to a subring of the ring 0' of a l l algebraic integers in K ,

curve of this type, and hence t o a complex torus.

which generates K.

Take a b a s i s {ul, w ) of L over 2. We may a s s u m e that 2 , Then one c a n easily show that w 1/ u 2

plications.

f

In this c a s e we say that E has complex multi-

In particular, if L = Zwl

i s isomorphic to

Q

.

P u t jo = j(wl/

+ Zw2 %).

i s an ideal in K ,

End(E)

F o r 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

0

i s a n algebraic number if E has complex multi-

plications. defines a one-to-one correspondence between m o r p h i s m - c l a s s e s of elliptic c u r v e s .

$11.

& a l l the

iso-

F u r t h e r m o r e we have an

Now write E a s E(&) if L = DL

for an ideal

6t in K.

We choose the equation f o r E(8L ) s o that g 2 , g3 c Q(jo). Suppose we could somehow prove that K(j ) i s an abelian extension of K.

important relation

(Anyway this i s not the m o s t difficult point of the theory. ) Take a p r i m e ideal

J7/

in K u n r a m i f i e d i n K ( j ),

andlet

(= the Frobenius automorphism of KG ) over K f o r

One should note that the right hand side can be obtained purely a l -

and g3 a r e meaningful.

gebraically f r o m the defining equation (3. 4) for E, while the l e f t

E(@L)O

i s defined analytically.

by

an analytic object h a s a deep meaning, though we know, f r o m (3. l ) , that g2 and g

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 i s obtained by u H Xu with a complex number X satisfying

2 ).

[T

, KGo)/K]

Then g2

Therefore we can define an elliptic c u r v e

This coincidence of an algebraic object with

depend analytically on wl and w2 . We call the 3 number e x p r e s s e d by ( 3 . 5 ) the invariant o_f E. Two elliptic curves

a =

Then one has a fundamental relation:

If we denote by j

( a)

the invariant of E(& ),

then (3.6) i s equi-

valent to

The answer is affirmative but not unique.

It may be said that the

world of mathematics is built with a g r e a t harmony but not always in the f o r m which

\r*e

expect before unveiling it.

applies to our present question.

This certainly

I s h a l l , however, f i r s t present a

comparatively simple answer which consists of the following t h r e e F r o m the relation (3.6) o r (3.7), one c a n easily derive Th. 2 and a l s o the r e c i p r o c i t y law in the extension KGo) of K.

H e r e I do

objects: (A1) abelian variety,

not go into d e t a i l of the proof of (3. 6), but would like to call the r e a d e r 1s

(B' ) Siege1 modular function,

attention t o the following point: Although no elliptic curves appear in

(C1) totally imaginary quadratic extension of a totally r e a l

Th. 2 , they conceal themselves in it through the above (3.6) and the following f a c t s . (3.8)

algebraic number field. At l e a s t this will include the above r e s u l t concerning elliptic c u r v e s

-

$/I?

The quotient

2

in o n e - 5 - 0 2 correspondence with all

a s a special case.

A different type of theory, which I feel r a t h e r un-

expected, and which a l s o generalizes Th. 2, will be discussed l a t e r .

t h e i s o m o r p h i s m c l a s s e s of elliptic c u r v e s . j(w 1w ) i s the invariant of an elliptic curve E isomorphic 1 2 to C / (Zwl Zw2). (3. 9)

+

-

2

(3.10)

--

Q(ul/ uZ)

2 imaginary quadratic,

End@) i s non-trivial.

4.

Relation between the points of finite o r d e r on

an elliptic curve and the modular functions of L e t u s now consider the question of generalizing Theorems 1 and

2 t o the fields of higher degree.

higher level.

We observe that t h e r e a r e three

objects: (A) elliptic c u r v e ,

(B) modular function, (C) imaginary quadratic field. Among many possible i d e a s , one c a n take the m o s t naive one, namely ask whether t h e r e exist generalizations of (A), (B), (C)whose i n t e r relation i s s i m i l a r to that of the original ones, a s described in (3.8-10).

Before talking about abelian v a r i e t i e s , l e t us discuss the topic given a s the title of this section.

Any hasty r e a d e r may skip this

section, and come back afterward. F i x a positive integer N.

Observe that any point t on E

such that Nt = 0 can be expressed a s

L e t K be a s above, and @ ,

Theorem 3. L,

@L

with integers a , b.

Now, f o r each o r d e r e d pair

(a, b ) of integers

(a, b) ? (0, 0) mod (N), we can define a meromorphic N function f (z) on by ab such that

#

and l e t

6 = Zw1 t Zw2 wifh

wl/ w2

an ideal in K. 6

6.

Take

Suppose that

g Z ( 6t ) g 3 ( 8 L ) # 0. Then the maximal abelian extension f K i s N g e n e r a t e d over K & j (wl/ w2) and the fab (wl/ w2) for a l l N, a , b ,

with a fixed ---

.

8L

.

N It should be observed h e r e 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 n e c e s s a r i l y be retained

in one of our l a t e r generalizations.

.

where z = w / w and L = Zwl t Zw2 This i s possible because the 1 2 Then right hand s i d e depends only on z = w / w 1 2' N N fab(z)=f z cd

(-C,

hT

The function field C ( i , f

a r SL2(Z),

the l a t t e r group i s isomorphic to S L (Z/ NZ)/ (21 ). Since our purpose 2 2 i s to construct number fields by special values of functions, i t i s meaningful to take Q,

Therefore, to

f

N N (2) = f (a ( 2 ) ) for a l l (a, b) if and only if a belongs ab ab

?I (N). { i J L } . It follows that j and the b:f

, for a l l (a, b ) ,

generate the field of a l l modular functions of level N.

ing, the modular functions of level N c a n be obtained f r o m the in-

we have the following r e s u l t which i s an analogue of Th. 1 for an imaginary quadratic field.

instead of C,

a s the basic field.

Now

Then we

obtain: Theorem 4.

N

Q(j, fab) i s a Galois extension o_f Q(j) whose

Galois group i s isomorphic t o GL2(Z/ NZ)/ {f12},

Roughly speak-

v a r i a n t of elliptic c u r v e s and points of o r d e r N on the curves.

N

), with a fixed N, is a Galois exab tension of C(j) whose Galois group i s isomorphic to r ( l ) / r ( N ) . {&12};

-d) mod (N).

By a simple calculation, we can show that, for every

m).

a)/

obtain the s a m e type of r e s u l t by modifying the definition of fLY ab suitably.

(a,b)=_(c,d)mod(N) o r (a, b )

a((-1t

(a

(a

) = 0 o r g3 ) = 0 according a s K = 2 2) o r K = Q ( In these special c a s e s , we c a n still

We note that g

statements

hold.

(i) F o r e v e r y a

E

G L 2 ( Z / NZ), the action of

of the Galois group is given by --(ii)

If

obtainedh

and the following

fib

a

a n element

-

f N with (c d) = (ab ) a . cd N y E S L ( Z ) , the action of y mod (N) Q(j, f ) i s 2 ab N 'f'(z)rj y ( y ( z ) ) for Q ( j , fab). I-+

YE

-

N ) contains ab det(a)

(iii) Q(j, f sends 5

2

6

5=

e 2nil N,

and

-

a r GL2 ( Z / N Z )

We call such {vl,

. . . , v 2n )

6

a Riemann form on Cn/ L.

Take a basis n and r e g a r d the elements of C a s

of L over Z ,

column vectors.

Then we obtain a m a t r i x

We shall l a t e r extend this theorem to the field of automorphic functions with r e s p e c t t o a m o r e general type of group. De-

of nX2n type, which may be called a p e r i o d m a t r i x for Cn/ L.

fine a m a t r i x P = (p..) of s i z e 2n by p.. = f * ( v i , v.). T h e n t h e U 1J J above a r e equivalent to the following (Ri-3): 5.

Abelian v a r i e t i e s and Siege1 modular functions

A non-singular projective v a r i e t y of dimension n , C,

defined over

(R;)

p..

5 )

An elliptic curve i s

nothing but a n abelian v a r i e t y of dimension one.

'P = - P ; Z;

1J

i s called a n abelian v a r i e t y if i t i s , a s a complex manifold, iso-

morphic to a complex torus of dimension n.

i )

52p-l

.

= 0,

&

-~

-

---

1 a ~ -t~I . i s a positive definite

hermitian matrix.

We know that any (or its i n v e r s e ) i s called a principal m a t r i x of 52

.

one dimensional complex t o r u s defines an elliptic c u r v e , but such

The m a t r i x P

i s not t r u e in the higher dimensional case.

Assuming these conditions, l e t A be a projective variety i s o n morphic t o C / L. Shifting the law of addition of cn/L to A , we

To explain the n e c e s s a r y

condition, l e t L be a lattice in the n-dimensional complex vector

.

a d i s c r e t e f r e e Z-submodule of rank 2n in C n n Then the complex t o r u s C / L h a s a s t r u c t u r e of projective variety, space

cn ,

i. e . ,

and hence b e c o m e s an abelian variety, if and only if t h e r e exists an R-valued R-bilinear f o r m properties:

c a n define a s t r u c t u r e of commutative group on A.

A X A 3(x, y ) C , x + y

E

Then the map

A

G ( x , y) on Cn with the following can be e x p r e s s e d rationally by the coordinates of x and y.

This i s

classically known a s the addition theorem of abelian functions. In general, a projective variety A,

'

defined over any field of

any c h a r a c t e r i s t i c , i s called an abelian v a r i e t y , if t h e r e exist rational mappings f : A X A -+ A and g: A t u r e on A by f(x, y) = x

+ y,

--t

A which define a group s t r u c -

g(x) = -x.

Additive notation i s used since any such

group s t r u c t u r e on a projective variety can be shown to be commutative.

corresponds t o such a n abelian variety.

Obviously

As a n analogue of S L (R), we introduce a group 2

It should be observed that such a variety defined over C ,

being a connected compact commutative complex Lie group, m u s t be isomorphic to a complex torus.

If n = 1, t h e r e i s a single universal family of elliptic curves

#.

p a r a m e t r i z e d by the point of

If n > I,

however, t h e r e a r e

infinitely many f a m i l i e s of abelian v a r i e t i e s depending on the elementary divisors of P ,

a s shown in the Supplement below.

F o r every U =

[:

1 1 6

Sp(n, R ) with a , b , c , d in Mn (R), we

define the action of U on

But we s h a l l f i r s t

fix our attention to one particular family by considering only abelian v a r i e t i e s f o r which P = J

n

.

where Put

When n > I, Under t h i s assumption, l e t

q

and w2 be the s q u a r e m a t r i c e s of s i z e

we can define an automorphic function with r e s p e c t to

Sp(n, Z ) to be a meromorphic function on Sp(n, Z).

invariant under

Fortunately, if n > 1, i t i s not n e c e s s a r y to impose any

n composed of the f i r s t and the l a s t n columns of respectively. -1 u1 If we change the One c a n show that w i s invertible. P u t z = w n coordinate s y s t e m of C by w2 , we may a s s u m e that 52 i s of the

f u r t h e r condition like that we needed in the c a s e n = 1. Such a

form

and level one).

.

function i s us'ually called a Siege1 modular function (of degree n

Put

r = Sp(n,

Z).

Now one can a s k whether the quotient

fillr

i s in one-to-one correspondence with all the isomorphism c l a s s e s of abelian v a r i e t i e s of type (5.1). This i s a l m o s t s o but not quite. Now it c a n be. shown (see Supplement below) that z i s s y m m e t r i c and

Im(z) i s positive definite. of degree n.

We denote by

$--

the s e t of a l l such z

Thus every abelian v a r i e t y , under the assumption that

, though z i s #n Moreover, e v e r y point of

P has the f o r m (5. l ) , c o r r e s p o n d s to a point of

not unique for a given abelian variety.

To

g e t a n exact answer, we define 2n r e a l coordinate functions x (u), 1 n n , x (u) (u E C ) by u = xi(u)vi , and consider a cohomology 2n c l a s s c on A represented by a differential f o r m

...

and (A, -

c ) respectively. ~

belong to of degree 2.

Such a c is called a polarization of A,

-of polarized

f

T&n

n

abelian v a r i e t i e s

of

(z)

In (iii), we of c o u r s e consider A a s a projective variety de-

and the s t r u c -

fined by s o m e homogeneous equations.

t u r e (A, c ) f o r m e d by A and i t s polarization c i s called a p o l a r i z e d abelian variety.

Then the coordinates of the point

~ ( z= )Y)(zl ~ ).

k, a n d

Now one can prove that the

r e p r e s e n t s a11 the isomorphism c l a s s e s

cohomology c l a s s c i s r e p r e s e n t e d by a divisor on A (i. e. an (n

type (5. l ) , the isomorphism being de-

dimensional algebraic s u b s e t of A).

O u r next question i s about the existence of s o m e functions s i m i l a r

1)-

If the defining equations f o r A

and such a divisor have coefficients in a field k ,

fined in a n a t u r a l way.

-

we s a y that

(A, c )

If o i s a s in (iii), the t r a n s f o r m s of the equations

i s defined over k.

to j and the analogue of (3.5). F i r s t one should note that t h e r e e x i s t s

by o define an abelian v a r i e t y together with a d i v i s o r , which t u r n s out

--

to be a polarized abelian v a r i e t y of type (5. l ) , which we write a s

a Z a r i s k i open s u b s e t V o_f 5 projective v a r i e t y V* and a holomorphic

of

mapping

$n/r

fn

Onto V.

Sn/r.

I

[:

We c a l l such a couple

(aj

+

p)/(yj

+ 6)

(V,

y)

?

for

(iv)

If

(A' , c 1)

corresponds

2

of -

cf(z))

((A, c ) ,

(A'

F u r t h e r m o r e , we would like t o have an analogue of

y)

i s a model for

fnfr

.

(A, c )

a polarized

(5. l ) , defined over a subfield k f into -

C. L e t z and z'

V onto V'

be points on

fn

f n , and

The couple

(V,

Namely, if

y) (V'

c a n b e c h a r a c t e r i z e d by

, cj?') i s another couple

with the s a m e p r o p e r t i e s , t h e r e e x i s t s a biregular isomorphism f of

abelian v a r i e t y with a P of type C, and

It i s analytic on

y ( z ) generate the whole field of Siege1 modular

these p r o p e r t i e s (i, i i , iii).

(ii) V is.defined over Q. (iji) k t

Q.

(A, c ) , a s explained in (iii). F r o m (i) it follows that

functions of degree n.

properties: (i) (V,

over

a t the s a m e t i m e , it i s a rational expression of the coefficients of defin-

the coordinates of ) with the following

(V,

c ' ),

plays a r o l e s i m i l a r to j.

Thus

ing equations f o r

a s follows: T h e r e e x i s t s a couple

,

2 specialization of (A, c ) o v e r Q, z' then ((A1, c 1), (zl )) specialization

F o r details we r e f e r the r e a d e r to the paper [ZO] in $12.

T h e r e f o r e a f u r t h e r refinement i s n e c e s s a r y , and can be given

T h e o r e m 5.

(A, cIa

We can actually prove a s t r o n g e r statement than (iii), which i s roughly

a model

In f a c t , in the

plays a r o l e of

of

a s follows:

GL2(C). Of c o u r s e one can not r e p l a c e j b y such a

function in Th. 2. (3.5).

.

This is-not sufficient f o r our purpose.

c a s e n = 1, the function any

V which induces a biregular isomorphism

T h i s was proved by W. L. Baily using the Satake

compatification of for

into

a

& isomorphism of

corresponding t_o

k

(A, c )

defined over Q such that

(iii), we s e e that the field Q ( 'f'(z))

y'

=f

0

(o

.

Moreover, from

has an invariant meaning f o r the

.

isomorphism c l a s s of

(A, c ) .

We c a l l i t the field of moduli of

(A, c ) .

Actually we c a n prove a l l these things without assuming P = J n F o r each choice of P (or r a t h e r f o r a choice of elementary divisors

(Vp ,

and a couple suitably.

yp)

r

(see Supplement below) acting on P with the properties (i, ii, iii) modified

of P ) , one obtains a group

.

%n

T h e r e f o r e , to discuss 52 satisfying (Ri-3), we may assume r0 -el with e a s in the above lemma. Let Y p be ol thatP=Le the space of a l l such 51,

and l e t

F u r t h e r , by considering the points of finite o r d e r on the

abelian v a r i e t i e s , one c a n generate the field of automorphic functions with r e s p e c t t o congruence subgroups of Sp(n, Z ) ; one then obtains a theorem analogous to Th. 4. The next thing t o do i s the investigation of special m e m b e r s of our family of abelian v a r i e t i e s , analogous t o elliptic c u r v e s with complex multiplications.

This will be done in s6.

In particular G then a U r Yp Now write

Supplement t_o s5. To d i s c u s s the families of abelian v a r i e t i e s

from

P

.

= Sp(n, R) if e = 1

n

.

If 52 c Yp

and U r G

P '

t

B J B = P, hence BG B-I = Sp(n, R). P n = (v v l ) with two elements v and v' of Mn(C). Then, Obviously

(R;), we s e e easily that

of a m o r e g e n e r a l type, for which P i s not n e c e s s a r i l y of the f o r m

(5. l ) , f i r s t we r e c a l l a well known Lemma. with e n t r i e s ---

L A P b_e

in 2.

invertible alternating m a t r i x of s i z e 2n

Then t h e r e e x i s t s an element U

that -

of

GL

2n

e l

(Z)

- ve -1 . t-v ' )

.

The l a s t fact implies that v and v1 a r e invertible. relations i t follows that e v l -'v definite imaginary p a r t , i. e. If z

$n

and U

e

i

-

a r e positive integers satisfying eitl

= 0 mod

(ei).

w

t

$n .

,

F r o m these

i s s y m m e t r i c and has a positive ev' - l v r

t]

=[:

hence by the above r e s u l t , where the --

i s positive definite.

n

.

r Sp(n, R ) , then

(z ln)U r Y

Jn'

ln)U = A(w

ln) with A r M (C) and n - 1 This shows Then one obtains w = (az t b ) ( c z t d) that

the action of U on

(z

.

-fn

can actually be defined.

Since the action

of U-I c a n b e defined,

In g e n e r a l , two d i v i s o r s X and Y a r e called algebraically

U gives a holomorphic automorphism of

equivalent, if t h e r e exist a divisor W and i t s specializations W

Now s e t

and W2 over an algebraically closed field such that X

-

1

-

Y = W

W

1

If t h e universal domain i s C , then the algebraic equivalence of div i s o r s coincides with the homological equivalence. It c a n e a s i l y be s e e n t h a t fn/Fp

rp

i s a d i s c r e t e subgroup of Sp(n,R).

Then

r e p r e s e n t s a l l the isomorphism c l a s s e s of p o l a r i z e d abelian

The notion of polarization can a l s o be defined in the c a s e of

X on A,

l e t L be

the l i n e a r s p a c e of a l l r a t i o n a l functions on A whose poles a r e contained in X (even with multiplicities).

Take a b a s i s {fo, fl,

. . . , fN)

Now a polarized abelian variety i s a couple a n abelian v a r i e t y A and a polarization

xt)

if it sends

. . . , fN ( x ) )projective ~

(i)

N-space.

X I . For

If

(A,

x)

0

(nu.

(2)

i s defined over k,

integers m

X' mt

belong to

such that mX

, then t h e r e a r e two positive

&

f_o (A,

.

i s o m o r p h i s m c l a s s of

m ' X t a r e algebraically

equivalent. . (3)

5 isomorphic

-

5 m a x i m a l s e t satisfying the above two conditions.

(A,

XI. This

P = J n .

a given (A,

(A, x ) to x),we can

then k i s contained & k. into the universal domain, 0

X)

if and only if --

If the universal domain i s C , k

contains a n ample divisor.

_II X

EO)

mappins on k

on A satisfying the following conditions:

(1)

This definition

with the following p r o p e r t i e s :

(ii) F o r a n isomorphism o of k We c a l l X ample if this i s a b i r e g u l a r embedding of A into the of d i v i s o r s projective space. Now a polarization of A i s a s e t

of A.

X ) formed by

i s called an isomorphism of to

prove that t h e r e e x i s t s a field k (fo(x),

(A,

i s equivalent to the previous one, if the universal domain i s C. An i s o m o r p h i s m of A of A' (A',

and consider the m a p

A 3 x

E v e r y abelian v a r i e t y , defined

h a s a n ample divisor.

Given a n abelian v a r i e t y A defined over a

field of any c h a r a c t e r i s t i c , and given a divisor

of L over k,

Riemann f o r m , then 3X i s ample.

over a field of any c h a r a c t e r i s t i c , h a s a polarization, since i t always

v a r i e t i e s with principal m a t r i x P.

positive c h a r a c t e r i s t i c .

Moreover, if a

divisor X r e p r e s e n t s the cohomology c l a s s c obtained f r o m a

(A,

o i s the identity

i s uniquely determined by the

x), and is called the field of moduli of

of c o u r s e coincides with Q (

Y, ( 2 ) )

in the special case

-2

6.

L e t k be a field of definition f o r A and the elements of End(A),

The endomorphism-ring of a n abelian variety;

and l e t D b e the vector space of a l l l i n e a r invariant differential forms

the field of moduli of a n abelian v a r i e t y

on A,

defined over k. If zl, n functions in C , then d zl,

with many complex multiplications

. . . , zn

. . . , dzn

f e r e n t i a l f o r m s on A,

F o r an abelian v a r i e t y A, we denote by End(A) the ring of a l l holomorphic endomorphisms of A.

a r e the complex coordinate

a r e considered a s invariant dif-

and one has

If A i s isomorphic to a

complex t o r u s Cn/ L , e v e r y endomorphism of A c o r r e s p o n d s to a n e l e m e n t T of M (C), r e g a r d e d a s a C-linear transformation n on Cn , satisfying T (L) L. T h e r e f o r e End(A) i s a f r e e Z-module

C

Now e v e r y A

(A) = End(A) 8 =Q,

and W = Q. L. Then W i s n a vector s p a c e o v e r Q of dimension 2n, which spans C over R, and

Then X H

End

we obtain

of finite rank.

Q

L e t End

Q

(A) i s isomorphic to the ring

E

*

<

End(A) a c t s on D a s usual; denote the action by A*. can be extended to an anti-isomorphism of End

into the ring of linear transformations in D.

--

(6. 2) This anti-isomorphism complex representation o_f End F o r each e l e m e n t of End

(A), consider the corresponding element T Q of Mn(C). Then we g e t a faithful r e p r e s e n t a t i o n of End (A) by complex Q m a t r i c e s of s i z e n , which we c a l l the complex r e p r e s e n t a t i o n of End (A).

Let

Q

(A)

F r o m the relation (6. l ) ,

equivalent to the t r a n s p o s e of t h e

Q

(A).

b e a Riemann f o r m on

cn/L.

For every T

E

End

Q

(A),

one can define a n element T~ of EndQ(A) by

Q

On t h e other hand, with r e s p e c t t o a b a s i s of W over Q (for example, {vl,.

. . , v 2n )

c o n s i d e r e d i n s5), we obtain a r e p r e s e n t a t i o n of End

by rational m a t r i c e s of d e g r e e 2n, sentation

of

End

Q

Q

(A)

which we c a l l the rational r e p r e -

(A). As an e a s y e x e r c i s e of l i n e a r a l g e b r a , one c a n

a n involution of an associative a l g e b r a S over Q (or R ) i s , by

prove : L e m m a . , The rational r e p r e s e n t a t i o n f End (A) to the d i r e c t s u m of the complex r e p r e s e n t a t i o n of -----complex conjugate.

H e r e we identify an element of End (A) with the corresponding element Q of M n ( C ) Then p $ a positive involutinn o_f EndQ(A). In g e n e r a l ,

Q

End

Q

equivalent (A) and i t s

definition, a one-to-one m a p p of S onto S such that

8

R a r e a l l r e a l , i. e. , F i s Q totally r e a l . F u r t h e r , if [K : F ] = 2, the d i r e c t f a c t o r s of K @ R Q a r e a l l C, i. e . , K i s totally imaginary. to s e e that the direct factors of F

Conversely, l e t F be a totally r e a l algebraic number field, P Such a p i s called positive if T r ( x x ) > 0 for 0 f x r S, where

a totally imaginary quadratic extension of F , automorphism of K over F.

T r denotes the t r a c e of a r e g u l a r representation of S over Q.

We fix such F , K, p,

If an a l g e b r a S over Q o r R h a s 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 e v e r y y c S, but this i s a contradiction, since ~r (xxP)> 0. It follows that S i s s e m i simple.

If e is the identity element of a simple component of s ,

then e e P f 0, hence e P = e. simple component of S.

It follows that p i s stable on each

Thus the classification of S and p can be

reduced t o the c a s e of simple algebras.

and p the non-trivial

Then p is a positive involution of K. and consider a triple

by a polarized abelian variety

(A, c , 8 ) formed

(A, c) and an isomorphism 8 of K

into End

(A) such that the map 8 (a) H 8 (aP) i s 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 m a y be l a r g e r than 8 (K).) We a s s u m e a l s o that 8 (1) i s the identity of End (A). Take cn/L and W a s 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

If S i s an a l g e b r a over Q with a positive involution p, c a n extend p t o a positive involution of S

K

we

8 *R . In particular ,

n = gm.

(6.4)

consider the c a s e where S i s a n algebraic number field, and use the l e t t e r K instead of S.

Put

Now r e s t r i c t the complex representation of End we obtain a representation In this situation, we s a y that

Then [K : F] = 1 o r 2.

By the g e n e r a l principle we just mentioned,

p i s extended t o a positive involution of the tensor product K BQR

which i s a d i r e c t s u m of copies of R o r C.

F r o m this fact i t i s e a s y

a field,

9

into C.

By our choice of K,

K into C ,

Q

(A) to 8 (K).

Then

of K by complex m a t r i c e s of s i z e n. (A, c , 8 )

of type

(K,

9).Since

i s equivalent to the d i r e c t sum of n i s o m o r p h i s m s of

K is K

there a r e exactly 2g isomorphisms of

which can be written a s

with a suitable choice of g isomorphisms rl, Let r

5

and s V be the multiplicity of

.. .,

7

g and p r V in

among them.

9

, re-

s

71

7

,. . . ,

s

.

6t of

Take any f r e e 2-submodule

K of rank 2g.

Put

spectively, o r symbolically, put

cg ,

It can easily be shown that L is a lattice in Note that a P u i s the complex conjugate of a

u

for e v e r y a r K and

a complex torus.

Take a n element

s o that c g / L is

5 of K s o that

e v e r y i s o m o r p h i s m a of K into C. F r o m the above l e m m a it follows

+

+

that ~ t = ~ s(Vr)(rV ~ p7 ) i s equivalent to a rational representation of K.

T h e r e f o r e we have Define a n R-valued alternating f o r m

i s of degree n and n = m g ,

Since

rv

+ s V= m

(v = I ,

..., g ) .

i s a positive integer.

6 In p a r t i c u l a r , if m = 1 (and hence n = g ) , either r i s 0.

or s

Exchanging r V and pry if n e c e s s a r y , we may a s s u m e that

- z;=~T~.

A.

,. . . , a

7~

(K, $1,

every s

E

K,

with

3

From

6

-

1'""

7g

, the existence of (A, c , 8 ) of

c ~ , ,~ can T be ~

shown a s follows.

l e t u ( s ) denote the element of

cg with

For

components

this way.

and t

cg/ L

hence

i s isomorphic

we obtain a polarization c of

the diagonal m a t r i x with diagonal elements

7

defines an element of Endo(A), aOt

C a , the

Thus we obtain

If 8t

m a t r i x sends L into L, (A, c , 8 ) of type

9)

(K,

i s a fractional ideal in K ,

ring of algebraic integers in K ,

which we write

B(a).

.-

one can prove that any (A, c , 8 ) of type 7

(resp. y).

F o r a suitable choice of t , we s e e e a s i l y that

F o r every a r K ,

8 ( a ) r End(A).

F o r a given K and type

component of x

becomes a Riemann f o r m on c g / L,

In p a r t i c u l a r , if

ip

i s the f h

t o an abelian variety A.

a

i.e.,

cg by

we have where x v ( r e s p . y

(6.7)

E ( x , y) on

and

C

(K,

hence

3).

Actually

i s constructed in

0

denotes the

then 8 ( 0 ) End(A). If n = 1, our

(A, c , 8 ) i s nothing but an elliptic c u r v e isomorphic to C/Bt

7.

The class-field-theoretical characterization

(provided that r1 i s the identity map of K). Now taking a period m a t r i x f o r A, a s in $5. 8(

H e r e we a s s u m e that

U ) C End(A).

K(y

(2))

point

y)

L e t (V,

of K' ( y ( z ) )

we obtain a point z of

(A, c ) i s such that P = J

n b e a couple a s in T h e o r e m 5.

$n

, and Let

b e the field g e n e r a t e d over K by the coordinates of the

y ( z ) . One m a y naturally a s k a question:

-

Is K (

40 (z))

L e t us f i r s t r e c a l l the fundamental t h e o r e m s of c l a s s field theory.

out of mode, since such will be m o s t convenient to d e s c r i b e the field K1 ( y ( z ) ) .

the m a x i m a l unramified abelian e x t e n s i o n f K? But if n > 1,

This i s s o if n = 1, a s a s s e r t e d by T h e o r e m 2. this i s not n e c e s s a r i l y t r u e . abelian extension of K,

On this topic, I s h a l l give a n exposition which i s somewhat

To c o n s t r u c t the maximal unramified

Z

L e t F be a n algebraic number field of finite d e g r e e , integral ideal in F , p r i m e s of F.

an

and .jL a (formal) product of r e a l archimedean

F o r an element a of F ,

we write

we s h a l l l a t e r d i s c u s s a function which i s

.

rather different from

(P

However, even though

is not a

function with the expected p r o p e r t y , y(z) h a s s t i l l an interesting number t h e o r e t i c a l p r o p e r t y , which i s roughly described a s follows:

-

T h e o r e m 6.

a

"

extension

for a

of

K1

.

L e t K' a c K.

be the field g e n e r a t e d

T h e n K1 (

(2))

&

different.

$

a = b / c , b =_ c =_ 1 mod

5

, and b , c a r e positive for e v e r y

archimedean p r i m e involved in

is a l s o a totally imaginary quadratic

E v e n the d e g r e e s of K and Kt

K f K'

over Q m a y be

ramified abelian extension of K1 . Then how big i s Kt ('-f'(z))?

1 mod*

T , and by P ( F , tg)

consisting of a l l principal ideals

We

ideal

Z U.

[g

in F unramified in M,

, M/ F ] i s meaningful.

M over F. s o that

(a) such

V

L e t M b e a finite abelian extension of F. can

The field K1 ( y ( z ) ) i s not n e c e s s a r i l y the maximal un-

s h a l l a n s w e r this question in the following section.

that a

Obviously

However, both c a s e s K = K1 and

&. We denote b y I ( F , Z ) the

group of a l l fractional ideals in F p r i m e to the subgroup of I ( F , 2')

I t c a n be shown that K'

happen if n > 1.

Q

unramified abelian

extension of a totally r e a l a l g e b r a i c number field. K' = K if n = 1.

over

if t h e r e e x i s t two algebraic integers b and c in F such that

Let

For every prime

the Frobenius automorphism

.?

Then we can define [BL

be the relative discriminant of

,

M/ F] for every

eL

e I(F,

3)

again by r v

.

Hrv , and

Put T =

H' = { y c G Then f r o m our definition of K'

We have now

Theorem 7.

The m a p (7.1) $ s u r j e c t i v e , and i t s k e r n e l con-

tains P ( F , 36) f o r -

some

G.

T h e r e f o r e , if Y i s the k e r n e l , I(F,

J ) /Y.

f c Y.

T h e o r e m 8.

fok s o m e

where

/Q

&,

The converse of Theorem 7 is given by Counting the number of e l e m e n t s , we s e e that [K' : Q] = 2h.

t h e r e e x i s t s a unique abelian extension M

of

M

C I(F,

&).

i s the r e l a t i v e d i s c r i m i n a n t

One c a n actually show that Y t

over

rl,.

. . , 7g

and K'

i s a n element (resp. ideal) in K .

the Galois group of S over Q.

More-

in K ' ,

This follows e a s i l y f r o m (7. 2).

Now l e t I' be the group of a l l ideals

/e

in K 1 such that

F. with an element f3 of K.

We c a l l this M

of $ 6 , l e t us take the

s m a l l e s t Galois extension S of Q containing K,

(resp. ideal ? )

of

F corresponding to Y t .

Coming back to K ,

ponding t o K.

o v e r , f o r e v e r y element a

n I ( F , $3)i s the k e r n e l of the m a p

the c l a s s field over ----

.

c a n find elements ol , . . , oh of G s o that

in F i s fully decomposed

For e v e r y g r o u p Y ' of ideals in F such that

7 a s

F such that Y t

(see Th. 6 ) , we observe that K t i s the subfield of S corresponding to H I . Since H t T -1 = T -1 , we

G(M/ F) i s isomorphic to

M o r e o v e r , a p r i m e ideal J

in M if and only if

Ty = T I .

2)

i s a homomorphism of I ( F , M over F.

I

into the Galois group G(M/ F) of

and denote by G

L e t H be the subgroup of G c o r r e s -

Extend each rV to an element of G,

and denote it

I t can easily b e s e e n that I t contains

P ( K t , (1)). Now Th. 6 i s refined in the following way: Theorem 6 ' .

Kt

correspondingt_o I t .

( 9 (2))

-

----

i s exactly the c l a s s field over K 1

F u r t h e r m o r e , we have an analogue of the relation (3.6). T o d e s c r i b e i t , l e t us denote by A ( & ) the abelian v a r i e t y isomorphic

to

cg/L

with L defined by (6.8) for a n ideal

in K.

T o prove (7.3), we have to introduce the notion of reduction of

Take a

a n algebraic v a r i e t y modulo a p r i m e ideal.

field k of definition f o r A ( a ) containing K' ( y ( z ) ) . L e t u b e an isomorphism of k into C such that u = K' (

(z)) f o r a p r i m e ideal

(7.3)

A ( a )a

8

in K'

5 isomorphic fo

.

[T

, K1 ( y ( z ) ) /K t ] on

b e a p r i m e ideal in k ,

Then we have

uA

%

A(O(,~-'), where

=

We consider the s e t

.

V,

F i r s t l e t us d e r i v e Th. 6' f r o m (7.3).

f c that

Let A = A ( 6 t ) and k be a s

an i s o m o r p h i s m of k into C.

;P -

of a l l homogeneous polynomials vanishing on

7 -integers.

F o r each

f r

,f ,

we

, which is a polynomial with coefficients in k ( P ). , denoted by V[ 1, to

To simplify the m a t t e r ,

8.

If V i s an abelian variety defined over k,

V[

i s a n abelian v a r i e t y defined over k( -

7]

finite number of

P.

F o r such a

P

f? f o r all then one c a n show

) for all except a

, reduction mod

of

l e t us a s s u m e that (iii) of Th. 5 i s t r u e f o r the p r e s e n t A even if we

e v e r y element of End(V) i s well defined, and gives an element of

d i s r e g a r d the polarization; namely a s s u m e

End (V[FI). We apply these f a c t s to the above A ( m ).

(7.4) A

isomorphic

to

A

7

.

~f and only if

y(z) =

~(2)'

(This is t r u e if g = 1, but not n e c e s s a r i l y s o if

.

g > 1. ) Now we observe

that A((%) and A ( c ) a r e isomorphic if and only if

that a p r i m e ideal if and only if

fl:=l

t

be-

By the principle (6. 2), we can find n l i n e a r l y independent l i n e a r invariant differential

forms

w,

... , wg

on A,

rational over k ,

s o that

) if and only if

b

( a € 0 ; v = I , . . . , g).

is a

Combining this fact with (7.4), we conclude in K '

decomposes completely in K' ( ( p ( z ) )

ih

i s a principal ideal in K.

the d e s i r e d r e s u l t , but not quite. about N ( -$?),

and

We a s s u m e

.

T h e r e f o r e , the notation being a s in (7. 3 ) ,

we s e e that A(UL )* i s isomorphic t o A ( principal ideal in K.

8t

It i s not difficult to

obtain A ( 8t ) defined over a n algebraic number field k. that k contains K' ('fJ (2))and i s Galois over K'

long to the s a m e ideal c l a s s .

This i s a l m o s t

We could not obtain the condition

since we d i s r e g a r d e d the polarization.

of polarization leads to Th. 6'

7

be the s e t of a l l common z e r o s of the polynomials f mod

L e t us b r i e f l y indicate how Th. 6 and (7. 3) c a n b e proved.

T

f

) the r e s i d u e field modulo

Let

Then we define the reduction of V modulo

by m e a n s

of the points of finite o r d e r on A.

above, and

and k(

whose coefficients a r e

consider f mod F u r t h e r we c a n obtain r a m i f i e d abelian extensions of K'

L e t V be a variety in

a projective s p a c e , defined over a n algebraic number field k.

.

A c a r e f u l analysis

L e t u s a s s u m e , f o r the s a k e of simplicity, that K i s n o r m a l over

Q, K = K' , and the c l a s s number of K is one, though Th. 6 ' comes somewhat t r i v i a l under the l a s t condition.

be-

By (7. 2 ) , we can

put a

=

A

-

1 A '

b

and l e t

2

7

=

Let

2

m:=l

u

be a p r i m e ideal in K of absolute degree one,

7A .

Take a p r i m e ideal

.

, and consider reduction modulo

b y putting tildes.

f?

in k which divides

Then it i s not difficult t o lift the isomorphism to that of A ( m ) O to ~ ( b - l m ) ,hence (7.3).

Indicate the reduced objects

F r o m (7.5) we obtain 8.

if

%

= (b) with a n integer b in K.

over k.

.-

As I mentioned in §3, there a r e s o m e other ways of generalizing

Let x be a generic point of A

Then the relation (7.6) shows that every derivation of %(x)

A further method of constructing c l a s s fields

Theorem 2.

F o r example this can be done by considering special

values of automorphic functions with r e s p e c t to a discrete subgroup of

-CV

vanishes on k(O (b)x), hence

S L (R) obtained f r o m a quaternion algebra. 2 A quaternion algebra over a field F i s , by definition, an algebra

F i s isomorphic to M2 @) , where F F denotes the algebraic closure of F. F o r our purpose we take F to B over F such that B 8

where p i s the rational p r i m e divisible by

2.

Since

be a totally r e a l algebraic number field of finite degree.

Then one

can prove that

we obtain

where D i s the division ring of r e a l Hamilton quaternions, On the other hand, if over K ,

then

AO

mod

p

(7. 7) shows that, if A = A (

7 8

A((~L)O

mod f'2

2

a i s the-Frobenius substitution for can be identified with

xp .

a), isomorphic f_o A(b

P

Therefore

[ F : Q], r > 0,

and r

is an integer such that

and r e g a r d B a s a subset of BR

0 <_ r <_ g.

.

- How many such B do

t h e r e e x i s t ? I shall answer this question afterward.

- 1 OL) mod

7.

g =

We assume that

(or i t s compactification when B = M2(Q)) i s a compact Riemann

a be the projections of a to the 1""' r t f i r s t r f a c t o r s M2(R). W e d e n o t e b y B t h e s e t o f a l l a in B For

a e B,

let a

s u r f a c e , and an automorphic function with r e s p e c t to

..

such that d e t ( a ) > 0 f o r v = 1 , . , r . Then e v e r y element a of V t B a c t s on the product -$r of r copies of the upper half plane on

-$

m e r o m o r p h i c function on

V

by NB, a l l a such that N ( a ) i s totally positive. B/ F O b s e r v e that B i s of dimension 4g over Q. B,

we understand a subring

f r e e Z-module of r a n k 4g.

O

of B,

$,

m a l o r d e r s in B.

r ( t)

In S2 we considered only d i s c r e t e subgroups of SL (R). 2 The action of a n element a of r ( ) on i s that of al, i. e . ,

5

the projection of a to the f i r s t factor M2(R) of BR

By a n o r d e r in

containing Z,

(satisfying

em ark.

.

The element

a l m a y not be contained in S L (R). But this does not produce any 2 difficulty, since we only have to consider

which i s a

An o r d e r i s called maximal, if i t i s not

contained p r o p e r l y in another o r d e r .

invariant under

a n additional condition when B = M2 (Q)).

being defined a s in §2. If we denote t i s the s e t of ( a ) the reduced n o r m of a to F, then B

the action of e a c h a

8

( 2. ) i s a

T h e r e a r e infinitely many m a x i -

We f i x a m a x i m a l o r d e r

0

in B ,

and put in place of

r ( Z ).

The group of the above type was f i r s t introduced by P o i n c a r 6 F u r t h e r , f o r e v e r y integral ideal

about 80 y e a r s ago in the c a s e F = Q,

1: in F , put

the g e n e r a l c a s e .

and l a t e r F r i c k e considered

They d i s c u s s e d t e r n a r y quadratic f o r m s instead of

quaternion algebras.

Then

r

show t h a t

t

and r ( T ),

a s subgroups of B , a c t on

/r(

) i s compact if and only if B

5' .

We have to define " s p e c i a l pointsn on

r

= S L 2 ( Z ) , and

Now l e t .us a s s u m e r = 1.

Then

has no z e r o -

em ma. - Let

%i/ sI not ? compact.

the quotient

$ . / I ?(

M

which i s isomorphic t o ? --

$/r (Z) i s compact unless

B = M2 (Q). T h e r e f o r e , a s was d i s c u s s e d in S 2 ,

relative to

,

analogous t o w / W of Th. 2, where we s h a l l examine the values of 1 2 automorphic functions. F o r this purpose, we notice:

o n e can

divisor other than 0. F o r example, if B = M2(Q), we can s e t

(?' = M 2 ( Z ) , hence

%

)

&a

totally imaginary quadratic

quadratic subfield of B

following a s s e r t i o n s hold.

over

-

extension of F

F.

Then the

If

(l) f i s _s F - l i n e a r isomorphism

f (M)

-

(0)

contained

2

o_f M into B ,

t

and z correspond to j and w / w of Th. 2. We 1 2 c a l l such a couple (V, y ) a canonical model f o r $ I T ( 7 ). If Thus

then

B , and e v e r y element of f(M)

-F

has exactly one fixed point on # which i s common to a l l elements of f(M) - F. (2) If 'ICM denotes the r i n g o f i n t e g e r s M, then t h e r e exists a n F - l i n e a r i s o m o r p h i s m f o_f M into B such that f ( % M) C --

point"

.

One can of c o u r s e a s k a question:

embedded in B ?

0

a s our

special

What kind of M can be

Leaving this question a s i d e for a while, we a r e now

.

non-singular -

y)

f o r m e d by & p r o j e c t i v e

c u r v e V and a holomorphic mapping

o_f

g i v e s a b i r e g u l a r isomorphism

(

9

s u r j e c t i v e u n l e s s B = M (Q).) 2 (ii) V i s defined over the c l a s s field k

to P(F, -

-

of

of F.

Z C ~ )w, h e r e

a0 " is

$/r (T

2

sertions:

$. into V ) into

V.

and

the product of &l archimedean p r i m e s

f be a s in (2) of the above L e m m a , a&

.

fixed point of t h e e l e m e n t s of f (M) - F of k ( and M i s exactly the c l a s s field over to P ( K Z ). -

f

(2))

T h e n t&

Z ).

isomorphism

.

0

In this s e n s e ,

(V,

(i7 )

It may be worth noting that the

z the

composite

M corresponding

B

8

&

Put B

2

= B @ F

(8. 2)

PB

(8.3)

F o r any finite s e t P with a n even number of archimedean

--

over

F,

(8.4) in -

.

Let PB b e t h e s e t o f a l l ? F ? ' i s a division algebra. Then we have the following a s -

of F.

a finite s e t with an even number of p r i m e s .

o r non-archimedean p r i m e s of F ,

F corresponding

(For notation, s e e § 7 . )

(iii) L e t M

$/I?(

6

T o answer the questions about B and M, denote the let Fk completion of F with r e s p e c t t o a n archimedean or a non-archimedean

B

over

of V onto

=

functions.

with the following p r o p e r t i e s : (i)

5

such that

m a x i m a l abelian extension of M can thus b e generated, over the maxi-

such that (V,

$ 1 (~7 ),

i s uniquely determined f o r

prime

T h e r e e x i s t s a couple

) a r e two canonical models f o r the s a m e

m a l abelian extension of F , by special values of some specific automorphic

ready t o s t a t e the main r e s u l t : T h e o r e m 9.

(PI

then we can show the existence of a biregular

a somewhat deep f a c t of a r i t h m e t i c of quaternion algebras.

-F

.

Vf , rational over k,

The f i r s t a s s e r t i o n i s quite e a s y to prove, but the second needs

We a r e going to take the fixed point of f (M)

'p ) and (V'

(V,

there ---

e x i s t s a quaternion a l g e b r a

--

unique up to F-linear i s o m o r p h i s m , such that P = P

B ' F can be F - l i n e a r l y embedded

_A quadratic extension M f

- ---

B if and only if M @ FF2 i s a field f o r e v e r y P P J 3 . These r e s u l t s a r e special c a s e s of H a s s e s f s t h e o r e m s on s i m p l e

a l g e b r a s over algebraic number fields.

Observe that g

-r

f a c t o r s of

(8.1) correspond to the archimedean p r i m e s of P

B ' The r e c i p r o c i t y law for the extension M. k( Cp (2)) over M c a n

b e described explicitly in t e r m s of the special points

(z). F o r

simplicity, l e t u s consider only the c a s e where the c l a s s number of

F in the n a r r o w s e n s e i s one, i. e . ,

p r i m e ideal

J2

let

.

= % 0

P(F,

Go) = I @ ,

9

F o r every

(1)).

(Such a n a

~ not unique. )

With such an element a , one has

? - i n t e g e r s in F

be the ring of

I '

3

Then

unitof

8

of U(

) consisting of a l l a

dividing to

16

i s a m a x i m a l o r d e r in B L e t J B 2 t 8' ) denote the g r o u p of a l l e l e m e n t s a of B such that a is a

and l e t U(

in F,

forall

8

r.

dividing

.Z

,

F o r every

b e the subgroup

ru

s u c h that a Z% I mod

I t c a n e a s i l y be shown that U(

( 0 l f C 7 ) ~ (see Notation).

t)

and l e t U (

3

t )/ U ( T )

a c U( t

y,

for all

H e r e we s a y that f

i s normalized if

2

i s isomorphic

(0 f a r M).

put If we define

-

or f

?

by ?(a) = f

6) f o r

a E M,

then we s e e that e i t h e r f

i s normalized. It should be observed that (iv) of Th. 10 i s a generalization of

Now we have

(3. 7 ) .

T h e o r e m 10.

---

T h e r e e x i s t s a s y s t e m of b i r e g u l a r i s o m o r p h i s m s

- v U b ),

R ( a ) of V t o 7

k,

defined f o r each a r U ( f )

and rational over

with the following p r o p e r t i e s :

(iii) R ( y )[

if

Example.

a - l p r Uo(E).

e2ni1

(z)] = 'p(y (2)) f o r e v e r y z r z be a s in (iii)

(iv) L e t M, f ,

$. if

of Th.

y r

Let 3

b e a n ideal in M p r i m e to

& the

Let us consider the c a s e F = Q(3

p r i m e s of F ,

sense

f o r which P

with

5

B c o n s i s t s of a l l but one archimedean

b e r one in the n a r r o w sense.

defined by the elements of B+ element a

fU ( Z

) such that f ( 8 ) O = a g .

.

-

The p r e s e n t F has the c l a s s num-

Moreover, a l l the maximal o r d e r s in

B a r e conjugate to each other under the inner automorphisms of

Then t h e r e e x i s t s an ----

=

the exception being the archimedean p r i m e of F c o r r e

sponding to the identity m a p of F.

Z. , and l e t

+ 5 -1)

f o r d = 7, 9, o r 11. By ( 8 . 3 ) . t h e r e e x i s t s a unique quaternion

a l g e b r a B over F ,

9 (still under the

condition f ( WM) C 0 ) . Suppose t h a t f & n o r m a l i z e d defined below. --

ro).

(9

positive integer N, N Q(i, f ) considered in 93. T h e r e f o r e the f i r s t t h r e e p r o p e r t i e s of ab R ( a ) in Th. 10 m a y be r e g a r d e d a s a generalization of Th. 4.

(i) R ( ~ ) o @ ) R(p) = R ( a p ) . (ii) R ( a ) = R ( p )

= M 2 ( Z ) , and r = (N)with a the function field of V i s exactly the field

F u r t h e r , if B = M2(Q),

.

Take a maximal o r d e r

73

B

in B ,

and

define I? = r(l) a s above. genus 0,

and

r

$1

Then one can prove that

extension K of F and consider

i s of

(A, c , 8 ) of type

3

s e n s e of 56 with a representation

modulo i t s center i s generated by t h r e e elements

(K,

9) in the

of K such that

y 2 y 3 y d = 1. which is the s m a l l e s t the m e a s u r e given by (3.4) is 11 42,

y 2 , y 3 , y d of o r d e r 2, 3, d, (If d = 7 ,

respectively, satisfying

value of (3.4) for a l l possible l?. ) These t h r e e elements have unique fixed points on

$,

one for e a c h , which we denote by w2. w3

Then t h e r e e x i s t s a unique m e r o m o r p h i c function

on

that C ( y ) is the field of a l l automorphic functions on r e s p e c t to J?, and

(i) (w2) = 1,

f' (w3) = 0,

by V the complex projective line, then

$/r

phism of

onto V.

canonical model for quadratic

(wd) =

.

UJ

.

.

such with

If we denote

gives a biregular i s o m o r -

Now we can prove that this

#/l7

5

, wd

$.

(V,

p)

$32

By (8.4), for e v e r y totally imaginary

extension M of F ,

t h e r e e x i s t s a F - l i n e a r isomorphism

f of M into B such that f ( % M)

C c.

L e t {zl,

. .. , z 9 }

be a s e t

of r e p r e s e n t a t i v e s for the r-equivalence c l a s s e s of the fixed points of f(M)

-F

f o r a l l such f.

Then q i s exactly the c l a s s number of M,

and f r o m (iii) of Th. 9 and (iv) of Th. 10, we obtain:

(8.5) The values conjugatesf

?(al),

?(z 1) over F,

...,

--

?(Z ) f o r m a complete s e t of 4 and for each i , M( y ( z i ) ) i s the maxi-

m a 1 unramified abelian extension

of

M.

where T,, and P a r e a s in 56.

Then i t can be shown that the

of this type a r e parameterized by the point on continuous group I?

acting on

$r

gr,

and t h e r e is a disis inone-to-one

$r/l"

such that

correspondence with a l l the isomorphism c l a s s e s of such Taking K suitably, we can identify J?'

r = 1). F o r this family of (A, c , 8 ) ,

#/rf with the properties

of

Q

of

(A, c. 8 ) .

with the above I? (in the case we can find a model (V' , )

analogous to those of Th. 5.

z a r e a s in (iii) of Th. 9, the corresponding End

(A, c , 8 )

If M and

(A, c , 8 ) i s such that

(A) contains a n isomorphic image of K 8 M.

F

The coordinates

(z) generate a n abelian extension of the nature described in Th. 6 ' .

This couple

(V1,

) is the f i r s t approximation to the desired

F r o m infinitely many such (V'

,

c a n construct a canonical model

), (V,

(V,

yl).

depending on the choice of K ,

y).

If B = M (Q), t h e r e i s a family of elliptic c u r v e s , for which the 2 value of j is the modulus. F o r the basic field F of higher d e g r e e , t h e r e is no such standard family of a b e l i a n v a r i e t i e s , though infinitely

Thus

has a s t r o n g resemblance t o the c l a s s i c a l modular

function j.

given

Unfortunately, the proofs of Theorems 9 and 10 a r e long and v e r y complicated.

many families of abelian v a r i e t i e s can be loosely associated with a

T h e r e f o r e I have to content myself with a rough

sketch of the m a i n ideas.

We take a totally imaginary quadratic

$ 1

.

It i s a n open question whether there exists any family

of geometric s t r u c t u r e s , other than the above canonical model (V,

y)

(A, c , 8 ), of which our i s a natural variety of moduli.

we

one, L e t V be a projective non-singular curve of genus h defined over an algebraic number field k.

F o r every p r i m e ideal J2'

2 .

l e t k ( 2 ) denote the residue field modulo equations for V modulo

&

almost all

J2

.

in k ,

Considering the

, we o b t a i n a c u r v e V [ p ] over k ( & '

(see §7). It can be shown that V[

2

(We c a l l such

T

where

9. T h e H a s s e zeta function of a n algebraic curve

] i s non-singular, and of genus h for

i. e.

=

0)

and the c l a s s number of F in the n a r r o w s e n s e i s

, every

ideal in F i s a principal ideal generated by a

totally positive element. there exists a n element

In this c a s e , f o r e v e r y right a in B'

8t ,

0 -ideal

.

such that

= a 8. Let us introduce cusp f o r m s and Hecke operators with r e s p e c t b d) c GL2 (R) with det (f) > 0, t o the group I? = r p ) . F o r e v e r y f =

6

Put

good. ) Therefore one can speak

of the z e t a function of V[f ] over k ( p ) which i s of the f o r m

Z

B

1 (u) = Z (u)/ [(I

8

- u)O - N ( ~ ) U ) ] ,

L e t m be a positive integer.

1 where u i s a n indeterminate, and Z (u) i s a polynomial of degree 2h.

Y

The H a s s e zeta function of V over k, plex v a r i a b l e s ,

denoted by Z ( s ; V/ k ) with com-

to r, we understand a holomorphic function -

f (2) on

$,

respect

satisfying

the following two conditions: (i) f ( y ( z ) ) j ( y , z)m = f ( z ) f o r a l l y c (ii)

i s now defined by

By a cusp f o r m of weight m w & h

f (P (z))j(P,

s

a cusp of I-', a n d p ,

4 holomorphic &

r.

q a r e a s in $ 2 ,

q = 2niz,

then

and vanishes a t ---

The l a t t e r condition is n e c e s s a r y only when B = M

2

q = 0.

(a), since

I? h a s no cusps otherwise. All such functions f o r m a vector s p a c e of the product being taken over a l l "good" sider also "badf1

&,

8.

Qt i s important to con-

which we s h a l l not discuss h e r e . )

finite dimension over C,

which we write S (r). If m = 2, m

the map

Now one

c a n m a k e the following Conjecture.

Z (s; V / k ) can be continued to the whole s -plane

and s a t i s f i e s 5 functional equation. The purpose of this section i s t o verify this conjecture for the curve which i s a canonical model for

f /r

(

7: ) in the s e n s e of

§8. F o r the sake of simplicity, we shall consider h e r e only the c a s e

gives an isomorphism of S ( r ) onto the space of differential f o r m s of 2 the f i r s t kind on $D, hence the dimension of S (r) i s equal t o 2 the genus of f/r.

F o r every a

t

E

B , we note that the double coset F a r can

(see (8.2-4)), and the second over the remaining p r i m e s in F.

(disjoint).

r a r = urZlair

Then for f c S (I?), we define m

More-

D (s) can be holomorphically continued to the whole s -plane, m and s a t i s f i e s a functional equation: over,

be decomposed into a finite number of one sided cosets:

( T a r ) f by m

-1 -1 (raTlmf = z d 1 f ( a i (z))j(ai ,

m 2)

.

( r a r ) m on Sm ( r ) , which i s of c o u r s e independent of the choice of the representatives In this way we obtain a l i n e a r transformation

a F o r e v e r y integral ideal 8t in F , i ' of a l l distinct ( r a r ) , such that a E 0 Then we define a Dirichlet s e r i e s D m endomorphisms of S (I?), by m

l e t T(BL),

denote the

.

(N ( a ) ) = 61. B/ F ( s ) , whose coefficients a r e linear

and

where

P;

few

stand for the usual gamma function.

means the s e t of non-archimedean

Let (V, y )

f o r a l m o s t a l l p r i m e ideals --over the residue field mod (9t

r u n s over a l l integral ideals in F.

in P

B ' The last

Now we have Theorem 11.

where

2

&a

3 &

2

,

canonical model for

f/r

.

the z e t a function o_f V z

F, coincides with

TZ, d

2

-

It can be shown that

D (s) converges for sufficiently l a r g e Re (s) and has a n Euler product: m

det[l Z

1

-

T(7I2u

+

2 N(J?)u I/ [(I

- u)(l - N(Z)u)]

(u) is the

B

,.,

(i. e . ,

--

Euler

2 -factor

-

of Dm (s) & y m = 2 and N(?)-' = u). Thus Z (s; V/ F) coincides & y det(D (s)) u p to a finite number 2 -factors.

2

where the f i r s t product is taken over a l l the p r i m e ideals

2

in

P

B

a

E

B

The proof of this theorem i s roughly a s follows. t define a s u b s e t X ( I ? a r ) of V X V by

.

For every

of

,

Therefore the equality

We s e e t h a t X(l?al?) is a n algebraic correspondence, which depends o n l y on F a r and not on the choice of a . = (rar),

with any e l e m e n t a

of

We o b s e r v e that T (

0 n Bt

7 I2

f = (NBI

such that

(a)).

should hold..

F o r such a n a , we w r i t e X ( I ' a r ) a s X L e t M, f , T a k e M s o that in M. Then

=

2

2

= NMl

(

7

= (NB1

Therefore

[q , M ( l (z))/ MI.

P

of

60Bt

(PI), hence

7

and

s o that f (

.

7'

=PW

(rPrI2= T (f) 2

~ ( ~ - ' ( z )X ) y ( z ) c X?

.

by the

Now l e t

~ ( 2 ) '= ~ ( ~ - ' ( z ) ) ,

By (iv) of Th. 10, we have

h e n c e , putting y = y ( z ) , we obtain

Then

and (iv) of Th. 10.

decomposes into two p r i m e ideals

We can find a n e l e m e n t

above r e m a r k . T

and z b e a s in (iii)

L e t A denote the diagonal of V X V.

of Th. 9 2

The relations (9.1) and (9. 2) show that any s y m m e t r i c function of and

tm

modulo

can be obtained f r o m a correspondence of V by reduction

8.

In p a r t i c u l a r , f o r e v e r y positive integer m ,

we have

Consider now

yT X y c X

2 .

r e d u c t i o n modulo a p r i m e divisor of a suitably l a r g e field, which h 7 divides , and denote the reduced object by tilde. Then y = y

2

f o r a polynomial

hence

P

m

--dud

, which i s determined by log (1

- XU + yu 2) = * x

Pm(x, y)um-l

m =I Let

rr

'V

rV

denote t h e Frobenius correspondence on V X V , i. e . , N

the l o c u s of x X x N ( f ) f o r x c V.

The above discussion shows that

with indeterminates x, y,

L e t I[Y] denote the number of fixed points of a correspondence

hl

h a s infinitely m a n y points in common with t m , t Since i s i r r e d u c i b l e , we have

X, of

trr

.

definition of X N

X

Y

2 '

.

+

.(xXV) N

the t r a n s p o s e

mcT8 .

F r o m our

Y,

i. e . , the intersection number of Y with

the zeta function of V mod i t can e a s i l y be s e e n that t~

P 8

Again f r o m the definition of X

= X

.

8 '

Therefore

we s e e that N

c o n s i s t s 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.

and u.

2

, then

A

.

If Z

2

(u) denotes

hl

But if Y i s a correspondence on V X V,

10.

we have I[Y] = 1[Y], and

Infinite Galois extensions with l -adic representations.

by the Lefschetz fixed point t h e o r e m , So f a r we have been interested only in the construction of abelian extensions.

$/TIf') can be employed to obtain m e a n i n g f d non-abelian ex-

for i where H (V) denotes the usual r e a l i-th cohomology group of V. Y =Xrar),

Now we a r e going to show that the above canonical model

tensions of a number field, with some pleasant features. For L e t us call (V,

it i s e a s y to s e e that

f', R ( a ) ) of Th. 10 a canonical s y s t e m of level

F o r every integral ideal tr(Y

1

&

1 H (V)) = 2 . ~ e [ t r( l ? a r ) 2 ] ,

,

and denote i t by (Va

In view of (9. 3), we obtain

hence our theorem.

(det[l du log

- T ( & )2u t N ( j )u2]-'}

t,

cussion c a n actually be done without this assumption. ) Let k a the c l a s s f i e l d o v e r F corresponding to P ( F .

,

a&), where

denote 4x0 is

the product of all archimedean p r i m e s of F (see $7). As i s stated in Th. 9, 10, Va

and R a ( a ) a r e defined over k a

c a n obtain a rational m a p

defined over kw

2

,

RoL( a ) ) . (We a r e still assuming that the c l a s s number of F i s one in the n a r r o w s e n s e , though our dis-

since S2 (I?) is isomorphic to the space of differential f o r m s of the f i r s t kind on V = $ 1 ~ . Also i t is obvious that 2 t r ( Y ( HOW)) = tr(Y ( H (V)) = the number of right c o s e t s in r a r . One can f u r t h e r show that t r (l?al?)2 is r e a l . Therefore we have

1 d du log 1 (u) =

.t.

8L in F, we fix a canonical s y s t e m of level

,

such that

.

If

eL C

d

, we

When

6L =

level

a.

ing.

b

on

, this m e a n s the uniqueness of canonical s y s t e m of

In the g e n e r a l c a s e , the m a p (10.1) defines a Galois c o v e r -

If we consider the c u r v e s over the u n i v e r s a l domain C ,

r ( $i )/r( cz )E% ,

the Galois group i s isomorphic t o

denotes the group of a l l units e of F such that e

4

then

.

If

Let y1 be an a r b i t r a r y point of Vm

y =

a&

m a y b e generic o r algebraic. ) Then k o, (y' ) depends

T b , &(yt ).

(y'

only on y,

and i s a finite Galois extension o_f k a. (y). F o r e v e r y

7 c

G(k.,

(Y)),

(yl ) / k

such that y1 --

7

-

r Z=

{y

&

= Rot,(P)(yl),

Proposition 2. let

t h e r e e x i s t s a n element T

Take a point z

I- ( 2)1 y (z) = z ) .

(i) r ( e e ) . r z = { Y c r ( b )

I

If

-

7 H

y is g e n e r i c on Vg surjective, a n d rZ= E b

that T1;

:

.

Take a point ya

P

.

s o that

L(z) = y' , and

G ( z ) . ) Let

a.

&

F o r every p r i m e ideal X

groups

0

one

ology.

~e't

( u l ) with

0;

such that u

F o r simplicity, we a s s u m e {Y r

(10.2)

1,

r(Z)1

$

, and Oy

BI

lt

consisting of the elements

= 1 mod f D for every

I-

v(z) = z ) = E

of the

with the usual product top-

atdenote the subgroup of a

ut c

. f.

F o r m the product

t

.

t

such that

c ( z ) = y. This i s satisfied f o r all except a finite number of points on Vr Under the assumption, we

.

a r e going to define an injection

-

' this isomorphism

i s

in F and an a r b i t r a r y for each

in F, l e t F1

(see Notation) f o r a l l

1

f o r a point z on

described in P r o p . 1,

f

be the localizations a s defined in g8.

where

-

E%

i s the closure of EI

C '2,

u ( an) u o ( t ) *

a ( ~) a = y% i f 01 C 1, and yr =

=

y

6t ) for a l l

(ya

over k (y) and the behavior of the Y Frobenius automorphisms (when y i s algebraic) with r e s p e c t t o

F o r every

on Va

andput

Y By Prop. 1, is a Galois extension of k r (y). Our purpose i s to investigate

hold:

(y)) into

k%

Let us now fix an integral ideal 2: point y on Vy

nuo( b )

R,(Y)(Y') = y t ) .

a n i s o m o r p h i s m o_f G(ka(y1 )/ k

(iii)

#

of

U(a,)

Then the following statements

(ii) F r o m the correspondence obtains --

P of

on k,

= o

f'*(z)=y,

c e r t a i n representations of the Galois group.

we take kOL a s the field of definition, we obtain: Proposition 1.

sothat

Y the Galois group of

where E %

=- 1 mod 1

f

denote the composite of the ka

6LC Z

the elements $ so

J . (For example, choose a point Z

OL

in

a.Let

s r G(&

we find, by P r o p . 1, an element '7

that YOL = Ra (ta)(y, s o that

1 k t (y)).

Y

em

=

5 ( a )of

1. It i s not difficult t o choose

f,

F o r each p r i m e ideal to an e l e m e n t u

the sequence { ( ( l n ) } n = l , 2 ,

C;

of

.

converges

...

We define J ( r ) to b e the element of

r e p r e s e n t e d 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

Er

{y }. But it i s unique up to inner automorphisms of OL F u r t h e r we have: Proposition 3.

If

o

~ kr tr

n u m b e r s ; our main i n t e r e s t i s of c o u r s e in this c a s e .

, J

M, f , and z b e a s in (iii) of Th. 9, and

@) p is rational over Q, B

algebraic groups over Q. m (2) p (a) = NF/ (a) independent

Let

XM the ring of integers i n

0. P u t

a s a subgroup of

contained

&

If

%.

y =

Tr(z)

(R

)/ E t

If M is not contained in k group which contains not go into details.

and A -

X

being considered

with an integer

for every a e

m,

a.

F o r e v e r y rational p r i m e I , i n t e g e r s , and l e t B

Then, f r o m (iv) of Th. 10, we

l e t QI

I = B QQQI, AI = A

aI

Let Proposition 4.

of

X

denote the field of

@,aI .

I -adic

F r o m p one can

naturally obtain a representation

obtain

of J -

consider a representation

issur-

F i r s t we con-

s ider the point fixed by an imaginary quadratic subfield of B.

and view

Take any c e n t r a l simple a l g e b r a A over Q , and

satisfying the following two conditions: y i s g e n e r i c on Vr

L e t u s now d i s c u s s the points y whose coordinates a r e algebraic

Then f ( X M ) C

.

y of Vr

.

jective.

M.

Next l e t us consider the c a s e of an a r b i t r a r y algebraic point

rb;?f) fir a s

M Ck

t

factors (y), then the image

r. (y), we have to consider a l a r g e r a subgroup of index 2 , but we shall

=

m,/B,

denote the product of the groups I of I , and Eg the c l o s u r e of Er

.

b to

0 in

f o r a l l prime

d.

Put

Combining J with a n a t u r a l homomorphism of I ET , we obtain a homomorphism

a:/

Now we m a k e the following assumption: e c Ey

(3) p ( e ) - 1 f o r a l l

Suppose this i s t r u e , and l e t f. of p,

.

o

J (cr

1 2

.

Then one

denote the principal polynomial

2 can define a

zeta function

k Yassociated with the representation

p

c(s; k y ; p) of

&

In view of (2), this i s automatically satisfied if either m i s even, a r e a l l totally positive.

o r the units of Er observe that P I

Under the assumption,

i s meaningful, and defines a representation of

J

x

G ( & ~ k/ r (y)) into AlI

.

Now i t i s important t o investigate the b e -

havior of the Frobenius automorphisms of

I -adic representations"

pI

o

JI

.

where the product i s extended over a l l p r i m e ideals

.gywith r e s p e c t to the

p r i m e t o W , and n i s the degree of f

F i r s t we notice:

7

8

in k z ( y )

'

Conjecture 11. 6(s; diy; P ) can be analytically continued f_o the Theorem 12.

There exists a n integral ideal p v ~

k

(y) s u c h

that e v e r y p r i m e ideal of k (y), p r i m e to I M , unramified & Ay which c o r r e s p o n d s t o the kernel of p I J I ' the subfield of ---

whole s-plane and satisfies

F u r t h e r , a s for the nature of

0

T h e r e f o r e , if

i s such a p r i m e ideal of k

divisor of

8,

&

,

dividing Y over k (y) for f2

Y determined only by

J2

.

(y),

5 functional

equation.

&

one may make

Y'

is a prime

is a Frobenius automorphism of and then the conjugacy c l a s s of p

'aT/ tr ,

Conjecture 111. The image of J i s a n open subgroup of unless y = -

yZ(z)

with a point z such that

In this setting, we have the following t h e o r e m ,

which m a y be r e g a r d e d a s the main r e s u l t of this section: T h e o r e m 13.

The roots of the principal polynomial

over QI are algebraic n u m b e r s fabsolute value -

of

p

0

J

I

(U

%

for s o m e totally imaginary quadratic F - l i n e a r isomorphism f

of

M

into

extension M f

F and a n

B.

N( It should be r e m a r k e d that the above representation p i s analo-

This fact leads u s t o a temptation of making the following conjectures.

gous t o the G r k s e n - c h a r a c t e r in Hecke1 s s e n s e , o r m o r e precisely, to the Grijssen-character of type (A ) in the s e n s e of Taniyama-Weil.

Conjecture I.

T h e principal polynomial

rational coefficients, and i s independent of I

of .

p

1

0

J

1

(U

2

)

has

If y =

yr(z) with a point

z fixed by f(M

Conjecture 111, and if M C k

-

(0)) a s excluded in

&

(y), then we can show that c(s; ; p) Y i s a product of s e v e r a l zeta functions of k 7 (y) with Grossen-characters.

One m a y a l s o notice that

b%r/Et.

i s analogous to the idsle c l a s s

group of a number field modulo the connected component.

Further,

if B = M (Q), 0 = M (Z), Z =N Z with a positive integer N, then 2 2 r (7 ) is the principal congruence subgroup of S L2 (Z) of level N, i s a l m o s t equivalent t o the choice and the choice of a point y on Vr of an elliptic curve. to itself.

gYy, p) i s the Hasse

((a,

= GL2(Q)

Thus, if p is the identity mapping of B'

zeta function of the c u r v e s o r r e s -

ponding to y.

A s i m i l a r fact holds a l s o when F = Q and B i s a divi-

sion algebra.

It is a n open question whether such a n interpretation

, P ) in the c a s e where F is of degree Y this connection, i t should be mentioned that the extension exists for c(s,

>1

&Y

.

One can show that a

*

t

B if

a

E

B,

and a H a

I

i s an involution

of B in the s e n s e of 46, which i s not n e c e s s a r i l y 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 a n involution of M (B). This involution c a n be n R-linearly extended to M (B ), where B = B @ R. Define a Lie n R R group G by

*

*

a

In

is

r a t h e r different f r o m the extension obtained f r o m the points of finite o r d e r on a n abelian v a r i e t y , if [ F : Q] > 1.

In view of (8. l ) , we have

11. F u r t h e r generalization and concluding r e m a r k s

We have obtained two different types of r e s u l t s : one i s r e p r e sented by Th. 5. Th. 6, and Th. 6'; the other by Th. 9 and Th. 10. They a r e however two special c a s e s of a m o r e g e n e r a l theorem.

According to this d i r e c t s u m decomposition, a d i r e c t product:

To

s e e t h i s , l e t us introduce discontinuous groups which include Sp(n, Z)

G = G X... 1

and F ( 7 ) a s s p e c i a l c a s e s . L e t F, B and r be a s in S8. Since B algebraic c l o s u r e

-F

BFT = MZ(F)

for the

of F, we c a n r e g a r d the elements of B a s b m a t r i c e s of d e g r e e 2 with e n t r i e s in F . F o r every a = d) c M ~ F ) , put

-

G can be decomposed into

One can easily show that

XGrXGr+lX...

XG

g

.

by a maximal compact subgroup i s a bounded s y m m e t r i c domain.

Then

one can speak of (meromorphic) automorphic functions and f o r m s on where the b a r m e a n s the quaternion conjugate in D. Gr+l,.

.. , Gg

space

fn,

a r e compact.

(p

in B ,

the action of U on the u-th factor U t o Gv. As in 58, take a maximal

If B = M2(Q) and

$8.

r

r

r

i s Sp(n, 2).

h

#nr

considered in

/ T i s compact in the following two c a s e s :

(ii) r = g, n = 1, and B i s a division algebra.

The group

was introduced by Siegel in his paper on symplectic geometry (under

s t r u c t a couple

(V,

y)

kr/r,

we c a n con-

with the properties analogous to those in Th. 9,

thus unifying the above mentioned two types of r e s u l t s .

A p a r t of the

r e s u l t s of $10 can a l s o be extended to such a g e n e r a l c a s e . One c a n investigate automorphic functions with r e s p e c t to a m o r e g e n e r a l type of group. group

Namely one takes a semi-simple algebraic

defined over Q and consider a Lie group

of the points with coefficients in R.

9. T h e r e a r e many interesting arithmetical problems But I should a t least

mention that almost a l l important questions a r e r e l a t e d to automorphic f o r m s and zeta functions explicitly o r implicitly, on which I have talked only in $9.

12. Bibliography

.

gnr

the r e s t r i c t i o n r = 1). Now, for this quotient

formed by integral

If n = I , the

i s a subgroup of finite index of the group

The quotient

(i) r < g;

0 = M2(Z),

acting on

qz

in this field, which I d a r e not enumerate h e r e .

and put

I n this way we obtain a discontinuous group

present

points on

Since Sp(n, R) a c t s on the Siege1

we c a n l e t e v e r y element U of G a c t on the product

, of r copies of n $, being that of the projection of order

this domain with r e s p e c t to the group I? =

Therefore,

gRconsisting

Suppose that the quotient of

Among a v a s t multitude of l i t e r a t u r e , I shall t r y to l i s t standard r e f e r e n c e books f r o m the view point of accessibility and (probable) comprehensibility, along with a few r e c e n t p a p e r s relevant t o the topics discussed in these lectures.

/

The r e a d e r 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 P u r e Mathematics, vol. 9, Amer. Math. Soc. 1966. This contains many interesting surveys of r e c e n t investigations, m o s t of which have abundant references.

F o r those who a r e m o r e

interested in the classical modular functions o r modular f o r m s , many important papers in [2] E. Hecke, Mathematische Werke, Gbttingen, 1959 will s e r v e a s standard references. e a s i e r t r e a t m e n t i s presented in

A m o r e systematic and somewhat

The l a t t e r will s e r v e a l s o a s an introduction to the theory of

[3] M. E i c h l e r , Einfiihrung in die Theorie d e r algebraischen Zahlen und Funktionen, B i r k h l u s e r , 1963 (the English r e v i s e d v e r s i o n

automorphic functions of s e v e r a l variables.

i s available).

r e l a t e d subjects, one can not m i s s

[ll] C. L. Siegel, Gesammelte Abhandlungen, 3 v o l . , Springer,

At t h e end of each chapter of t h i s book, t h e r e a r e plenty of references.

As a textbook on the c l a s s i c a l theory of elliptic functions,

the following m a y b e recommended: [4] C. J o r d a n , C o u r s dl analyse de 1' i c o l e polytitchnique. P a r i s ,

On this topic and other

1966. Especially f o r Siege1 modular functions, the standard knowledge can be obtained f r o m [12] H. Maass, L e c t u r e s on Siegel' s modular functions, Tata

vol. 11, Ch. VII. As f o r the theory of complex multiplication of elliptic functions, I pick h e r e only two, old and (relatively) new: [5] H. Weber, Lehrbuch d e r Algebra 111, 2nd e d . , 1908,

[ 6 ] M. Deuring, Die klassenkijrper d e r komplexen Multiplication. Enzyclopadie d. math. Wiss. neue Aufl. Bd. 12, Heft loII, Stuttgart,

Institute, 1954-55, [13] SBminaire H. C a r t a n , 19571 58, Fonctions automorphes. A detailed account of the r e s u l t s 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 v a r i e t i e s and i t s applications to number theory, Publ. Math.

1958. The fundamental m a t e r i a l of abelian v a r i e t i e s i s p r e s e n t e d by [7] A. Weil, V a r i b t e s abeliennes e t courbes alg'ebriques , Hermann, P a r i s , 1948.

Soc. Japan, No. 6, Tokyo, 1961. A r e c e n t volume [15] A. Weil, B a s i c number theory, Springer, 1967

[a] S. Lang, Abelian v a r i e t i e s , Interscience, New York, 1959.

contains a m o d e r n t r e a t m e n t of c l a s s field theory, a s well a s t h e

The analytic t h e o r y of theta functions and abelian v a r i e t i e s i s

s t r u c t u r e t h e o r e m s of simple a l g e b r a s over number fields, which

systematically treated in [9] A.. Weil, Introduction a 1' btude d e s variittes kahlhriennes, Hermann, P a r i s , 1958, [lo] C. L. Siegel, Analytic functions of s e v e r a l complex variables, lecture

notes, Institute f o r Advanced Study, 1948,

reprinted with c o r r e c t i o n s , 1962.

generalize (8. 2-4).

The l a t t e r subject, in a concise style, c a n be

found in [16] M. Deuring, Algebren, Ergebn, d e r Math., B e r l i n , 1935. As f o r the general theory of arithmetically defined discontinuous groups, I mention h e r e only t h r e e papers: [17] A. Bore1 and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), 485-535.

connection of the H a s s e zeta function of an elliptic curve with the

[18] G. D. Mostow and T. Tamagawa, On the compactness of arithmetically defined homogeneous s p a c e s , Ann. of Math. 76 (1962),

Diophantine problems on the curve i s discussed in a survey a r t i c l e

446-463.

(of c o u r s e with numerous r e f e r e n c e s )

[19] W. L. Baily and A. B o r e l , Compactification of a r i t h m e t i c

r e f e r e n c e t o elliptic c u r v e s , J. London Math. Soc., 41 Q966), 193-291.

quotients of bounded s y m m e t r i c domains, Ann. of Math. 84 (19661, 442-528. The compactness c r i t e r i o n , which g e n e r a l i z e s that f o r of s11, i s given in [17] and [18].

#nrlr The l a s t paper 1191 proves the existence

of a Z a r i s k i open s u b s e t of a projective v a r i e t y isomorphic t o a given quotient like

$

1

in g e n e r a l .

F o r t h e s e topics, s e e a l s o the

a r t i c l e s in [I]. T h e o r e m s 5, 9, 10, 11 and t h e i r generalizations a r e proved in my papers: [20] G. S h i m u r a , Moduli and f i b r e s y s t e m s of abelian v a r i e t i e s . Ann. of Math. 83(1966), 294-338. [21] G. Shimura, Construction of c l a s s fields and zeta functions of algebraic c u r v e s , Ann. of Math. 85 (1967), 58-159. [22] G. S h i m u r a , Algebraic number fields and symplectic discontinuous groups, Ann. of Math. 86 0967), 503-592. The l a s t section of [22] i s a weaker v e r s i o n of the r e s u l t s s t a t e d in §lo, for which a full account will b e d i s c u s s e d in a forthcoming paper.

Some b a s i c concepts of I -adic r e p r e s e n t a t i o n s can b e found in [23] Y. Taniyama, L-functions of number fields and zeta functions of abelian v a r i e t i e s , J . Math. Soc. Japan, 9(1957), 330-366. This v e r i f i e s a l s o the H a s s e conjecture f o r abelian v a r i e t i e s with sufficiently many complex multiplications.

a l s o [14], and l e c t u r e

F o r this topic, s e e

notes by J. -P. S e r r e to b e published soon.

[24] J. W. S. C a s s e l s , Diophantine equations with s p e c i a l

The