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
SpringerVerlag 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 (NSFGP 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 SpringerVdag Berlin. Hddelberg 1%8 .I,ibmty of Congress Catalog G r d Number 682>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 fieldtheoretical 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 or 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 nonabelian 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 noncompact
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 nontrivial. ) 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 wellknown 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 onetoone 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,
[email protected]) i s nontrivial.
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 (3.810).
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 (Ri3): 5.
Abelian v a r i e t i e s and Siege1 modular functions
A nonsingular 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 )
52pl
.
= 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 ndimensional complex vector
.
a d i s c r e t e f r e e Zsubmodule 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 Rvalued Rbilinear 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 onetoone 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 isnot 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 (Ri3), 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 BI = 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 . tv ' )
.
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 UI 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)
Nspace.
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 endomorphismring 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 Clinear 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 Zmodule
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
*
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 nontrivial
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 2submodule
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 Rvalued 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 classfieldtheoretical 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 theFrobenius 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