APPLICATIONS TO

OF

NUMBER O.

THE

THEORY

OF MODULAR

FORMS

THEORY M.

UDC 511.334515.178

Fomenko

The survey is devoted to arithmetic questions in the theory of modular forms and, in particular, to arithmetic applications of modular functions; mainly only elementary and analytic aspects of this topic, as a rule, for the case of a single variable are presented. The present s u r v e y is devoted to a r i t h m e t i c questions of the theory of modular f o r m s . This theory, which attracted the attention of mathematicians both of the nineteenth century and the f i r s t half of this century, has recently experienced intense development. Connections are being developed with algebraic number theory and algebraic g e o m e t r y , the theory of r e p r e s e n t a t i o n s , functional a n a l y s i s , and p-adie methods. The work of the school on modular functions of one variable in Antwerp is c l e a r c o r r o b o r a t i o n of this [457]. An appropriate article covering the m a t e r i a l of Ref. Zh. Matematika will probably be forthcoming in due time. The present s u r v e y - the f i r s t in "Itogi Nauki" on a r i t h m e t i c applications of modular functions - presents mainly only the e l e m e n t a r y and analytic aspects of this topic and, as a rule, for the case of one variable. The c a s e of s e v e r a l variables is only touched on; this mainly c o n c e r n s w o r k on Hilbert modular f o r m s which naturally extends the problems we consider. Although we have mainly c o n s i d e r e d papers f r o m the last 10 y e a r s , in a number of c a s e s it was n e c e s s a r y for us to r e t u r n to older results as well. 1.

The

Heeke

Theory

There a r e a large number of books and papers which p r e s e n t various aspects of the foundations of the theory of modular f o r m s [120, 229, 237, 253, 280, 284, 353, 404, 417, 419, 499, 502,539,584, 591,594, 617]. For the r e a d e r ' s convenience we p r e s e n t in this section some definitions and c l a s s i c a l facts. We begin with some basic definitions. Let F be a d i s c r e t e subgroup of the group SL(2, R); F will f r e quently denote a congruence subgroup, i.e., a subgroup of the group SL(2, Z) containing a {homogeneous) principal congruence subgroup of degree (or level) N:

w h e r e N > 0 is an integer. The most important examples of groups F a r e :

F = SL(2, Z);

(the Hecke subgroup); F = F(N). Below H denotes the upper half plane H = {z6C I lm (z) > 0}. The group SL(2, R) acts on H according to the rule .

az +b

fa b~6s L (2,

g ~z; = c-F~-~ ' g = ic Ifg=~azb~OL+(R ) (the group of real 2 •

matrices withdetg>0),

R) t h e n w e set

] (g, z) = ( c z + d) (det g)-~:2,

(/[~ g) (z) = / [ ~\c-7~) + bl J (g,

z)-<

Points z I and z 2 are called F-equivalent if ?(z I) = z 2 for s o m e T C F. T h e concept of a fundamental d o m a i n for F is introduced in the usual m a n n e r . A point s belonging to R ~ {co} is called a F-parabolie vertex (cusp) if

Translated f r o m Itogi Nauki i Tekhniki, Algebra, Topologiya, G e o m e t r i y a , Vol. 15, pp. 5-91, 1977.

0090-4104/80/1404-1307507.50 9 t980 Plenum Publishing Corporation

1307

t h e r e e x i s t s a p a r a b o l i c e l e m e n t 7 E F w i t h f i x e d point s . If H*=H~ { F - p a r a b o l i c v e r t i c e s }, t h e n the s t r u c t u r e of a c o m p a c t R i e m a n n s u r f a c e c a n be i n t r o d u c e d on F \ H * . If F is a n a r b i t r a r y d i s c r e t e s u b g r o u p of the g r o u p SL(2, i t ) , then F is c a l l e d a F u c h s i a n g r o u p of f i r s t kind if F \ H * is c o m p a c t . Such g r o u p s a n d , in p a r t i c u l a r , F(N) have a f i n i t e n u m b e r of F - n o n e q u i v a l e n t p a r a b o l i c v e r t i c e s . L e t F be a F u c h s i a n g r o u p of f i r s t k i n d , and l e t k > 0 be a n i n t e g e r . A c o m p l e x - v a l u e d f u n c t i o n f(z) is c a l l e d a F - a u t o m o r p h i c f o r m of w e i g h t k if f(z) is d e f i n e d in H and s a t i s f i e s the f o l l o w i n g c o n d i t i o n s : 1) /a b ~ F f [az + b~=(cz_t_d)~ f (z) f o r a l l 7= lc d) ; 2) f(z) is h o l o m o r p h i c i n H; 3) f(z) i s h o l o m o r p h i c a t e a c h r - p a r a b o l i c v e r t e x . W e d e n o t e by Mk(F) the s p a c e of s u c h f u n c t i o n s . If F is a c o n g r u e n c e s u b g r o u p , t h e n the e l e m e n t s of Mk(F) a r e c a l l e d m o d u l a r f o r m s [or m o d u l a r f o r m s of d e g r e e N i f F = F(N)]. L e t r be a D i r i c h l e t c h a r a c t e r m o d N [ i . e . , a h o m o m o r p h i s m @ :(Z?NZ)*-->C* ]. W e s a y that r is e v e n ( r e s p e c t i v e l y , odd) i f ~ ( - 1 ) = i [ r e s p e c t i v e l y , ~ ( - 1 ) = - 1 ] . W e s h a l l a s s u m e t h a t ~ and k have the s a m e p a r i t y : r = ( - 1 ) k . If f(z) s a t i s f i e s the c o n d i t i o n

c--Z'~'d)= (a) (cz + d)kf (Z) for all

7=(ab]6Po(N) and c o n d i t i o n s 2) and 3) f o r m u l a t e d a b o v e , t h e n f(z) is c a l l e d a m o d u l a r f o r m f o r F~(N)

of w e i g h t k and c h a r a c t e r r [or a m o d u l a r f o r m of t y p e (k, r f o r F0(N) o r a m o d u l a r f o r m of type (k, N, ~)1. The s p a c e of s u c h f o r m s is d e n o t e d by Mk(F0(N), r o r Mk(N, '$). L e t f(z) ~ M k ( F ) . I n a n e i g h b o r h o o d of e a c h F - p a r a b o l i c v e r t e x s f(z) c a n be e x p a n d e d i n a F o u r i e r s e r i e s

2 a , e 2'a''/~ ( h e r e ~ = e27riz/h is the l o c a l u n i f o r m i z i n g v a r i a b l e in a n e i g h b o r h o o d of the point s). If r = F0(N), r~=O

t h e n the F o u r i e r e x p a n s i o n of f(z) in a n e i g h b o r h o o d of ~r h a s the f o r m oo

/ (z)=~ ~,nz. n=0

A F - a u t o m o r p h i e f o r m f(z) ~ Mk(F) w h i c h v a n i s h e s a t a l l r - p a r a b o l i c v e r t i c e s is c a l l e d a F - p a r a b o l i c f o r m (cusp f o r m ) . The s p a c e of s u c h f o r m s i s d e n o t e d by S k ( F ) . The s u b s p a c e of p a r a b o l i c f o r m s in Mk(F0(N) , ~) = Mk(N , r is d e n o t e d by Sk(F0(N), ~) o r Sk(N , r I n the c a s e o f f o r m s of p r i n c i p a l t y p e [ i . e . , ~ = 1 (modN)] the n o t a t i o n is s o m e t i m e s s i m p l i f i e d : Mk(F0(N) , 1) = Mk(F0(N)) = Mk(N); s i m i l a r l y Sk(F0(N) , 1) = Sk(F0(h0) = Sk(N). I f f a n d g b e l o n g t o Sk(F) , t h e n i t is p o s s i b l e to d e f i n e the s c a l a r p r o d u c t (the P e t e r s s o n i n n e r p r o d u c t ) (f, g) = (f, g)k,r = I f f (z) g (z) V~ ax ~v F

w h e r e z = x + iy a n d F is a F - f u n d a m e n t a l d o m a i n . It is e a s y to s e e t h a t t h i s is w e l l d e f i n e d . It c a n be p r o v e d t h a t Sk(F) i s a f i n i t e - d i m e n s i o n a l H i l b e r t s p a c e . We give several examples. 1) W e c o n s i d e r the f u n c t i o n f i r s t i n v e s t i g a t e d by R a m a n u j a n c~

r

A (Z)= e2:uzl~ (I - - e2~inz)24=Zz (n) e2n'nz; n=l

n=I

t h i s is a n S L ( 2 , Z ) - p a r a b o l i c f o r m of w e i g h t 12. 2) L e t F be a s u b g r o u p of f i n i t e index in SL(2, Z ) , and l e t F 0 be the t r a n s l a t i o n s u b g r o u p of the g r o u p F g e n e r a t e d by the " l e a s t " t r a n s l a t i o n z - - z + q. F o r the i n t e g e r k > 0 and the i n t e g e r v -> 0 w e d e f i n e the P o i n c a r e s e r i e s of w e i g h t k and c h a r a c t e r (index) v

o~ (z) = ~ e 2~';-~ (ez + d)~, Y

w h e r e 3' r u n s t h r o u g h a s y s t e m of r e p r e s e n t a t i v e s

1308

7=

d

of F 0 \ F .

Gv(z) i s a F - a u t o m o r p h i c f o r m of w e i g h t

2k and is a F - p a r a b o l i c f o r m if ~ -> 1. It c a n be p r o v e d t h a t any p a r a b o l i c f o r m f ~ S2k(F) is a l i n e a r c o m b i n a t i o n of the s e r i e s G~(z), v _> 1. I

3) L e t F = SL(2, Z). The f u n c t i o n Go(z)=2r.(2k)Ek ( z ) = ~

(cz+af~ w h e r e (c, d) r u n s t h r o u g h ai1 p a i r s of

(c, d)

i n t e g e r s e x c e p t (0, 0) is c a l l e d the E i s e n s t e i n s e r i e s ,

and oo

Ek (z) = 1 -~

(-1)~4k ~ ~2k-~(n ) e 2~I~" B~

is c a l l e d the n o r m a l i z e d E i s e n s t e i n s e r i e s ; g(s), R i e m a n n z e t a function; and ~k ( n ) = Z

dL

din

4) L e t A = (aij) be a n e v e n i n t e g r a l 2k x 2k m a t r i x , i . e . , aij ~ Z, aij ~ 2 Z ; k -> 2. L e t Q(X) = 1/2XtAX be a n i n t e g r a l p o s i t i v e q u a d r a t i c f o r m in x l , x2 . . . . . theta series 0 (z; Q ) = ~

X2k. W i t h Q(X) we a s s o c i a t e the

e ~Q(:~), M

w h e r e M r u n s t h r o u g h a l l i n t e g r a l c o l u m n v e c t o r s w i t h c o m p o n e n t s ml . . . . , m2k. L e t N be the d e g r e e of the f o r m Q(X), i . e . , the s m a l l e s t i n t e g e r >0 f o r w h i c h NA -1 is a n e v e n i n t e g r a I m a t r i x . It c a n be p r o v e d that the f u n c t i o n 0(z; Q) is a F 0 ( N ) - m o d u l a r f o r m of w e i g h t k and c h a r a c t e r a(d) = ( A / d ) , w h e r e A is the d i s c r i m i n a n t of the f o r m Q(X). 5) The function co

j (z) = A (z) =

@ 744-~-

C (rt) qn rt~l

c (n)GZ,

q = e 2~/~,

is F - i n v a r i a n t and m e r o m o r p h i c i n H and a t ~; h e r e F = SL(2, Z). It is a n e x a m p l e of a F - m o d u l a r f u n c t i o n and is c a l l e d the m o d u l a r i n v a r i a n t . I n 1937 H e c k e [284] c o n s t r u c t e d his t h e o r y of o p e r a t o r s f o r F ( N ) - p a r a b o l i c f o r m s .

For simplicity we eo

s h a l l d i s c u s s the c a s e of the g r o u p F0(N). The H e c k e t h e o r y a s s i g n s to e a c h f o r m f ( z ) = ~ the D i r i e h l e t s e r i e s D (s, f ) =

2 a~n-~

a~e~"~'~ES~ (N, qJ)

w h i c h e x t e n d s a n a l y t i c a l l y to the e n t i r e s p l a n e and s a t i s f i e s a s i m p l e

I

f u n c t i o n a l e q u a t i o n . M o r e o v e r , in the H e c k e s e n s e the s p a c e Sk(N , 9) p o s s e s s e s a b a s i s c o n s i s t i n g of f u n c t i o n s w i t h m u l t i p l i c a t i v e p r o p e r t i e s of the F o u r i e r c o e f f i c i e n t s . W e r e c a l l that the H e c k e T - o p e r a t o r is d e f i n e d on Sk(N , 9) a s f o l l o w s [in p l a c e of Tk(n) we s h a l l o f t e n w r i t e T(n)]: d--1

T k ( n ) ] f ( z ) =nk-~ Z a>O ad=n

~ , * ( a ) f , (~z+b a !td-~. b=0

We s h a l l r e s t r i c t o u r s e l v e s to c o n s i d e r i n g o p e r a t o r s T(p) w h e r e p is a p r i m e n u m b e r ; the g e n e r a l c a s e r e d u c e s to t h i s . P e t e r s s o n [509] i n t r o d u c e d on the s p a c e Sk(N , ~b) a m e t r i c (the P e t e r s s o n i n n e r p r o d u c t ) a n d s h o w e d t h a t the a l g e b r a of o p e r a t o r s g e n e r a t e d by T(p) [(p, N) = 1] is a c o m m u t a t i v e a l g e b r a of n o r m a l o p e r a t o r s on Sk(N , 9) a n d h e n c e t h e r e e x i s t s a b a s i s f o r Sk(N, ~) c o n s i s t i n g of the e i g e n I u n c t i o n s of the o p e r a t o r s T(p) f o r a l l (p, N) = 1, i . e . , T(p) l f = Xpf. M o r e o v e r , the F o u r i e r c o e f f i c i e n t s a n of t h e s e e i g e n f u n c t i o n s f(z) p o s s e s s m u l t i p l i c a t i v e p r o p e r t i e s : If (p, N) = 1, t h e n anp + ~ ( p ) p k - l a n / p = Xpa n (we w r i t e a n / p f o r p ~'n) ; in p a r t i c u l a r , if a(1) = 1 = N, t h e n ap = Xp and aqp = a q a p f o r a l l p r i m e p, q, p ~ q. If (p, N) = 1, t h e n the o p e r a t o r T(p) is not n e c e s s a r i l y nor real.

a,e 2 ~ ,

Let f(z)=~

a n d l e t D ( s , f ) = ~ _ ~ a,,n-~ b e t h e M e l l i n t r a n s f o r m of t h e f u n c t i o n f ( z ) .

n~l

I f ( r , N) = 1 and

n~l

?( is a p r i m i t i v e c h a r a c t e r rood r w e s e t

1309

r--1

g (~0 : ~-J Z (x) e2~x/', x~O 0o

D (s,/, x ) = ~ x (~) a~n-~, L (s, f, X) = (r2N) ~/2(2u) -s F (s) O (s, f , Z)A m a j o r a c h i e v e m e n t of Hecke w a s the p r o o f of the following t h e o r e m : 1) E a c h D i r i e h l e t s e r i e s L(s, f, ~), f ~ Sk(N, ~) c o n v e r g e s in s o m e half plane, extends a n a l y t i c a l l y to the e n t i r e plane as an e n t i r e function w h i c h is bounded in v e r t i c a l s t r i p s , and s a t i s f i e s the functional equation L (s, f , X) = ik~ (r) Z (N) g (Z)2 r -1- L (k-- s, f Ik o, ~), 0--1 where ~=(N 0)" 2) T(p) l f = apf for all p if and only if D (s, f) ~ I 2 (1 -- app -s + ~, (p) pk-l-2~)-~ p (it is a s s u m e d that f is a n o r m a l i z e d f o r m , i.e., a 1 = 1). C O R O L L A R Y . L e t fl, f2 in Sk(N, ~) be eigenfunctions of T(p) for all p, and let fl, f2 have the s a m e c o r r e s p o n d i n g e i g e n v a l u e s . Then fl and f2 a r e m u l t i p l i e r s of one a n o t h e r . oo

Weil [675] p r o v e d a c o n v e r s e t h e o r e m . We fix positive i n t e g e r s N and k and s u p p o s e that D (s) ~ . ~ a,,n-* c o n v e r g e s a b s o l u t e l y for s = k - 6, 6 > 0; s u p p o s e that f o r any p r i m i t i v e c h a r a c t e r X m o d r , (r, N) = 1, the series co

L* (s, z) = (2~)-~ r (s) extends to an entire function of s which the form

is bounded

~ a~z (n)n-~

in any vertical strip and satisfies a functional

equation

of

L* (s, Z) ~ (const) L* (k -- s, ~) (for the e x a c t e x p r e s s i o n f o r c o a s t s e e , for e x a m p l e , Well [675] and a l s o the books of Ogg [499] and G e l b a r t [253]). Then oo

f (z)=~.~ a , , e 2 ~ S k (N, ~,). tz=l

M o d u l a r i t y c a n thus be c h a r a c t e r i z e d in t e r m s of D i r i c h l e t s e r i e s . A t k i n and L e h n e r [134] m a d e a n o t h e r i m p o r t a n t c o n t r i b u t i o n to the Hecke t h e o r y . This is connected with the c o n s t r u c t i o n of a s a t i s f a c t o r y t h e o r y of Hecke o p e r a t o r s T(p) on the s p a c e of F 0 ( N ) - p a r a b o l i c f o r m s not only for (p, N) = I but a l s o for p[ N. We begin with a s i m p l e e x a m p l e (we follow the p r e s e n t a t i o n of G e l b a r t [253]). We c o n s i d e r the twod i m e n s i o n a l s p a c e S~2(F0(2)) containing f2(z) = A(2z), fl(z) = A(z). T h e s e f o r m s have the s a m e eigenvalues for all T(p), p ;~ 2, but they a r e l i n e a r l y independent. I n c o n n e c t i o n w i t h this e x a m p l e the following q u e s t i o n a r i s e s : let f ~ Sk(N , r W h a t additional r e s t r i c tions m u s t be i m p o s e d on f in o r d e r that the F o u r i e r c o e f f i c i e n t s an, (n, N) = 1, c o m p l e t e l y d e t e r m i n e f ? If T ( p ) l f = 0 f o r all (p, h0 = 1, is it p o s s i b l e to conclude that f = 0 ? H e c k e p r o v e d that if ~ is a p r i m i t i v e c h a r a c t e r , then a n = 0 for (n, N) = 1 i m p l i e s that f -= 0 (in this e a s e e a c h f o r m f ~ Sk(N , $) is a "new f o r m " in the s e n s e of A t k i n and L e h n e r [419]; s e e below). The w o r k of Atkin and L e h n e r [134] (see a l s o [419]) deals with the c a s e ~ = 1 (mod N). We fix m (any p r o p e r d i v i s o r of N) and d (a d i v i s o r of N / m ) . We c h o o s e a b a s i s { g j ) for Sk(F0(m)) c o n s i s t i n g of eigenfunctions of T(p) for (p, N) = 1. L e t S~(F0(N)) be the s u b s p a c e of Sk(F0(NO) spanned by the functions gj(dz), and let S~(F0(N)) be its o r t h o g o n a l c o m p l e m e n t in 8k(F0(N)) (relative to the P e t e r s s o n m e t r i c ) . We c h o o s e a basis {fi} for 8~r c o n s i s t i n g of eigenfunctions f o r T(p) with (p, N) = 1 [this is p o s s i b l e , s i n c e the s u b s p a e e s S~(F0(N)) go o v e r into t h e m s e l v e s 1310

under the action of T(p), (p, N) = 1]. We say that two f o r m s in Sk(F0(N)) belong to the s a m e equivalence class if they have the s a m e eigenvalues for all T(p) with (p, N) = 1. Any f o r m fi we call a new form; the old class is the class of forms in Sk(F0(N)) of the f o r m f(dz), w h e r e f is a fixed new f o r m relative to F0(m). Elements of the old class are called old f o r m s . The r e s u l t s of Atkin and Lehner can be formulated as follows: 1) the space Sk(F0(N)) has a basis consisting of old and new c l a s s e s of f o r m s ; 2) each new class consists of a single f o r m which is an eigenfunction for each T(p) with eigenvalue equal to 0 if p21N and equal to • if p iN, p2 ~N; 3) elements of the basis are equivalent only in the cases that they a r e old f o r m s and belong to the same c l a s s . Li [429] partially generalized the results of Atkin and Lehner to the case of the group F(N). Before Atkin and Lehner the Hecke o p e r a t o r s T(p), pl N, w e r e studied by Ogg [495, 498]; see also the w o r k of Asai [124]. H e r r m a n n [288] dealt with the t r a n s f e r of the Hecke theory to the ease of Hilbert modular f o r m s . We give the definition of the Hilbert modular group F a s s o c i a t e d with a completely r e a I field of algebraic numbers F of degree n. Let I-I n = {Z ~ ( Z , . . . . .

Zn)EC n

]Im z~ > 0 . . . . . Im z~ > 0}.

We consider the group of 2 • 2 m a t r i c e s F with determinant one, the elements of which are algebraic integers

I:

ooo.

kY

u]

"'

\y(1)zl+6(11,

..,,

7(n)zn+6(n )

where a (1) = cg, . . . . a (n) are numbers conjugate to a ; ~(1) = # . . . . . 8, etc. (the e n u m e r a t i o n is consistent).

~(n) a r e numbers conjugate to the number

Shimura [611] developed a theory of Diriehlet s e r i e s with E u l e r product and a functional equation for groups containing the case of the Hilbert modular groups by invoking the adele method of Iwasawa and Tare. A f u r t h e r and v e r y profound g e n e r a l i z a t i o n of the Hecke theory (together with a c h a r a c t e r i z a t i o n of the Weft m o d u l a r f o r m s mentioned above) to the case of automorphtc f o r m s on GL(2) over a global field was obtained by Weft [678] and Jaequet and Langlands [316] (there is also an exposition of these r e s u l t s in the book of Gelbart [253]). Weft [678] studied automorphic forms on GL(2) over an a r b i t r a r y global field F (i.e., a number field or a field of functions on a curve over a finite field), defining them as functions on GA = GL(2, A), where A is the adele ring of F, which satisfy special conditions. An automorphic f o r m on GL(2) in the sense of Jacquet and Langlands [316] is a c e r t a i n special irreducible r e p r e s e n t a t i o n ~ of the group GA. Although the theories of Well and J a c q u e t - L a n g l a n d s a r e parallel in many w a y s , their language is altogether different. We shall not p r e s e n t the content of these r e m a r k a b l e investigations in such an e l e m e n t a r y survey. In the very useful book of Gelbart [253] the r e a d e r will find an a c c e s s i b l e exposition of this subject and a d i s c u s s i o n of the connection of the c l a s s i c a l and new objects of the theory of automorphic f o r m s . We r e m a r k only that the concept of an automorphic f o r m in the sense of J a c q u e t - L a n g l a n d s includes as special e a s e s : 1) the c l a s s i c a l e11iptic f o r m s ; 2) the nonanalytic wave functions of Maass [433]; 3) the modular forms of Hilbert; 4) the real analytic t

E i s e n s t e i n s e r i e s of the type ~

ys

l cz+d

ps ; 5) the Hecke L - s e r i e s with G r o s s e n c h a r a k t e r e n ; 6) automorphie

C,d

f o r m s on quaternion a l g e b r a s , etc. We note that the r e s u l t s of Atkin and Lehner have been c a r r i e d over to the case of the J a c q u e t - L a n g lands theory by C a s s e l m a n [190], Miyake [456], and Novodvorskii [100, 101]. The w o r k of Miyake [456] in application to the c l a s s i c a l case of elliptic modular f o r m s g e n e r a l i z e s some r e s u l t s of Atkin and Lehner to the case of the space of parabolic f o r m s Sk(F0(N) , ~), w h e r e r is an a r b i t r a r y c h a r a c t e r of the group ( Z / N Z ) * . Miyake uses a r g u m e n t s of Atkin and Lehner in the proofs, while C a s s e l m a n operates with the theory of r e p r e s e n t a t i o n s . We shall make s e v e r a l r e m a r k s concerning e s t i m a t i o n of the eigenvalues of Hecke o p e r a t o r s . For the case of the function A(z) Ramanujan supposed that its F o u r i e r coefficients satisfy the inequality IT(p)[ --< 2p 11/2. If Cp is an eigenvalue of the o p e r a t o r T(p) on Sk(N , r w h e r e k is an integer -> 2, then P e t e r s s o n conjectured that I Cp] _< 2p(k-0/2 for (p, iN) = 1. This implies that the F o u r i e r coefficients of the function

1311

co

f (z) = ~ a~e2~"'eSk(N, ~) m u s t have the following e s t i m a t e : a~-=O ~,n 2

j.

Many p a p e r s w e r e devoted to this conjecture. Analytic methods w e r e only able to give the e s t i m a t e (cf. Raakin [551], S e l b e r g [589])

The f i r s t m a j o r r e s u l t in the proof of the P e t e r s s o n conjecture is due to E i c h l e r [2301 who obtained a proof in the c a s e k = 2 (see a l s o S h i m u r a [608], I g u s a [305]). I m p o r t a n t investigations by Kuga and S h i m u r a [392] and I h a r a [307] then followed. Finaliy, Deligne [216] f i r s t reduced the P e t e r s s o n c o n j e c t u r e to a known eonjeeture of Weft and then proved the l a t t e r [218]. The methods of these p a p e r s belong to a l g e b r a i c g e o m e t r y , and we t h e r e f o r e l i m i t o u r s e l v e s to what has been said. In e o n n e e t i o n w i t h n u m b e r - t h e o r e t i c applications we briefly f o r m u l a t e the following eonjeeture of Weft [675]. Let E be an elliptic c u r v e defined o v e r Q; iet N be the e o M u e t o r , and let L(s) be the z e t a function of the c u r v e E. Weft c o n j e c t u r e d that L(s) is the Dirieb_let s e r i e s a s s o c i a t e d (by means of the Metlin t r a n s f o r m ) with s o m e new F0(N)-parabolie f o r m of weight 2. Langlands conjectured an anatogous connection between c e r t a i n A r t i n L - s e r i e s and I"0(N)-parabolie f o r m s of weight 1. The r e v e r s e connection was c o n s i d e r e d by Deligne and S e r r e [219]. Let N -> 1 be an i n t e g e r , let ~ be a c h a r a c t e r of the group ( Z / N Z ) * with the condition ~(-1) = - 1 , and let f(z) E MI(N , r Suppose that f(z) is an eigenfunetion of the Heeke o p e r a t o r s T(p) for all p.~N and s e t

T (p) [f = c6f. We denote by Q the a l g e b r a i c c l o s u r e of the field Q and by G the group Gal (Q--/Q). In the basic t h e o r e m of their w o r k Deligne and S e r r e show that t h e r e exists a l i n e a r r e p r e s e n t a t i o n p : G ~ GL(2, C) with the following p r o p e r t i e s : 1) p is u n r a m i f i e d at all p f N ; 2) T r (Fp, p) = ap and det (Fp, p) = ~(p) for all p~N, w h e r e Fp, p is the pi m a g e of the F r o b e n i u s substitution for p. I r r e d u c i b i l i t y of p is equivalent to p a r a b o l i e i t y of f. COROLLARY. The R a m a n u j a n - P e t e r s s o n conjecture is valid for weight 1: l apl _< 2. As an application of the m a i n r e s u l t , 'the authors prove the following t h e o r e m : suppose that co

f ( z ) = Z a~e2~GSI (N, 4) IZ~I

is a new f o r m ; let p be the c o r r e s p o n d i n g r e p r e s e n t a t i o n of G. Then: 1) the A r t i n conductor of the r e p r e s e n t a oo

t i o n p is equal to N; 2) the A r t i n L-function L(s, p) is equal to

D(s,f)=~_~ a,,,-.

The following a s s e r t i o n is a e o r o l l a r y of this t h e o r e m : the function L(s, p) is entire (i.e., the A r t i n conj e c t u r e is valid for p). S e l b e r g [586] and E i c h l e r [234] w e r e able to find a m e a n s of defining the t r a c e of the Hecke o p e r a t o r T(n) on the s p a c e Sk(N, $). S e l b e r g ' s method is a l s o applicable in multidimensional e a s e s ; in p a r t i c u l a r , it is applicable to H i l b e r t modular f o r m s (see Shimizu [604,605]). E i c h i e r ' s method is based on the theory of gene r a l i z e d Abelian i n t e g r a l s connected with automorphie f o r m s of one v a r i a b l e and the application of this theory to m o d u l a r c o r r e s p o n d e n c e s ; a m o r e a b s t r a c t v e r s i o n of it can be found in the w o r k of E i c h l e r [237] and in the p a p e r of Kappus [324]. Many p a p e r s have been devoted to calculating t r a c e f o r m u l a s of Heeke o p e r a t o r s T(n) under v a r i o u s conditions; s e e , in p a r t i c u l a r , E i c h l e r [ 2 3 1 , 2 3 4 - 2 3 6 , 2 3 9 , 2 4 0 ] , Ishikawa [312], Hijikata [292], S a g o [571], and Shimura [621]. The m o s t f a r - r e a c h i n g r e s u l t was obtained by Hijikata [292] who proved an explicit f o r m u l a for the t r a c e of the o p e r a t o r T(n) on the s p a c e Sk(N , r w h e r e k -> 2 is an integer; N, an a r b i t r a r y natural number; and ~, an a r b i t r a r y c h a r a c t e r (meal 1'0. In his pioneering w o r k [586] Selberg a l r e a d y obtained the following d i r e c t a r i t h m e t i c c o r o l l a r i e s of his f o r m u l a for the t r a c e ~k(T(n)) of the Heeke o p e r a t o r Tk(n) on the s p a c e Sk(SL(2, Z)) = S k. It is well known that 1312

for k = 2, 4, 6, 8, 10, and for the number of c l a s s e s , 6, 8, 10, 14 we obtain new for the Ramanujan function

14 the space S k is empty. F r o m ( ~ 2 ( T ( n ) ) = 0 there follows the K r o n e c k e r relation and f r o m %(T(n)) = 0 there follows an analogous relation of E i c h l e r [233]; for k = relations. If k = 12, then the t r a c e formula gives a simple a r i t h m e t i c e x p r e s s i o n r (n}.

E i c h l e r [232] f i r s t applied considerations r e l a t e d to the computation of t r a c e s of Hecke o p e r a t o r s to the c l a s s i c a l Hecke p r o b l e m of the r e p r e s e n t a b i l i t y of parabolic f o r m s by theta s e r i e s . We c o n s i d e r the space of F0(N)-parabolic f o r m s of weight k of principal type which we denote by Sk(N). Hecke ([284, p. 884]) conjectured that for prime p the space $2(p) is spanned by linear combinations of theta s e r i e s of d e g r e e p. This conjecture was proved by E i c h l e r [232] by computing and c o m p a r i n g the t r a c e s of Hecke o p e r a t o r s and Brandt m a t r i c e s (the analogues of Hecke o p e r a t o r s for algebras of definite quaternions over Q). We shall now briefly d e s c r i b e E i c h l e r ' s method. Let co

~ (z) = ~

at (n) e2~"~, a~ (1) = 1, i = 1,2 . . . . . g,

tt~I

be an o r t h o n o r m a l basis of the space $2(p) consisting of the eigenfunctions of all Hecke o p e r a t o r s T(n), (n, p) = 1, acting on this space. We c o n s i d e r the r e p r e s e n t a t i o n Rl(n) of the ring of Hecke o p e r a t o r s in the space S2(p): T (n) l ~ = R ~ ( n ) ~ , (n, p ) = l , w h e r e 9 is a column v e c t o r with components ~l(z), . . . , ~g(Z). Computation of the t r a c e of this r e p r e s e n t a t i o n was a l r e a d y d i s c u s s e d above. Let B be the a l g e b r a of definite quaternions over Q of d i s c r i m i n a n t p2; let .~ be the o r d e r in B of level p; let ~ (~=1 . . . . . H) be a s y s t e m of r e p r e s e n t a t i v e s of the c l a s s e s of left . ~ - i d e a l s ; let .~, be the right o r d e r s of the ideals ~ , ; ~ ~ . runs through the s y s t e m of r e p r e s e n t a t i v e s of all c l a s s e s of left . ~ - i d e a l s . Let rrt,v(n)be the number of all integral ideals of n o r m n left-equivalent to @ ~ - ~ , and let wu be the number of units in . ~ . We consider P(n) = ( , ~ ( n ) ) , the Brandt m a t r i x of o r d e r H (Anzahlmatrix [232]; ideal number matrix [235]),

(w~-~ ... wz~ p (o)=\;r; :.: Let O(z) be the matrix with elements which are q u a t e r n a r y theta s e r i e s co

n=0

We have

.

,.

.=

(<;

(n) J

0' (z) = ~ p' (n) e~~ = 0 L (z)) tz~l

which is a matrix of o r d e r H - 1 with elements which are the differences of q u a t e r n a r y theta s e r i e s of degree p. E i c h l e r c o n s i d e r e d the r e p r e s e n t a t i o n of the ring of Hecke o p e r a t o r s in the subspace | of the space $2(p) generated by parabolic f o r m s the differences of theta s e r i e s ; proving that T (n) I 0' (z) = P ' (n) 0' (z), (n, p) = l, he found [232] the t r a c e of this r e p r e s e n t a t i o n . It was found that t r a c e P'(n) = traceRl(n) , (n, p) = 1. This (accounting with the complete reducibility) implies the equivalence of the r e p r e s e n t a t i o n s considered. Thus, | = 82(p), and E i c h l e r ' s t h e o r e m is proved. In the s a m e w o r k [232] E i c h l e r showed that in the case of even k >- 2 the f o r m f(z) 6 Sk(p) can be r e p r e s e n t a t e d by a linear combination of theta s e r i e s of 2k variables of levels 1 and p if p is sufficiently large. Starting f r o m other c o n s i d e r a t i o n s , Kitaoka [339] proved this last r e s u l t of E i c h i e r for s m a l l levels p = 2, 3, 5, 11. F o r a d i s c u s s i o n of the p r o b l e m of the r e p r e s e n t a b i l i t y of modular f o r m s belonging: to the space M2(p, (p/)) by theta s e r i e s see the w o r k of Kitaoka [343] and P o n o m a r e v [531]. In the w o r k [236] E i c h l e r generalized his r e s u l t s to the case of s q u a r e - f r e e N; in p a r t i c u l a r , h e obtained a r e l a t i o n between the t r a c e s of Hecke o p e r a t o r s and the t r a c e s of various g e n e r a l i z e d Brandt m a t r i c e s c o r r e s p o n d i n g to the divisors of N.

1313

co

"~ a ~e2ninz~sk (N). A c o r o l l a r y of this is the following theorem: let k >- 2, let N be s q u a r e - f r e e , and let f(z)=z..~ c~

n~l

Then it is possible to find a f o r m g ( z ) ~ . ~ b,,e~i,,~~S~ (N) which is a linear combination of s e v e r a l generalized tt~l

(i.e., with s p h e r i c a l functions of weight k - 2) q u a t e r n a r y theta s e r i e s of various levels dividing N such that an = bn for any (n, N) = 1. If f(z) is an eigenfunction of all the Hecke o p e r a t o r s T(n) with (n, N) = 1, then for g(z) it is also possible to take the eigenfunction of such o p e r a t o r s . E i c h l e r ' s results can be r e f o r m u l a t e d in the following manner (Eichler [240], Hijikata and Saito [293]): let S~(N} be the "essential part" (i.e., the subspace of new forms) of the space Sk(N). Then the following a s s e r tion holds (Eichler): 1) for s q u a r e - f r e e N S~(N) is spanned by theta s e r i e s of level N; 2) hence Sk(N) is spanned by theta s e r i e s 0(z) of level M, MI N, and their "translations" 0(dz) by divisors d l ~ .

Hijikata and Saito [293]

(see also Kalinin [46]) generalize this r e s u l t of E i c h l e r in two directions: 1) in place of N being s q u a r e - f r e e it is a s s u m e d that N = pM, (p, M) = 1; 2) the subspace of theta s e r i e s c o r r e s p o n d i n g to S~(N) is c h a r a c t e r i z e d . Generalizations of E i c h l e r ' s results to the case of automorphic f o r m s on GL(2) in the spirit of the J a c q u e t - L a n g l a n d s theory can be found in the w o r k of Jacquet and Langlands [316], Gelbart [253], and Shimizu [607]. The last w o r k is in a c e r t a i n sense an adele analog of the w o r k [293]. Witt was the f i r s t to consider the linear dependence between theta s e r i e s [684]. His work was continued by Kneser [352]. Kitaoka [341] studied conditions under which two quadratic forms have identical theta s e r i e s . The results on the linear dependence between theta s e r i e s can be formulated in t e r m s of the linear dependence of the Epstein zeta functions a s sociated with the theta s e r i e s . Applying t r a c e formulas, Shimizu [606] obtained c e r t a i n linear dependences between zeta functions of definite quaternion algebras. The action of Hecke o p e r a t o r s on the space of theta s e r i e s was studied by E i c h l e r [229], Kitaoka [340], and Freitag. 2.

The

Rankin

of

One

and

Convolution.

Several

Connection

of Modular

Forms

Variables

Rankin [550] (see also Selberg [585]) studied the scalar product of Dirichlet series which are Mellin transforms of modular forms by a method which has recently become an important tool in the investigation of Dirichlet series and modular forms (Andrianov [ii], Doi and Naganuma [224], Jacquet [315], Naganuma [473], Shimura [619]). We consider the Dirichlet s e r i e s

2 ann -~, ~ n~l

b~n-s a s s o c i a t e d by means of the Mellin

n=l

t r a n s f o r m with parabolic f o r m s f(z) and g(z) relative to some subgroup F of the group SL(2, Z). Rankin r e p r e sented the s c a l a r product mentioned above, namely, the Dirichlet s e r i e s

~ a~b-~n-.,

(1)

as an integral convolution (with the E i s e n s t e i n s e r i e s as kernel) of the modular forms f(z) and g(z). Utilization of the modular and analytic p r o p e r t i e s of both the f o r m s f(z) and g(z) and of the kernel makes it possible to analytically extend the s e r i e s (1) as a m e r o m o r p h i c function to the entire complex s plane and to obtain a suitable functional equation. Rankin himself treated the c a s e s F = SL(2, Z), F(N). The case F = F0(N) was cons i d e r e d by Ogg [496] (for f o r m s of principal type) and Asai [124] (for f o r m s of nonprincipal type). A slight generalization of Ogg's results is given in the w o r k of Li [429]. Jacquet [315] gave a profound generalization of Rankin's r e s u l t s to the case of automorphic f o r m s on GL(2) over any global field. Shimura [622] recently obtained an interesting r e f i n e m e n t of Rankin's results. Let f(z) ~ Sw(F0(M) , )0, w h e r e w > 0 is an integer of the s a m e parity as X. Let f (z)= ~ c (n) e2~'~

be the F o u r i e r expansion of f(z).

n=l

We a s s u m e that c(1) = 1 and that f(z) is an eigenfunction of all Hecke o p e r a t o r s of level M. Then by the Hecke theory

~ C (n) n -s = 1-[ [1 -- c (p) p-s _~_Z (t9)pw-l-2s]-l, n~l

p

where the product is taken over all prime numbers p; ~((n) is taken equal to z e r o if (n, M) ~ 1. We expand each factor: 1314

1 - - c (p) p - s @ z (p) pw-~-2s= (1 - - %p-s) (1 - - ~pp-s), w h e r e C~p, /3p a r e c o m p l e x n u m b e r s .

F o r any p r i m i t i v e D i r i c h l e t c h a r a c t e r r we d e f i n e a new 2 u l e r p r o d u e t

D (s) = D (s, f , ,r = I-[ I(1 - - q) (p) a~ p-s)(1 - - ~ (p) apB,pp-~) (1 - - @(p) ,~ p-O]-h

(2)

P

I t is e a s y to s e e that D (s) = L (2s - - 2w + 2, X2}2) ~ ] } (n) c (n ~) n -s ~- I-[ [1 --'5 (p) c (p2) p-s + + (p)2 Z (P) c (p~) pw-~-2s_ ~ (p)a X (p)apaw-a-as]-!, P

where L (s, Z2~2) =~__~ Z (n)2~ (n) 2n-~. n=l co

It is c l e a r t h a t D(s) and ~

c (n2)n -~ c o n v e r g e a b s o l u t e l y for Re (s) >> 0. T h e p r i n c i p a l r e s u l t of S h i m u r a [622] I

is the f o l l o w i n g T h e o r e m 1 of his w o r k : W e s e t (s~(s+l\

~ (s)= =-~/2r l-~) ~ i - ~ - I r (

s--w+2--Lo

2

)D(s),

w h e r e X0 = 0 o r 1 a c c o r d i n g to w h e t h e r X ( - 1 ) ~ ( - 1 ) = 1 o r - 1. T h e n R(s) e x t e n d s a s a m e r o m o r p h i c function to the e n t i r e s plane w h i c h is h o l o m o r p h i c e v e r y w h e r e w i t h the e x c e p t i o n of p o s s i b l e s i m p l e p o l e s a t the p o i n t s s =wands =w-1. The f u n c t i o n R(s) is o f t e n e n t i r e . C o n d i t i o n s w h e n this is s o a r e g i v e n in T h e o r e m 2. F o r e x a m p l e , R(s) is a n e n t i r e f u n c t i o n if M = r = 1 (r is the c o n d u c t o r of the c h a r a c t e r ~). The e q u a l i t y co

co

~.~ X (n) @(n) n ~-l-s D (s) = L(2s -- 2w + 2, X2~2) ~., ,~ (n) c (n)2n -s .'I=I

shows the close relation of the results c(n2). If M= r = i, then

n~l

of Shimura

to the work

of Rankin

[550]. We

note that c(n) 2 differs from

co

(s--w+

I) D (s) --[ (2s--2w-~- 2) ~

c (n)2n -s,

n=l

where

~ is the Riemann

zeta function.

Rankin

showed

thatthe function co

R* (s) = (2~)-2* F (s) P (s - - w + 1) ~ (2s-- 2w + 2) ~_, c (n)2n -~ n=I

e x t e n d s m e r m o p h i e a l l y to the s p l a n e and is h o l o m o r p h i c e v e r y w h e r e w i t h the e x c e p t i o n of s i m p l e p o l e s a t the p o i n t s s = w and s = w - 1; f u r t h e r , it s a t i s f i e s the f u n c t i o n a l e q u a t i o n

R* (2w-- 1 - s ) = R * (s). I t f o l l o w s f r o m R a n k i n ' s r e s u l t s t h a t in the e a s e M = r = 1 the f u n c t i o n R is m e r o m o r p h i c on the e n t i r e p l a n e and s a t i s f i e s the f u n c t i o n a l e q u a t i o n

R (2~- 1 - s) = R (s), s i n c e R (s)=2~+~(~+~)/2R* (s)/~ ( s - - m + 1) , w h e r e ~ ( s ) ~ - * / 2 1 ? (s/2) ((s). D i v i s i o n by ~(s - w + 1) d o e s not p r o v i d e the a s s e r t i o n t h a t R(s) is h o l o m o r p h i c ( a c c o r d i n g to R a n k i n ) . It f o l l o w s f r o m the r e s u l t s of S h i m u r a t h a t R(s) is h o l o m o r p h i c . S h i m u r a o b s e r v e s that in the e a s e M r > 1 his t h e o r e m s a r e not f o r m u l a t e d in the b e s t f o r m . The b e s t f o r m u l a t i o n w a s o b t a i n e d by G a l b a r t and J a c q u e t [255] in the f r a m e w o r k of the J a e q u e t - L a t g l a n d s t h e o r y of a u t o m o r p h i e f o r m s on GL(2) o v e r g l o b a l f i e l d s . G e n e r a l i z a t i o n s of the R a n M n c o n s t r u c t i o n p l a y a n i m p o r t a n t r o l e in a n a p p r o a c h to the p r o o f of c o n j e c t u r e s of S a t o - T a t e t y p e (Ogg [501], S e r r e [592, 593]). W e s h a l l f o r m u l a t e the S a t o - : r a t e c o n j e c t u r e f o r the 1315

R a m a n u j a n function r(n). Let oe

I I (1 If we s e t

(z) = D (e~**9, Im (z) > 0, then the f u n c t i o n A up to a c o n s t a n t muitiple is the n a t u r a l p a r a b o l i c f o r m of weight 12 r e l a t i v e to SL(2, Z). In p a r t i c u l a r , f o r all p r i m e n u m b e r s p the function A is an eigenfunetion of all Hecke o p e r a t o r s T(p), and the c o r r e s p o n d i n g e i g e n v a l u e s a r e T(p). By the Hecke t h e o r y

L~ (s) = ~ 9 (n). n =

I I Hp (p-S) , ,

u~l

p

where /L/; (X) = 1 -- ~ (p) X @ pI122, and LT(S) extends to the e n t i r e c o m p l e x plane as a n e n t i r e function; the function (2=)-~ P (s) L~ (s) is i n v a r i a n t u n d e r the c h a n g e of v a r i a b l e s ~ 12 - s. By the R a m a n u j a n c o n j e c t u r e p r o v e d by Deligne (see Sec. 1 above) f o r all p r i m e s p t h e r e is the e s t i m a t e I r(p)l < 2p ~1/2. We have

Hp (X) = (1 - % X ) (1 - ~pX), where

The S a t o - T a t e c o n j e c t u r e (which w a s o r i g i n a l l y f o r m u l a t e d f o r the e a s e of elliptic c u r v e s without c o m p l e x multiplication) is that the a n g l e s ~0p a r e u n i f o r m l y d i s t r i b u t e d in the i n t e r v a l (0, zr) with r e s p e c t to the m e a s u r e (2/~) sin2~0 9 d~0. This q u e s t i o n is r e l a t e d (Serre [592, Chap. 1, A. 2]) to the a n a l y t i c e x t e n s i o n of the D i r i c h l e t series ) ' m=-1,2,....

L m (s) = Hp ,=o (1 - - %n% - . . . . .;

It is r e q u i r e d to p r o v e that Lm(s) a d m i t s an a n a l y t i c e x t e n s i o n to e n t i r e function of s, and L,~ 1 ~--if-- • 0 . The e x i s t e n c e of a functional e q u a t i o n of the usual type is a l s o c o n j e c t u r e d (Serre [593]). The e a s e m = 1 is c l a s s i c a h the function Ll(s) c o i n c i d e s with L r ( s ) ; the e a s e m = 2 w a s c o n s i d e r e d by R a n k t n [550] and S h i m u r a

[6221 (see above). New i m p o r t a n t r e s u l t s w e r e r e c e n t l y obtained in this d i r e c t i o n . G e l b a r t and J a e q u e t (see the cited w o r k [255]) showed that the E u l e r p r o d u c t D(s) [see (2)] with suitable F - f a c t o r s c a n be c o n s i d e r e d as the Mellin t r a n s f o r m of a n a u t o m o r p h i c f o r m on GL(3). T h i s , in p a r t i c u l a r , m a k e s it p o s s i b l e to extend to the e n t i r e s oo

oo

plane the functions ~ ~a (n) n -s and ~ ~4 (n) n -s . This w o r k of G e l b a r t and J a e q u e t is c l o s e l y r e l a t e d to the w o r k 1

1

[317,318,319]. We note that Y o s h i d a [691] p r o v e d a n a n a l o g of the S a t o - T a t e c o n j e c t u r e , and n u m e r i c a l v e r i f i c a t i o n of it w a s c a r r i e d out by L e h m e r [409]. A beautiful c o n s t r u c t i o n belonging to this c i r c l e of ideas w a s s u g g e s t e d by Doi and N a g a n u m a [224] (for f o r m s of p r i n c i p a l type) and N a g a n u m a [473] (for f o r m s of n o n p r i n c i p a l type). We s h a l l p r e s e n t the l a t t e r c o n s t r u c t i o n . Let p - 1 (mod4) be a p r i m e n u m b e r with the condition that the n u m b e r of c l a s s e s of the field K = vo

( z /"~-- -- ~z.~ a ne 2 ~ n z E S , (P,)l) be a n o r m a l i z e d e i g e n Q ( f ~ ) be equal to 1, and let • = (p/) be a c h a r a c t e r of K. L e t f ~

f u n c t i o n of all the Hecke o p e r a t o r s T(n). We c o n s i d e r the p r o d u c t ~n~-s \n~l

1316

~

- -

I ann

-s "

Naganuma erties:

proved that this product is the Mellin transform

of a function f of two variables possessing the prop-

i__ I

)(~2z,+~, ~"z2+~')=)(z,, z2) (~ is an integer of the field K, s is a unit in K, tr is a conjugate element o f # e K over Q). If the field K is Euclidean (i.e., as is well known, D= 5, 13, 17, 29, 37, 41,73), then the m a t r i c e s (~ :-1,~)_: and ( _ 10 1) generate %

SL (2, O) (O is the ring of integers in K), and { is a Hilbert modular form. By the fine t h e o r e m of V a s e r s h t e i n [22] on g e n e r a t o r s of the group SL (2, O) the modularity of f holds even without the a s s u m p t i o n that the field K c~

is Euclidean. In the w o r k [224] Doi and Naganuma s t a r t e d f r o m the parabolic f o r m .~ (z)=N~

a~e2~,,~ of weight

I

k relative to SL(2, Z) which is an eigenfunetion of all the Hecke o p e r a t o r s , and they c o n s i d e r e d the product ann-~

~ z ( n ) ann-~ , / \n=l

n=l

w h e r e X is a c h a r a c t e r of the real quadratic field Q0fD). The r e m a i n d e r is analogous to what has been p r e sented above. Application of the Rankin convolution plays an important role in the a r g u m e n t s of Doi and Naganuma [in the proof of formulas of type (3)]. Zagier [693,695] gave a completely different proof of their results. We fix a r e a l quadratic field K = Q0/D) with d i s c r i m i n a n t D; let k > 2 be an even number, let b be the different of the field K, and let N(x) = xx' be the n o r m of the element x ~ K. We c o n s i d e r the functions for integers m >0, k > 2

~.~ (z. z2) = ~

"

1

(~,~,+~z, +~'~,+b)k (zl, z2~H),

(t,b~z

~6b-~

N(~.)--ab=m/D

w h e r e the s u m m a t i o n goes over all triples (a, b, )0 satisfying the conditions indicated, while the triple (0, 0, 0) is excluded. The functions :0m(Z~, z 2) c o n s t r u c t e d are modular f o r m s of weight k for the Hilbert modular group SL (2, O). The f o r m ~0 is a multiple of the H e c k e - E i s e n s t e i n s e r i e s for K, and the remaining f o r m s ~ m are parabolic. Z a g i e r proves that for fixed z 1, z 2 ~ H the function co

(z. z2; z ) = ~ mk-%~(z, z2) e2"~ (zGH) (considered as a function of z) belongs to the space Sk(D , • w h e r e X = (D/) is a c h a r a c t e r of K. Z a g i e r further shows that the function of Naganuma {(z 1, z 2) (see above) is equal (up to a s c a l a r multiple) to the P e t e r s s o n s c a l a r product of f(z) and ~t(zi, z2; z). This implies (without applying the r e s u l t of L. N. Vasershtein) that f(zl, z 2) is a parabolic f o r m for SL (2, O). Zagier studies the linear mapping f ~ { of the space Sk(D , • into the space of parabolic f o r m s of weight k for SL (2, O) ; the image of this mapping has dimension 1/2 dimSk(D , )0 and is spanned by the f o r m s Wm(Zt, z 2) (m = 1, 2, . . . ). Finally, Z a g i e r considers the usual modular f o r m s of weight 2k for SL(2, Z): win(z, z). Cohen [203] proposed another c o n s t r u c t i o n of f o r m s of two variables. Asai [124] extended the r e s u l t s of Naganuma to the case of parabolic f o r m s f(z) ~ Sk(N , X), w h e r e N is an odd number which is not n e c e s s a r i l y prime. Here (in place of the t h e o r e m of L. N. Vasershtein) the Well c h a r a c t e r i z a t i o n [678] of Hilbert modular f o r m s in t e r m s of the c o r r e s p o n d i n g Dirichlet s e r i e s is used. More general r e s u l t s on the a r i t h m e t i c relations between Hilbert parabolic f o r m s over completely real fields and parabolic f o r m s of one variable w e r e obtained by Saito [572,573]. Let F be a completely real number field, let O be the ring of i n t e g e r s , and let Sk(F) be the space of Hilbert parabolic f o r m s of weight k (k is an even number) relative to the group F ~ O L ( 2 , O)+ consisting of elements of the group GL (2, O) with positive determinants. For a (Archimedean or non-Archimedean) n o r m , of the field F let F , denote the c o m pletion of F with r e s p e c t to v. For a n o n - A r c h i m e d e a n n o r m ~ (=~) let O~ be the ring of ~ - a d i c integers of the field E v. Let FA be the adele ring of the field F; we consider the group GL(2, FA). Let ~ r = ] ~ OL (2, O~)}( non-Arch 1-[ GL (2, F~) [an open subgroup of the group GL(2, FA)]. We consider the Hecke ring ~ / ~ ( ~ / s , OL(2, Fn)) and Arch its action T( ) on Sk(F). It is a s s u m e d that F is a cyclic extension of the field Q of prime degree l, the class number of F is equal to 1, and the conductor of the extension F / Q is equal to the prime number q. For the 1317

ordinary modular group SL(2, Z) we consider its adelization ~Q=I-[GL(2, Zp)~(GL(2, R) and the He(3ke ring R--/~ (~o, GL (2, QA)). The ring R acts on the spaces of parabolic forms Sk(SL(2 , Z)) and Sk(F0(q) , X), X is a character of the group (Z/qZ)* of order I. It therefore has a representation T 1 on Sk(SL(2 , Z)) and T X on Sk(F0(q), X). The operator T: Tf(zl, .... zD = f(z 2 ..... z/, z I) acts on the space Sk(F). Using the operator T, we define a new subspa(3e S~ (F)cSk (F) (the space of symmetric Hilbert parabolic forms) as follows: Sk(F) = {f ~ Sk(F) I T(e)Tf = TT(e)f for any eE/~ (~F, GL (2, FA))}. Obviously, S~(F) is stable relative to the action of g~, ! and we arrive at a new representation of the Heeke ring 9 on the space Sk(F). The main objective of Saito's work [573] is to show that the representation T S of the ring ~ on S~(F) can be obtained from the spaces of parabolic forms Sk(SL(2 , Z)) and Sk(F0(q) , • for various characters X of the group (Z/qZ)* of order I. The author defines a homomorphism k:~-~/~ such that any R-module becomes an 91 -module. The main result of the work [573] (Theorem 3) asserts that for k -> 4 there exists a subspace Sc| (P0 (q), Z), where • runs z

through all characters

of order l of the group (Z/qZ)*,

such that S'k (F) ~ Sk (SL (2, Z))|

and

|

(F0 (q), Z)

x

S| as an 9 -module. This result is deduced from an equality for the traces of operators which generalizes certain results of Hirzebruch [298,299,301] and Busam [173]. In the case l = 2 the result of Doi and Naganuma [224] indicated above is obtained from Saito's theorem. Similar results were obtained by Jacquet [315] from the point of view of representation theory. Saito's results were reinterpreted and extended first by Shintani and then by Langlands [see R. P. Langlands, Base Change for GL(2). The Theory of Saito-Shintani with Applications, Lecture Notes, Inst. Adv. Study, Princeton, N. J., 1975]. Work of Hirzebrueh and Zagier [301] recently appeared (preliminary publications: Hirzebru(3h [299], Zagier [696]) inwhich new problems in the topic of interest to us are studied. Let p = 1 (rood4) be a prime number, and let 9 be the ring of integers of the field Q(~p). The factor space X=H2/SL(2, O) is a noncompact complex surface (the modular surface of Hilbert) with a finite set of singularities. Let T N be the image in X of the curve in H 2 given by the equation a V~-z~ ~ + zz~ - z'z~ + b Vp=

0,

where a, b6Z, kE(-9, kk'q-abp=N . If MN is not a square, then the number of intersections TMT N of the curves T M and T N is well defined; the authors compute it. For example, if N is not a square, then TIT N = Hp(N), where

Np ( N ) = ~ H ((4N--x~-)/p), x~4N x ~ 4 N ( m o d p)

and H(k) is the n u m b e r of c l a s s e s of positive definite binary quadratic f o r m s of dis(3riminant - k . Let ~( be the c o m p a c t s u r f a c e obtained by adding to X the parabolic v e r t i c e s (regarding the r e s o l u t i o n of s i n g u l a r i t i e s at p a r a b o l i c v e r t i c e s , see Hirzebru(3h [298]). The c o m p a c t i f i c a t i o n of the c u r v e T N defines a cycle in the h o m o l ogy group H 2(X). This group d e c o m p o s e s canonically into the d i r e c t s u m of the image of H 2(X) and the subs p a c e g e n e r a t e d by c u r v e s a r i s i n g f r o m the r e s o l u t i o n of s i n g u l a r i t i e s at the parabolic v e r t i c e s ; let T~q be the component of T N in the f i r s t t e r m . Then

(TOrTeN)~ =Hp (N) -q-Ip (N), where

Io(N)~- #

~_~rain(k, k'). L>>O, ~ ' ~ N

e

(3

Computation of (TMTN) X in the g e n e r a l e a s e was also c a r r i e d out. The authors f u r t h e r show that the function

~, (z) ~- - - ~ § ~, (I-Ip(N) + Ip (N)) e2~iNz (z~H) N~I C

C

is a F0(p)-modular f o r m of weight 2 and c h a r a c t e r X = (P/). The n u m b e r (TMTN) ~ (if p4~M) is the N - t h F o u r i e r c o e f f i c i e n t of the modular f o r m T(M) l~Op, w h e r e T(M) is the Hecke o p e r a t o r on the s p a c e M2(P, (p/)). The p a p e r a l s o contains i n t e r e s t i n g c o n j e c t u r e s . In the proof of m o d u l a r i t y of ~pp(Z) the modular behavior of the function

~,H(m)e ~'~'nz is used. E a r l i e r w o r k was a l s o devoted to this l a s t question (Mordell [459, 460], Siegel [629], .,n~0

Eichler [2331; see a l s o Golubeva and Fomenko [42]). 1318

The Rankin convolution has found application in the following problem posed by Yu. V. Linnik. Let Fi (i = 1 . . . . , m) be fields of algebraic n u m b e r s , and let ~i be the Hecke c h a r a c t e r on the divisor group of the field Fi;

L~, (s, ~)-- ~

~ r/,(u,__) (Re s > 1) s

N f'i/ QU/=n

is the Hecke L - s e r i e s of the field Fi; here 11~ runs through the integral divisors of the field Fi. It is well known that the Hecke L - s e r i e s extends to the entire s plane and satisfies the usual f o r m of functional equation (Hecke [284]). Yu. V. Linnlk introduced the interesting new o b j e c t - the s c a l a r product of Hecke L - s e r i e s

L& ," ' - , e.~(s; q,

""~

~.,)=

= -/./s

NFi/Qlli=r~,

1~1,2

.....

(Res>l).

(4)

ra

l.-~rt
Yu. V. Linnik conjectured that the scalar product he introduced is extendable to the entire plane and has a functional equation (of usual type). In the work of Vinogradov [31] (see also [97]) and Draxl [225] the scalar product (4) was extended, respectively, to the lines Re s = 1/2 and Re s = 0. These investigations are not based on a modular technique. The Rankin convolution is applied to the proof of the Linnik conjecture in the work of Fomenko [116]; its applicability (for the case of two Hecke L-series of quadratic fields) is based on the classical result of Hecke and Maass (Hecke [284], Maass [433]) according to which the Hecke L-series of a quadratic field is the Mellin transform of some modular form. The Linnik conjecture for two quadratic fields was considered also in the work of Gaigalas [35] in which in some cases the Linnik conjecture is also proved for the case of three quadratic fields. The Hecke-Maass result was generalized by Well, Jacquet, and Langlands [316] to the case of a quadratic extension of a global field. In combination with the results of Jaequet [315], this gives a proof of an analog of the Linnik conjecture for two Heeke L-series of the quadratic extension of any global field. Cases of nonquadratic extensions have so far not yielded to this method, since no analogs of the Hecke-Maass result are known for them, although the recent results of Gelbart and Jacquet mentioned above lend hope for progress here as well. 3.

Modular

Forms

of H a l f

Integral

Weight

We are concerned with modular f o r m s on the upper half plane with a factor of automorphicity (cz + d)kJ2, w h e r e k > 0 is an odd number. C l a s s i c a l examples are the theta function 0 (z)=

e 2,~i....

~

and the Dedekind eta function co

~ (z) = e~,~/12 ] ] (1 - e 2 ~ ) Hecke was a l r e a d y aware of the difficulties in developing his theory of T - o p e r a t o r s for f o r m s of half integral weight. In his last paper ([284, pp. 919-940]) he c o n s i d e r e d linear relations of the type

(~+~)= 0

F(pz)+~

F T l rnod p

for functions F = rlaob with a + b - 1 (rood2). Hecke proved that linear relations of the type considered exist for any prime p > 3 for the functions ~, 0, ~3 and V40-1 (van Lint [426,428] extended the study of such linear relations). Heeke's w o r k gave impetus to the investigations of Wohlfahrt [685] who c o n s i d e r e d the definition of Hecke o p e r a t o r s on the space Mr(F , v) of modular f o r m s relative to a group F of real weight r > 0 with s y s t e m of multipliers v; here F is a subgroup of finite index of the modular group SL(2, Z). Wohlfahrt f i r s t introduces on the space Mr(F , v) c e r t a i n o p e r a t o r s TA(Q) and then shows that the Hecke o p e r a t o r s on the entire space Mr(F, v) can be defined [by means of the o p e r a t o r s T~(Q)] only if the functions ,.-,

:ab~

IIQ ~ [Qt~ (cz + d ) - ' / (Qz), ~d= ~c d)'

1319

a r e l i n e a r l y independent; h e r e Q runs through a complete s y s t e m of p r i m i t i v e F-nonequivalent m a t r i c e s with I QI = n. F o r functions f with l i n e a r l y dependent ftQ it is possible to introduce other o p e r a t o r s , but they will not n e c e s s a r i l y be "Hecke o p e r a t o r s " for the r e m a i n i n g functions of the s p a c e M r ( F , v). At the end of his p a p e r Wohlfahrt d i s c u s s e s s p e c i a l l y modular f o r m s of half i n t e g r a l weight and obtains in s o m e c a s e s f o r m u l a s for the number of r e p r e s e n t a t i o n s of i n t e g e r s by quadratic f o r m s with an odd number of v a r i a b l e s . W o h l f a h r t ' s exposition is ponderous and s o m e w h a t f o r m a l . The i m p o r t a n t works of S h i m u r a [618, 619] s p e c i a l l y devoted to f o r m s of half integral o r d e r clarify the s u b s t a n c e of the m a t t e r and go c o n s i d e r a b l y further. We give a detailed exposition of S h i m u r a ' s r e s u l t s . Let G be a g r o u p w i t h e l e m e n t s which a r e p a i r s (a, ~), w h e r e ~=(cab)GsL(2, R),

92=t(cz+d) (t is a c o m -

plex n u m b e r with I tl = 1). The multiplication law of the p a i r s is (a, r (fi, $') = ( a . fl, r For a function f on the upper half plane H and ~ = (a, r we s e t f]k[[] = f(a(z)) 9 $(z) -k. A holomorphic function f on H is called an entire modular f o r m of weight k / 2 r e l a t i v e to a d i s c r e t e subgroup A of the group G if flk[6 ] = f for all 6 E A and conditions of " h o l o m o r p h i e i t y " a r e s a t i s f i e d at all p a r a b o l i c v e r t i c e s of the group A. We denote the s p a c e of such f o r m s by M~/2(A); let S~/:(A) be the s u b s p a c e [in Mk(A) ] of p a r a b o l i c f o r m s r e l a t i v e to A. Let N be an integer which is a multiple of 4, let

/

it

1, aa=

i,

it if

dEl(mod4), d~3(mod4).

We c o n s i d e r the subgroup A0(N) [ r e s p e c t i v e l y , AI(N)] of the group G consisting of p a i r s (~, j(3')), 7 E F0(N) [ r e s p e c t i v e l y , ,/E F~(N)]. Let X be a c h a r a c t e r of the group ( z / N Z ) * , and let M~/2(N, X) [ r e s p e c t i v e l y , S~/2 (N, • be the s u b s p a c e of M~/2(AI(N)) [ r e s p e c t i v e l y , S~/2(A~(N))] f o r m e d by e l e m e n t s f such that f[~ [~l = Z (d) f for all

= (" ]

/ ab = (c d)

("

(N)..

The d i m e n s i o n of the s p a c e s M~/2(A) and S 1/2 (A) is computed by means of the R i e m a n n - R o c h t h e o r e m . For details and f u r t h e r c l a s s i c a l facts we r e f e r to the s u r v e y of Shimura [618] and concentrate our attention on s o m e r e c e n t r e s u l t s . We fix a positive N, a multiple of 4, and for each integer m > 0 we s e t A = A0(N) , a = (10 m0/, ~ = (c~, m 1/4) and consider the d e c o m p o s i t i o n ~ =

U . ~ , 9 We define the Hecke o p e r a t o r T(m) on M~/2

(N, X) by setting T ( r a ) l f = t'~k/4--1Z ~((av)f) k V-,l, v 1 a.d,

where ~=!l*

*)' *)" The o p e r a t o r T(m) maps M1/2(N'k •

[and s J Z ( N , •

not a s q u a r e , then T(m) = 0. Shimura proved the following r e s u l t s : 1) Let p be a p r i m e n u m b e r , and let fE M~/2(N, •

We s e t

f ( z ) = ~ a ( n ) e 2"Lnz, n~O

oo

(T (p2)]/) (z) = ~ a b (n) e2~/'~.

1320

into itself. If (m, N) = 1 and m is

'n P ~- ' a (n) -k Z ( / ) P k-2a (n/P 2) , whe re k = (k -- 1)/2, Zl (m) = Z (m) (,_~_~)--1 ~ ; we set a ( n / p ~) : Then b (n) = a ( / n ) -k X1(P) [~-) 0 if p2-~n and x(m) = •

= 0 if (m, N) ~ 1.

2) Suppose that f satisfies the previous conditions; we a s s u m e that f is a c o m m o n eigenfunction of the o p e r a t o r s T(p 2) for all prime p: T(p2){ f = Wpf. Let t be a positive integer not divisible by p2 with the condition (p2 N) = 1. Then there is the decomposition oo

n=l

p

w h e r e the product is taken over all p r i m e p. The main r e s u l t of Shimura is the following proposition 3) in which modular f o r m s of integral weight a r e constructed f r o m modular forms of half integral weight. c~

3) Let f ( z ) = ~

a (n)e2"~i"~S~/2 (N, Z), k > 3, and let t be s q u a r e - f r e e . We define the c h a r a c t e r Xt modulo tN

a=I

by setting xt(rn)=x(tn)

t~J

c >,=(k--l)/2,

and we define the function Ft on H by setting F~ (z) = ~ A~ (n) e 2 ~ , n~I

At (n) n-~ = n=l

x~ (m) m ~-~-~ rn~I

a (trn~) rn-~ I

We a s s u m e that f is a c o m m o n eigenfunction of the o p e r a t o r s T(p 2) for all prime divisors p of the number N which are relatively p r i m e to the conductor of the c h a r a c t e r Xt. Then Ft is an o r d i n a r y modular f o r m of weight k - 1; more p r e c i s e l y , Ft(z) ~ Mk_l(F0(Nt), • with some positive integer Nt depending on N, • t. If k -> 5, then F t is a parabolic form. Shimura conjectured that Nt = 2N. (This conjecture was proved by Niwa [492]; see below.) S h i m u r a ' s proof of the main r e s u l t 3) is based on the Wei[ c h a r a c t e r i z a t i o n [675] of modular forms. Shimura [619,621] also advanced the following interesting conjecture. Let B1/2 be the space of entire modular forms of weight 1/2 relative to a congruence subgroup F of the group SL(2, Z). Can the space B~/2 be spanned by theta s e r i e s of the f o r m ~ (n) exp (2~in~rz), n

w h e r e r ~ Q and r is a c h a r a c t e r ? This c o n j e c t u r e w a s recently solved in the positive sense by Deligne and S e r r e (unpublished). For other open questions, see Shimura ([619], Sec. 4). Niwa [492] gave a more direct proof of the main r e s u l t of Shimura. His proof is based on the r e p r e s e n tation of a modular f o r m of integral weight as the P e t e r s s o n s c a l a r product of the indeterminate theta s e r i e s of Siegel and Well 0(z, g} (see Weft [673]) and a parabolic f o r m of half integral weight. The t r a n s f o r m a t i o n formula for a modular f o r m of integral weight is a c o r o l l a r y of the t r a n s f o r m a t i o n f o r m u l a for the theta s e r i e s ; the latter follows f r o m the Well representation. Such a d i r e c t proof of S h h n u r a ' s r e s u l t enabled Niwa to prove also the Shimura conjecture: Nt = 2N. The connection of the theory of S i e g e l - W e i l theta s e r i e s with S h i m u r a ' s r e s u l t s on modular forms of half integral weight was f i r s t d i s c o v e r e d by Shintani [626] who c o n s t r u c t e d parabolic f o r m s of half integral weight on the basis of parabolic f o r m s of integral weight. Let N, k be natural n u m b e r s , let X be a c h a r a c t e r rood N, and let S2k(F0(N) , X) be the space of parabolic forms of weight 2k for the group F0(N) with c h a r a c t e r X; if 41 N, tet S~/~(F0(N},o~_ • be the space of parabolic f o r m s of half integral weight k + 1 / 2 for F0(N) with c h a r a c t e r X. F o r each N, k, X a linear mapping 2

1/2

S2~ (P0 (N), Z ) ~ S2~+~(1~0(4N), X'):f (z)--> O(z, f), is constructed where X' is a c h a r a c t e r m o d 4 N which is given by the formula • (d) = x ( d ) ( N / d ) ( - 1 / d ) k. The parabolic f o r m s 0(z, f) are the S i e g e l - W e i l theta s e r i e s . The mapping f(z) -- 0(z, 0 commutes with the actfon of the Hecke o p e r a t o r s .

1321

We note a l s o that the c o n n e c t i o n b e t w e e n m o d u l a r f o r m s of half i n t e g r a ~tegral weights w a s a l s o lsed by A n d r i a n o v [8] in his i n v e s t i g a t i o n of the r e p r e s e n t a t i o n of i n t e g e r s by q u a d r a t i c f o r m s with an odd n u m )er of v a r i a b l e s . A m o n g the p a p e r s c o m p l e t e d b e f o r e S h i m u r a ' s i n v e s t i g a t i o n s , we m e n t i o n the p a p e r of K10ve [350] who ~arried over to the s p a c e of p a r a b o l i c f o r m s of half i n t e g r a l w e i g h t S1/2(SL(2, Z), v) c e r t a i n r e s u l t s of the ?etersson-Hecke theory and studied the Fourier coeffieients of the eigenfunetions of the Heeke operators [,(p2)of this space.

Estimates of the Fourier eoeffieients and arithmetic applications of modular forms of half integral weight are presented below. 4.

Application

to Quadratic

Forms

The t h e o r y of m o d u l a r f o r m s has b e e n s u c c e s s f u l l y applied to a c l a s s i c a l p r o b l e m of n u m b e r t h e o r y : the r e p r e s e n t a t i o n of i n t e g e r s by q u a d r a t i c f o r m s . The a n a l y t i c c i r c l e method is a l s o v e r y applicable to this p r o b l e m . We f i r s t f o r m u l a t e w h a t this method d e l i v e r s . Let k >- 4, let f(x~, . . . , x k) be a positive, i n t e g r a l q u a d r a t i c f o r m of d e t e r m i n a n t d, and let n be a positive i n t e g e r . Then (Malyshev [89]) f o r the n u m b e r of r e p r e s e n t a t i o n s (x 1. . . . . x k) of n u m b e r s n by the f o r m f r(f; n) t h e r e is the a s y m p t o t i c f o r m u l a (for n - - ~)

r(f;n)

~k/2

k

m ~k--I 2 H ( f ; n ) + O (d~_ +3/

2

.nkf4_ii4+e) '

(1)

where H(f; n) is the special series of the problem, and the 0 constant depends only on k, e. If certain congruence conditions are satisfied this implies the representability of n by the form f if n > C(d). The modular technique is applied to obtain analogous results in the following manner. We consider the theta series 0(z) = 0(z; f) associated with the form f: +oo

0(z,f)-We

expand

it as a sum

of an Eisenstein

series

~

e ~ ( ~ ....... ~), t m z > O .

xl,... ,xk~eo

and a parabolic

0 (z; f)-----E (z; f ) + 3

form:

(z; f);

(2)

comparison of the corresponding n-th Fourier coefficients gives the equality

r ( f ; n)--~ (/; n)-~ R (;; n),

(3)

while the P e t e r s s o n c o n j e c t u r e p r o v e d by Deligne (see Sec. 1) (k = 2k 1 is even and >-4) gives e f t ; n) = O (nk/r (e > 0). The leading t e r m is e x p r e s s e d in v a r i o u s explicit f o r m s depending on the method applied. A s m e n t i o n e d in Sec. 1, E i c h l e r [230] a l r e a d y in 1954 p r o v e d the P e t e r s s o n c o n j e c t u r e for weight 2; his w o r k w a s extended by S h i m u r a [608] and A n d r i a n o v [7] in the f r a m e w o r k of the a p p l i c a t i o n to q u a t e r n a r y q u a dratic forms. F o r odd k >- 5 the m o d u l a r t h e o r y has s o far given no i m p r o v e m e n t of (1) (See. 3 above). The c a s e k = 3 in w h i c h deep r e s u l t s have b e e n obtained by a n a l y t i c m e t h o d s (the w o r k of Linnik and Malyshev) [89] is a l m o s t i n a c c e s s i b l e to the t h e o r y of m o d u l a r f o r m s . A s y m p t o t i c f o r m u l a s in w h i c h the r e m a i n d e r t e r m is e s t i m a t e d w e r e mentioned above. The n u m b e r of p a p e r s devoted to " e x a c t " f o r m u l a s for the n u m b e r of r e p r e s e n t a t i o n s by q u a d r a t i c f o r m s is v e r y l a r g e . This topic w a s a l r e a d y p o p u l a r in the l a s t c e n t u r y (Gauss, E i s e n s t e i n , Liouville, etc.). H e r e we s h a l l s p e a k only of a few types of e x a c t f o r m u l a s , r e f e r r i n g to the p a p e r of M a l y s b e v [91] f o r the p r o b l e m a t i c s in this a r e a . A l a r g e n u m b e r of c o n c r e t e r e s u l t s a r e contained in the w o r k of Kogan (see, in p a r t i c u l a r , his book [53]) and G. A. L o m a d z e and his students (T. V. Vepkhvadze, G. P. Gogishvili, etc.). The c a s e in w h i c h S(z; f) - 0 i d e n t i c a l l y in (2) d e l i v e r s an i m p o r t a n t c l a s s of e x a c t f o r m u l a s , i.e., 0(z; f) = E(z; f), r(f; n) = p(f; n). The o l d e s t e x a c t f o r m u l a s a r e c o n n e c t e d with the c a s e f = x} + . . . + x~. We denote by rk(n) the n u m b e r of r e p r e s e n t a t i o n s of n by the s u m of k s q u a r e s . F o r m u l a s (2) and (3) f o r the c a s e f = x~ + . . . + x ~ have the f o r m ~ (z)=Ek ( z ) + S k (z), (4)

1322

r~ (n) = p~ (n) + R~ (n); h e r e 03 (z) = ~3 (01 z) =

~

(5)

e "~n'~ , and Pk(n) is the " d i v i s o r f u n c t i o n . " It is w e l l known ( s e e , e . g . , B a t e m a n [144],

n=--~

R a n k i n [556]) t h a t f o r k - 8, Sk(z) ~ 0. The m o s t f a m o u s c a s e of s u c h e x a c t f o r m u l a s is G a u s s ' t h e o r e m on t h r e e s q u a r e s (see [144]). R a n k i n [557] p r o v e d t h a t f o r k > 8 the c o r r e s p o n d i n g p a r a b o l i c f o r m Sk(z) ~ 0. T h i s r e s u l t w a s g e n e r a l i z e d by G o g o s h v i l i [41] a n d K o g a n [56] to a r b i t r a r y p r i m i t i v e p o s i t i v e f o r m s f w i t h a n u m b e r of v a r i a b l e s ->4. Many p u b l i c a t i o n s of K o g a n to w h i c h w e r e f e r the r e a d e r a r e a l s o r e l a t e d to R a n k i n ' s w o r k . V o r o n e t s t d i and M a l y s h e v [34] c o n s i d e r e d the w e l l - k n o w n p r o b l e m of the r e p r e s e n t a t i o n of a p a i r of n u m b e r s by s u m s of i n t e g e r s and t h e i r s q u a r e s and o b t a i n e d the f o l l o w i n g a n a l o g of R a n k i n ' s r e s u l t . L e t r s (m, n) be the n u m b e r of s o l u t i o n s of the s y s t e m of D i o p h a n t i n e e q u a t i o n s x~@ . . . @ x s = r n , x~ +

in i n t e g e r s x~ . . . . . Xs; s -> 3. i n v e s t i g a t e d . By the e f f o r t s of 3 -< s -< 8, r s ( m , n) = Ps(m, n). precisely, for each integer m

2

... ~x~=n

The q u e s t i o n of w h e n r s (m, n) = Ps (m, n) (the s i n g u l a r s e r i e s of the p r o b l e m ) is m a n y a u t h o r s (see the b i b l i o g r a p h y in [34] and a l s o [600]) i t w a s s h o w n that for V o r o n e t s k i i and M a l y s h e v p r o v e d t h a t f o r s -> 9 this is a l r e a d y not so. M o r e t h e r e e x i s t s a n i n t e g e r n, s n - m 2 > 0, s u c h t h a t r s ( m , n) ~ Ps(m, n).

R e t u r n i n g to (5), w e note t h a t in a n u m b e r of c a s e s it is p o s s i b l e to g i v e e x a c t f o r m u l a s f o r rk(n) f o r k > 8 a l s o . W e s h a l l d e s c r i b e s o m e of the r e s u l t s o b t a i n e d . It i s w e l l known t h a t ~k(0] z) ~ M k / a ( F ~, vk), w h e r e F ~ (the s o - c a l l e d t h e t a g r o u p ) c o n s i s t s of a l l T ~ S L ( 2 , Z) f o r w h i c h T ~- I o r T - V (rood2)

and the m u l t i p l i e r v is a f u n c t i o n on F ~ w i t h the p r o p e r t y ~3(01Tz) = (cz + d)~/2v(T)~3(0t z); h e r e Tz~az+ b cz+cr

zfIt,

--~
L e t Sk/2 = S k / 2 ( F o , v k) be the s u b s p a e e of p a r a b o l i c f o r m s i n M k / 2 ( F ~ , vk)o It is known t h a t d i m S k / 2 = [ 1 / 8 ( k 1)] = ~k- T h u s , in c o m p l e t e c o r r e s p o n d e n c e w i t h w h a t h a s b e e n s a i d a b o v e , ~ k = 0 f o r k <- 8. F o r k > 8 the f~.~ Ak-sro4r~4r ....... 3 ~2 ~4 (r = 1 , 2 . . . . . ~k), where

co

~4=~4(01z)=l+2~(-1)~q

n~, q=e ~ ,

z~,

n=l

f o r m a b a s i s of Sk/2; this w a s a l r e a d y s h o w n in old w o r k s of M o r d e l l . L o m a d z e [69] c o m p u t e d the c o n s t a n t s k - 8 ~254. 42 Ck, 9 -< k -< 16 in the e x p a n s i o n ~ = Ek(z) + Ck~Q3 T h i s m a k e s it p o s s i b l e to o b t a i n e x a c t f o r m u l a s . F o r ~ k > 1 it is m o r e c o n v e n i e n t to u s e o t h e r b a s e s (see L o m a d z e [69, 70, 71], R a n k i n [556]). F o r e x a m p l e , R a n k i n [556] i n the c a s e k = 20 ( ~ 0 = 2) e x p l i c i t l y i n d i c a t e d a b a s i s c o n s i s t i n g of p a r a b o l i c f o r m s w i t h m u l t i p l i c a t i v e F o u r i e r c o e f f i c i e n t s ( e i g e n f u n e t i o n s of the H e c k e o p e r a t o r s )

(z)= *

.

!

/

08 , ~ 4 ~ 4

We have , 16 r20 (n) = , o 2o(n) -T ~3 (155T1 (n) -~ 76~* (n)),

w h e r e ~l(n}, ~ ( n ) a r e the c o r r e s p o n d i n g n - t h F o u r i e r c o e f f i c i e n t s of the b a s i s f u n c t i o n s . In the w o r k of C o h e n [200] the f u n c t i o n rk(n) , k -> 5 and odd, is s t u d i e d by m e a n s of the t h e o r y of m o d u l a r f o r m s of h a l f i n t e g r a l w e i g h t ; h e r e the r e s u l t s of S b A m u r a ' s w o r k [619] a r e u s e d . The n u m b e r s of r e p r e s e n t a t i o n s rk(n) a r e r e l a t e d to the v a l u e s of L ( s , X) a t n e g a t i v e i n t e g r a l p o i n t s ; X = Xd(m) = ( d / m ) is the J a c o b i K r o n e e k e r s y m b o l . The m a i n r e s u l t s a r e a s f o l l o w s : i f d' = ( - 1 ) ( k - 0 / 2 d is the d i s e r i m i n a r t t of a q u a d r a t i c f i e l d , 1323

then rk (d) = A (k) L ((3 -- k)12, ~(~,)+

0

(d k14)

with an explicitly written constant A(k); the r e m a i n d e r t e r m is identically z e r o for k = 5, 7. In p a r t i c u l a r , if d is the disoriminant of a real quadratic field Q((d), then r 5 (d) = 480 (5 -- 2 (d/2)) ~Q(V~)(-- 1). the value of ~Q(~Fd)(-1) can be e x p r e s s e d in t e r m s of the values of the function Cohen [201,204]); for example, i f n -= 2, 3 (mod4) and is s q u a r e - f r e e , then 1

~Q(W)(--1)=-~

~]

~(rn)=~_~d

(see Siegel [636],

dim

o(n--S~).

Is/< V~-

It is thus possible to obtain still another e x p r e s s i o n for rs(d). Similarly, if d is d i s c r i m i n a n t of an imaginary quadratic field Q ( ( - d ) , then r~ (d) = -- 28 (41 -- 4 (d/2)) L (-- 2, X-a). Siegel [628] showed that if a positive q u a d r a t i c f o r m f with number of variables >2 belongs to singiec l a s s type, then the s u m of the singular s e r i e s a s s o c i a t e d with f gives the exact value for the number of r e p r e sentations of natural numbers by the f o r m f. In the work [81] Lomadze obtained exact formulas for the number of r e p r e s e n t a t i o n s of integers by each of 82 primitive positive t e r n a r y diagonal f o r m s belonging to s i n g l e - c l a s s types. Many papers of Lomadze have been devoted to finding exact formulas for the numbers of r e p r e s e n t a t i o n s of numbers by positive t e r n a r y f o r m s belonging to m a n y - c l a s s types: we note, for example, [82, 88,431]. Binary quadratic f o r m s are c o n s i d e r e d in technically s i m i l a r papers (Lomadze [76, 79], Vepkhvadze [28, 29]). Analogous results a r e obtained also for quadratic f o r m s with a number of variables ->4 (Lomadze [80, 87], Vepkhvadze [27, 30], Gogishvili [40]). These papers and a number of other works of Lomadze and his students use the following consideration: If the leading t e r m s a r e the F o u r i e r coefficients of the c o r r e s p o n d i n g E i s e n s t e i n s e r i e s , then the s o - c a l l e d auxiliary t e r m s a r e e x p r e s s e d in t e r m s of the F o u r i e r coefficients of products of theta functions with c h a r a c t e r i s t i c s (or their derivatives). The c o r r e s p o n d i n g computations a r e s o m e t i m e s v e r y complicated, although the auxiliary t e r m s themselves in a number of c a s e s a r e simple n u m b e r - t h e o r e t i c functions (see, e.g., Lomadze [82, 88]). In the w o r k [86], Lomadze explicitly c o n s t r u c t s a basis of the space of parabolic f o r m s S6(N, 1) for N = 3, 5, 7, 11 in the f o r m of generalized q u a t e r n a r y theta s e r i e s with spherical functions. This enables him to obtain exact f o r m u l a s for c e r t a i n quadratic f o r m s with 12 v a r i a b l e s . Kogan [52] studied questions of the r e p r e sentability of parabolic f o r m s S (z; f) by a linear combination of binary theta s e r i e s with spherical functions and deduced a number of c o r r e s p o n d i n g exact formulas. The r e s u l t s of Eiehler [230], Shimura [608], and Igusa [305] made it possible to obtain exact formulas for the case of q u a t e r n a r y quadratic f o r m s of level N and d i s c r i m i n a n t N2 under the condition that dimS2(N , 1)= 1. The latter holds only for N = 11, 14, 15, 17, 18, 19, 20, 21, 24, 27, 32, 36, 49. Indeed, the space S2(N , 1) is generated by the normalized eigenfunction q~N(Z) = ~ on=l s (n) exp (2rizn) of the Hecke o p e r a t o r s T(p), p~ N. F r o m the E i c b l e r - S h i m u r a reduction f o r m u l a it follows that for all YfN s (p) = Tr ~p, where ,~p is the F r o b e n ius e n d o m o r p h i s m of the curve VN reduced modulo p; here VN is an elliptic curve over Q with function field Q(j(z), j(Nz)). It is easy to show that for p~fho (an explicitly defined integer) p--1 x=0

P

w h e r e y2 = 4x 3 _ g2x _ g3 is the model of VN in W e i e r s t r a s s form; an explicit and v e r y simple a r i t h m e t i c exp r e s s i o n is thus obtained for s (p). Application of the e x p r e s s i o n indicated for s (p) to exact formulas is given in the book of Kogan [53] and in [115]. Andrianov [9] for N = 27, 32, 36 (in these c a s e s VN is a curve with c o m plex multiplications) obtained the formula indicated above for s (p) without using the results of E i c h i e r and Shimura. We shall p r e s e n t one of the exact formulas of Andrianov. Let f = x z + y2 + 9(z 2 + t~); then 0(z; f) e M2(36,1). For prime p, (p, 6) = 1, we have

,,)=

1324

4

(,, +

8 /p-1 [ x ~ + l

~

J)

S i m i l a r f o r m u l a s a r e a l s o obtained in other c a s e s (Abdullaev and Kogan [3, 4], Abdullaev [2]), for e x a m p l e , for c e r t a i n f o r m s with six v a r i a b l e s . The Weft conjecture [675] that the zeta function of the elliptic c u r v e E / Q is the conductor A a s s o c i a t e d with a F0(A)-parabolic f o r m of weight 2 m a k e s it possible to c o n s t r u c t parabolic f o r m s explicitly. This finds application in the theory of exact f o r m u l a s . F o r e x a m p l e , in the book of Kogan [53] an exact f o r m u l a is given for r(f; p), f = x 2 + y2 + 32(z 2 + t2); 0(z; f) ~ M2(128,1) , dim $2(128 fl) = 9. The f o r m u l a obtained is unconditional, since for the conductor A = 27 the Well c o n j e c t u r e has b e e n proved (see, e.g., [304]). R e g a r d i n g analogous r e s u l t s , see [1, 5, 55, 56]. 5.

Congruences

for

the

Fourier

Coefficients

of Modular

Forms

The l a r g e s t number of investigations is devoted to the congruences connected with the R a m a n u j a n Tfunction. We c o n s i d e r the unique parabolic f o r m of weight 12 r e l a t i v e to SL(2, Z) co

A(z)~e"'fi(i--e2""0'4=~ ~(n)e ~'''', I m z > 0 . n=l

n=l

R a m a n u j a n o b s e r v e d that t h e r e e x i s t c o n g r u e n c e s connecting T(n) and ~ , ( n ) = ~ d" modulo powers of s m a l l a[n p r i m e n u m b e r s . We e n u m e r a t e the r e s u l t s p r e s e n t l y known. 1) mod 2 ~. Kolberg [367] showed that

(n) ~ zL1 (n) (rood 21'), if n~- 1 (rood 8); 1217a~i (n) (rood 213), if n-~3(mod8); 1537ai~ (n) (rood 2'2), if n ~ 5 (rood 8); 705an (n) (rood 214), if n ~ 7(rood 8). 2) m o d 3 ~. In the paper of Swinnerton-Dyer

~(n)=-n-61~

[655] there is the result , .((mod36),

if n~l(mod3), n--=2 (rood 3).

if

3) rood 5 T. In the paper [655] (see also Lahiri [396]) there is the result

(n) ~ n-30~71 (n) (rood 53),

if

(n, 5) = 1.

4) rood75. The paper [655] (see a l s o [366]) with r e f e r e n c e to D. H. L e h m e r p r e s e n t s the r e s u l t ~(mod7), ~(n)~n%(n)[(mod72),

if if

n--O,t,2,4(mod7), n_~3, 5, 6 (rood 7).

5) rood 23. Wilton [681] proved congruences p r e s e n t e d by S e r r e [593] and in [655]. 6) rood691. R a m a n u j a n [546] proved that T ( n ) -- o l l ( n ) (rood691).

It is recently shown that congruences

for the Ramanujan

T-function are related to deep properties of co

/-adic representations.

This theory is due to Deligne [216] and S e r r e [593]. Let f ( z ) = ~

a~e2~i~z be a p a r a -

tz~l

bolie f o r m of weight k for SL(2, Z), let a 1 = 1, and suppose that each coefficient a n E Z. Six values of k a r e known for which t h e r e e x i s t p a r a b o l i c f o r m s with the p r o p e r t i e s indicated and for which the s p a c e s of parabolic f o r m s of weight k a r e o n e - d i m e n s i o n a h k = 12, 16, 18, 20, 22, 26. S e r r e conjectured and Deligne proved that for each of the six parabolic f o r m s e n u m e r a t e d above and any p r i m e l there exists a continuous r e p r e s e n t a t i o n ~t :Oal (KJQ) -> OL (2, Z~) (KI is the m a x i m a l a l g e b r a i c extension of Q r a m i f i e d only at t) such that the image of the Frobenius p - e l e m e n t F r o b (p) for p ~ l has c h a r a c t e r i s t i c polynomial X 2 - apX + p k - J w h e r e ap is the p-th coefficient of the c o r r e sponding parabolic f o r m and k is its weight. S w i n n e r t o n - D y e r [655] (see also S e r r e [595]) with the help of this r e s u l t investigated the existence of congruences for the coefficients a n of the parabolic f o r m s indicated above. It is found that if l is exceptional in the c a s e w h e r e Imp l is not v e r y l a r g e [does not contain SL(2, Z!)] between reduction rood l of the t r a c e of the m a t r i x p/(Frob (p)) and its d e t e r m i n a n t pk-~ there m u s t be c e r t a i n relations which lead to c o n g r u e n c e s between ap and pk-1 rood I. The p r o b l e m of finding all exceptional p r i m e s l for f thus g e n e r a l i z e s the p r o b l e m of finding c o n g r u e n c e s between T(n) and ~u(n). More p r e c i s e l y , S w i n n e r t o n - D y e r p r o v e s the following result: 1325

A s s u m e that I is exceptional. Then: a) there exists an integer m such that a n - nmc~k_l_2m(n) (mod l) for all n relatively prime to l; b) a m - 0 (mod l) if n is a quadratic non_residue modulo I; c) pl-ka~ -= 0, 1, 2, 4 (mod/) for all p r i m e p ~ l. The set of exceptional prime numbers is finite and can be found explicitly. The exceptional p r i m e s for the case of the Ramanujan function (k = 12) are 2, 3, 5, 7, 23, 691 and no congruences for "r(n) modulo any other p r i m e number exist. Swinnerton-Dyer also c o n s i d e r s possible generalizations of the results obtained to the case of congruences modulo powers of l. In p a r t i c u l a r , he showed that the last three congruences for 7(n) mod2 c~ a r e best possible. Panchishkin [102] showed that congruences for T(n) rood6912 do not exist. Katz [334] completely solved the problem of determining all congruences modulo a r b i t r a r y powers of p r i m e numbers for the F o u r i e r coefficients an of the parabolic f o r m s indicated above. The results of S e r r e [595] and SwirmertonDyer w e r e g e n e r a l i z e d by Ribet [563]. In p a r t i c u l a r , in his w o r k the r e p r e s e n t a t i o n P/,24 a s s o c i a t e d with the space of parabolic f o r m s of weight 24 relative to SL(2, Z) is analyzed, and the c o r r e s p o n d i n g exceptional p r i m e s l a r e found. This is the s m a l l e s t weight to which the r e s u l t s of S e r r e and Swinnerton-Dyer do not apply. Ramanujan conjectured and Watson proved [G. N. Watson, Math. Z., 39, 712-731 (1935); see also Hardy ([283, See. 10.6)] that r(n) - 0 (mod691) for a l m o s t all integers n. S e r r e [597] generalized this r e s u l t in tl~e following way. Suppose that co

f (Z) -- E Cne2ninz/M' M > 1, n~O

is a modular f o r m of integral weight k ~> 1 relative to some congruence subgroup in SL(2, Z). It is a s s u m e d that Cn, n = 0, 1, 2, . . . lie in the ring OK Of integral elements of a finite extension K / Q . For integral m >- 1 let El, m denote the set of those n for which c n ~- 0 (rood m), i.e., e~tnOK; Ef, m(X) is the set of n ~ Ef, m not exceeding x. S e r r e proves that there exists c~ > 0 such that

x--E~,~(x)~O(x/log~x)

for x - ~ c ~ .

In p a r t i c u l a r , Cn = 0 (modm) for a l m o s t all n. F o r the case m = I (a prime number) and e n = 7(n) the following r e s u l t is obtained: let Sl(x) be the set of integers n -< x such that ~(n) ~ 0 (mod l). We have

$2 (x) ~ I X~/2 and Sz(x) ~ ctx/log~(Z)x, if I _> 3; here e l > 0 and the exact e x p r e s s i o n for c~(l) is presented in the paper. The proof makes use of analytic a r g u m e n t s of Landau and Wintner applied to the L-functions a s s o c i a t e d with the l-adic r e p r e s e n t a t i o n s of Deligne [216]. In the w o r k of S e r r e [598] these and analogous questions a r e studied s y s t e m a t i c a l l y with application, for example, to j(z). By s i m i l a r methods Radoux [541] proved the uniform distribution of the values of r(n) in the l - 1 nonzero residue c l a s s e s modulo the prime l. Manin [93, 94,440] obtained v e r y interesting formulas for the eigenvalues of the Hecke o p e r a t o r s . These papers deal, r e s p e c t i v e l y , with F0(N)-parabolic f o r m s of weight 2, F ' - p a r a b o l i c forms of weight 2 (F' is a congruence subgroup of level N), and F - p a r a b o l i c f o r m s of e v e n w e i g h t k (F = SL(2, Z) / + 1 ) . We shall describe some of the r e s u l t s of the paper [94]. Let F = SL(2, Z) / • let w -> 0 be an even integer; let Sw+2 be the space of F - p a r a b o l i c f o r m s of weight w + 2. We consider the f o r m if(z) ~ Sw+2; the numbers tco

rk (~) =~ ~(z)z~dz, O~
a r e called the periods of ft. The author, using the technique of his previous w o r k [93,440] and proceeding c~

f r o m the work of Shimura [609], proves f i r s t the following t h e o r e m on the periods: let ~ ( z ) = ~ ~e2.~z~GS~§ n~l

be a nonzero f o r m which is an eigenfunetion for all Hecke o p e r a t o r s T(n) l r = hn~. Then the ratios (ro (~):r2 (~):... : r , (~)),

(rl (~):... :r~_, (~))

are rational over the field of algebraic numbers O(k~, )~2, . . . . kn . . . . ). F u r t h e r , a t h e o r e m is proved in which [for r0(~) ~ 0] an explicit formula is deduced for the coefficients kn of the f o r m ~.

1326

[(w--2)/4] ZO

n=A~s '-t-S8'

l=i

here

r~=&(~),

%+, ( n ) ~ - ~ ] d ~'§ din

w h i l e in the o u t e r s u m on the r i g h t the s u m m a t i o n g o e s o v e r a l l i n t e g e r s o l u t i o n s of the e q u a t i o n n = AA' + bO' w h i c h s a t i s f y the c o n d i t i o n s : A > 5 > 0 and e i t h e r A' > 6' > 0 o r A t n , A' = n / A , b' = 0, 0 < ~ / A _< 1 / 2 . M o r e o v e r , the t e r m s w i t h 0 / A = 1 / 2 a r e t a k e n w i t h c o e f f i c i e n t 1 / 2 . F o r the c a s e w = 10 w e find

o,~(~)_~ ( ~ ) = ~ ,

691.,

n=AA'+66"

~.~

T8-(~ ~

_

A~) + ~ (~o~4_~4~o).

T h i s y i e l d s a c o m p l e t e l y new p r o o f of the c o n g r u e n c e r(n) - %t(n) (rood 691). In a s i m i l a r w a y it is p o s s i b l e to o b t a i n c o n g r u e n c e s of R a m a n u j a n t y p e f o r the o t h e r p a r a b o l i c f o r m s i n d i c a t e d a b o v e of w e i g h t w + 2 = 16, 18, 20, 22, 26 m o d u l o l a r g e p r i m e n u m b e r s (Manin [94, p. 390]). F i n a l l y , w e note w o r k r e l a t e d to this topic of v a n d e r P o l [663], L e v i n [64, 65, 66], S c h o e n e b e r g [579], and R e s n i k o f f [562] in w h i c h f o r m u l a s a r e o b t a i n e d f o r r(n) i n c l u d i n g Crk(n). On the b a s i s of f o r m u l a s of this type N e w m a n [488] p r o v e d that r(p) =- 0 (modp) f o r p = 2, 3, 5, 7, 2411. N i e b u r [489] o b t a i n e d a s o m e w h a t different formula n--1

(n) = n4: (n) - 24 ~ , 0 5 k 4 - 5 2 k 3 . + 18~n2). o (~) o ( n - k), k=l

by m e a n s of w h i c h K. F e r g u s o n c o m p o s e d a t a b l e of r e s i d u e s of r ( p ) m o d p f o r 3 -< p -< 65,063 but found no new s o l u t i o n s of the c o n g r u e n c e r(p) -= 0 (rood p). F o r c o m p l e t e n e s s we note a l s o the w o r k of M c C a r t h y [ 4 4 2 , 4 4 3 ] . A l a r g e n u m b e r of i n v e s t i g a t i o n s a r e d e v o t e d t o c o n g r u e n c e p r o p e r t i e s of the F o u r i e r c o e f f i c i e n t s of the m o d u l a r i n v a r i a n t j(z) w h i c h i s g i v e n , f o r e x a m p l e , by (x = e 27rlz) ](z)=

/

1-}-240

/7

oa (/t) x ~ lz=l

X~(1--xr)24~X-1+744@ r=l

c(n) xn" n=I

C l a s s i c a l l y the f u n c t i o n J(z) = j ( z ) / 1 7 2 8 w a s c o n s i d e r e d . It is known ( L e h n e r [417, pp. 349-350]} that c(m) b(m -a/4 ae m ; a, b, c o n s t a n t s . T h e r e f o r e , t h e r e e x i s t s no m u l t i p l i c a t i v e taw of the t y p e c(mn) = c ( m ) c ( n ) , (m, n) = 1. L e h m e r [407] f i r s t d i s c o v e r e d c o n g r u e n c e s f o r the F o u r i e r c o e f f i c i e n t s c(n) of the m o d u l a r i n v a r i a n t j (z). L e h n e r [ 4 1 2 , 4 1 3 , 4 1 9 ] p r o v e d that f o r a n y p o s i t i v e i n t e g e r s a and n t h e r e a r e the c o n g r u e n c e s c (2~n) =-0 (mod23~+s), c (3~n) ~ 0 (mod32~+a),

c (5an) =--0(rnod5~+~), c (7~n) ~ 0 (mod7~). L e h n e r a l s o p r o v e d t h a t if a = 1, 2, 3 and n > 0, t h e n c ( l l a n ) ---- 0 (mod l l a ) . A t k i n [127] s t r e n g t h e n e d this r e s u l t by s h o w i n g t h a t it is t r u e f o r any a -> 1. K o l b e r g [363] p r o v e d t h a t 2 aa+~ is the p r e c i s e p o w e r of two d i r i d i n g c(2 a) ( L e h n e r ' s c o n j e c t u r e ) ; t h i s f o l l o w s f r o m the K o l b e r g c o n g r u e n c e c (2an) =----23~+s3~-%rz(n) (mod2a~+~a).

K o l b e r g a l s o o b t a i n e d c o n g r u e n c e s f o r c(8n + 1), e(8n + 3), e(Sn + 5), r e s p e c t i v e l y , m o d u l o 27, 23, 28. In the w o r k [364] K o l b e r g p r o v e d that c ( 3 ~ n ) ~ + 3e=+alO=-'a (n)/n (rood 32'~+6), n ~ +_ 1 (rood 3). Aas

[118,

119] r e f i n e d L e h n e r ' s r e s u l t s i n the f o l l o w i n g w a y :

a) c (5~n) ~ --3~-~5'~+~n~ (n) (rood 5~+2); this p r o v e s L e h n e r ' s c o n j e c t u r e that c (5 a) r 0 (rood 5a+2); b) c (7an) :-- -- 5'~-~7~n% (n) (rood 7~+'); c (7~n) ~- 0 (rood 7~+1), if ( n / 7 ) = - 1.

1327

F o r m o d u l i of the t y p e q a , w h e r e q -> 13 is a p r i m e n u m b e r the c o n g r u e n c e s f o r c(n) a r e m o r e c o m p l i c a t e d . N e w m a n [481] d i s c o v e r e d the b e a u t i f u l c o n g r u e n c e c(13n) -: --T(n) (mod 13), w h e r e 7(n) is the R a m a n u j a n function. By m e a n s of the i d e n t i t y

t/El

~k ( " ) ) ~ n = X~ H (1 - - x n ) 24/~ n~l

K o l b e r g [365] i n t r o d u c e s the f u n c t i o n Tk(n) (a g e n e r a l i z a t i o n of the R a m a n u j a n function) and o b t a i n s f o r n > 0 the c o n g r u e n c e s

c(17n)---7~4(17n) (motif7), c(19n)------4~3(19n) (rnodl9), c (23n) ------13~11 (23n) (rnod23). In the w o r k [481] c i t e d a b o v e N e w m a n a l s o s h o w e d t h a t e (132n) = 8c(13n) (mod 13) and t(nl) - t(n)t(l) + l - l t ( n / l ) 0 (mod 13); h e r e t(n) =- c ( 1 3 n ) / c ( 1 3 ) (rood 13) and 1 r 13 is a p r i m e n u m b e r . U s i n g the f a c t t h a t c(91) --- 0 (rood 13), he d e d u c e d the c o n g r u e n c e s c ( 7 . 1 3 a ) =- 0 (rood 13) if a -> 1; c(91n) ~ 0 ( r o o d 1 3 ) i f (n, 7 ) = 1 . It f o l l o w s f r o m t h e s e c o n g r u e n c e s that c(n) -= 0 (rood 13) w i t h p o s i t i v e d e n s i t y and that c(n) is c o n t a i n e d in e a c h r e s i d u e c l a s s (rood 13) i n f i n i t e l y often. N e w m a n ' s r e s u l t s a r e g e n e r a l i z e d in the p a p e r of A t k i n and O ' B r i e n [133], and i t is p r o v e d that: 1) f o r any a _> 1 t h e r e e x i s t s a c o n s t a n t k s , 13-~k~ , s u c h t h a t f o r a l l n t h e r e is the c o n g r u e n c e c (13~+1n)------k~c (13~n) (rood 13~); 2) c (n) - 0 (mod 133) f o r a n i n f i n i t e s e t of n; 3) f o r a l l c~ _> 1 and a l l a w i t h the c o n d i t i o n (a, 13) = 1 t h e r e e x i s t i n f i n i t e l y m a n y n s u c h that c (n) ~= a (rood 13~). In this p a p e r the f o l l o w i n g c o n j e c t u r e is a d v a n c e d : l e t & _> 1 and t(n) - c(13C~n)/e(13 c~) (rood 13~). Then t h e r e i s the c o n g r u e n c e

t (nl) -- t (n) t (l) + l-~t (nil) -~ 0 (rood 13~), w h e r e l r 13 is a p r i m e n u m b e r . A p r o o f of this c o n j e c t u r e is p r e s e n t e d in the book of L e h n e r [419] w i t h r e f e r e n c e to a n u n p u b l i s h e d p a p e r of A t k i n (see a l s o Koike [358] and r e s u l t s p r e s e n t e d below). A t k i n [130, 131] m a d e s t i l l a n o t h e r c o n j e c t u r e . L e t p -< 23 be a p r i m e n u m b e r , and l e t l ~ p be p r i m e . We s e t t a ( n ) = c(pC~n)/c(p~ w h e r e n and c~ a r e a n y p o s i t i v e i n t e g e r s . A t k i n c o n j e c t u r e s t h a t t~ ( n l ) - t~ (n) t~ (1)+ L-~t~ (~-) ------0 (rood p~),

t~ (np) -- t~ (n) t~ ( p ) ~ O (mod p~), w h e r e t a ( n / l ) = 0 if l ~ n . A c o n j e c t u r e f o r p r i m e s p > 23 has a n a n a l o g o u s f o r m . A t k i n a n n o u n c e d the p r o o f of the c o n j e c t u r e f o r p = 2, 3, 5, 7, 13, but i t a p p e a r s that he h a s not p u b l i s h e d it. T h e l a r g e w o r k of K o i k e [358] is d e v o t e d to c o n g r u e n c e s b e t w e e n m o d u l a r f o r m s and f u n c t i o n s [in p a r t i c u l a r , to a g e n e r a l i z a t i o n of the N e w m a n c o n g r u e n c e for c(13n)] and the A t k i n c o n j e c t u r e . L e t p be a f i x e d p r i m e n u m b e r , and l e t !~ be the s p a c e of a l l m e r o m o r p h i c f u n c t i o n s on the u p p e r h a l f plane i n v a r i a n t u n d e r z ~ z + 1. W e d e f i n e on the s p a c e !~ a n o p e r a t o r U (p): If F (z) E!~, t h e n p--I

L e t M be the Z - m o d u l e g e n e r a t e d by the f u n c t i o n s {fl U(p)n; f ~ Z [ j ( z ) ] , n -> 0}; l e t S k be the s p a c e of p a r a b o l i c f o r m s of w e i g h t k r e l a t i v e to S L ( 2 , Z ) , and l e t Sk, z be the s u b m o d u l e of Sk c o n s i s t i n g of a l l e l e m e n t s of S k w i t h F o u r i e r c o e f f i c i e n t s w h i c h a r e r a t i o n a l i n t e g e r s . The m a i n t h e o r e m of p a r t I of the w o r k of K o i k e [358] is the f o l l o w i n g : L e t c~ be a p o s i t i v e i n t e g e r . T h e n f o r e a c h g ~ M t h e r e e x i s t h ~ Z[j(z)] a n d F(z) ~ Spc~_pC~_~, z s u c h t h a t g - h - F(z) (rood pC~). T h e s e h and F(z) a r e unique rood pC~. S e t t i n g in this t h e o r e m p = 13, c~ = 1, co

g ( z ) = ] (13) ] U (13)= ~ c (13n) e2~l.~, n~0

w e o b t a i n a new p r o o f of the N e w m a n c o n g r u e n c e c(13n) = - - T ( n ) (rood 13), s i n c e

1328

g (z) - - a (0) ~ c (13) a (z) (mod 13), c (13)_~ - - 1 (rood 13). T h e p r o o f of the m a i n t h e o r e m is b a s e d on c e r t a i n r e s u l t s of I h a r a and D e l i g n e ' s t h e o r e m on p - a d i e " r i g i d i t y " of the m a p p i n g j(z) - - j ( p z ) [ s e e B. 13work, P u b l . M a t h . I n s t . H a u t e s E t u d e s S c i . , 3__77, 2 7 - 1 1 5 (1969)]. The A t k i n c o n j e c t u r e is s t u d i e d in p a r t II of K o i k e ' s w o r k [35Sl. The p - a d i e H e c k e o p e r a t o r s a r e d e f i n e d on c e r t a i n p a d i c B a n a e h s p a c e s . The A t k i n c o n j e c t u r e is found to be c l o s e l y r e l a t e d to the e x i s t e n c e a n d c o n s t r u c t i o n of the e i g e n f u n e t i o n s of p - a d i c H e e k e o p e r a t o r s on s u c h s p a c e s . K o i k e c o m p l e t e l y p r o v e s the A t k i n c o n j e c t u r e for p = 13. The p r o c e d u r e of the p r o o f f o r the e a s e p = 13 a l s o g o e s t h r o u g h in the g e n e r a l c a s e u n d e r c e r t a i n a s s u m p t i o n s (which he did not p r o v e ) . F r o m the r e s u l t s p r o v e d by K o i k e in c o n n e c t i o n w i t h a t t e m p t s to p r o v e the A t k i n c o n j e c t u r e w e note the following: for 13 -< p -< 97 e(p) is not d i v i s i b l e by p. F o r the t e c h n i q u e a p p l i e d by K o i k e in p a r t II w e r e f e r to the p a p e r s of K o i k e [358, 359] and I>a~ork [226]. In the w o r k [227] D w o r k s k e t c h e s a p r o o f of the A t k i n c o n j e c t u r e in g e n e r a l f o r m . S e r r e [598] a p p l i e s his t e c h n i q u e to e(n) and o b t a i n s a g e n e r a l i z a t i o n of c e r t a i n of the r e s u l t s f o r m u l a t e d a b o v e : l e t 1 -> 7 be a p r i m e n u m b e r , and l e t r > 0 be a n i n t e g e r . F o r e a c h i n t e g e r a t h e r e e x i s t s a n i n f i n i t e s e t of i n t e g e r s n s u c h that

c(n) --=a(mod #) and

Congruences for the Fourier coefficients of the functions ( j ( z ) - 1728)I/2, (j(z))I/3 are considered in the work of Kl0ve [348] and Erevik [242]. Many papers have been devoted to the classical partition function p(n), where p(n) is the number of partitions of the natural number n, i.e., the number of ways in which n can be written as a sum of positive integers, while two such ways are considered equivalent if they differ only in the order of the terms. ,Gupta [281] has written a detailed survey of the results together with a very extensive bibliography. Here we shall r e s t r i c t o u r s e l v e s to the s t u d y of c o n g r u e n c e p r o p e r t i e s of p(n) o b t a i n e d by m e t h o d s of the t h e o r y of m o d u l a r f u n c t i o n s . T h e w e l l - k n o w n f o r m u l a of E u l e r co

00

1

II(1-x~) rt~l

2

p (n)x~ t~=l

c a n be t r a n s f o r m e d to the f o r m (x = e 2niz) co

e.,z/12.~-i (z)= 1 + ~ p (n) e2~'~, n=!

w h e r e ~(z) is the D e d e M n d e t a function. T h u s , p(n) is r e l a t e d to m o d u l a r f o r m s . R a m a n u j a n [547] c o n j e c t u r e d t h a t i f q = 5, 7, or 11 and 2 4 m - 1 ( m o d q n ) , t h e n p(m) - 0 ( m o d q n ) . He p r o v e d his c o n j e c t u r e f o r n = 1, 2 [547, 548]. W e c o n s i d e r q = 5, n = 1. R a m a n u j a n d e d u c e d his c o n g r u e n c e f r o m a n i d e n t i t y e q u i v a l e n t to the f o l l o w i n g i d e n t i t y w i t h the D e d e k i n d e t a f u n c t i o n : 4

l=0

\ ~ (z) j

( R a d e m a e h e r [533, 539]). A n a n a l o g o u s i d e n t i t y h o l d s f o r q = 7. K o l b e r g [360] o b t a i n e d a n u m b e r of new i d e n t i t i e s c o n t a i n i n g p(qn + s ) , w h e r e q = 3, 5, 7 and s = 0, 1, . . . , q - 1. In [368] K o l b e r g c a r r i e d out a s y s t e m a t i c s t u d y of i d e n t i t i e s of R a m a n u j a n t y p e . W a t s o n [672] p r o v e d the c o n j e c t u r e f o r q = 5 and a l l n and (in s u i t a b l y m o d i f i e d f o r m - the e x p o n e n t n in the c o n g r u e n c e p(m) - 0 ( m o d q n) is r e p l a c e d by [(n + 2 ) / 2 ] ) f o r q = 7 and a l l n. W a t s o n ' s m e t h o d is b a s e d on the c o r r e s p o n d i n g m o d u l a r e q u a t i o n s a n d s e r v e s p o o r l y a l r e a d y in the e a s e q = 11 due to i t s c o m p l e x i t y . L e h n e r [ 4 1 1 , 4 1 4 ] d e v e l o p e d a n o t h e r m e t h o d in a p p l i c a t i o n to t h i s p r o b l e m ; in p a r t i c u l a r , he p r o v e d [414] the R a m a n u j a n c o n j e c t u r e for q = 11 and n = 3. F o l l o w i n g the m e t h o d of L e h n e r , A t M n [127] p r o v e d the c o n j e c t u r e f o r q = 11 and a l l n. A t k i n and S w i n n e r t o n - D y e r [135] p r o v e d s o m e of the R a m a n u j a n c o n g r u e n c e s w i t h o u t m o d u l a r c o n s i d e r a tions. 1329

N e w m a n [486] p r o v e d t h a t the d e n s i t y of i n t e g e r s n s u c h that 51 p(n) is s o m e w h a t l a r g e r t h a n 1 / 5 . In [484] N e w m a n a d v a n c e d a n i n t e r e s t i n g c o n j e c t u r e : f o r any i n t e g e r r , 0 -< r -< m - 1, the c o n g r u e n c e p(n) ~ r (mod m) h a s i n f i n i t e l y m a n y s o l u t i o n s in n o n n e g a t i v e i n t e g e r s n. N e w m a n v e r i f i e d i t f o r m = 2, 3, 13. In the w o r k [485] N e w m a n v e r i f i e d the c o n j e c t u r e for m = 65 and e s t a b l i s h e d c e r t a i n c o n g r u e n c e s f o r p(n). I n the p a p e r of A t k i n and O'Brien [133] the study of e(n) and p(n) is conducted in parallel. Let P(N) = p(n) if N = 2 4 n - i ; let P(N) = 0 if N < - 1 or N ~ - 1 (rood24) or N is nonintegral. The authors prove the following theorem: For all a -> 1 there exists an integral constant Ka not divisible by 13 such that for all N P (13~+2N) --=I(~P (13~N) (rood 13~). The a u t h o r s c o n j e c t u r e the f o l l o w i n g : s u p p o s e t h a t a >- 1 and p ~ 13 is a p r i m e n u m b e r >-5. T h e n t h e r e e x i s t s a c o n s t a n t k = k(p, c~) s u c h t h a t f o r a l l N p (p2.13aN) - - {k - - ( - - 3.13~N/p) p-~}. P (13~-N) § p-ap (13~N/p2) ~ 0 (rood 13~), w h e r e ( a / b ) is the L e g e n d r e s y m b o l . The c o n j e c t u r e i s v e r i f i e d in the c a s e s a = 1 , 2 . the f o l l o w i n g r e s u l t s :

The a u t h o r s f u r t h e r p r o v e

1) P(593. 13 N) --- 0 (mod13) if (N, 59) = 1; 2) p(n) = 0 (mod 134) f o r a n i n f i n i t e s e t of n; 3) f o r a l l a >- 1 a n d a l l a w i t h the c o n d i t i o n (a, 13) = 1 t h e r e e x i s t s a n i n f i n i t e s e t of n s u c h that p(n) --a (rood 1 3 a ) ; 4) p(3373n 2 - (n2 - 1 ) / 2 4 ) = P ( 1 3 2 . 4 7 9 n 2) ~- 0 (rood132) if (n, 6) = 1; 5) P(972" 1032. 132N) = 0 (rood 132) if ( N / 9 7 ) = ( N / 1 0 3 ) = - 1 . The a u t h o r s ' m e t h o d i s e l e m e n t a r y and is b a s e d on u s i n g the m o d u l a r e q u a l i t y b e t w e e n ~?(169z)/~(z) and ~2(169z)/~2(13z). T h e r e s u l t s of A t k i n and O ' B r i e n c o n f i r m in c e r t a i n ( v e r y s p e c i a l , to be s u r e ) c a s e s the f o l l o w i n g s t r e n g t h e n i n g of the N e w m a n c o n j e c t u r e due to t h e m : for g i v e n a, m the c o n g r u e n c e p(n) = a (mod m) is s o l v a b l e f o r v a l u e s of n w i t h p o s i t i v e d e n s i t y . (The w o r k [302] is a l s o d e v o t e d to t h i s c o n j e c t u r e and i t s a n a l o g s . ) S o m e of the r e s u l t s of A t k i n and O ' B r i e n w e r e o b t a i n e d e a r l i e r by N e w m a n [ 4 8 0 , 4 8 1 ] . W e s h a l l now d e s c r i b e the c o n t e n t of A t k i n ' s w o r k [129]. L e t q = 5, 7 , o r 13, and l e t a = 6, 4 , o r 2 c o r r e s p o n d i n g l y . S u p p o s e a l s o t h a t p >- 5 is a p r i m e n u m b e r d i s t i n c t f r o m q. T h e n

tp (N) ~ pap (Np2) + p ( _ 3 N /p) P (N) -~ P (Nil -2) ~ 7;P (N) (rood q~), i f ( - N / q ) = - 1 ; h e r e "yp is a n i n t e g r a l c o n s t a n t not d e p e n d i n g on N, and 7p - P(P + 1 ) ( 3 / p ) ( m o d q ) . A s a n a p p l i c a t i o n of t h e s e r e s u l t s , A t k i n e s t i m a t e s the q u a n t i t y

x~co

n~
)~0(odm)

in the cases m = 5, 7, or 13. It is asserted that the results indicated above lead to a proof that p(n) belongs to each class of residues rood (56. 74. 132) infinitely often. An entire sequence of congruences for P( ) and lower bounds for d( ) in the case of other moduli is also announced. Kolberg

[361,362]

and Klgve

[349] proved

a number

of congruences

for p(n).

The work of K_I0ve [348] contains a large number of congruences for the Fourier coefficients of various modulo functions and, in particular, j(z); moreover, the Newman conjecture mentioned above regarding an infinite number of solutions of the congruence p(n) -= r (rood m) is verified in this work for m = 17 and certain other large primes m = p. In the work of Klove [351] the Newman conjecture is verified for m = 112. Moreover, the following estimatea are obtained: d(5) >0.20194, these improve

somewhat

the corresponding

d(7) >0.15069, d(13) >0.00204;

estimates

of Atkin [129].

Analogs of the function p(n) have also been investigated. For example, let q(n) be the number of partitions into n distinct parts, and let q0(n) be the number of partitions into n distinct odd parts. The congruence properties of these functions are similar in many ways to those of p(n) (see [566,567] and the bibliography given there).

1330

We

note, finally, the work

[683] in which theta analogs of the Ramanujan

identities are found, for example,

co

Z ( - - 1)mQ (3m) e €

=- ~ (Jz) ~(z);

m~0

the c o e f f i c i e n t s Q(m) a r e g i v e n by

0-~(z)= ~ ( - 1)~Q (m) e"~"z, ~ ( z ) = ~ (Oiz). trt~0

6.

Kronecker We

Limit

Formulas.

Dedekind

Eta

Function

first recall classical facts. Let z ~ H, and let y(z) be the imaginary

part of z, i.e., y(z) = y > 0 for

z = x + iy; l e t F = S L ( 2 , Z ) , and l e t F1 be s u b g r o u p c o n s i s t i n g of , = (~) c d GP w i t h the c o n d i t i o n e = 0; l e t or(z) = (az + b) (cz + d) -~. The n o n h o i o m o r p h i c E i s e n s t e i n s e r i e s r e l a t i v e to F i s d e f i n e d by

e* (z, s)= ~ {y(~(z))}~ (Res> 1). ff~Pt\P

It is e a s y to s e e that

E* (z, s)= ;--i-NE (z, s), where

1~

gs i mz + n l2S.

E (Z, S ) = -~

ttz , r ~ - - o o (m,n)@(O,O)

In o t h e r w o r d s , E * ( z , s) a s a f u n c t i o n of s is e s s e n t i a l l y the E p s t e i n z e t a f u n c t i o n of a p o s i t i v e d e f i n i t e b i n a r y q u a d r a t i c f o r m . T h e s e r i e s E (z, s) e x t e n d s to the e n t i r e s p l a n e w i t h a s i n g l e s i n g u l a r i t y : a s i m p l e pole at the point s = i.

T h e f i r s t l i m i t f o r m u l a of K r o n e e k e r a s s e r t s Iim g"

that =2~(C--Iog2--1og(JfT[~(z)[2)),

m q - n z ] - 2 ~ - S---I ~

(1)

s~l [(m,n)~(0,0)

w h e r e z ~ H, C is E u l e r ' s c o n s t a n t , and O(z) is the D e d e l d n d e t a f u n c t i o n 0o

(z) = e~i~/I~I I (1 - e 2~t~) (zCH). D e d e t d n d p r o v e d t h a t (if the p r i n c i p a l b r a n c h of the l o g a r i t h m is taken) az + + db'l= log-q |~.cz ) log ^q (z) T 2I l o g ( c z + d ) + ~ i S ( a , 1 c

4 let

b, c, d),

s(G d); s(c, d} is c a l l e d the D e d e k i n d s u m and h a s the

elementary expression Jc]--I ~=0

'

w h e r e ((x)) = x - [x] - 1 / 2 , Ix] i s the g r e a t e s t i n t e g e r - x . K r o n e e k e r a l s o c o n s i d e r e d the s e r i e s e2,~l(mu+nv) ,

F(s, z; u, v ) = g ~

Im§ rtt,n~-oo ( ra,n)4qO,O)

(Re S > 1),

(2)

1331

w h e r e (u, v) ~ R 2 but z ~ H. The s e r i e s (2) extends analytically to the entire s plane. In the case (u, v) E Z 2 we a r r i v e at the s e r i e s (1); for (u, v) ~ Z 2 the function F(s, z; u, v) is analytic at the point s = 1. In the second limit f o r m u l a of Kronecker (which we do not p r e s e n t here) the value of F(1, z; u, v) is computed. For these and other c l a s s i c a l facts see Siegel [632] and R a d e m a c h e r [539]. As concerns the s u m s(c, d), the following r e m a r k a b l e r e c i p r o c i t y t h e o r e m is due to Dedekind: s(c,d)+s(d,c)--

I

1 9 e

.

1

. d

if (c, d) = 1. It is interesting that the quadratic law of r e c i p r o c i t y of Gauss follows immediately f r o m the r e c i procity t h e o r e m for s (d, c). Indeed, let c be an odd number. The following formula holds [here (d/c) is the Legendre symbol] : d ~

1

c--I

-~]=(--1)~

, c~-I (rood2).

Regarding this and other facts we r e f e r to the s u r v e y of Zagier [697]. The l i t e r a t u r e devoted to the study and g e n e r a l i z a t i o n of Dedekind sums is enormous (see R a d e m a c h e r and Grosswald [540]}; here we therefore limit ourselves to a d e s c r i p t i o n of some r e c e n t results. Goldstein [261] g e n e r a l i z e d K r o n e c k e r ' s limit formulas to the case of E i s e n s t e i n s e r i e s relative to any Fuchsian group of f i r s t kind F iF is a d i s c r e t e subgroup of the group SL(2, Z) with finite invariant volume]. Here a nonzero F - a u t o m o r p h i c f o r m of weight 1 / 2 [the analog of ~(z)] and c e r t a i n sums generalizing the Dedekind sums a r i s e . The author studies their a r i t h m e t i c properties and advances s e v e r a l conjectures. Many authors have c a r r i e d over the K r o n e c k e r formulas to fields of algebraic numbers. The foundation was laid by Hecke in his w o r k of 1917 [284]; his w o r k was continued by Herglotz [287] and Meyer [448]. These works devoted to real quadratic fields w e r e s y s t e m a t i z e d and extended by Zagier [694]. We shall briefly p r e sent the r e s u l t s they contain. Let K be a number field, and let ~K(S) be its Dedekind zeta function; we expand ~K in the finite s u m (s) =

(s, A), A

where A runs through the group of ideal c l a s s e s of the field K and 1

N~

(Re s > 1).

aEA The function ~(s, A) extends analytically to the entire s plane with only one singularity which is a simple pole at s = 1 with residue ~ (not depending on A). In a neighborhood of the point s = 1 (s, A) = s ~ l

+ p (A) + ?1 (A) (s-- 1) + . . . .

In the s i m p l e s t ease (K is an imaginary quadratic field) an e x p r e s s i o n for p(A) is given by the f i r s t Kronecker limit formula. It is easy to explain the interest in computing p(A). Indeed, suppose X is any nontrivial c h a r a c t e r on the group of ideal c l a s s e s ; it is easy to show that L (1, X)= ~ )C(A) p (A)

(Z =#=X0).

A

Thus, it becomes c l e a r that the limit formulas can be applied to obtain formulas for the class numbers of ideals of fields of algebraic numbers. Many authors have concerned themselves with such problems; we r e f e r the r e a d e r to the books of Meyer [448] and Siegel [632] and the papers of R a m a e h a n d r a [544], Schertz [575], etc. Hecke [284] computed p(A) in the case of a real quadratic field K; Hecke's e x p r e s s i o n for p(A) is an integral containing the function log I ~(z) l. Meyer [448] applied H e c k e ' s method to ideal c l a s s e s B in the r e s t r i c t e d s e n s e of a real quadratic field K. It is c l e a r that M e y e r ' s results a r e new only when all units of the field K have positive n o r m s . In this case each class of ideals A in the broad sense is the union of two c l a s s e s of ideals B and B* in the r e s t r i c t e d s e n s e , whence p(B) + p(B*) = p(A). Meyer [448] computed the difference p(B) - p(B*) in t e r m s of Dedekind sums. Z a g i e r [694] computes p(B) in different t e r m s than Meyer. As a c o r o l l a r y of his r e s u l t s he obtains the following result: Let p -= 3 (mod 4) be a prime number with the condition h(p) = 1 [h(q) is the number of c l a s s e s of the field Q(~q)]. Let l+ and l_ be the period lengths of the continuous fractions for ~p and -~fp. Then l_ - l+ = 3h(-p).

1332

We remark that still early modular considerations led Hirzebruch to an analogous formula for h(-p) (see Hirzebruch [298,300], Zagier [697]) which generated an entire sequence of papers (see the paper of Lang [402] and the literature cited there). Limit formulas for fields of higher degrees were obtained by Konno [370], Katayama [326], Asai [122], and Goldstein [260,262]. Kormo [370] considered an imaginary quadratic extension K of a totally real field of algebraic numbers k. Let n be the degree of k/Q; let ~ be the absolute class of ideals in K. Konno showed that in the expansion

~K (s; ~)~-a~/(s--l)+ao+a~ (s--1)i... the constant a 0 is expressed in terms of the special value log T~ (z(~).... , z(,l; m, n) with some analytic function ~k defined on H x . . . x H = H n [generalized functions N(z)]; m,n are some integral ideals in k. Let h K be the number of classes of ideals of the field K. As an application of his results, Konno obtained a formula for hF/hK, where F is the absolute field of classes over K. Goldstein [260] extended Konno's investigations by establishing a Kronecker limit formula for nonanalytic Eisenstein series of the Hilbert modular group for a totally real field k. He applied his investigations to the following well-known conjecture of Hecke: If K is an imaginary quadratic extension of the field k, then there exists an "elementary" formula for hK/h k. Goldstein computes the ratio hK/h k in terms of the periods of certain complex differential forms connected with the generalized q-function. It follows from Siegel's work [637] that these periods are effectively computable rational numbers. However, Goldstein did not prove the Heeke conjecture. This was done recently by Shintani [627] who found for h K /h k an elementary arithmetic expression in terms of the relative discriminant. Asai [122] obtained a Kronecker limit formula in the case of an arbitrary field of algebraic numbers F (which is assumed to be a single-class field for simplicity). Let 9 be the ring of integers in F; let rl, r 2 be the number of real and complex norms of the field F. The product ~--/-/~, >(/-/~, is taken as the analog of the upper half plane; here H is the ordinary upper half plane and Hq is the quaterr~ion upper half space. The Hilbert modular group F~SL(2, 9 acts on the space ~ . Let E(z, s), where z ~$4, be the Eisenstein series for the group F. This series extends holomorphically to the entire s plane with one singularity which is a simple pole at the point s = I. For the series E(z, s) Asai finds a limit formula analogous to the first Kronecker limit formula. This formula contains a harmonic modular form h(z) [the analog of log I ~)(z)I] the properties of which are studied in some detail. In the work [123] Asai deduces an analog of the function 77(z) associated with a cyclotomic field. There are a n u m b e r of proofs of the Dedekind formula

l o g ~ (= ( z ) ) ~ l o g ~

( z ) - } - y1l o g

l1~~+7d- -~! - p =_ t ~ .-~- ~+z~s

" (d,c)

(c>O), 539], Chowl,a [591],

Weft [676], Knopp [354], Goldstein and de la Torre [265]). The proof of Chowla [591] and Well [676] of the formula log ~ (-I/z) -- log ~ (z) + I/2 log z/i is based on the connection of modular forms and Dirichlet series with a functiomal equation by means of the Mellin transform. Indeed, let ~(s) = (2~)-SF(s)~0(s), where ~(s) = ~(s)~(s + i). It is easy to see that the Mellin transform of the function 9 is o+i~

f

G--too

w h e r e ~ > 1, I m z > 0. By d i s p l a c i n g the line of i n t e g r a t i o n to the left, computing the r e s i d u e s at the points s = 0, ~:1, and applying the functional equation ~(s) = ~ ( - s ) , we a r r i v e at the d e s i r e d f o r m u l a for log 7 ( - 1 / z ) . In the w o r k [265] this method of p r o o f is a p p l i e d to any a E SL(2, Z). Analogous c o n s i d e r a t i o n s a r e used in the W e l l c h a r a c t e r i z a t i o n of m o d u l a r f o r m s [675]. It is i n t e r e s t i n g that Hecke h i m s e l f in w o r k of the y e a r 1924 applied such arguments in considering the modular properties of the generalized Dedekind ~-function associated with the Eisenstein series of the Hilbert modular group of a real quadratic field (Hecke [284], No. 20). The work of Hecke was recently extended by Goldstein and de la Torre [266]. These authors studied the modular properties of the generalized eta function introduced in the work of Konno [370] and Goldstein [260]. The formulations of the results of Goldstein and de la Torre are very complicated, and we sl~all not present them in the present survey. In the work [262] Goldstein obtained a Kronecker limit formula for the case of the Lseries of the class of ideals with a Hecke character (Grossencharakter)of an arbitrary field of algebraic numbers. His formula is rather complicated.

1333

T e r r a s [657] studied the Epstein zeta function Zn

(S, ~)=-~ Z (laSa)-9' a~Zn-O

where S is the n • n matrix of a positive definite {real) quadratic f o r m , Rep > n / 2 , and the sum is taken over all column n - v e c t o r s with integer coefficients which do not vanish simultaneously. She obtained an expansion of the function Zn in t e r m s of Bessel functions by generalizing the known r e s u l t of Selberg and Chowla [590] pertaining to the case n = 2, and by investigating the Laurent expansion of the function Zn in a neighborhood of p = n / 2 , she a r r i v e d at the f i r s t Kronecker limit formula for Zn(S , P). Her formula also contains a c e r t a i n generalization of ~?(z). In [659] T e r r a s applied the r e s u l t s of the previous w o r k to obtain formulas for Z2(I , 3/2) and ~(3). For further results in this direction see T e t r a s [658,660]. There are a large number of papers in which the ~?-function and the Dedekind sum are generalized f r o m various points of view. See, e.g., Berndt [149-151], Bodendieck and Halbritter [166], Carlitz [181, 182, 184187], Lang [399], Lewittes [423], Meyer [449,450], R a d e m a c h e r [534, 537,539], Schoeneberg [ 5 8 1 , 5 8 3 , 5 8 4 ] , Z a g i e r [692], etc. We note a paper of Lehner [418] devoted to the multipliers of 77(z). Let s = SL(2, Z). We recall that ~(z) is a F - m o d u l a r f o r m of weight 1 / 2 with a s y s t e m of multipliers v. Let G be the set of all A ~ F for which v(A) -1. It is c l e a r that G is not a group, but there exist groups ' for example, {S24}, where S = (101) 1 , which a r e subsets of the set G. Lehner proves the following conjecture of R a d e m a e h e r : each subgroup of the group F contained in G is cyclic. It is also proved that G is contained in the eommutant of the group F. We pass now to questions connected with the powers of ~(z). Let e 2'7iz = x, and let ~(z) = xl/24~p(x), where co

(x) = I I 0 - x.). rt~l

We expand ~d~) in powers of x: oo

~oa (x)=

~ p,~(rn)x".

Many authors have undertaken finding formulas for Pd(m) for specific d (see the interesting r e p o r t of Dyson [228]). These formulas have been found in the following c a s e s : d=3, 8, 10, 14, 15, 21, 24, 26, 28, 35, 36 . . . .

(3)

We present some of them: 1) ~ a ( x ) = ~ ( - - 1)k (2k+ 1) X~ ( k + ~ 2)

~2O(x)~ Z

(Jacobi);

(1/6)mn(m+n)(rn+2n) x(u~2)1(m+'~

(Winquist [682]);

m~-l(mod6) n----~l(mod6)

3) for d = 24 there is the r e m a r k a b l e f o r m u l a (Dyson [228])

, (n) -~f_~ (a-b) (a--c) (a--a) (a -e) (b--r)it213!4I(b- 6 ) (b-e) (c--d)(c--e) (a-e) w h e r e the s u m extends over all sets of integers a, b, c, d, e satisfying the conditions a, b, e, d, e ~ 1, 2, 3, 4, 5(rood5), a + b + c + d + e = 0 , a 2 + b 2 + c 2 + d 2 + e 2 = 10n. Macdonald in the important w o r k [ 4 3 6 ] gave a formula for pd(m) if d is the dimension of a simple Lie algebra. The numbers d in (3) are p r e c i s e l y (if d = 26 is dropped) the dimensions of the simple algebras A1, A2, B~, G~, A3, B~, A4, D4, As, B4. . . . . Macdonald obtained his identities r e g a r d i n g the ~-funetion by specializing more general formulas for an affine root s y s t e m . Macdonald's paper generated a s e r i e s of works within the f r a m e w o r k of the theory of Lie algebras (Garland [251], Verma [665], Moody [458], Kats [47], Garland and Pepowsky [252], Kostant [ 3 7 1 ] ) . Van Aseh [125] and van Aseh [126] proved the Macdonald identity for vd using modular forms. Let V be a vector space of

1334

d i m e n s i o n l _> 1, and l e t R(~V) be a r e d u c e d o r un_reduced r o o t s y s t e m ; w e e q u i p V w i t h a s c a l a r p r o d u c t <~*, ~2) i n v a r i a n t u n d e r W{R), the W e y l g r o u p of the s y s t e m R; l e t q(~) = 1/2<~, {), ~ ~ V. S u p p o s e t h a t { ~ , . . . . . o~l} is a b a s i s of the s y s t e m of r o o t s R , and l e t r be the n u m b e r of p o s i t i v e r o o t s . L e t L(cV) be a l a t t i c e s u c h t h a t the q u a d r a t i c f o r m q ( L ) ~ Z and s L = L f o r a l l s e W ( R ) ; l e t L* be the l a t t i c e d u a l to L. L e t ~ ~ L*. W e consider the f u n c t i o n (4)

0(z, L , ~ ) = ~ II < ~+~,~ > d "'~+t) ~,~L cx>o

This is a theta function with spherical function

~-~II < ~, a > relative to the quadratic

form q. It is a modular

a>0

form of weight d/2 relative to a congruence subgroup of the modular group; here d = l + 2r. To each simple Lie algebra it is possible to assign a theta series (4) of special form. Using the modularity of the series (4) and of the function 7)2d(z), it is possible to show that the function 82(z, L, ~)/N2d(z) is bounded and hence constant. This implies the formula for 7?d(z) of Macdonald ([436, formula (8.9)]). In the work [125] this method is applied to obtain new identities. Polynomials are considered which are skew symmetric with respect to the Weyl group. Using a theorem of Chevalley, the author determines the dimension of the space of homogeneous, skew-symmetric spherical polynomials of any degree. On the basis of these results, the author obtains several new identities which include, in particular, together with N (z) also the Eisenstein series. Identities for ~73~ and 774~ are presented as examples. F i n a l l y , it is to be n o t e d that D e d e k i n d s u m s a r e c o n n e c t e d w i t h c e r t a i n t o p o l o g i c a l p r o b l e m s ~ F o r this w e r e f e r to the s u r v e y s of H i r z e b r u c h [ 2 9 6 , 2 9 7 ] and the p a p e r of Z a g i e r [692]. 7.

Analytic

Number

Theory

and

Modular

Forms

W e b e g i n w i t h e s t i m a t e s of F o u r i e r c o e f f i c i e n t s . In 1939 R a n k i n ([550], P a r t ID p r o v e d t h e e s t i m a t e a n = o(nk/2-1/5); an, n - t h F o u r i e r c o e f f i c i e n t of a F - p a r a b o l i c f o r m of i n t e g r a l w e i g h t k > 0; F , a c o n g r u e n c e s u b g r o u p of l e v e l N. It has b e e n m e n t i o n e d a b o v e t h a t by D e l i g n e ' s r e s u l t s [ 2 1 6 , 2 1 8 ] the F o u r i e r c o e f f i c i e n t s of the f u n c t i o n c~

f (z)=~.a ane2~inzfS~ (N, ~), k is an iateger > 2 , n~I

' ~-!+~ a r e e s t i m a t e d by

a~=O \n ~ /.

E a r l i e r the e s t i m a t e an = O (n ~/2-1/4+~)

(1)

w a s known (see R a n k i n [551, 560], S e l b e r g [589], and M a l y s h e v [90]). The e s t i m a t e (1) is a l s o v a l i d in a m u c h m o r e g e n e r a l s i t u a t i o n : f o r the F o u r i e r c o e f f i c i e n t s of p a r a b o l i c f o r m s of i n t e g r a l w e i g h t r e l a t i v e to a c o n g r u e n c e s u b g r o u p of the H i l b e r t m o d u l a r g r o u p (Gundlach [277]). The p r o o f of e s t i m a t e s of the t y p e (1) is b a s e d on the r e p r e s e n t a t i o n of the p a r a b o l i c f o r m a s a l i n e a r c o m b i n a t i o n of P o i n c a r e s e r i e s . The e x p r e s s i o n f o r the F o u r i e r c o e f f i c i e n t s of the P o i n c a r e s e r i e s c o n t a i n s K l o o s t e r m a n s u m s . U s i n g the known e s t i m a t e s of W e f t f o r K l o o s t e r m a n s u m s , w e o b t a i n the d e s i r e d r e s u l t . F o r e s t i m a t e s of the F o u r i e r c o e f f i c i e n t s of p a r a b o l i c f o r m s r e l a t i v e to h y p e r - A b e l i a n g r o u p s s e e the w o r k of G u n d l a c h [278]. ~ a - R e s u l t s for the F o u r i e r c o e f f i c i e n t s of m o d u l a r f o r m s of i n t e g r a l w e i g h t a r e a l s o known (of the l a t e r w o r k s e e R a n k i n [560], J o r i s [323]). F o r e x a m p l e , in the c a s e of the f u n c t i o n A(z) J o r i s p r o v e d t h a t

,(n)=~2(nn/2exp (c

(I~ n)1122 (,og log n)23/22 )) '

w h e r e c > 0. Such r e s u l t s show t h a t the e s t i m a t e of D e l i g n e is b e s t p o s s i b l e . E s t i m a t e s of the F o u r i e r c o e f f i c i e n t s of m o d u l a r f o r m s of a r b i t r a r y w e i g h t r e l a t i v e to any d i s c r e t e g r o u p w i t h a p a r a b o l i c v e r t e x a t ~o have b e e n o b t a i n e d by P e t e r s s o n [515] ( L e h n e r [416] o b t a i n e d s o m e r e f i n e m e n t ) ; s e e a l s o Ogg [500]. W e s h a l l d e s c r i b e the e s t i m a t e s of the F o u r i e r c o e f f i c i e n t s of p a r a b o l i c f o r m s of h a l f i n t e g r a l w e i g h t o b t a i n e d by P a r s o n [507]; l e s s g e n e r a l r e s u l t s w e r e o b t a i n e d e a r l i e r by Knopp and S m a r t [355]. L e t F be a c o n g r u e n c e s u b g r o u p of l e v e l N; l e t r _> 1 / 2 be a h a l f i n t e g e r . We c o n s i d e r the s p a c e S r ( F , v) of F - p a r a b o l i c f o r m s of w e i g h t r w i t h s y s t e m of m u l t i p l i e r s v. L e t v = vlv u, w h e r e v~ is a c h a r a c t e r and v 2 is a s y s t e m of m u l t i p l i e r s f o r U-l(z) p r o v e d that f o r n - - ~

u

4(-r/2

+ i-r/2]).

LetF(z)--~.~a,,e((nq-~)z/

)GS~(I, v); h e r e e(z)

e 2~iz. P a r s o n

~+,>0

1335

a== 0 (n r/2-b4 log3,~n ~-1/~ (nt + ~.t)). The method of proof is the usual one - reduction of the p r o b l e m (by means of Poincare series) to an estimate of generalized K l o o s t e r m a n sums. As the example of the function ~3(z) shows, without additional r e s t r i c t i o n s the estimate obtained is best possible. Linnik [425] advanced the following conjecture: let N be a large integer; let

T (N, g ) = ~

exp 2hi ~- (x' + N x)

X mod

g"

be a K l o o s t e r m a n sum, and suppose that gl > N1/~-e~ (a0 > 0 is a r b i t r a r i l y small). Then

T (N, g) = 0 (g]+~) g-~gx

for each a > 0. A proof of this conjecture would make it possible to move forward in a number of problems (see Linnik [425]). The excellent s u r v e y of Selberg [589] contains, together with a discription of various methods of estimating the F o u r i e r coefficients of modular f o r m s , a d i s c u s s i o n of the Linnik conjecture and its generalizations to K l o o s t e r m a n sums a s s o c i a t e d with various d i s c r e t e groups. Let a(n) be an a r i t h m e t i c function, and let A o ( x ) ~ ]

a(n)(x--n)o be the Riesz sum. C h a n d r a s e k h a r a n and

n~:X_ _

N a r a s i m h a n [193, 194] established that (see also Berndt ([146], part VII; K e l l e r [335]) for a broad class of a r i t h m e t i c functions Ap(x) is e x p r e s s e d as an infinite s e r i e s of Bessel functions. In p a r t i c u l a r , this is true for the F o u r i e r coefficients of modular forms. For example, for a(n) = T(n)

P(p_l+l),~
co

1

= . 2[ L T ~~162 .=~ ~ .

},

:

w h e r e J~(x) is the usual Bessel function of o r d e r #. Here x > 0, p > - 1 / 2 , and s means that for n = x , z(x) is t a k e n w i t h multiplicity 1 / 2 . This identity for p > 0 was proved by Wilton [680] and for p = 0 by Hardy [282]. There are also other identities with the coefficients of modular f o r m s ; e.g. (see [193]), for R e s > 0

In the w o r k [146, part II] Berndt studied identities for sums of the f o r m

F o r simplicity, below we shall consider the Ramanujan function co

h (Z)=~.~ "~(n) e 2~nz. t~l

Rankin [550, part II] proved that

Using this result, he [550, part Ill] proved that 6

1

At p r e s e n t the best r e s u l t is

( 6-~-+81

T (x) = O t x

j,

obtained by Walfiez [669] on the basis of the Ramanujan conjecture (proved by Deligne). The optimal result should probably be s t r o n g e r , since according to Walfiez [669, 20] T (x) ~ ~ (x23/4).

This estimate of Walfiez has now been somewhat

improved

(see Joris [320], [321]):

T (x) ~-~+ (x 23/4log log log x), 1336

w h i l e in the p r o o f the f o l l o w i n g e s t i m a t e ( f i r s t p r o v e d by W a l f i c z [21]) w a s u s e d : t h e r e e x i s t s a c o n s t a n t c > 0 such that n<:x

I n the p a p e r of J o r i s [322] e x t e n d i n g the w o r k of C h a n d r a s e k h a r a n and N a r a s i m h a n [194] the ~ ] - r e s u l t s i n d i c a t e d a r e c a r r i e d o v e r to a b r o a d c I a s s of m u l t i p l i c a t i v e f u n c t i o n s . S i n c e the f u n c t i o n ~ ~2 (n)n -~ s a t i s f i e s a f u n c t i o n a l e q u a t i o n of the s a m e t y p e a s in the w o r k of C h a n nel d r a s e k h a r a n and N a r a s i m h a n (see t l a n k i n [550, p a r t II], t h e i r r e s u l t s a r e a l s o a p p l i c a b l e to the f u n c t i o n a(n) = r2(n) (see K e l l e r [335]). H a r d y [283] w a s a l r e a d y i n t e r e s t e d in the p r o b l e m of c a r r y i n g o v e r the t h e o r e m on prime numbers ~ logp~x

to the c o e f f i c i e n t s of m o d u l a r f o r m s . He noted t h a t the p r i m e - n u m b e r

theorem

p~
f o r the R a m a n u j a n f u n c t i o n 9 (p) log p = 0 (x~: ~) p<;x

w o u l d follow f r o m the f a c t t h a t

~.(s)~.~(n)n-s%O

for

Res=13/2

[the s t r i p 1 1 / 2 -< Re s -< 1 3 / 2 is c r i t i c a l f o r ~(s)]. R a n l d n p r o v e d the a b s e n c e of z e r o s of ~(s) on the l i n e R e s = 1 3 / 2 [550, p a r t t]. M o r e n o [465] p r o v e d t h a t q)(s) ~ 0 f o r s = a + i t in the d o m a i n ~ >- 1 3 / 2 - B / l o g (I tl + 2), B > 0. T h i s i m p l i e s t h a t 9 (2) tog p << x~a/2 exp ( - - A (log

x)~/2),

p~
w h e r e A > 0. W e c o n s i d e r the f u n c t i o n ?~ ( s ) = ~ - . 2 ( n ) n - n - ~ ; i t c a n be e x p a n d e d in a n E u l e r p r o d u c t ; the s t r i p 0 -< R e s _< 1 is c r i t i c a l f o r it. It is p o s s i b l e to s h o w (lVIoreno [466]) t h a t ~l(s) has no z e r o s in the d o m a i n ~ --1 - C / l o g (2 + I tl), C > 0; this l e a d s to the a s y m p t o t i c f o r m u l a

"~ :~(p)p-**logp=x-f-O(xe-~

D>0,

x - + oo.

p~
It is a l s o p o s s i b l e to s h o w ( M o r e n o [466]) t h a t f o r s o m e 0 < 1 and a l l x -> x 0 92 (P)P-lI l o g p >) x~ x~p-.
M o r e o v e r , it w a s p r o v e d in [117] t h a t if 0 = 1 - 103, t h e n in the i n t e r v a l ix, x + xO], x > x 1 t h e r e is a l w a y s a p r i m e p w i t h the c o n d i t i o n r2(p)p - n > 1 / 2 . By D e l t g n e ' s t h e o r e m , f o r any p r i m e p w e have r2(p)p - ~ < 4. oo

The R i e m a n n h y p o t h e s i s is c o n j e c t u r e d to be t r u e f o r the f u n c t i o n ~ ( s ) ~ ~

~ (n) n -~ : n i l the z e r o s of q0(s)

in the c r i t i c a l s t r i p l i e on the l i n e R e s = 6. G o i d s t e i n [259] and M o r e n o [467] found n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r the v a l i d i t y of this c o n j e c t u r e . W e s h a l l p r e s e n t the r e s u l t of M o r e n o : the R i e m a r m h y p o t h e s i s f o r ~o(s) is e q u i v a l e n t to the e s t i m a t e ~ (p) <
Good [268-270] o b t a i n e d a n a p p r o x i m a t e f u n c t i o n a l e q u a t i o n f o r
~ - f 1~ ( ~ + i 0 pdt,wh~re

6
0

He p r o v e d , m o r e o v e r ,

t h a t qo(6 + it) = O(I i l l / 2 ) , J t[ - - ~.

1337

8.

Various

Results

I. The Values of Z e t a Functions at I n t e g r a l P o i n t s . Various methods have been a p p l i e d to p r o b l e m s of this type. As a l r e a d y mentioned a b o v e , the K r o n e c k e r l i m i t f o r m u l a s have b e e n s u c c e s s f u l l y used in computing v a l u e s of the z e t a functions at the point s = 1 (ef. Sec. 6). Heeke a l r e a d y noted two w a y s of proving the t h e o r e m that the z e t a function of the i d e a l c l a s s of a r e a l q u a d r a t i c field at the points s = 0, - 1 , - 2 , . . . has r a t i o n a l v a l u e s . One of t h e s e was d e v e l o p e d by Klingen [345] and Siegel [636] who obtained an analogous r e s u l t for a t o t a l l y r e a l field of a l g e b r a i c n u m b e r s k. L e t r be the d e g r e e of the field k, and let ~k(S) be the z e t a function of the field k. The e s s e n c e of the p r o o f of Klingen and S i e g e l is that ~(1 - m), w h e r e m > 0 is an i n t e g e r , is c o n t a i n e d in the z e r o t h c o e f f i c i e n t of the F o u r i e r e x p a n s i o n of the E i s e n s t e i n s e r i e s r e l a t i v e to the H i l b e r t m o d u l a r group a s s o c i a t e d with the field k. Gundlach [279] a l s o made use of this fact. S e r r e [595] e s t i m a t e d the d e n o m i n a t o r of the n u m b e r ~k(1 - m) and i n d i c a t e d c o n g r u e n c e s connecting the v a l u e s ~k(1 - m) by i m proving the c o r r e s p o n d i n g r e s u l t s of S i e g e l [636]. See a l s o the p a p e r s of S i e g e l [635,637] devoted to this topic. Shintani [627] found a n o t h e r method for obtaining the r e s u l t s of Klingen and Siegel. The r e s u l t s d e s c r i b e d above play an i m p o r t a n t r o l e in p - a d i c methods (see S e r r e [596]) which we do not d i s c u s s . R e g a r d i n g o t h e r r e s u l t s on this topic which a r e t r a d i t i o n a l in c h a r a c t e r s e e Koshlyakov [61], B a r n e r [139], B e r n d t [148, 152], v a n d e r Blij [163], Cohen [ 2 0 1 , 2 0 2 , 2 0 4 ] , G r o s s w a l d [272,275], Gueho [ 2 7 6 , 6 6 6 , 667], K a t a y a m a [330, 332], Kenku [337], Land [399,400], Rideout [564], S c h e r t z [575], S m a r t [641], S t a r k [651], T e r r a s [659, 660]. II. Diophantine A p p r o x i m a t i o n s . A r i t h m e t i c Minima. D i r i c h l e t p r o v e d that if w is a r e a l i r r a t i o n a l numb e r , then the i n e q u a l i t y I

q

q2

is s a t i s f i e d by an infinite s e t of r e a l r a t i o n a l n u m b e r s p / q if k = 1. The p r o b l e m of finding the m i n i m a l k with this p r o p e r t y was s o l v e d by Hurwitz who showed that if k = 1 / ~ 5 , then t h e r e e x i s t s an infinite s e t of f r a c t i o n s s a t i s f y i n g i n e q u a l i t y (1); if, on the o t h e r hand, k < 1/~/5, then in e a c h i n t e r v a l of the r e a l axis t h e r e is an co for which i n e q u a l i t y (1) is s a t i s f i e d for only a finite s e t of r a t i o n a l n u m b e r s . The method of Hurwitz depends on continuous f r a c t i o n s . In w o r k of 1917 F o r d [245] p r o v e d this r e s u l t using the s t r u c t u r e of the fundamental d o m a i n of the m o d u l a r group, and in [246] he i n v e s t i g a t e d by an analogous method a s i m i l a r p r o b l e m in the c o m p l e x domain. R a n k i n [554] u n d e r t o o k the g e n e r a l i z a t i o n of these r e s u l t s to the c a s e of a F u c h s i a n group of f i r s t kind with a p a r a b o l i c v e r t e x at ~o and, in p a r t i c u l a r , r e p r o v e d a n u m b e r of r e s u l t s obtained e a r l i e r by continuous f r a c t i o n s (W. T. Scott, Bull. Am. Math. Soc., 46, 124-129 (1940); T o r n h e i m [661]). Newman [487] s t u d i e d the p r o p e r t i e s of i s o m e t r i c c i r c l e s of c o n g r u e n c e s u b g r o u p s . Let F = SL(2, Z), z ' = Tz = (az + b ) / ( c z + d), T 6 F. The i s o m e t r i c c i r c l e of this t r a n s f o r m a t i o n is the c i r c l e l cz + dl = 1 in the z plane; t cz + dl -< 1 is the i s o m e t r i c d i s k of the t r a n s f o r m a t i o n T. If G is a s u b g r o u p of the group F of finite index, then t h e r e e x i s t s a l e a s t p o s i t i v e i n t e g e r r = r(G) s u c h that the r e a l axis is c o v e r e d by i s o m e t r i c disks of e l e m e n t s of G with r a d i i _>1/r. In p a r t i c u l a r , if n is a p o s i t i v e i n t e g e r and G = F(n), then I~ankin showed [552] that r _< n 11/2. This i m p l i e s that e a c h r e a l , i r r a t i o n a l ~ a d m i t s a r a t i o n a l a p p r o x i m a t i o n h / k such that k>0, I~-h/kl < 2 / n k , k - < n ~/2 and e i t h e r 0an+ 1, k) = 1 o r ( h n - l , k ) = 1. Newman shows that such a p p r o x i m a t i o n s always e x i s t w i t h k = O(n l+e) for e a c h a > 0. He a l s o shows that r(F(n)) < n 2+a for n > C(e) for any e > 0 and that for a s u i t a b l y c h o s e n c o n s t a n t c > 0 , r(F(n)) > cn 2. The r e s u l t is thus c l o s e to b e s t p o s s i b l e . F i n a l l y , we m e n t i o n that Cohn r e l a t e d the known w o r k of A. A. M a r k o v on a r i t h m e t i c m i n i m a of i n d e t e r m i n a t e b i n a r y q u a d r a t i c f o r m s to the study of m o d u l a r g r o u p s and g e o d e s i c s on p e r f o r a t e d t o r i (a t o r u s minus a disk). In this c o n n e c t i o n we r e f e r the r e a d e r to the w o r k of Cohn [ 2 0 5 , 2 1 3 , 2 1 4 , 2 1 5 ] . III. The W o r k of H e e g n e r . B a k e r [137] and S t a r k [643] s o l v e d the p r o b l e m of finding a l l i m a g i n a r y q u a d r a t i c fields of c l a s s n u m b e r one. They a l s o s o l v e d the analogous p r o b l e m of finding a l l i m a g i n a r y q u a d r a t i c f i e l d s of c l a s s n u m b e r two (Stark [650,654], Baker [138]). S e v e r a l a u t h o r s noted that the old w o r k of H e e g n e r [285] contains a r e m a r k a b l e method which r a p i d l y l e a d s to a s o l u t i o n of the s i n g l e - c l a s s p r o p e r t y . The m e r i t for c l a r i f y i n g this w o r k of H e e g n e r which is difficult to u n d e r s t a n d belongs to B i r c h [156, 157], S t a r k [645, 6 4 7 , 6 4 9 , 6 5 2 ] , S i e g e l [634], and D e u r i n g [220]; s e e a l s o M e y e r [452] and Chowla [199]. The e s s e n c e of H e e g n e t ' s method is the following. We c o n s i d e r an i m a g i n a r y q u a d r a t i c field Q(,Fd) of d i s c r i m i n a n t d < 0. Let j(z) = q-~ + 744 + . . . be the m o d u l a r i n v a r i a n t (q = e2~iz); l e t 72(z) = [j(z)]~/3; 6 = (1 + ~ d ) / 2 if d = 1 (rood4),

1338

2=i

and 6 = i / 2 f d if d - 0 (rood4). We set ~=e--Y-f2(S),

1

co (

1 \

f ( z ) = q - F S I I ~lq-q'Z-~-). The number f = f(vrd) is a root

of the equation f24 + ?f16 _ 256 = 0. Birch [156] noted that H e e g n e r ' s a r g u m e n t s imply the Weber conjecture: f is an algebraic number of degree 3h(d), w h e r e h(d) is the number of c l a s s e s of the field Q((d). This r e s u l t makes it possible to reduce finding s i n g l e - c l a s s i m a g i n a r y quadratic fields to a simple Diophantine equation. H e e g n e r ' s method has so far not provided a solution to the p r o b l e m of finding all t w o - c l a s s fields. He gave a w e a k e r r e s u l t - finding t w o - c l a s s fieids with even d i s c r i m i n a n t s (see A b r a s h k i n [6]). Kenlva [336] previously proved this r e s u l t by a method of Stark [643]. IV. The K u m m e r Conjecture. Let p be a prime rational number, let ~ = e2~i/P, and let • be a c h a r a c t e r of o r d e r k_> 3. The Gauss s u m of o r d e r k i s P

(z) = ~ z (x) ~; Jr

the cubic Gauss s u m is c a l l e d the K u m m e r sum. We c o n s i d e r the case k = 3. Let p - 1 (rood6). I t is easy to see that 7-3(X) = p% w h e r e ~ is a p r i m e d i v i s o r of the n u m b e r p i n the f i e l d Q ( p ) , p = ( - 1 + ( : - 3 ) / 2 ; ~ = (a +

3b -J=-3)/2, a - 1 (rood 3);we shall a s s u m e that b > 0. The K u m m e r s u m lies inside the f i r s t , third, or fifth sextant of the complex plane. Thus, all p r i m e s p = 6t + 1 decompose into 3 c l a s s e s . K u m m e r conjectured that each c l a s s contains an infinite set of prime n u m b e r s , and the densities of the c l a s s e s c o r r e s p o n d i n g to the fifth, third, and f i r s t sextants are in the ratios i : 2 : 3. Recent computations (see Newman and Go!dstine [474], L e h m e r [410], F r h b e r g [247]) have not c o n f i r m e d the K u m m e r conjecture. Computations of F r h b e r g c a r r i e d out for p = 6t + 1 < 200,000 showed that the densities indicated above are in the ratios 4 : 5 : 6. What law actually holds is so far unclear. Several w o r k s have been devoted to Gauss sums of o r d e r k -> 3. C a s s e l s [191] (see also McGettrick [445]) found a hypothetical e x p r e s s i o n for the K u m m e r s u m in the f o r m of a product of the values of the W e i e r s t r a s s elliptic functions. For the case k = 4 McGettrick [444] conjectured an analogous f o r m (withthe Jacobi elliptic f u n c tions). Kubota proposed a new a p p r o a c h to the study of Gauss sums based on the theory of modular functions. In [383,384] he c o n s i d e r e d the E i s e n s t e i n s e r i e s c o r r e s p o n d i n g to a r i t h m e t i c groups of the type of the P i e a r d modular group SL(2, J), w h e r e J is the ring of integers of an imaginary quadratic field, acting on t h r e e - d i m e n sional hyperbolic space. The coefficients of the F o u r i e r expansions of such E i s e n s t e i n s e r i e s contain Dirichlet s e r i e s with coefficients which are biquadratic Gauss sums. The analytic continuation and the functional equation of the E i s e n s t e i n s e r i e s imply the extendability and the functional equation for the Dirichiet s e r i e s with biquadratic Gauss s u m s . M o r e o v e r , the analytic p r o p e r t i e s of these Dirichlet s e r i e s make it possible to prove some r e s u l t s on the distribution of the arguments of the Gauss sums. The case of the K u m m e r sum was investigated in the w o r k [388] in s i m i l a r fashion. Unfortunately, this a p p r o a c h also has so far not afforded a final investigation of the K u m m e r conjecture. The methods used by Kubota a r e essentially beyond t:he scope of our s u r v e y ; we t h e r e f o r e r e f e r the r e a d e r to the w o r k s of Kubota [383-385, 387-389], Selberg [588], Weft [673], Gelbart [254], and Gelbart and Sally [256]. We indicate s e v e r a l other investigations r e g a r d i n g Gauss sums of o r d e r ->3: Reshetukha [105], Krgtzel [374], Loxton [432], Moreno [468]. V. Some A r i t h m e t i c Sums. B i r c h [154] studied the sums i)--I a,b~O

p--1

--

--~.

=

w h e r e R is a positive integer and p is a prime number. Using the method of I h a r a [307], he showed that

l

p--1

Sp,(p)-- 1= ~ ~

E2~H (E ~ - 4p),

w h e r e H ( - D ) is the number of c l a s s e s of quadratic f o r m s ax 2 + bxy + cy 2 with d i s c r i m i n a n t - D . Let r be the t r a c e of the Heeke o p e r a t o r T(p) on the space S2k(SL(2, Z)). F r o m the S e l b e r g - E i e h l e r f o r m u l a it follows that

Se(P)=

R!(R+D!2R!p e + ~ ( p - - 1 ) _

(2~-F1) ( R - - k ) ! ( R + k + l ) !

pe-k (p--I)(~2k+~(T (P))-F 1) 4 - ( p - I),

1339

This gives Sn/n~ 2R!. p ~ + 2 1 R ~ ) . xr~ RI(R+I)! M o r e o v e r , for p -> 5 we have S1 (p) = p2,

$2 (p) = 2p a - 3p, S 3 (p) = 5p 4-- 9p 2-- 5p, S 4 (p) = 14p 5 - - 28p a - 20p 2-- 7p, Ss (p) ~ 42p6-- 90p 4 - 75p a -- 35p2 - 9p-- 9 (p). The values of SR(p) , R = 1, 2, 3 w e r e found long ago by Mordell [463]. The value of $5(p) gives an e x p r e s s i o n for r(p) in t e r m s of the number of points of a c e r t a i n algebraic variety. In this connection see also the w o r k of Kuga and Shimura [392]. Birch also considered the s u m V~(p)= 2

7fa~ ~_~ exp,-}-

~ 24

-24 D

a=0 b=0 x=0

Below, p_> 5. The sum VR(P) , 1 _< R _< 4 has a completely e l e m e n t a r y e x p r e s s i o n , V6(p) is e x p r e s s e d in t e r m s of Vs(p), and V5 ( p ) = 4 2 p ~(p -- 1) (p --2) §

(p _ 1) ^~(p),

w h e r e IT(p)] -< 2p 3/2. Atkin [128] conjectured that y(p) = - ( - 1 / p ) M p ) , w h e r e MP) is the eigenvalue of the Hecke o p e r a t o r T(p 2) acting on the eigenfunetion F(z) = ~4(z)~(5z). Let Fp be a prime field of c h a r a c t e r i s t i c p (>2); let Fp = Fp - ( 0 ,

1}; let

ix ( x - 1) (x--~,) I

Yamauehi [689] c o n s i d e r e d the s u m

S~(p) ~ ~ %24 (~) ~W'p

and showed that

$I (p)=p2-2p-3, $2 (P) = 2p a -- 4P 2 -- 9p -- 3 -- bp,

S 3(p) = S p 4 - 10p3-- 27p 2 - - 15p -- 3 -- 5pbp -- 2cp, w h e r e bp and ep a r e the eigenvalues of the Hecke o p e r a t o r s T(p) acting on the s p a c e s of F0(4)-parabolic forms of weight 6 and 8, r e s p e c t i v e l y . L I T E R A T U R E 1.

2.

3.

4.

5.

6.

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1341

36. 37. 38. 39. 40. 41. 42.

43. 44. 45. 46. 47. 48. 49. 50. 51.

52. 53. 54. 55. 56.

57. 58. 59.

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tions. I. The z e r o s of the function

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1358

x(n)n-~ on the line ~ s = ~ .

[I. The o r d e r of the F o u r i e r coefficients

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