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Additional info for 1975 U.S. - JAPAN SEMINAR - APPLICATIONS OF AUTOMORPHIC FORMS TO NUMBER THEORY

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B Lemma 1, t h e r e e x i s t s a l i f t i n g Hecke operators. p So must be a cusp form. Let such t h a t 1 + x(p)p q sx f = RE* f of f k- 1 x (p) $ . Let a be an odd prime - ) which i s a common eigenfunction k-1 $ 0 + p k- 1 (mod L-1) , there (mod 11) : hence of a l l e x i s t s some F(E 2 s x) + F i r s t we assume 2). Then we have L 4 q. I t is clear t h a t f u n c t i o n with r e s p e c t t o q r 1 Put (mod 4) x(-1) = 1 = (-1) g = - f. and l e t k . Then we have E2,X k dJ('))-T , which i s denoted by go (z dz rl(q) I t is clear t h a t be an k r 2 g 5 - B k2k , ~ T, i s a modular a- 1 and s = L 1 .

Let 1m b e a (N,R)=l b e t h e normalized a d d i t i v e FN is t h e f i e l d o f a l l t h e modular f u n c t i o n s o f l e v e l N whose F o u r i e r expansions with r e s p e c t t o a r d = . be K l e i n l s modular f u n c t i o n . p r i m i t i v e , d e t a = N) j = 0, k) (resp. n a t u r a l l y considered a s a s u b f i e l d o f RN a& i s t h e a l g e b r a i c c l o s u r e of t h e prime f i e l d i n c h a r a c t e r i s t i c k00 Then Put the restriction of QN denote t h e r e s i d u e f i e l d o f k ) Q of QN denote t h e f i e l d generated and l e t 0.

ZTinz/ 00 To e a c h Hecke d i f f e r e n t i a l f(z) if f ( z ) = 2 ~ r i A z /+~ aoe 9 n =0 let us associate the Dirichlet series, @ ( s ) satisfies t h e following properties: Then (q) -S I. The f u n c t i o n 8(s) = I'(s)$(s) can be a n a l y t i c a l l y c o n t i n u e d t o a meromorphic f u n c t i o n s u c h t h a t i s an e n t i r e f u n c t i o n of f i n i t e genus. 1: The c o r r e s p o n d e n c e f + @ i s-a b i j e c t i o n from - t h es et of a d 1 Hecke d i f f e r e n t i a l s of s i g n a t u r e - {h,~,B,C,yj a n d th e se t of a l l convergent D i r i c h l e t series s a t i s f y i n g --a n d 11.

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1975 U.S. - JAPAN SEMINAR - APPLICATIONS OF AUTOMORPHIC FORMS TO NUMBER THEORY by Participants and Observers


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