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1975 U.S. - JAPAN SEMINAR - APPLICATIONS OF by Participants and Observers

By Participants and Observers

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4 (1958) 217-239. [4] The b a s i s problem f o r modular forms and t h e t r a c e s of t h e Hecke , o p e r a t o r s , L e c t u r e Notes i n Math. 320 (1972) 75-151. [5] H. H i j i k a t a , On c e r t a i n i d e n t i t i e s between t h e t r e c e s of Hecke o p e r a t o r s , Proc, J. Acad. 48 No. 8(1972) 585-588. 161 , E x p l i c i t formula of t h e t r a c e s of Hecke o p e r a t o r s f o r J. Math. Soc. Japan, 26 (1974) 56-82. [7] -, - 0 (N), and H. S a f t o , On t h e r e p r e s e n t a b i l i t y of modular forms by t h e t a s e r i e s , Number Theory, a l g .

P i s a n o r d e r of and + R p V g(L) f i x e d maximal unramified f i e l d B P B oP4s 0 H (R) C; H ( R') i s indef by 0 T (pV ) lJ = 0, 1, 0 0 T (n, R) p , T (n, R'). 4. Sk(R) i s d e f i n i t e . ( c f . Shimizu [ 9 ] ) B ) b e , a n a n t i - r e p r e s e n t a t i o n d e f i n e d a s t h e composite of t h e , an B : injection symmetric t e n s o r r e p r e s e n t a t i o n %= i n t h e sense of [ 1 0 ] , (R) H(B~, X B*. + GL2 ( :B and t h e t r a n s p o s e of k-ply C) GL2( C)-)GL(V ) .

Alsc R e P. L a n g l a n d s has explained t c me how % one should be a b l e t o check t h e g e n e r a l i zed Ramanujan-Petersson hypothesis f o r holomorphic cusp forms u s i n g t h e methods i n t r o d u c e d by Deligne f o r t h e almost everywhere r e s u l t , I I I I I REFERENCES 1. W. Casselman, "on some r e s u l t s of Atkin and ~ e h n e r " ,-Math. I Ann. 201 ( 1 9 7 3 ) . i I 2. II P. Deligne, La c o n j e c t u r e de w e i l f f , Publ. I. H. Eo S. (1974); a l s o : A-adiques 3 11 S.

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