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

By Participants and Observers

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B e rdie Bestinimung Dirichletscher Reihen dusch ihre Funktionalgleichung", Math. Ann. 112 (19361, pp. 664-699. 2. Ogg, A. A. Benjamin, 1969. 3. Weil, A. "Sur une formule classiqueftJ. Math. Soc. Japan, 20(1968), pp. 399-402. On t h e t h e t a s e r i e s obtained from c e r t a i n o r d e r s of q u a t e r n i o n a l g e b r a s Hiroaki 8 0. I n t r o d u c t i o n . a theta series number H i j ikata S t a r t i n g w i t h a q u a d r a t i c fapm 2 ( z , F) 2 ~ ( n ,F) e2 n=O c ( n , F) a s c o e f f i c i e n t s , t h e n : (0) $ ,(' , F) .

O d u l a rforms of h a l f - i n t e g r a l weight and t h e i n t e g r a l of c e r t a i n f u n c t i o n s ", Nagoya J. of Math,, 56 ( 1 9 7 5 ) . 9. J. S h a l i k a , R e p r e s e n t a t i o n s of t h e two by two unimodular group over l o c a l f i e l d s , 10. G. orms of h a l f - i n t e g r a l weight", Ann. , 11. T. Shintani, Thesis, Jghns Hopkins (1966). 97 ( 1 9 7 3 ) . "on t h e c o n s t r u c t i o n of holornorphic cusp forms of h a l f - i n t e g r a l w e i g h t f f , Nagoya J.

H E VB - above decomposition. S i n c e lf and r y(hr) = 0 argument t o 8 of W (i) (ii) W1. for 2 2. 0 r=l h(h,l~) z 0 1) (mod a P if = bl(p) ... +1' r-1 For h 3 gi W o ,r E for r r 2. 0 By t h e same argument, it i s proved t h a t h Hence < i = n ll (T(p) - bi ( p ) I w ) r r=2 r IW i s t h e i d e n t i t y o p e r a t o r on W. If uniquely. where r' -** a r a' a ' , it holds We s h a l l show W Wr = Wr n W . with r r=l with t h e v e c t o r subspace generated by Wr 0 Put (p)1 A1 = {bl ( p ) , S belong t o t h e same s u b s e t .

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