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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) . = F, one can c o n s t r u c t inT w i t h t h e r e p r e s e n t a t i o n i s a modular form.

I E K and 2 hi = K 1 K 6 2 2. o f dimension be t h e Put n. K - l i n e a r endomorphism. 2 Z bi(n)zn nrO E V, we put W We is t L ( L aibi(n))z n 'L h belongs t o Kt[[z]] and t h e mapping I : W + Kt[[z]] n>0 i = l 'L ~ ( h )= h i s e a s i l y proved t o be well-defined and t o be which i s d e f i n e d by . h = r\, injective. that h = For X bnzn E W , we d e f i n e n10 s a t i s f i e s a l l c o n d i t i o n s i n Lemma. h assumption, t h e r e e x i s t s a b a s i s o f of a l l Hecke o p e r a t o r s .

W ( S(7)) = f' But w(s(F) occurs i n r . Therefore so does i s whether 1 Then - . ~ ( 7 )should occur i n of some r e c e n t r e s u l t s of D e l l g r d s . e. Let fip denote t h e r e g u l a r i n L2 ( G ) . dm t h e F,arr:arlujarl-Petersson r e p r e s e n t a t i o n of G~ P hypothesis proved by Deligne i n [;I]i t follows tiiat Twill occur P in f o r almost every p. Thus c r c~o r ~ j e c t u r . t sm a t t h e same Gk with I I q Q. D. To s e e why on I 03 % is true for a l l -rr . G r m t e d t h i s , i t s k ~ c ~ : l idc l l o w d i r e c t l ] t h a t P If p = W , this f a c t i s already implicit T occursin as P i n the literature.

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