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Sample text

PN Put 00 and l e t Let J(T) km . = {f = v~ t h e mapping : % + IR I t i s obvious t h a t denoted by VN. clear that V 1 VN "' uw t o ON. with r e s p e c t t o Let p kN ( r e s p . pm) N kN i s = Q N ( J ( ~ ) J, ( a r ) I a M2(Z), E eN- > Then -a). RN Put i s a subring of we d e f i n e i s uniquely extended t o a v a l u a t i o n o f N' Q(J(T)) *IF, FN, which i s a l s o NIN'. for =VN with respect t o FN with r e s p e c t t o VN. J(T) . Let k_*FN i s isomorphic Then in characteristic N Itisalso i n the sense of The r a m i f i c a t i o n t h e o r y o f t h e s e f i e l d s i s e s t a b l i s h e d by I g u s a , and rl(N) 1 FN For each f = I ane 2nnnT'N, i s t h e Gauss v a l u a t i o n o f j = J ( T ) (mod V1).

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. , t o appear. 12. A. Weil, " ~ u rc e r t a i n e s groupes d ' o p e r a t e u r s u n i t a i . , Cornell University I t h a c a , New York 14853 June 1975 111 (1965).

P . c o n t a i n s t h e maximal compact s u b r i n g of P c o n t a i n s t h e maximal compact s u b r i n g of a maximal s p l i t semi-simple commutative s u b f i e l d of Lemma. (i) (I. of R L i t s maximal o r d e r P' (iii) w i t h ( I ) , t h e n c l a s s number of B Define t h e l e v e l (R) = (p1 S(R) = Hecke r i n g s . 1. of > = a T i z a t i o n of ) = R(V( fl maximal o r d e r , and 0 T (n) = H0 (Rp) X , BA) Cusp-forms If is indefinite, R = type number of , B~ a=M ( R ) = ngX R = 1.

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