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51 1-1. Preliminaries First we r e c a l l t h e f a c t s about t h e modular function f i e l d s o f l e v e l characteristic 0 and R > 0 N in from Igusa [ I ] , [2] and Shimura [ 9 ] . We s h a l l f i x a prime number R with (R, 6) = 1 once and f o r a l l . Let N be a p o s i t i v e i n t e g e r with by a p r i m i t i v e +me Q00 valuation of l y i n g on L a&. 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.

6 ) : Put lz3 L. F~ FN. t o t h e modular f u n c t i o n f i e l d o f l e v e l Igusa 121. (resp. such t h a t {-) FN denote t h e r e s i d u e f i e l d o f of m e 2nfl 'IN have c o e f f i c i e n t s i n / N I Min{uN(an)) z ane ~ T J - - ~ - ~E TFN n and i t s q u o t i e n t f i e l d c o i n c i d e s with E p u ON. 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 = .

Then t h e r e e x i s t s of s i g n a t u r e - {A,A,B,C,-1). The - s i g n a t u r e s f o r which Hecke d i f f e r e n t i a l s e x i s t are {X,O,O,C,-11, X = Pcos(n/q), except i n case -25 - q an even i n t e g e r - L 3, in which - I signature case there exist Hecke differentials of (X,A,O,C,-1). It is possible to explicitly describe the Hecke differentials also in the case y = -1, but such a description is beyond the scope of this report. Also worth mentioning, but beyond the scope of this report, are the arithmetic applications of the functions fX(z), Hecke's group G(f) whose transformation properties under embody a generalization of Dirichletts class number formula for imaginary quadratic fields.

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