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Additional info for 1975 U.S. - JAPAN SEMINAR - APPLICATIONS OF AUTOMORPHIC FORMS TO NUMBER THEORY

Example text

For t h e modular equation of order p. Thia admits a generalization of the Galois theory related t o modular fmctiona of level pn ( n 3 ~ ) . (ii) We need a new way of defining the Deuring's represcntativea (in the We o b 8 e ~ e of <12]). unique l i f t i n g of an ordinsry Qtible point m a upecial ' fiber which is not normal on the total surface. -. ( i i i ) We must prove that evexy subgroup t o some member of r* *% with f i n i t e index corresponds of . *&I The use of Abbyankar's lama wrramificd coverings.

T h e r e e x i s t s t h e a l g e b r a i c formula of [$IT L-1 (mod R), z = e 2 ~ r f l r This i s due t o Ihara. 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 .

We may assume Put 0 = 0 Take e Wr. 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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1975 U.S. - JAPAN SEMINAR - APPLICATIONS OF AUTOMORPHIC FORMS TO NUMBER THEORY by Participants and Observers


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