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

**Example text**

Let a be an odd prime - ) which i s a common eigenfunction k-1 $ 0 + p k- 1 (mod L-1) , there (mod 11) : hence of a l l e x i s t s some F(E 2 s x) + F i r s t we assume 2). Then we have L 4 q. I t is clear t h a t f u n c t i o n with r e s p e c t t o q r 1 Put (mod 4) x(-1) = 1 = (-1) g = - f. and l e t k . Then we have E2,X k dJ('))-T , which i s denoted by go (z dz rl(q) I t is clear t h a t be an k r 2 g 5 - B k2k , ~ T, i s a modular a- 1 and s = L 1 . Put t = 9 (W-1) k t = (11-1) s and t and s a r e coprime.

On modular curves over finite fields; Papers presented a t the Internat. Colloq. on Discrete Subgroups; ~ahuary,1973; m a y ; 161-202. Egc. Sci. Univ. Tokyo, IA 21-3 (1974) 309-332. fi l]Ihara, P and H M i ; Criteria related t o potential unraaifiednes8 snd reduction of unranified coverings of c w e s . To appear In J. F'ac. Sci. Miv. To-)qro, , 8. , Elliptic f'tmctions, Addison-Wesley . 33 ~ o r i t a , , ~ . Ibaravsconjectures and moduli space of n2] Lang abelian oarietic8; Mim. note(1970). @&hmforb, D o , An analytic construction of degenerating curves over complete local rings; Comp.

If, B . @ H(B: , P ,'q) OJL) B ) by some P and a prime by : R -rrZp(R), P a p ( ~ ) 1) B~ ( H ( B ~, Y(R) 0 0 T(n) = T(n, R), T (n) = T (n, R) - B p p g ( ~ ) (conjugate i n B of . B . P i s a n o r d e r of and + R p V g(L) f i x e d maximal unramified f i e l d B P B oP4s 0 H (R) C; H ( R') i s indef by 0 T (pV ) lJ = 0, 1, 0 0 T (n, R) p , T (n, R'). 4. Sk(R) i s d e f i n i t e . ( c f . Shimizu [ 9 ] ) B ) b e , a n a n t i - r e p r e s e n t a t i o n d e f i n e d a s t h e composite of t h e , an B : injection symmetric t e n s o r r e p r e s e n t a t i o n %= i n t h e sense of [ 1 0 ] , (R) H(B~, X B*.