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Our results can be applied t o the Shinnua rrrlw33 at , see 46. i$;[7j. 5 and L9l. Sow other con~rlbutionat o this problem are i n 16]+0] work solves t h i s conjecture when B. About ten year6 ago, we gave a conjecture on the uxistence of such connections ( $ A ( i s ) when cible l a t t i c e of %(R) all and Morita [13]. 4 - and l a a good prime with respect t o the congruence relation. C. i s t h e complex n-er field. We put N p = q, 8 8' -adic f i e l d % v * the q-th b& power mrphirsn. It I8 easy to see f n o m these conditions that thezxn-phim (yl, y2): V 0 + V x V t men image of V* in V X V * .

Put For convenience, we use t h e same n o t a t i o n s a s i n . of t h e valuation r i n g o f K a F FI; by using. c o i n c i d e s with t h e r e s i d u e f i e l d o f LI; q = e 2 n f l ~ Here and FI; = Put denote t h e r e s i d u e f i e l d We r e c a l l t h e f a c t s about t h e a l g e b r a i c formula f o r [6] except f o r d instead of d R' be t h e maximal ideal Let Through t h e F o u r i e r expansions with r e s p e c t t o V*. 9 , LN V =VN. i s n a t u r a l l y considered a s a s u b f i e l d o f Put , FI; Then it i s proved t h a t I h a r a [3], [4], [S] and Koike [6].

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. Math. 29-174. 6 53 Shinnua, G . Math. Soc Japan 131961) 275-331.

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