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Let NO, be the image of O + NT and consider the deg (17 fl 17')-dimensional Fq-module HO(N,/NO,). Then 2 forms a principal homogeneous space of HO(NT/NO,),and 1 turns out to be equivariant with the canonical group-homomorphism HO(NT/WT) Obs. ) But this is not yet sufficient for our purpose, as this describes our mapping ,8 only up to unknown translations in Obs. The determination of 13, itself seems to offer a serious arithmetic question, except in the trivial case where R, = Fq[[t]]/tn+land (C,-,, Ck-,; T,-,) is the obvious extension of (C, C'; T).

On(-K,)q) in where the restriction homomorphism &(- K) Cf ,,(- qKnt)(resp. 0(-Kr)-+ &,(qK,))l). This definition of F is independent of thz choice of x ; in fact, F can also be defined as thz subsheaf of O whose stalk at P = (Q, Q') (Q E C, Q' E C') is the module of all such derivations 6 of Oc,c,qp that, for any I E O,,, (resp. t)to 17' (resp. 6 ( p z j ) to Z7) is a q-th power element of O,,,, (resp. O,,,), whenever P E 17' (resp. P E 17). It is clear that E c F c O. Although F is not an O,,,,-Submodule of O, it is a coherznt O~xc,-Module.

Put a = ul-". U-I ( = 1 (mod n)), and w' = w/u. Then the equation is transformed to (w')O-' = a. We shall construct a sequence {Y,)~= in~ such that y,,, = y, (mod xi+') and y;-l = a (mod ;ri+l). Then w' = y, would give a solution of the above equation. Put yo = 1, and suppose that yo, . , y, have been constructed. r"lB(i)). Then + + < < < Theorem 2'). Let a, be a Frobenius mapping of gn with ~ ( a , )= Y < m , and choose any c E R, with cR, = ;ruR, (as a normalizing constant). n-,/B,-,. 1) A parallel result for 9, = lim 9, was given in my previous mimeographed note t "Non-abelian invariant differentials (1971)".

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Algebraic Number Theory by IYANAGA

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