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By a general argument (5 9), we can prove that there exists a differential on (resp. oi) of @, not divisible by p, such that - an a, ~ ; ; n= Pwn (resp. ;w: = pwi) . Moreover, if on-, (resp. wi-,) denote the differentials of @,-, obtained by reduction of w, (resp. wi) modulo pn, then on-, (resp. oh-,) are uniquely determined modulo multiples of elements of R;,, and they are independent of the choice of r,. Consider the ordered pair (an-,, wk-,) as a differential invariant of (C,, Ch ; T,(U,)). In particular, we can associate to each solution (C,, Ck ; T,) of Problem A, its invariant (w,-,, ok-,), and this defines the map (*).
This existence turns out to be equivalent with the vanishing of the obstruction class $1) which is an element of a 4(q - l)(y - 1) dimensional Fq-module up Obs = Ker (H2(E)--+ H2(8)) ($6) . (I) (5 7). 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.
But stronger than the coincidence of the local class of (U;, TA) with that of (U;, T;) at each point of Uiu fl 17 fl 17'. ,, are called equivalent if T i and T: are congruent mod F. e.. the local classes of (Ui, T3,) and (U:, T:) coincide at each point of II f l 17'). (Use the affine cover C: x C:' = U2,2. ) Therefore? non-equivalent F-intimate families determine distinct local conditions. Corollary 2. Let (C,-,, Ci-, ; T,-,) and C,: C:' be fixed as above, and let 1 be a local condition on (C,-,, C',-, ; T,-,).
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