Dr. Katz DEq Homework Solutions 78

Dr. Katz DEq Homework Solutions 78 - braic extension L/K...

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78 N.M. Katz: after the base change Spec(Fq)--~ T VQFq- >~cJ# I 1 Spec(Fq)- . , T the p-curvature of the free (_9~ | Fq-module with integrable connection ((M, 17)1~/) | F 0 is zero. (5.4.3.2) A standard "passage to the limit" shows that if the property (5.4.3.1) holds for one set of choices (A, q/, (M, V)) then it holds for every choice. It is thus an intrinsic property of the germ of (Mc, Vc) at the generic point of Sc which we call having p-curvature zero for almost all primes peS. Thus, given a function field K/C (i. e., a finitely generated field exten- sion of C), a differential equation (N, V) over K (i. e., a finite dimensional K-space N together with an integrable connection V: N~f2~/cQKN ), and an infinite set 27 of prime numbers, it makes sense to say that (N, V) has or has not p-curvature zero for almost all primes pc27. (5.4.3.3) Because p-curvature is a differential invariant, its vanishing is of a local nature for the 6tale topology. It follows that (N, 17) has p- curvature zero for almost all p~27 if and only if there exists a finite alge-
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Unformatted text preview: braic extension L/K such that (N, V)| L has p-curvature zero for almost all peZ. K Putting together (5.4.1) and (5.4.3.3), we have the obvious (5.4.4) Proposition. Let (N, 17) be a differential equation over a function field K/C (5.4.3.2). If (N, 17) has a full set of algebraic solutions, then it has p-curvature zero for almost all primes p. (5.4.5) Let K/C be a function field, and let Ux be a smooth quasi-projec- tive K-variety. By Hironaka ([18]), we can find a projective and smooth K-variety Xx which contains UK as an open set, such that the complement DK = XK UK is a union of smooth divisors D~, ~ in XK which have normal crossings. By "general nonsense", we can find a finitely generated sub- ring A of C, a smooth Spec(A)-scheme S with geometrically connected fibres such that K is the function field of its complex fibre So, a projective and smooth S-scheme f: X--* S and divisors D~ in X which are smooth...
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