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Dr. Katz DEq Homework Solutions 113

Dr. Katz DEq Homework Solutions 113 - A lgebraic Solutions...

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Algebraic Solutions of Differential Equations 113 tible sheaf ~Q on EQ extends uniquely (up to isomorphism) to an inver- tible sheaf ~ on E which fibre by fibre has degree zero (in fact, ~Q~ (gEQ([PQ]--[0]), for a unique PQ~EQ(Q), and, denoting by P its unique prolongation to a section of E over T, we have ~=(9~([P]-[0]) (cf. ( (7.5.6) At the expense of enlarging the integer n, i.e. localizing on Spec(Z [1/n]), we may suppose that the conncetion [7Q on s extends to a T-connection 17 on ~e. The hypothesis that the original d. t.k. o~ on EQ was locally logarithmic mod p for almost all p is equivalent to the hypo- thesis that, for almost all p, the inverse image of (~, V) on Ep has p- curvature zero. (7.5.7) Suppose we view (5r V) as an element in H I(E, ~/T)O, which sits in the short exact sequence ( ( O--~ F(E, g2~/T)--~HI(E,Q~/T)O--~ E(T)--+O. Reducing modulo p, we find an element (~p, 17p) in H 1 (Ep, * O~p/vp)o which sits in the short exact sequence (
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