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Dr. Katz DEq Homework Solutions 79

# Dr. Katz DEq Homework Solutions 79 - Algebraic Solutionsof...

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Algebraic Solutions of Differential Equations 79 over S and cross normally relative to S, such that the fibre of X (respec- tively of Di) over the generic point of S c is XK (resp. Di, K). (5.4.5.1) The stalk over the generic point of S c of the coherent (9 s- module with integrable connection relative to T (5.4.5.2) (R"J, ((2~/s(log D)), V) is the differential equation over K (5.4.5.3) (H~R (UKIK), V). Combining (5.4.1), (5.4.3.2), (5.4.4) and (5.4.5.1), we may restate Theorem 5.5 birationally: (5.5) Theorem (= 5.1 bis). Let K be a function field over C, U a smooth quasi-projective K-variety, and n > 0 an integer. Then the Jollowing condi- tions are equivalent. (5.5.1) (HgR(U/K), V) has a full set of algebraic solutions (5.5.2) There exists an infinite set Z of prime numbers such that (H~R(U/K), V) has p-curvature zero for almost all p~E. (5.5.3) (H~R(U/K), F) has p-curvature zero for almost all primes p. 5.6. In the situation of (5.4.5), suppose that a finite group G acts as a group of K-automorphisms of U K. By Hironaka ([18, 19, 41]) we can find a compactification XK of
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