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
quasiprojective Kvariety, 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 pcurvature zero for almost all p~E.
(5.5.3)
(H~R(U/K), F) has pcurvature 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 Kautomorphisms of U K. By Hironaka ([18, 19, 41]) we can
find a compactification XK of
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 Fall '11
 NormanKatz
 Natural number, Prime number

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