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.
(7.4.0.3)).
(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 Tconnection 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 (7.4.1.5)
(7.5.7.1)
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
(7.5.7.2)
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 Fall '11
 NormanKatz
 Algebraic Topology, Homological algebra, Abelian category

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