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114
N.M. Katz:
Taking this difference simultaneously for all primes > 7,
p,~n,
we get a
group homomorphism
(7.5.8.1)
H 1 (E, 12~/r) o , II
F(Ep,
Q~p/r,).
p~7
p~n
(7.5.8.2)
Proposition. The
kernel of this homomorphism consists precisely
of those (SO, V) which have pcurvature zero for all p > 7, pXn. The inverse
image of the torsion subgroup of the target (those
"tuples"
having almost
all components zero) consists precisely of those (SO, I 7) which have p
curvature zero for almost all primes p.
Proof
This follows from (7.5.7.4) and the definition of(7.5.8.1). Q.E.D.
(7.5.9)
The restriction of the homomorphism (7.5.8.1) to the subgroup
F(E, f2~/r)
of Tconnections on the structural sheaf (9 E is just the diagonal
embedding via simultaneous reduction mod p:
(7.5.9.0)
F(E, ~/r)~ 1] F(Ep,
1
O~,tr,).
p>7
p~'n
Passing to the quotient, (7.5.8.1) induces a homomorphism
II r(Ep, 1
f2E,/r~)
(7.5.9.1)
EQ(Q)=E(T)___,
,== 7.p~,,
F(E,
O~/r)
(7.5.9.2)
Proposition.
A point
Po~EQ(Q)
lies in the kernel of
(7.5.9.1)/f
and only if there exists on the invertible sheaf SO
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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