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Dr. Katz DEq Homework Solutions 114

# Dr. Katz DEq Homework Solutions 114 - 114 N.M Katz T aking...

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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 ---, I-I 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 p-curvature 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 T-connections 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 I-I 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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