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According to Milne's Arithmetic Duality Theorems, Proposition I.6.4: $0 \to H^1(G_S, A[m]) \to H^1(K, A[m]) \to \oplus_{v \not\in S}H^1(K_v, A)$ for an abelian variety $A$ and a nonempty set of primes $S$ containing all infinite primes and the primes of bad reduction.

I want to prove this using étale cohomology. My idea was to extend $A$ to an abelian scheme and use the (excision) long exact sequence, but this lead me nowhere. Can someone give me some hints?

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  • $\begingroup$ Have you looked at II Section 5 of the book? $\endgroup$
    – mephisto
    Commented Apr 1, 2011 at 16:35

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Just so that we're clear here, I think you're referring to Proposition I 6.5 in Milne's book. Also, the proposition is stated for the $m$-primary component $A(m)$ of $A$ rather than the $m$-torsion $A[m]$.

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