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For an abelian variety $A$, the $p$-adic Selmer group is defined to be the subset of $H^1(G_k,H_1^{et}(A;\mathbb{Q}_{p}))$ whose restriction to $G_{k_v}$ is in the image of $A(k_v)$ for all places $v$ of $k$.

Minhyong Kim's paper states on its third page (p.91) that his Selmer varieties generalize $\mathbb{Q}_p$ Selmer groups of elliptic curves. As I understand it, his Selmer variety would refer to the subset of $H^1(G_k,H_1^{et}(A;\mathbb{Q}_{p}))$ that is unramified for $v \nmid p$ and crystalline at $v \mid p$ (assuming the abelian variety has good reduction there).

Is it clear that these two definitions are equivalent? Why is this true?

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    $\begingroup$ I think you are asking why the Bloch-Kato Selmer group $H^1_f$ is equal to the usual Selmer group defined by the local image of the Kummer map. If so this is Example 3.11 in Bloch-Kato in Grothendieck Festschrift. $\endgroup$
    – Chris Wuthrich
    Sep 23, 2016 at 14:56

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You should read the Bloch--Kato paper in the Grothendieck Festschrift. This was, I believe, the first paper to consider Selmer groups of Galois representations defined by local conditions coming from p-adic Hodge theory (e.g. crystalline at p).

The definition you quote from Kim is exactly the Bloch--Kato definition of the Selmer group; and Bloch and Kato prove in their paper that in the case of abelian varieties their definition recovers the classical Selmer group.

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