TITLE: Selmer Group versus Selmer Variety
QUESTION [5 upvotes]: 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?

REPLY [9 votes]: 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.