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](http://www.ucl.ac.uk/~ucahmki/alb.pdf) 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?



Selmer Group versus Selmer Variety
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.


