Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T\_p A \otimes \mathbf{Q}\_p$ be the $\mathbf{Q}\_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. (added in edit)


I keep seeing a natural map $A\to H^1(G,V)$. How is this map constructed? What does it have to do with "Kummer theory"?


What is the image of this map? That is, how can one describe it? Does it have to do with Selmer groups?


Sorry for the vagueness.



Construction of Kummer map for abelian variety 
Does Silverman, The Arithmetic of Elliptic Curves, X.1 or Cornell-Silverman-Stevens, p. 33 help? Form the long exact sequence of $0 \to A[\ell^n] \to A \to A \to 0$ (analogue of the Kummer sequence if you replace $A$ by $\mathbf{G}\_m$) and take the inverse limit.


Relation to the Selmer group: $\mathrm{Sel}(A/K)\_m \subseteq H^1(K,A[m])$ and I suspect $\mathrm{Sel}(A/K)\_m = H^1(\mathrm{Spec}\,\mathcal{O}\_K,\mathcal{A}[m])$ (disregarding the archimedean places).


