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.