Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$

Question

I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when:

(1.) $q=p$

and/or

(2.) $E$ has multiplicative reduction at $q$.

Here, $E$ is an elliptic curve or, more generally, an Abelian variety.

Any suggestions on literature?