Let $K$ be a local field (in fact, finite extension of $\mathbb{Q}_p$) and let $A$ and $B$ be abelian varieties over $K$.

Associated to $A$ and $B$ are the Tate-modules $T_p(A)$ and $T_p(B)$.

Both of these are canonically $G_K:=\mathrm{Gal}(\bar{K}/K)$-modules (or $K[G_K]$-modules if someone prefers this).

What I want to do is compute $\mathrm{Ext}^1_{K[G_K]}(T_l(A),T_l(B))$ for all $l$, even $l=p$ (and if possible $\mathrm{Ext}^2_{K[G_K]}(T_l(A),T_l(B))$).

Are there any results out there in this direction?

If not, does anyone have a clue how to compute a projective resolution of a Tate-module (as a $K[G_K]$-module$?

Also, would it matter if I instead considered the completed group algebra $K[[G_K]]$?

**Aside** Yesterday I attended the ceremony where the named mathematician collected his Abel-price, and today, the Abel-lectures by said mathematician and Richard Taylor. Unfortunately, I realized I needed to know the answer to this question when I got back to my office, otherwise I could've pestered recipient and Taylor. :P
But hey, now you all get to think about this instead!