Diferent abelian varieties over local field with the same p-adic representation? Let $K$ be a local field with residue field of char $p$, denote $G$ its Galois group. Is it possible that we have two Abelian varieties $A_1$ and $A_2$, defined over $K$, such that  they are  not isogeny （over $K$ or $\bar{K}$ ）, but have  isomorphic p-adic Galois representation of $G$?  
 A: Yes, this can happen. Here is a counterexample (which is probably not the simplest possible, but it's the one that first came to mind). 
There are not very many 2-dimensional representations of the Galois group of $\mathbf{Q}_p$ which are "crystalline" in Fontaine's sense. Fontaine's functor $\mathbf{D}_{\operatorname{cris}}$ classifies them by linear data: 2-dimensional vector spaces over $\mathbf{Q}_p$ with a filtration and a linear operator $\varphi$ (the Frobenius) satisfying some compatibility properties. If $V$ is the $p$-adic Galois representation coming from an elliptic curve over $\mathbf{Q}_p$ with good reduction, then $\varphi$ has characteristic polynomial $X^2 - a_p(E) X + p$, where as usual $a_p(E) = p + 1 - \# \overline{E}(\mathbf{F}_p)$.
If $a_p = 0$, then this uniquely determines $\mathbf{D}_{\operatorname{cris}}(V)$ as a $\varphi$-module, and the conditions on the filtation ("weak admissiblity") mean that if $a_p(E) = 0$ then there is (up to isomorphism) a unique possibility for the filtration. So, in other words, all elliptic curves over $E$ with good supersingular reduction have isomorphic $p$-adic Galois representations [edit: if $p>3$, at least]. But they certainly aren't all isomorphic (or even isogenous) to each other, so that gives a counterexample.
