Let $p$ be a prime number, and $E/\mathbf{Q}_p$ and elliptic curve with good reduction $\bar E_p$. Can the Galois representation of $\mathbf{Q}_p$ given by the $p$-adic Tate module of $E$ be recovered from the Dieudonn\'e module associated to the $p$-divisible group $\bar E_p[p^\infty]$ of the reduction $\bar E_p$?

In case of ordinary reduction, can the [EDIT: possible] splitting of the $p$-torsion $E[p]$, as Galois module, be understood in terms of the Dieudonn\'e module of the $p$-torsion $\bar E_p[p]$?

Thanks!

plusa certain direct summand $Fil^1M\subset M$. Roughly speaking, $M$ is the crystalline cohomology of $\overline{E}_p$ (or its dual, depending on your conventions), and it is canonically identified with the de Rham cohomology of the Neron model of $E$ over $\mathbb{Int}_p$. The direct summand $Fil^1M$ is then obtained from the de Rham filtration and completely determines the lift. – Keerthi Madapusi Pera Oct 12 '11 at 3:19