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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 $\bar E_p[p]$?


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I guess you want to assume good ordinary reduction, since in the supersingular case we have $\overline{E}_p[p^{\infty}]=0$. – François Brunault Oct 11 '11 at 23:39
Your question is basically: can you recover a $p$-divisible group over $\mathbb{Z}_p$ from its reduction mod $p$? In general, the answer is no, since the deformation space of a $p$-divisible group over $\mathbb{F}_p$ is positive dimensional. But if you impose additional conditions, like the curve being CM, then you might be able to cut out a $0$-dimensional sub-space of the deformation space and then find a unique lift over $\mathbb{Z}_p$. – Keerthi Madapusi Pera Oct 12 '11 at 1:41
It's essentially the same, by Tate's theorem on the full faithfulness of Tate module functor. – Keerthi Madapusi Pera Oct 12 '11 at 3:13
I should also say, you can recover the $p$-divisible group over $\mathbb{Z}_p$ from the Dieudonne module, let's call it $M$, plus a 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
No, this is a different Tate's theorem! It says that the functor that takes a $p$-divisible group over $\mathbb{Z}_p$ to its generic fiber is fully faithful. You cannot in general recover the $p$-divisible group of the Neron model just from the Dieudonne module associated with the special fiber; as I said, you need to take into account the extra information given to you by the Hodge filtration. – Keerthi Madapusi Pera Oct 12 '11 at 4:12

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