5
$\begingroup$

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!

$\endgroup$
12
  • $\begingroup$ I guess you want to assume good ordinary reduction, since in the supersingular case we have $\overline{E}_p[p^{\infty}]=0$. $\endgroup$ Oct 11, 2011 at 23:39
  • 3
    $\begingroup$ 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 $\mathbb{Z}_p$. $\endgroup$ Oct 12, 2011 at 1:41
  • 1
    $\begingroup$ It's essentially the same, by Tate's theorem on the full faithfulness of Tate module functor. $\endgroup$ Oct 12, 2011 at 3:13
  • 3
    $\begingroup$ 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. $\endgroup$ Oct 12, 2011 at 3:19
  • 2
    $\begingroup$ 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. $\endgroup$ Oct 12, 2011 at 4:12

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.