Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the reduction is ordinary or supersingular? I guess this is true when $K=\mathbb{Q}_p$ because in the ordinary case the representation is reducible, while in the supersingular case, the representation is irreducible.

In general I know that the representation is reducible in the ordinary case because it has a $1$-dimensional unramified quotient. But I'm not sure whether or not the representation is irreducible in the supersingular case for arbitrary $K$.

The reason I ask this question is because I was wondering whether or not two curves over a number field with good reduction above $p$ that have isomorphic $p$-torsion representations necessarily have the same reduction type at primes above $p$ (ordinary or supersingular, that is).


This can't be the case for all $p$-adic fields. Indeed, start with $E_1{/\mathbb{Q}_p}$ an elliptic curve with good ordinary reduction and $E_2{/\mathbb{Q}_p}$ an elliptic curve with good supersingular reduction.
Let $K = \mathbb{Q}_p(E_1[p](\overline{K}), E_2[p](\overline{K}))$. Then upon base extension to $K$, both $E_1[p]$ and $E_2[p]$ have the same $\mathbb{F}_p[\operatorname{Gal}_K]$-module structure: namely they are both isomorphic as abelian groups to $(\mathbb{Z}/p\mathbb{Z})^2$ and both have trivial Galois action. (Moreover the ordinary/supersingular dichotomy does not change upon base extension: this depends only on the $j$-invariant of $E$ modulo $p$.)

There is something to be said in the positive direction though coming from restrictions on torsion in the formal group of $E_{/K}$ depending on the ramification index $e(K/\mathbb{Q}_p)$. Let me know if you want to hear more details about that...

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  • $\begingroup$ Thank you Pete! Your example makes perfect sense. I thought that the absolute ramification index might be relevant, since certain related facts that I've needed to use depend upon $e(K/\mathbb{Q}_p)$ being less than $p-1$. If you have the time to add some more details, I'd definitely appreciate it. $\endgroup$ – Keenan Kidwell Feb 8 '12 at 20:07
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    $\begingroup$ Well, what I know about torsion in formal groups appears in $\S 3.1$ here: math.uga.edu/~pete/cx6.pdf. These bounds will allow you to show that if $e(K/\mathbb{Q}_p)$ is sufficiently small then in the case of supersingular reduction you will never have nontrivial $p$-torsion over the maximal unramified extension, whereas in the ordinary reduction case you will always have nontrivial $p$-torsion over the maximal unramified extension. (And this goes for abelian varieties as well...) $\endgroup$ – Pete L. Clark Feb 8 '12 at 20:13
  • $\begingroup$ Awesome! Thanks for the reference. I can use this because I'm already assuming the bound on the ramification index. $\endgroup$ – Keenan Kidwell Feb 8 '12 at 20:22
  • $\begingroup$ @Pete I was trying to prove, along the same lines, that if $e<p-1$, then $E_1[p]\cong E_2[p]$ implies that $\tilde{E_1}[p]\cong \tilde{E_2}[p]$. Given Raynaud's result, I am only left with showing that the $p$-torsion of the generic fibre of the Néron model, i.e. $E_i[p]$ is the generic fibre of the $p$-torsion of the Néron model, and that the same holds if I replace "generic" by "special" (making do with finite flat rather than étale group schemes). This must be a triviality, but I have been unable to find a reference or an argument. Any help would be greatly appreciated! $\endgroup$ – Filippo Alberto Edoardo Nov 20 '19 at 8:43

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