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Unfortunately, the following question is somewhat ill-posed. However, I hope to make what I'm looking for sufficiently clear.

Given two elliptic curves (or Abelian varieties) over $\mathbb{C}$, one can tell whether or not they are isogenous by looking at their rational Hodge structures. Similarly, over $\mathbb{Q}$ or a finite field, it suffices to look at $\mathbb{Q}_\ell$-adic Tate modules as Galois representations.

My question is what the analogue is over a $p$-adic field.

I'm looking for an answer which produces an abelian category $\mathscr{C}$ linear over a field of characteristic zero and which can be defined (in an imprecise sense) without knowing what a variety is, plus a contravariant functor from smooth projective varieties to $\mathscr{C}$ such that two elliptic curves over a $p$-adic field are isogenous if and only if their images in $\mathscr{C}$ are isomorphic.

For example, this can be done over $\mathbb{C}$ or a field finitely generated over its prime field using Hodge theory or the Galois action on $\ell$-adic cohomology respectively.

So the question can be phrased another way: what is the $p$-adic version of the Hodge/Tate conjectures? I would like a category which should conjecturally receive a fully faithful functor from the category of pure motives.

Note that etale cohomology is not a fine enough invariant over the $p$-adics (though I would be interested to hear any positive results where it is!). Moreover, ignorant of $p$-adic Hodge theory as I am, I'm told that $(\varphi, \Gamma)$-modules and the like are similarly not fine enough to distinguish isogeny classes of elliptic curves (even with good reduction).

I should say I'm very much only an amateur in these $p$-adic things. I'm hoping to find an answer that motivates stepping into some of this morass of $p$-adic Hodge theory.

This question can clearly be formulated (equally imprecisely) over any field. Other fields over interest certainly include positive characteristic local fields, $\overline{\mathbb{Q}}_p$ and $\mathbb{C}_p$. Though I'd certainly be glad to hear of any other fields where such a thing is known. (Of course, anything which involves choosing an isomorphism with $\mathbb{C}$ is very much against the spirit of the question).

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I don't have an answer, but you might find this interesting: mathoverflow.net/questions/53014/… –  Tommaso Centeleghe Nov 18 '11 at 0:37
    
Presumably etale cohomology is involved? –  Will Sawin Nov 18 '11 at 2:23
    
Dear Moosbrugger, Do you know Serre--Tate theory? (I.e. are you hoping to get an answer that goes beyond Serre--Tate theory, or would Serre--Tate theory, which is about deforming a fixed curve mod p to a curve over a p-adic field, already be of interest?) Regards, Matthew –  Emerton Nov 18 '11 at 3:38
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Perhaps you might find Maja Volkov's Les représentations l-adiques associées aux courbes elliptiques sur ${\bf Q}_p$ (reference-global.com/doi/abs/10.1515/crll.2001.046) interesting. –  Chandan Singh Dalawat Nov 18 '11 at 4:02
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Very vague comment: if you take as your element of $\mathcal{C}$ both the etale cohomology and the de Rham cohomology of the curve, plus a comparison isomorphism, then aren't you constructing a gadget that knows that cohomology of the reduction plus the location of the Hodge filtration, and hence you can reconstruct the curve up to isogeny by Serre-Tate? –  Kevin Buzzard Nov 18 '11 at 8:04
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