Let $K$ be a local field with residue field of char $p$, denote $G$ its Galois group. Is it possible that we have two Abelian varieties $A_1$ and $A_2$, defined over $K$, such that they are not isogeny (over $K$ or $\bar{K}$ ), but have isomorphic p-adic Galois representation of $G$?

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    $\begingroup$ See mathoverflow.net/questions/53014/… $\endgroup$ – Damian Rössler Jan 15 '13 at 17:00
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    $\begingroup$ Yes. Consider elliptic curves $E$ over $K=\mathbf{Q}_p$ with good ss reduction, so $V_p(E)$ is irreducible and crystalline with Hodge-Tate weights $0$ and $1$. By a calculation in $p$-adic Hodge theory (for 2-dimensional representations and distinct Hodge-Tate weights), $V_p(E)$ is determined up to abstract isomorphism by the char. polynomial of the linear (!) Frobenius on the Dieudonn\'e module of the reduction $E_0$. But geometric isogeny classes mod isom. are countable and the deformation ring of $E_0$ has uncountably many $\mathbf{Q}_p$-points, so lots aren't $\overline{K}$-isogenous. $\endgroup$ – user30379 Jan 15 '13 at 17:05
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    $\begingroup$ Addendum to previous comment: one can make a variant in the good ordinary reduction case, except that the case of decomposable $V_p(E)$ (i.e., Serre-Tate canonical lifts) needs to be excluded. In the multiplicative reduction case there is an affirmative result near the end of Serre's book "Abelian $\ell$-adic representations" (using calculations with Tate curves). The omitted details in my comments are good exercises, so I'll leave it at that. $\endgroup$ – user30379 Jan 15 '13 at 17:16
  • $\begingroup$ @pranavk: Sorry, didn't see your comment -- you said considerably more than I did in less space! $\endgroup$ – David Loeffler Jan 15 '13 at 17:20

Yes, this can happen. Here is a counterexample (which is probably not the simplest possible, but it's the one that first came to mind).

There are not very many 2-dimensional representations of the Galois group of $\mathbf{Q}_p$ which are "crystalline" in Fontaine's sense. Fontaine's functor $\mathbf{D}_{\operatorname{cris}}$ classifies them by linear data: 2-dimensional vector spaces over $\mathbf{Q}_p$ with a filtration and a linear operator $\varphi$ (the Frobenius) satisfying some compatibility properties. If $V$ is the $p$-adic Galois representation coming from an elliptic curve over $\mathbf{Q}_p$ with good reduction, then $\varphi$ has characteristic polynomial $X^2 - a_p(E) X + p$, where as usual $a_p(E) = p + 1 - \# \overline{E}(\mathbf{F}_p)$.

If $a_p = 0$, then this uniquely determines $\mathbf{D}_{\operatorname{cris}}(V)$ as a $\varphi$-module, and the conditions on the filtation ("weak admissiblity") mean that if $a_p(E) = 0$ then there is (up to isomorphism) a unique possibility for the filtration. So, in other words, all elliptic curves over $E$ with good supersingular reduction have isomorphic $p$-adic Galois representations [edit: if $p>3$, at least]. But they certainly aren't all isomorphic (or even isogenous) to each other, so that gives a counterexample.

  • $\begingroup$ I think that for $p=2$ you can have supersingular reduction and $a_p=2$... $\endgroup$ – user30035 Jan 15 '13 at 21:38
  • $\begingroup$ @wccanard: And similarly for $p=3$, right? $\endgroup$ – user30379 Jan 16 '13 at 1:38
  • $\begingroup$ So if we require that these two Abelian varieties are ordinary, then the answer will be no ? $\endgroup$ – TOM Jan 16 '13 at 9:34
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    $\begingroup$ In the ordinary case there are a few more possibilities because the Galois rep is reducible and it may or may not be split, but there are still only finitely many possible Galois reps for each p and they fall far short of distinguishing the infinitely many isogeny classes of elliptic curves. $\endgroup$ – David Loeffler Jan 16 '13 at 21:22

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