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Let $E_1$ and $E_2$ be elliptic curves over a field $k$, and let $l$ be a prime coprime to the characteristic of $k$ (if $char(k) \ne 0$). Let $\varphi$ denote the canonical map

$Hom(E_1,E_2)\otimes_{\mathbb{Z}} \mathbb{Z}_l \rightarrow Hom_{G_k}(T_lE_1,T_lE_2)$ `.

For any field $k$, $\varphi$ is easily shown to be injective, and (a case of) the Tate conjecture says that if $k$ is a finite field or a number field then $\varphi$ is surjective (though the proof is hard). What can we say if instead $k$ is a local field?

More precisely, my questions are:

1)Is there a reason/counterexample explaining why $\varphi$ will not be surjective for $k$ a local field?

2)If the answer to (1) is `yes', is there a weaker statement along the lines of the above which is (or is expected to be) true for $k$ a local field? (Sorry this is rather vague).

Thanks, David

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    $\begingroup$ See mathoverflow.net/questions/53014/… $\endgroup$ – Felipe Voloch Aug 20 '11 at 12:25
  • $\begingroup$ Thanks Felipe Voloch, the answers there are exactly what I was looking for. Sorry for duplicating your question, I didn't think to search for the name `isogeny theorem'. $\endgroup$ – David Holmes Aug 20 '11 at 13:27
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Hi David,

A nice question. The map $\varphi$ can be very far from surjective! One way to see this is as follows. Let us work over $Q_p$, and suppose first that $l\neq p$. Working with elliptic curves with good reduction, the corresponding Galois representation is determined by $a_p$ (as this determines the characteristic polynomial of Frobenius).

Now suppose instead that $l=p$, and consider elliptic curves with good ordinary reduction. Then the Galois representation is again determined (up to a finite number of possibilities) by $a_p$. More generally, the classification of crystalline Galois representations by weakly admissible modules shows that the local Galois representation is always determined up to a finite number of possibilities by $a_p$.

Now thinking about cardinality shows that there must be many pairs of elliptic curves over $Q_p$ which are not isogenous, but have isomorphic $l$-adic Galois representations.

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  • $\begingroup$ Thanks very much JT, that seems to do it (combined with Felipe's link for part (2)). $\endgroup$ – David Holmes Aug 20 '11 at 13:26

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