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



Tate conjecture for elliptic curves local fields
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.


