Frobenius elements in Galois representations with SL_n image

Abstract

Suppose we have a elliptic curve over a number field whose mod $l$ representation has image isomorphic to $S{L}_2({\mathbb{F}}_l)$. We present a method to determine Frobenius elements of the associated Galois group which incorporates the linear structure available. We are able to distinguish $S{L}_n({\mathbb{F}}_l)$-conjugacy from $G{L}_n({\mathbb{F}}_l)$-conjugacy; this can be thought of as being analogous to a result which distinguishes $A_n$-conjugacy from $S_n$-conjugacy when the Galois group is considered as a permutation group.

5 pages