Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves

Abstract

We show that a two dimensional $\ell $-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a $GL(2)$-$\ell$-adic representation $ρ$ at a set of places of $K$ of positive upper density is potentially equivalent to $ρ$. For an elliptic curver \( E \) defined over a number field \( K \) and a finite place \( v \) of \( K \) of good reduction for \( E \), let \( F(E,v) \) denote the Frobenius field of \( E \) at \( v \), given by the splitting field of the characteristic polynomial of the Frobenius automorphism at \( v \) acting on the Tate module of \( E \). As an application, suppose \( E_1 \) and \( E_2 \) defined over a number field \( K \), with at least one of them without complex multiplication. We prove that the set of places \( v \) of \( K \) of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if \( E_1 \) and \( E_2 \) are isogenous over some extension of \( K \). We show that for an elliptic curve \( E \) defined over a number field \( K \), the set of finite places of \( K \) such that the Frobenius field \( F(E, v) \) at $v$ equals a fixed imaginary quadratic field \( F \) has positive upper density if and only if \( E \) has complex multiplication by \( F \).