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

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 $F$.

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