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

We show that a two dimensional ℓ  -adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a GL(2)GL(2)-ℓ-adic representation ρ at a set of places of K of positive upper density is potentially equivalent to ρ.As an application, for E1E1 and E2E2 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 E1E1 and E2E2 are isogenous over some extension of K.For an elliptic curve E defined over a number field K, we show that the set of finite places of K   such that the Frobenius field F(E,v)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.