On a question of Lang and Trotter

In axiomatizing their study of Frobenius distributions in GL₂-extensions of the rationals, Lang and Trotter list a few standard properties of the Galois representations on the Tate modules of an elliptic curve over Q, and they ask in passing whether these properties characterize such representations among all strictly compatible families of l-adic representations of Gal(Q/Q). The aim of the talk is to refine their question and to formulate it as a consequence of Serre's conjecture.