Compatible families of elliptic type

Abstract

In axiomatizing their study of Frobenius distributions [5], Lang and Trotter introduce the notion of an adelic Galois representation of elliptic type, and they ask in passing whether every such representation arises from an elliptic curve (see pp. 5 and 19 of [5]). Formulated in the language of `-adic representations [7], their question is as follows. Put G = Gal(Q/Q), let p denote a prime, and write σp for any Frobenius element at a prime ideal of Q over p. Let {ρ`} be a two-dimensional strictly compatible family of integral `-adic representations ofG with exceptional set S, and for p / ∈ S put a(p) = tr ρ`(σp) with any ` 6= p. Also, let ω` : G→ Z` denote the `-adic cyclotomic character. Although ρ` is a priori a map into GL(2,Q`), after a conjugation in GL(2,Q`) we may regard it as a map G→ GL(2,Z`).