Constructing abelian varieties from rank 2 Galois representations

Let U be a smooth affine curve over a number field K with a compactification X and let {L} be a rank 2, geometrically irreducible lisse { {Q}}_ -sheaf on U with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field E {Q}, and has bad, infinite reduction at some closed point x of X U. We show that {L} occurs as a summand of the cohomology of a family of abelian varieties over U. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when E= Q, then {L} is isomorphic to the cohomology of an elliptic curve EU U.