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⊂ Q̄ , 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 .