Constructing abelian varieties from rank 2 Galois representations

Abstract

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let ${\mathbb {L}}$ be a rank $2$ , geometrically irreducible lisse $\overline {{\mathbb {Q}}}_\ell$ -sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \overline {\mathbb {Q}}_{\ell }$ , and has bad, infinite reduction at some closed point $x$ of $X\setminus U$ . We show that ${\mathbb {L}}$ 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 $E=\mathbb {Q}$ , then ${\mathbb {L}}$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$ .