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 ${\bar{\mathbb{Q}}}_{\ell }$-local system on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset {\bar{\mathbb{Q}}}_{\ell }$, 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-Tsimerman, who show that when $E=\mathbb{Q}$, then $\mathbb{L}$ is isomorphic to the cohomology of an elliptic curve $E_U\to U$.

arXiv comment23 pages