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Let $U$ be a smooth affine curve over a number field $K$ with acompactification $X$ and let $\mathbb L$ be a rank $2$, geometricallyirreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomicdeterminant that extends to an integral model, has Frobenius traces all in somefixed number field $E\subset \bar{\mathbb Q}_\ell$, and has bad, infinitereduction 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 ofSnowden-Tsimerman, who show that when $E=\mathbb Q$, then $\mathbb L$ isisomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.

Constructing abelian varieties from rank 2 Galois representations