Constructing abelian varieties from rank 2 Galois representations

TL;DR

This paper demonstrates that certain rank 2 Galois representations with specific properties can be realized as summands of the cohomology of families of abelian varieties over a curve.

Contribution

It extends Snowden-Tsimerman's result to more general number fields, constructing abelian varieties from rank 2 Galois representations with particular local and global properties.

Findings

Rank 2 Galois representations are realized in the cohomology of abelian varieties.

The method generalizes previous results from rational to arbitrary number fields.

Provides a new link between Galois representations and algebraic geometry over number fields.

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 $E_U\rightarrow U$.