Constructing abelian varieties from rank 3 Galois representations with real trace field

Abstract

Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 ${\overline{\mathbb{Q}}}_{\ell }$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$ extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking $U$, there exists an abelian scheme $f:B_U\to U$ such that $L$ is a summand of $R^2{f}_{\ast }{\overline{\mathbb{Q}}}_{\ell }(1)$. The key ingredients are: (1) the totally real assumption implies $L$ admits a square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over $U_{\bar{K}}$ geometrically realizing $L$; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration.

3 pages, comments welcome!