Local systems which do not come from abelian varieties

TL;DR

The paper constructs local systems on punctured smooth curves over finite fields that are not derived from abelian varieties, providing a criterion to distinguish such systems from those of geometric origin.

Contribution

It introduces a new criterion to identify local systems not arising from abelian varieties, expanding understanding of local systems in positive characteristic.

Findings

Constructed explicit examples of non-abelian origin local systems

Developed a criterion inspired by Hodge theory for characteristic zero

Demonstrated existence of local systems beyond abelian variety families

Abstract

For each smooth curve over a finite field, after puncturing it at finitely many points, we construct local systems on it of geometric origin which do not come from a family of abelian varieties. We do so by proving a criterion which must be satisfied by local systems which do come from abelian varieties, inspired by an analogous Hodge theoretic criterion in characteristic zero.