Motivic local systems on curves and Maeda's conjecture

TL;DR

This paper proves finiteness results for rank two local systems on certain algebraic curves and punctured spheres, providing counterexamples to a conjecture and advancing understanding related to Maeda's conjecture in the context of function fields.

Contribution

It establishes finiteness of rank two local systems of geometric origin on genus two curves and four-punctured spheres, and extends these results to positive characteristic, contributing to Maeda's conjecture.

Findings

Finitely many genus two curves admit such local systems.

Counterexamples to Esnault and Kerz's conjecture are provided.

Results support Maeda's conjecture over function fields.

Abstract

We show that only finitely many complex genus two curves and four punctured spheres admit rank two local systems of geometric origin, and moreover each carries finitely many. This gives further counterexamples to a conjecture of Esnault and Kerz: counterexamples over very general curves were recently obtained by Landesman and Litt. In the second part we prove an analogue of this result in positive characteristic, namely that over $\overline{\mathbb{F}}_p$, only finitely many genus two curves admit non-trivial rank two local systems pulled back from a fixed quaternionic Shimura variety, and the same for $\mathbb{P}^1$ minus four points; conjecturally, every rank two local system arises as such a pullback. This provides results towards Maeda's conjecture on Galois orbits of eigenforms over function fields. The proofs make use of ideas from the work of Landesman and Litt such as isomonodromy, as well as crucially the description of the Goren-Oort strata due to Tian and Xiao.