Infinitely many Shimura varieties in the Jacobian locus for $g \leq 4$

TL;DR

This paper classifies specific families of Galois covers of curves that produce Shimura subvarieties within the Jacobian locus for genus up to 4, revealing infinitely many such subvarieties in these genera.

Contribution

It proves that only six families satisfy the numerical condition for generating Shimura subvarieties in low genus, and shows these families have fibrations with infinitely many Shimura fibers.

Findings

Only six families satisfy the condition in genus 2, 3, or 4.

These families admit fibrations with totally geodesic subvarieties.

The Jacobian locus contains infinitely many Shimura subvarieties for g ≤ 4.

Abstract

We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these families admit two fibrations in totally geodesic subvarieties, generalizing a result of Grushevsky and M\"oller. Countably many of these fibres are Shimura. Thus the Jacobian locus contains infinitely many Shimura subvarieties of positive dimension of any $g \leq 4$.