A bound on the dimension of a totally geodesic submanifold in the Prym locus

TL;DR

This paper establishes upper bounds on the dimension of totally geodesic submanifolds within the Prym locus, linking geometric properties of curves to the structure of Shimura varieties.

Contribution

It provides new bounds on the dimension of such submanifolds, connecting gonality and genus to geometric constraints in the Prym locus.

Findings

Bound depending on gonality k

Genus-dependent bound

Implications for Shimura varieties

Abstract

We give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura variety of A_{g-1}, contained in the Prym locus. First we give such a bound for a germ passing through a Prym variety of a k-gonal curve in terms of the gonality k. Then we deduce a bound only depending on the genus g.