# Shimura curves in the Prym locus

## TL;DR

This paper investigates the presence of Shimura curves within the Prym locus of abelian varieties, providing criteria and computational verification for their existence in specific cases up to genus 28.

## Contribution

It establishes a criterion for identifying Shimura curves in the Prym locus and computationally finds numerous examples in both ramified and unramified cases up to genus 28.

## Key findings

- 43 Shimura curves in the unramified Prym locus
- 9 Shimura curves in the ramified Prym locus
- Most Shimura curves are not in the Jacobian locus

## Abstract

We study Shimura curves of PEL type in $\mathsf{A}_g$ generically contained in the Prym locus. We study both the unramified Prym locus, obtained using \'etale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is 1-dimensional and the quotient of the base curve by the group is $\mathbb{P}^1$. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples gotten in this way up to genus 28. We obtain 43 Shimura curves generically contained in the unramified Prym locus and 9 families generically contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.