Shimura curves in the Prym loci of ramified double covers

TL;DR

This paper constructs and identifies 184 Shimura curves within ramified Prym loci of abelian varieties, extending previous unramified cases to ramified double covers with specific group actions.

Contribution

It generalizes the construction of Shimura curves to ramified double covers, providing explicit examples and computational evidence of their presence in Prym loci.

Findings

184 Shimura curves found in ramified Prym loci

Construction compatible with fixed group actions on base curves

Extension of unramified case to ramified double covers

Abstract

We study Shimura curves of PEL type in the space of polarised abelian varieties $A^{\delta}_p$ generically contained in the ramified Prym locus. We generalise to ramified double covers, the construction done in [10] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is ${\mathbb P}^1$. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci.