Explicit formulas for infinitely many Shimura curves in genus 4

TL;DR

This paper constructs infinitely many Shimura curves within the genus four Jacobian locus, explicitly computing their period matrices and revealing new families of Jacobians as special covers of elliptic curves.

Contribution

It provides explicit constructions of infinitely many Shimura curves in genus four, including a Shimura-Teichmuller curve, using Shimura's original method.

Findings

Constructed infinitely many Shimura curves in genus four.

Explicitly computed period matrices for these Shimura curves.

Identified Jacobians as ${f Z}/3$ and ${f Z}/6$ covers of elliptic curves.

Abstract

In this paper we construct infinitely many Shimura curves contained in the locus of Jacobians of genus four curves. All Jacobians in these families are ${\mathbb Z}/3$ covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura-Teichmuller curve that parameterizes Jacobians that are suitable ${\mathbb Z}/6$ covers of ${\mathbb P}^1$. We compute explicitly the period matrices of the Shimura curves we construct using the original construction of Shimura for moduli spaces of abelian varieties with automorphisms.