Explicit formulas for infinitely many Shimura curves in genus 4

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 $\mathbb{Z}\mathrm{/}3$ 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 $\mathbb{Z}\mathrm{/}6$ 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.