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 ℤ/3 covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura-Teichmüller curve that parameterizes Jacobians that are suitable ℤ/6 covers of P¹. 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.