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.
Categories: math.AG