Shimura Curves within the Locus of Hyperelliptic Jacobians in Genus 3

We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian three-folds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian three-folds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained within a divisor in the moduli space of abelian varieties.