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On Shimura curves in the Schottky locus
We show that a given rational Shimura curve U in the moduli space of principally polarized abelian varieties Ag,1 can only lie in the closure of the Schottky locus Mg for finitely many genera g. We achieve this by using a result of Viehweg and Zuo which says that the corresponding families of Jacobians parameterized by U have to be isogenous to the g-fold product of a modular family of elliptic curves. Reducing the situation to characteristic p, we will see that the genus of such families is bounded where the bound depends only on the cardinality of the bad locus.
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