On Shimura curves in the Schottky locus

TL;DR

The paper demonstrates that for large genus g, rational Shimura curves with maximal Higgs fields do not intersect the Schottky locus, using a combination of algebraic geometry and number theory tools.

Contribution

It establishes a non-intersection result between Shimura curves with maximal Higgs fields and the Schottky locus for large g, extending understanding of the geometry of moduli spaces.

Findings

Shimura curves with maximal Higgs fields do not intersect the Schottky locus for large g.

The use of Viehweg and Zuo's result links families of curves to modular elliptic curves.

Number theoretic conjectures imply the non-existence of certain families for large genus.

Abstract

We show that a given rational Shimura curve Y with strictly maximal Higgs field in the moduli space of g-dimensional abelian varieties does not generically intersect the Schottky locus for large g. We achieve this by using a result of Viehweg and Zuo which says that if Y parameterizes a family of curves of genus g, then the corresponding family of Jacobians is isogenous over Y to the g-fold product of a modular family of elliptic curves. After reducing the situation from the field of complex numbers to a finite field, we will see, combining the Weil and Sato-Tate conjectures, that this is impossible for large genus g.