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On Shimura curves in the Schottky locus

Stefan Kukulies

Number 1
Author Stefan Kukulies
Year 2007

We show that a given rational Shimura curve Y in the moduli space of g-dimensional abelian varieties does not intersect the Schottky locus for large g if its Higgs field is strictly maximal. We achieve this by using a result of Viehweg and Zuo which says that if Y parameterizes a family of curves, 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.

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