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On Shimura curves in the Schottky locus

Author(s): Stefan Kukulies
Journal: J. Algebraic Geom.
Posted: August 18, 2009
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Abstract | References | Additional information

Abstract: We show that a given rational Shimura curve $ Y$ with strictly maximal Higgs field in the moduli space of $ g$-dimensional principally polarized abelian varieties does not map to the closure of the Schottky locus for large $ g$ if the generic point is the jacobian of a smooth curve.

We achieve this by using a result of Viehweg and Zuo which says that the corresponding family of abelian varieties over $ Y$ is $ Y$-isogenous 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 for large $ g$ no such family can become the jacobian of a family of curves.

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Additional Information:

Stefan Kukulies
Affiliation: Universität Duisburg-Essen, Mathematik, 45117 Essen, Germany
Email: Stefan.Kukulies@uni-due.de

PII: S 1056-3911(09)00528-1
Received by editor(s): March 20, 2008
Received by editor(s) in revised form: December 26, 2008
Posted: August 18, 2009
Additional Notes: This work was financially supported by the Deutsche Forschungsgemeinschaft

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