Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On Shimura curves in the Schottky locus


Author: Stefan Kukulies
Journal: J. Algebraic Geom. 19 (2010), 371-397
DOI: https://doi.org/10.1090/S1056-3911-09-00528-1
Published electronically: August 18, 2009
MathSciNet review: 2580680
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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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Stefan Kukulies
Affiliation: Universität Duisburg-Essen, Mathematik, 45117 Essen, Germany
Email: Stefan.Kukulies@uni-due.de

Received by editor(s): March 20, 2008
Received by editor(s) in revised form: December 26, 2008
Published electronically: August 18, 2009
Additional Notes: This work was financially supported by the Deutsche Forschungsgemeinschaft