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
Published electronically: August 18, 2009
MathSciNet review: 2580680
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

Additional Information

Stefan Kukulies
Affiliation: Universität Duisburg-Essen, Mathematik, 45117 Essen, Germany

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

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
AMS Website