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Veech holomorphic families of Riemann surfaces, holomorphic sections, and Diophantine problems


Author: Yoshihiko Shinomiya
Journal: Trans. Amer. Math. Soc. 366 (2014), 3161-3190
MSC (2010): Primary 30F60; Secondary 32G15
DOI: https://doi.org/10.1090/S0002-9947-2014-06056-X
Published electronically: January 21, 2014
MathSciNet review: 3180743
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Abstract: In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their holomorphic sections by some points of corresponding flat surfaces. The construction gives us concrete solutions for some Diophantine equations over function fields. Moreover, we give upper bounds of the number of holomorphic sections of certain holomorphic families of Riemann surfaces.


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Yoshihiko Shinomiya
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Email: shinomiya.y.aa@m.titech.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2014-06056-X
Received by editor(s): April 10, 2012
Received by editor(s) in revised form: October 25, 2012
Published electronically: January 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.