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Schwarzian differential equations and Hecke eigenforms on Shimura curves

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  31 October 2012

Yifan Yang
Yifan Yang*
Affiliation:
Department of Applied Mathematics, National Chiao Tung University and National Center for Theoretical Sciences, Hsinchu, Taiwan 300, Taiwan (email: yfyang@math.nctu.edu.tw)
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Abstract

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Let X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation \[ _2F_1\biggl (\frac 1{24},\frac 7{24};\frac 56; -\frac {2^{10}\cdot 3^3\cdot 5}{11^4}\biggr )=\sqrt 6 \sqrt [6]{\frac {11}{5^5}} \] and other similar identities.

Keywords

Hecke eigenformsHecke operatorsSchwarzian differential equationsShimura curves

MSC classification

Secondary: 11F12: Automorphic forms, one variable 11G18: Arithmetic aspects of modular and Shimura varieties 33C05: Classical hypergeometric functions, $_2F_1$

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 1 - 31
Copyright
Copyright © The Author(s) 2012

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