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Computing fundamental domains for the Bruhat–Tits tree for ${\rm GL}_2(\mathbf{Q}_p)$, $p$-adic automorphic forms, and the canonical embedding of Shimura curves

Part of: Other groups of matrices Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 April 2014

Cameron Franc  and
Marc Masdeu
Cameron Franc
Affiliation:
Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064, USA email cfranc@ucsc.edu
Marc Masdeu
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA email masdeu@math.columbia.edu

Abstract

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We describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.

MSC classification

Secondary: 11F06: Structure of modular groups and generalizations; arithmetic groups 20H10: Fuchsian groups and their generalizations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 1 - 23
Copyright
© The Author(s) 2014 

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