Fundamental domains for Shimura curves.
type: Item
instance of: scholarly article
title: Fundamental domains for Shimura curves.
zbMATH Open document ID: 1044.11052
author: David Kohel
author: H. A. Verrill
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/232390
review text: Let \(A\) be an indefinite quaternion algebra over \(\mathbb Q\) with discriminant \(D\), and let \(\mathcal O\) be a maximal order in \(A\). An Eichler order of index \(N\) in \(\mathcal O\) defines a discrete subgroup \(\Gamma^D_0 (N)\) of \(\text{PSL}_2 (\mathbb R)\), and the quotient \(X^D_0 (N) = \Gamma^D_0 (N) \backslash \mathcal H\) of the Poincaré upper half plane \(\mathcal H\) by \(\Gamma^D_0 (N)\) is a Shimura curve. In this paper the authors describe a process for determining a fundamental domain for \(X^D_0 (N)\) in \(\mathcal H\) and show that such a fundamental domain realizes a finite presentation of the quaternion unit group modulo its center. They also provide explicit examples for the curves \(X^6_0 (1)\), \(X^{15}_0 (1)\), and \(X^{35}_0 (1)\).
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 2058865
zbMATH Keywords: Shimura curves
zbMATH Keywords: fundamental domains
zbMATH Keywords: modular curves
zbMATH Keywords: quaternion algebras
describes a project that uses: Magma
describes a project that uses: ecdata
MaRDI profile type: Publication
OpenAlex ID: W2051143616
cites work: Q4341641
cites work: Q3840149
cites work: Q4541479
cites work: Q3864636
cites work: Courbes de Shimura hyperelliptiques
cites work: Arithmétique des algèbres de quaternions
DOI: 10.5802/JTNB.398
description: scientific article; zbMATH DE number 2058865