Schwarzian differential equations associated to Shimura curves of genus zero
type: Item
instance of: scholarly article
title: Schwarzian differential equations associated to Shimura curves of genus zero
zbMATH Open document ID: 1308.11047
author: Fang-Ting Tu
published in: Pacific Journal of Mathematics
publication date: 2014-12-30T00:00:00Z
review text: Let \(X_0^D (N)\) be the Shimura curve associated to a division quaternion algebra over \(\mathbb Q\) of discriminant \(D \neq 1\), and level \(N\) with \((D,N) = 1\). Let \(X = X_0^D (N) / W_D\), where \(W_D\) is the group of Atkin-Lehner involutions \(W_m\) on \(X_0^D (N)\) with \(m | D\). Suppose \(X\) has genus \(0\), and \(t(\tau)\) generates the field of automorphic functions on \(X\). Then \(t'(\tau)^{\frac{1}{2}}\) is a solution to a Schwarzian differential equation of the form \(f'' + Q(t)f = 0\), where \(Q(t)\) is a rational function of \(t\). The author shows that there are exactly \(13\) pairs \((D, N)\) such that \(X = X_0^D (N) / W_D\) has genus zero and \(N>1\); these pairs are explicitly listed. In addition there are \(8\) pairs \((D, 1)\) such that \(X = X_0^D (M) / W_D\) has genus zero for some \(M > 1\). For each of these \(21\) pairs, a generator \(t(\tau)\) of the field of automorphic functions on \(X\) is found, and the rational function \(Q(t)\) is explicitly determined, thus making explicit the Schwarzian differential equations associated to automorphic forms for these Shimura curves.
reviewed by: Salman Abdulali
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11G18
zbMATH DE Number: 6382655
zbMATH Keywords: automorphic form
zbMATH Keywords: modular form
zbMATH Keywords: Shimura curves
zbMATH Keywords: Schwarzian differential equation
zbMATH Keywords: Atkin-Lehner involution
MaRDI profile type: Publication
full work available at URL: pjm.2014.269.453
OpenAlex ID: W2119261355
DOI: 10.2140/PJM.2014.269.453
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description: scientific article; zbMATH DE number 6382655