Schwarzian differential equations associated to Shimura curves of genus zero

Fang-Ting Tu

Vol. 269 (2014), No. 2, 453–489

DOI: 10.2140/pjm.2014.269.453

Abstract

Let $X_0^D\left(N\right)$, where $\left(D,N\right)=1$, denote the Shimura curve associated to an Eichler order of level $N$, in an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$. Let $W_{D,N}$ be the group of all Atkin–Lehner involutions on $X_0^D\left(N\right)$ and $W_D$ the subgroup consisting of Atkin–Lehner involutions $w_m$ with $m÷D$. In this paper, we will determine Schwarzian differential equations associated to Shimura curves $X_0^D\left(N\right)∕W_D$ of genus zero in the cases where there exists a squarefree integer $M>1$ such that $X_0^D\left(M\right)∕W_D$ is of genus zero.

Keywords

automorphic form, modular form, Shimura curves, Schwarzian differential equation

Mathematical Subject Classification 2010

Primary: 11F12

Secondary: 11G18

Milestones

Received: 1 December 2013

Revised: 27 March 2014

Accepted: 31 March 2014

Published: 26 July 2014