Schwarzian differential equations and Hecke eigenforms on Shimura curves

TL;DR

This paper characterizes automorphic forms on genus zero Shimura curves via Schwarzian differential equations, develops a method to compute Hecke operators, and derives new hypergeometric identities as by-products.

Contribution

It introduces a novel approach linking Schwarzian differential equations to automorphic forms on Shimura curves and provides a computational method for Hecke operators.

Findings

Explicit characterization of automorphic forms via Schwarzian equations

A new method for computing Hecke operators on these forms

Derivation of new hypergeometric identities

Abstract

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 $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}} $$ and other similar identities.