Canonical models of arithmetic (1; e)-curves

Abstract

The Fuchsian groups of signature (1; e) are the simplest class of Fuchsian groups for which the calculation of the corresponding quotient of the upper half plane presents a challenge. This paper considers the finite list of arithmetic (1; e)-groups. We define canonical models for the associated quotients by relating these to genus 1 Shimura curves. These models are then calculated by applying results on the \({\mathfrak{p}}\)-adic uniformization of Shimura curves and Hilbert modular forms.