Moduli of punctured tori and the accessory parameter of Lamé’s equation

Abstract

To solve the problems of uniformization and moduli for Riemann surfaces, covering spaces and covering mappings must be constructed, and the parameters on which they depend must be determined. When the Riemann surface is a punctured torus this can be done quite explicitly in several ways. The covering mappings are related by an ordinary differential equation, the Lamé equation. There is a constant in this equation which is called the “accessory parameter". In this paper we study the behavior of this accessory parameter in two ways. First, we use Hill’s method to obtain implicit relationships among the moduli of the different uniformizations and the accessory parameter. We prove that the accessory parameter is not suitable as a modulus-even locally. Then we use a computer and numerical techniques to determine more explicitly the character of the singularities of the accessory parameter.