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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Moduli of punctured tori and the accessory parameter of Lamé’s equation
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by L. Keen, H. E. Rauch and A. T. Vasquez PDF
Trans. Amer. Math. Soc. 255 (1979), 201-230 Request permission


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.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 255 (1979), 201-230
  • MSC: Primary 30F10; Secondary 14H15
  • DOI:
  • MathSciNet review: 542877