Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

On the structure of the Whittaker sublocus of the moduli space of algebraic curves

Ernesto Girondoa1 and Gabino González-Dieza2

a1 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain ernesto.girondo@uam.es

a2 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain gabino.gonzalez@uam.es

Abstract

We prove the compactness of the Whittaker sublocus of the moduli space of Riemann surfaces (complex algebraic curves). This is the subset of points representing hyperelliptic curves that satisfy Whittaker's conjecture on the uniformization of hyperelliptic curves via the monodromy of Fuchsian differential equations. In the last part of the paper we devote our attention to the statement made by R. A. Rankin more than 40 years ago, to the effect that the conjecture ‘has not been proved for any algebraic equation containing irremovable arbitrary constants’. We combine our compactness result with other facts about Teichmüller theory to show that, in the most natural interpretations of this statement we can think of, this result is, in fact, impossible.

(Received January 08 2004)

(Accepted July 27 2005)