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

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.