On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

Abstract:

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface $y^2=\prod _{i=1}^{2g+2}(x-a_i)$ as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values $a_i$. Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker’s conjecture cannot be true in its full generality. Recently, numerical computations have shown that Whittaker’s prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms. The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker’s conjecture.