Real Schottky uniformizations and Jacobians of May surfaces

Abstract

Given a closed Riemann surface R of genus p >= 2 together with an anticonformal involution tau : R -> R with fixed points, we consider the group K(R, tau) consisting of the conformal and anticonformal auto-morphisms of R which commute with tau. It is a well known fact due to C. L. May that the order of K(R, tau) is at most 24(p - 1) and that such an upper bound is attained for infinitely many, but not all, values of p. May also proved that for every genus p > 2 there are surfaces for which the order of K(R, tau) can be chosen to be 8p and 8(p + 1). These type of surfaces are called May surfaces. In this note we construct real Schottky uniformizations of every May surface. In particular, the corresponding group K(R, tau) lifts to such an uniformization. With the help of these real Schottky uniformizations, we obtain (extended) symplectic representations of the groups K(R, T). We study the families of principally polarized abelian varieties admitting the given group of automorphisms and compute the corresponding Riemann matrices, including those for the Jacobians of May surfaces.