Real Schottky Uniformizations and Jacobians of May Surfaces

Abstract

Given a closed Riemann surface $R$ of genus $p \geq 2$ together with an anticonformal involution $\tau:R \to R$ with fixed points, we consider the group $K(R,\tau)$ consisting of the conformal and anticonformal automorphisms 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 $p \geq 2$ 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 \textit{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 $K(R,\tau)$. 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.