# Real Schottky Uniformizations and Jacobians of May Surfaces

### Rubén A. Hidalgo

Universidad Técnica Federico Santa María, Valparaíso, Chile### Rubí E. Rodríguez

Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

## Abstract

Given a closed Riemann surface $R$ of genus $p≥2$ together with an anticonformal involution $τ:R→R$ with fixed points, we consider the group $K(R,τ)$ consisting of the conformal and anticonformal automorphisms of $R$ which commute with $τ$. It is a well known fact due to C. L. May that the order of $K(R,τ)$ 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 $K(R,τ)$ 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 $K(R,τ)$ lifts to such an uniformization. With the help of these real Schottky uniformizations, we obtain (extended) symplectic representations of the groups $K(R,τ)$. 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.

## Cite this article

Rubén A. Hidalgo, Rubí E. Rodríguez, Real Schottky Uniformizations and Jacobians of May Surfaces. Rev. Mat. Iberoam. 20 (2004), no. 3, pp. 627–646

DOI 10.4171/RMI/403