JournalsrmiVol. 20, No. 3pp. 627–646

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
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Given a closed Riemann surface RR of genus p2p \geq 2 together with an anticonformal involution τ:RR\tau:R \to R with fixed points, we consider the group K(R,τ)K(R,\tau) consisting of the conformal and anticonformal automorphisms of RR which commute with τ\tau. It is a well known fact due to C. L. May that the order of K(R,τ)K(R,\tau) is at most 24(p1)24(p-1) and that such an upper bound is attained for infinitely many, but not all, values of pp. May also proved that for every genus p2p \geq 2 there are surfaces for which the order of K(R,τ)K(R,\tau) can be chosen to be 8p8p and 8(p+1)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)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,τ)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.

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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