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.