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SCHOTTKY UNIFORMIZATIONS OF GENUS 3 AND 4 REFLECTING ${\mathcal S}_{4}$
Published online by Cambridge University Press: 20 July 2005
Abstract
Schottky uniformizations are provided of every closed Riemann surface $S$ of genus $g \in \{3,4\}$ admitting the symmetric group ${\mathcal S}_{4}$ as group of conformal automorphisms. These Schottky uniformizations reflect the group ${\mathcal S}_{4}$ and permit concrete representations of ${\mathcal S}_{4}$ to be obtained in the respective symplectic group $\mbox{Sp}_{g}({\mathbb Z})$. Their corresponding fixed points, in the Siegel space, give principally polarized Abelian varieties of dimension $g$. For $g=3$ and for some cases of $g=4$ they turn out to be holomorphically equivalent to the product of elliptic curves.
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- The London Mathematical Society 2005