SCHOTTKY UNIFORMIZATIONS OF GENUS 3 AND 4 REFLECTING S4

Schottky uniformizations are provided of every closed Riemann surface S of genus g ∈ {3, 4} admitting the symmetric group S4 as group of conformal automorphisms. These Schottky uniformizations reflect the group S4 and permit concrete representations of S4 to be obtained in the respective symplectic group Spg (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.