Geometric uniformization in genus 2.

Abstract

Compact Riemann surfaces with large automorphism groups have been studied extensively. Here we treat genus 2 case in a geometric way. The three surfaces with exceptional symmetry are Wiman curves of types I and II and an Accola--Maclachlan surface. By considering the location of Weierstrass points, we exhibit Dirichlet fundamental polygons for these surfaces and calculate their period matrices. A list of orbifolds covered by closed surfaces of genus 2 is also obtained, leading to the list of triangle groups having a normal subgroup of genus 2 and to their arithmeticity.