On automorphism groups of compact Riemann surfaces of genus 4 (Q1073968)

Let X be a compact Riemann surface of genus g. Since a conformal automorphism of X acts on the space of holomorphic differentials on X, a subgroup of the conformal automorphism group Aut(X) of X can be represented as a subgroup of GL(g,\({\mathbb{C}})\). In this paper, in case \(g=4\), the authors determine the subgroups of GL(4,\({\mathbb{C}})\) whose conjugates come from subgroups of Aut(X). The main theorem is mentioned in terms of a Riemann-Hurwitz type relation and an Eichler type formula.