On compact Riemann surfaces with a maximal number of automorphisms
A finite group \(G\) is a Hurwitz group if it acts as a group of automorphisms on a compact Riemann surface \(X\) of genus \(g\) and has maximal possible order \(84(g- 1)\). As such it is a quotient of the Fuchsian triangle group \((0;2,3,7)\) so that \(X/G\) has genus zero. The author introduces the notion of a genus 1 Hurwitz group with the same maximality property but subject to the restriction that \(X/G\) has genus 1. As such, \(G\) has order \(4(g-1)\) and is a quotient of the Fuchsian group of signature \((1;2)\). On a large subclass of the class of all finite groups, simple necessary and sufficient conditions are obtained for a group to be a genus 1 Hurwitz group in which case, the quotient of genus 1 can be shown to have prescribed \(j\)-invariant.
scientific article; zbMATH DE number 492795
scientific article