Automorphisms group of generalized Fermat curves of type (k, 3)

Abstract

The determination of the full group of automorphisms of a closed Riemann surface is in general a very complicated task. For hyperelliptic curves, the uniqueness of the hyperelliptic involution permits one to compute these groups in a very simple manner. Similarly, as classical Fermat curves of degree k admit a unique subgroup of automorphisms isomorphic to Z(k)(2), the determination of the group of automorphisms is not difficult.

In this paper we consider a family of non-hyperelliptic Riemann surfaces, obtained as the fibre product of two classical Fermat curves of the same degree k, which exhibit behaviors of both elliptic and hyperelliptic curves. These curves, called generalized Fermat curves of type (k, 3), are the highest regular abelian branched covers of orbifolds of genus zero with four cone points, all of the same order k. More precisely, a generalized Fermat curve of type (k, 3) is a closed Riemann surface S admitting a group H < Aut(S), called a generalized Fermat group of type (k, 3), so that H congruent to Z(k)(3) and S/H is an orbifold with signature (0, 4; k, k, k, k). In this paper we prove the uniqueness of generalized Fermat groups of type (k, 3). In particular, this allows the explicit computation of the full group of automorphisms of S. (C) 2012 Elsevier B.V. All rights reserved.