Automorphisms group of generalized Fermat curves of type
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 ≅ 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 .