The automorphism groups of a family of maximal curves

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmáros introduced a curve C3 which is maximal over Fq6 and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves Cn, indexed by an odd integer n⩾3, such that Cn is maximal over Fq2n. In this paper, we determine the automorphism group Aut(Cn) when n>3; in contrast with the case n=3, it fixes the point at infinity on Cn. The proof requires a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point.