Automorphism groups of a family of maximal curves

Abstract:

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 Korchmaros introduced a curve C_3 which is maximal over F_{q^6} and determined its automorphism group.  Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves C_n, indexed by an odd integer n >= 3, such that C_n is maximal over F_{q^{2n}}.  In this talk, I will first explain why this family of maximal curves is interesting and then describe recent work with Guralnick and Malmskog in which we determine the automorphism group

Aut(C_n) when n > 3; in contrast with the case n = 3, the automorphism group fixes the point at infinity on C_n.  The proof uses ramification theory and a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point.