The Automorphism Groups of a Family of Maximal Curves

TL;DR

This paper determines the automorphism groups of a family of maximal algebraic curves over finite fields, extending previous results for specific cases and revealing new structural properties of these groups.

Contribution

It generalizes the automorphism group determination to all curves C_n with n > 3, showing they fix the point at infinity unlike the n=3 case.

Findings

Automorphism groups of C_n for n > 3 are explicitly determined.

New structural result about automorphism groups with Sylow p-subgroups fixing exactly one point.

Automorphism groups for these maximal curves differ from the n=3 case by fixing the point at infinity.

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 greater than or equal to 3, such that C_n is maximal over F_{q^{2n}}. In this paper, we determine the automorphism group Aut(C_n) when n > 3; in contrast with the case n=3, it fixes the point at infinity on C_n. 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. MSC:11G20, 14H37.