Automorphism group and subfields of the generalized Giulietti-Korchmáros function field

Official URL: http://192.168.1.20/record=b1378265 (Table of Contents)

A function field over a finite field which has the largest possible number of rational places, with respect to Hasse-Weil bound, is called maximal. The most important example of a maximal function field is the Hermitian function field H. It has the largest possible genus among maximal function fields defined over the same finite field, and it is the unique function field with this genus, up to isomorphism. Moreover, it has a very large automorphism group. Until recently there was no known maximal function field which is not a subfield of H. In 2009, Giulietti and Korchm áros constructed the first example of a maximal function field over the finite field Fq6 , where q is a prime power, which is not subfield of H over the same finite field. They also determined the automorphism group of this example. Later, a generalization of Giulietti and Korchmáros construction to Fq2n for any odd number n≥3 was given by Garcia, Güneri and Stichtenoth and was shown to be maximal. In this thesis, we determine the automorphism group of the generalized Giulietti- Korchmáros function field. Moreover, some subfields of the generalized Giulietti- Korchmáros function field and their genera are also determined.