CONTRIBUTIONS TO THE THEORY OF FUNCTION FIELDS
IN POSITIVE CHARACTERISTIC
BURÇİN GÜNEŞ
Mathematics, PhD Dissertation, 2019
Thesis Jury
Prof. Dr. Cem Güneri (Thesis Advisor),
Asst. Prof. Nurdagül Anbar Meidl (Thesis Co-advisor),
Assoc. Prof. Kağan Kurşungöz,
Prof. Dr. Özgür Gürbüz,
Prof. Dr. Massimo Giulietti,
Assoc. Prof. Ekin Özman,
Date &Time: July 19th, 2019 – 15:30
Place: Sabancı University - Karaköy Minerva Palace
Keywords : automorphism group, function field, Galois extension, Hermitian function field, Hurwitz’s genus formula, nilpotent subgroup, maximal curve, positive characteristic
Abstract
In this thesis, we consider two problems related to the theory of function fields in positive characteristic. In the first part, we study the automorphisms of a function field of genus g≥2 over an algebraically closed field of characteristic p > 0. We show that for any nilpotent subgroup G of the automorphism group, the order of G is bounded by 16(g-1) when G is not a p-group and by 4pg2/(p-1)2 when G is a p-group. We also provide examples of function fields attaining these bounds; therefore, the bounds we obtained cannot be improved. In the second part, we focus on maximal function fields over finite fields having large automorphism groups. More precisely, we consider maximal function fields over the finite field Fp4 whose automorphism groups have order exceeding the Hurwitz’s bound. We determine some conditions under which the maximal function field is Galois covered by the Hermitian function field.