Automata and finite order elements in the Nottingham group

Abstract

This thesis was written by Djurre Tijsma from October 2017 until July 2018 as part of the master's programme Mathematical Sciences at Utrecht University. The research was supervised by prof. dr. Gunther Cornelissen and the second reader is prof. dr. Frits Beukers. We use automata theory to study the finite order elements of the Nottingham group N over the finite field with p elements, p a prime number. From 2010 the only known elements of order not a prime number were three different elements of order 4. After introducing some preliminaries and the theory of p-automata, we present a method for constructing algebraic equations for finite order elements of N over the rational function field. In the specific case p=2 we use a 2-automaton to construct for five different elements of order 4 an explicit power series. These five power series are different from the three known power series found in the literature.