Non-existence of mod 2 and mod 3 Galois Representations

Abstract

This project studies the non-existence of irreducible Galois representations of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ unramified outside $p = 2$ and $p = 3$. This topic was studied mainly by John Tate ([22]) and J.P. Serre ([18] and [20]), with further results obtained in recent years, but the literature for the original problem consists on very disperse, informal and outdated notes that rely on deep topics or that do not go over the details. Our goal is to provide a simplified proof using everything that has been studied after Tate and Serre introduced it, while including the details needed to follow the proof. The main tool for this project is the study of local fields and how they ramify, to simplify the problem through an upper bound for the discriminant of extensions introduced by Tate. In Chapter 1 we introduce everything necessary for the proof of Tate's bound, focusing on local field theory and class field theory. In Chapter 2 we present the problem and how to work with it locally, and we state and prove Tate’s bound (actually a refinement of Tate’s original bound by Moon and Taguchi, introduced in [12]). Chapter 3 focuses on the applications of Tate’s bound to $p = 2$ and $p = 3$ and analysing the particular Galois groups and corresponding extensions that arise, to finally prove the non-existence of Galois representations unramified outside these values of $p$.