Topics in Galois representations

Embargo until
2025-08-01
Date
2021-05-10
Journal Title
Journal ISSN
Volume Title
Publisher
Johns Hopkins University
Abstract
In this thesis, we explore two conjectures about Galois representations. The first one is the Tate conjecture. Using Calegari's theorem on the Fontaine-Mazur conjecture and a trace formula, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus 3. This result is joint work with Lian Duan. The second one is the Serre weight conjecture. By a careful choice of uniformizer, we prove the weight elimination part of the Serre weight conjecture for rank two unitary groups when p=2. Then we study the weight existence part of this conjecture.
Description
Keywords
Galois representations, Elliptic surfaces, the Tate conjecture, Serre weights
Citation