Dimensions of Deformation Rings and Automorphy Lifting Theorems

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

This research project studies questions in algebraic number theory. The connection between number theory and automorphic forms, which are generalizations of periodic functions, is a very active area of investigation and has led to solutions of longstanding problems in number theory. Such work has applications to solving Diophantine equations -- algebraic equations with integer coefficients for which integer solutions are sought -- which play a role in a wide range of applications from encryption protocols to error correction. Additionally, methods developed in the area of algebraic number theory can have other surprising applications to cryptography. This project aims to deepen knowledge in this important subject. The questions that the project studies are related to classical problems in number theory, including the Leopoldt conjecture and higher dimensional analogs. The methods to be used relate to automorphy lifting theorems, which have played a crucial role in important developments such as the proof of Fermat's Last Theorem and the proof of Serre's conjecture. The project aims to augment these methods and to make them more flexible, which will lead to more applications in the field of solutions of polynomial equations and its relation to automorphic forms.