Galois Representations and Arithmetic Questions

Cornell University, Ithaca NY

Investigators

Abstract

The investigator will study questions in the deformation theory of global Galois representations. Let a two dimensional mod p Galois representation be given. Serre has conjectured this representation "comes from" a modular form. His conjecture necessarily implies the mod p Galois representation is the mod p reduction of some p-adic Galois representation. The author plans to continue his study of whether such deformations exist independently of Serre's Conjecture, and if they can be arranged to be potentially semistable in the sense of Fontaine. Such constructions, especially combined with recent work of Taylor, could be regarded as providing evidence for Serre's Conjecture. There are interesting cases that are still open, such as reducible residual representations and representations of the absolute Galois group of totally real fields. The proposed research falls under the general area of Number Theory, which has its roots in the study of whole numbers and their various properties. Natural questions that arise in this field, such as properties of prime numbers and how frequently they occur, can have applications to coding theory and cryptography. The proposed research is also related to Galois Theory, the study of symmetries of solutions to equations. Studying these symmetries can shed light on the understanding their solutions, a basic mathematical question.