Galois Representations and Arithmetic Questions

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The investigator will study questions in the deformationtheory of global Galois representations. Let a twodimensional 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-adicGalois representation. The author plans to continue hisstudy of whether such deformations exist independently ofSerre'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 reducibleresidual representations and representations of the absoluteGalois 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 proposedresearch is also related to Galois Theory, the study of symmetriesof solutions to equations. Studying these symmetriescan shed light on the understanding their solutions, abasic mathematical question.