Galois Representations and Arithmetic Questions

调查员将研究问题的deformationtheory的全球伽罗瓦表示。设给定一个二维模p Galois表示，Serre证明了这个表示“来自“一个模形式。他的猜想必然意味着mod p伽罗瓦表示是某些p-adicGalois表示的mod p约化。作者计划继续研究这样的变形是否独立于Serre猜想而存在，以及它们是否可以被安排为方丹意义下的潜在半稳定。这样的结构，特别是结合最近的工作泰勒，可以被视为提供证据塞尔猜想。有有趣的情况下，仍然是开放的，如reducibleresidual表示和representations的absoluteGalois群的totally真实的fields。拟议的研究福尔斯数论的一般领域，它有其根源的研究整数和它们的各种性质。在这个领域中出现的自然问题，例如素数的性质以及它们出现的频率，可以应用于编码理论和密码学。这项研究也与伽罗瓦理论有关，伽罗瓦理论研究方程解的对称性。对这些对称性的研究有助于理解它们的解这一基本数学问题。

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.