Galois Representations and Arithmetic Questions

研究者将研究全局伽罗瓦表示的变形理论中的问题。给出一个二维模p伽罗瓦表示。Serre推测这种表示“来自”模形式。他的猜想必然意味着模p伽罗瓦表示是某个p-adi伽罗瓦表示的模p化简。作者计划继续研究这种变形是否独立于serre猜想而存在，以及它们是否可以被安排成Fontaine意义上的潜在半稳定。这些构造，特别是与泰勒最近的工作相结合，可以被认为是为塞尔猜想提供了证据。有一些有趣的情况仍然是开放的，比如可约残差表示和完全实数域的绝对伽罗瓦群的表示。拟议的研究属于数论的一般领域，其根源在于对整数及其各种性质的研究。在这个领域中出现的自然问题，例如素数的性质及其出现的频率，可以应用于编码理论和密码学。所提出的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。研究这些对称可以帮助我们理解它们的解和基本的数学问题。

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.