Galois Representations and Modular Forms

在过去的50年里，数论中的一个大主题是自同构型、伽罗瓦表示和代数几何中的对象之间的关系。有一个巨大的猜想网络(例如Artin猜想、Shimura-Taniyama猜想、朗兰兹猜想、Serre猜想和Fontaine-Mazur猜想)将这三个看似非常不同的学科(分别与分析、代数和几何有关)联系在一起。这些猜想目前的研究进展是非常令人兴奋的。在过去的几年中，国家自然科学基金会与许多合作者一起，完成了Shimura-Taniyama猜想的证明；证明了P-adad域上GL(N)的局部Lang-land猜想；证明了全实域上椭圆曲线前的Sato-Tate猜想；证明了任意维伽罗瓦表示的第一个广义自同构提升定理和位势自同构定理；证明了CM域上任何极化的、正则的、不可约动机的L函数在整个复平面上具有亚纯连续性，并满足预期的函数方程。PI建议继续改进目前可用的自同构提升和势自同构定理；将Rapoport-Zink空间的上同调与GL(N)以外的群的局部朗兰兹猜想联系起来；与Kevin Buzzard和Joe Rabinoff一起证明奇度Artin猜想；以及考虑与Galois表示和自同构形式有关的更多投机问题，例如如何理解非常退化的Hodge-Tate数/无穷小特征标的情况。此外，PI将继续他的博士后工作，特别是研究生的工作。这个思想圈导致了安德鲁·威尔斯在300多年后对费马最后定理的著名证明。事实证明，这种结合非常富有成效。

A big theme in number theory in the last 50 years has been the relationshipbetween automorphic forms, Galois representations and objects from algebraicgeometry. There is an extensive web of extraordinary conjectures (for instancethe Artin conjecture, the Shimura-Taniyama conjecture, Langlands' conjectures,Serre's conjecture and the Fontaine-Mazur conjecture) linking thesethree seemingly very different subjects (which relate to analysis, algebra andgeometry respectively). Progress on these conjectures is currently very exciting.Under previous NSF grants the PI, with various collaborators, completedthe proof of the Shimura-Taniyama conjecture; proved the local Langlandsconjecture for GL(n) over a p-adic field; proved the Sato-Tate conjecture forelliptic curves over totally real fields; proved the first general automorphy liftingtheorems and potential automorphy theorems for Galois representationsof arbitrary dimension; and proved that the L-function of any polarized, regular,irreducible motive over a CM field has meromorphic continuation to thewhole complex plane and satisfies the expected functional equation. The PIproposes to continue to improve the currently available automorphy liftingand potential automorphy theorems; to relate the cohomology of Rapoport-Zink spaces to the local Langlands conjecture for groups other than GL(n); withKevin Buzzard and Joe Rabinoff to prove the Artin conjecture for odd degreetwo representations of the Galois group of a totally real field in which 5 splitscompletely; and to think about more speculative problems relating Galois representations and automorphic forms, for instance how to understand the case of very degenerate Hodge-Tate numbers/infinitesimal character. In addition the PI will continue his work with post-docs and, particularly, with graduate students.This circle of ideas is the one that led to Andrew Wiles' celebrated proof ofFermat's last theorem after over 300 years. They fall into the general area ofarithmetic geometry - a subject that blends two of the oldest areas of mathematics:number theory and geometry. This combination has proved extraordinarilyfruitful. Among its many consequences are new error correcting codes.Such codes are essential for both modern computers (hard disks) and compactdisks.