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。在这三个看似截然不同的学科(分别与分析、代数和几何有关)之间，存在着广泛的非同寻常的猜想(例如Artin猜想、Shimura-Taniyama猜想、朗兰兹猜想、Serre猜想和Fontaine-Mazur猜想)。关于这些猜想的进展目前非常令人兴奋。在以前的美国国家科学基金会的资助下，PI与不同的合作者一起完成了Shimura-Taniyama猜想的证明；证明了p-ady域上GL(N)的局部朗兰兹猜想；证明了全实域上椭圆曲线的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、正则、不可约运动的L函数在整个复平面上亚纯连续且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位势自同构定理；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Artin猜想，证明5完全分裂的全实域的Galois群的奇二次表示；并考虑与Galois表示和自同构形式相关的更多投机问题，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。它们属于算术几何的一般领域--这是一个融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合是非常富有成效的。在其众多后果中，有一种是新的纠错码。这种代码对于现代计算机(硬盘)和光盘都是必不可少的。

A big theme in number theory in the last 50 years has been the relationship between automorphic forms, Galois representations and objects from algebraic geometry. There is an extensive web of extraordinary conjectures (for instance the Artin conjecture, the Shimura-Taniyama conjecture, Langlands' conjectures,Serre's conjecture and the Fontaine-Mazur conjecture) linking these three seemingly very different subjects (which relate to analysis, algebra and geometry respectively). Progress on these conjectures is currently very exciting. Under previous NSF grants the PI, with various collaborators, completed the proof of the Shimura-Taniyama conjecture; proved the local Langlands conjecture for GL(n) over a p-adic field; proved the Sato-Tate conjecture for elliptic curves over totally real fields; proved the first general automorphy lifting theorems and potential automorphy theorems for Galois representations of arbitrary dimension; and proved that the L-function of any polarized, regular, irreducible motive over a CM field has meromorphic continuation to the whole complex plane and satisfies the expected functional equation. The PI proposes to continue to improve the currently available automorphy lifting and potential automorphy theorems; to relate the cohomology of Rapoport-Zink spaces to the local Langlands conjecture for groups other than GL(n); with Kevin Buzzard and Joe Rabinoff to prove the Artin conjecture for odd degree two representations of the Galois group of a totally real field in which 5 splits completely; 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 of Fermat's last theorem after over 300 years. They fall into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.