Galois Representations and Modular Forms

在过去的50年里，数论的一个重要主题是自守形式、伽罗瓦表示和代数几何对象之间的关系。有一个广泛的网络非凡的猜想（例如阿廷猜想，志村谷山猜想，朗兰兹猜想，塞尔猜想和丰泰恩马祖猜想）连接这三个看似非常不同的主题（分别涉及分析，代数和几何）。在美国国家科学基金会的赠款PI与多个合作者完成了Shimura-Taniyama猜想的证明;证明了p-adic域上GL（n）的局部Langlands猜想;证明了全真实的域上椭圆曲线的Sato-Tate猜想;证明了第一个一般的自同构提升定理和任意维Galois表示的势自同构定理;证明了CM域上任意极化正则不可约运动的L-函数对整个复平面具有亚纯延拓，并满足期望的函数方程。本项目提出继续改进现有的自同构提升定理和势自同构定理，将Rapoport-Zink空间的上同调与GL（n）以外的群的局部Langlands猜想联系起来，与Kevin Buzzard和Joe Rabinoff一起证明5完全分裂的全真实的域的Galois群的奇次二表示的Artin猜想;并考虑更多的有关伽罗瓦表示和自守形式的投机性问题，例如如何理解非常退化的Hodge-Tate数/无穷小特征的情况。此外，PI将继续他的工作与博士后，特别是与研究生。这一圈的想法是一个导致安德鲁怀尔斯著名的证明费马最后定理后超过300年。它们属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的学科。事实证明，这种结合非常富有成效。在它的许多结果中，有新的纠错码。这种码对现代计算机（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 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.