Galois Representations and Modular Forms

在过去的50年里，数论的一个大主题是自同构形式、伽罗瓦表示和代数几何对象之间的关系。这三个看似截然不同的学科（分别与分析、代数和几何有关）之间存在着大量的非凡猜想（例如，Artin猜想、Shimura-Taniyama猜想、Langlands猜想、Serre猜想和Fontaine-Mazur猜想）。这些猜想的进展目前非常令人兴奋。在以前的NSF资助下，PI与各种合作者一起完成了志村-谷山猜想的证明；证明了p进域上GL(n)的局部langlands猜想；证明了全实数域上的前椭圆曲线的Sato-Tate猜想；证明了任意维伽罗瓦表示的第一个一般自同构提升定理和潜在自同构定理；证明了CM域上任意极化正则不可约动机的l函数对整个复平面具有亚纯延拓性，并满足期望的泛函方程。pii建议继续改进目前可用的自同构提升和潜在的自同构定理；将非GL(n)群的Rapoport-Zink空间的上同性与局部Langlands猜想联系起来；与kevin Buzzard和Joe Rabinoff一起证明了奇次的Artin猜想，证明了5完全分裂的全实数场的伽罗瓦群的两个表示；并思考更多关于伽罗瓦表示和自同构形式的思辩问题，例如如何理解非常简并的霍奇-泰特数/无穷小字符的情况。此外，PI将继续与博士后，特别是研究生一起工作。这种思想循环在300多年后导致了安德鲁·怀尔斯著名的证明费尔马特最后定理。它们属于算术几何的一般领域——这门学科融合了数学中两个最古老的领域：数论和几何。事实证明，这种组合非常富有成效。其诸多后果之一是新的纠错码。这些代码对现代计算机（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.