Geometric Methods in the Theory of Automorphic Forms

自守形式理论中的几何方法

• 批准号: 0003249
• 负责人: Alexander Braverman
• 金额: $ 6.38万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0003249&HistoricalAwards=false
• 关键词: Geometric; Methods; Theory; Automorphic; Forms

Geometric methods in the theory of automorphic forms. Abstract.This proposal consists of 2 parts. In the first (and main) partwe introduce certain functions on the set ofisomorphism classes of irreducible representations of a reductivegroup over a local field. We call them gamma-functions. The existenceof such functions is a corollary of the local Langlands conjecture.We formulate some conjectures about explicit construction of thesegamma-functions and explain how they can be applied to study theproblem of Langlands lifting itself (both locally and globally).In the second part we suggest how to generalize the results ofauthor's joint paper with D.Gaitsgory entitled "GeometricEisenstein series". In particular we plan to study intersectioncohomology sheaves on parabolic Drinfelds' spaces and also generalizeour previous results to the case of tamely ramified Eisenstein series. This proposal is in the part of mathematics known as the Langlands program.The Langlands program is part of number theory. Number theory is the studyof the properties of the whole numbers and is the oldest branch ofmathematics. From the beginning problems in number theory have furnished adriving force in creating new mathematics in other diverse parts of thediscipline. The Langlands program is a general philosophy that connectsnumber theory with calculus; it embodies the modern approach to the studyof whole numbers. One aspect of this proposal is to explore theapplications of geometric techniques within the Langlands program.

自同构形式理论中的几何方法。摘要。本建议书由两部分组成。在第一部分（也是主要部分）中，我们引入了局部域上约化群的不可约表示的同构类集合上的某些函数。我们称之为函数。这种函数的存在性是局部朗兰兹猜想的必然结果。我们提出了关于这些函数的显式构造的一些猜想，并解释了如何将它们应用于研究朗兰兹提升问题（局部和全局）。第二部分提出了如何推广作者与D.Gaitsgory合著的“GeometricEisenstein系列”论文的结果。特别地，我们计划研究抛物型drinfeld空间上的相交上同轴，并将我们以前的结果推广到纯分枝的Eisenstein级数的情况。这个建议属于数学中被称为朗兰兹纲领的部分。朗兰兹程序是数论的一部分。数论研究的是整数的性质，是数学中最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分中创造新的数学提供了推动力。朗兰兹纲领是一种将数论与微积分联系起来的普遍哲学；它体现了研究整数的现代方法。这个建议的一个方面是探索几何技术在朗兰兹程序中的应用。