Geometric Methods in the Theory of Automorphic Forms

自守形式理论中的几何方法。 摘要.本提案由两部分组成。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。我们称之为伽马函数。这类函数的存在性是局部Langlands猜想的一个推论.本文给出了这类函数的一些显式构造方法，并说明了如何应用它们来研究Langlands提升问题（局部和整体）.在第二部分中，我们提出了如何推广作者与D.Gaitsgory合著的题为“GeometricEisenstein级数”的论文中的结果.特别地，我们计划研究抛物Drinfelds空间上的交上同调层，并将我们以前的结果推广到Tamely分歧的Eisenstein级数的情况。这个建议是在数学中被称为朗兰兹纲领的部分。朗兰兹纲领是数论的一部分。数论是研究整数性质的学科，是数学最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分创造新的数学提供了驱动力。朗兰兹纲领是一种将数论与微积分联系起来的一般哲学;它体现了研究整数的现代方法。这个建议的一个方面是探讨几何技术在朗兰兹程序中的应用。

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.