Geometric Methods in Automorphic Forms

在Langlands的程序中，出现了一个顽固的问题，到目前为止，这个问题一直没有得到普遍的证明：Langlandsand Shelstad猜想，它通常被称为“基本引理”。 在它的原始公式中，基本引理是两个不同代数群上的轨道积分之间的等式。 在这个项目中，基本引理被翻译成一个声明有关的cohomologyof两个不同的pro-algebraic簇。 然后发展了计算这些上同调群的技术，使它们可以互相重叠。 这个过程应该导致在“unramified情况”下的基本引理的证明，并且很可能最终导致在一般情况下的证明。在过去的二十年中，数学家们一直致力于证明一个集合，由于R。朗兰兹，它统一了几个显然非常不同的领域：数论，群表示论和自守形式。 这些领域中的每一个都将受益于能够从其他领域引进技术、问题和结果。 这些问题（一度被认为几乎没有希望）的进展是惊人的：其中许多问题已经完全解决或得到验证，所有问题都在许多特殊情况下得到了检查。 即使这些部分结果也是非常重要的：例如，A.怀尔斯最近著名的费马猜想的证明，从“朗兰兹纲领”中得到了一些重要的结果。“研究人员现在乐观地认为，这个计划可能会在未来十年内成功完成。

9900324GoreskyIn Langlands' program, one stubborn problem has emerged which, so far, hasdefied all attempts at a general proof: the conjecture of Langlandsand Shelstad which is generally known as "the fundamental lemma". In itsoriginal formulation, the fundamental lemma is an equality betweenorbital integrals on two different algebraic groups. In this project,the fundamental lemma is translated into a statement relating the cohomologyof two different pro-algebraic varieties. Techniques are then developedfor computing these cohomology groups in such a way that they may becompared with each other. This procedure should result in a proof ofthe fundamental lemma in the "unramified case", and may well eventuallylead to a proof in the general case.During the last twenty years, mathematicians have directed tremendousefforts towards proving a collection of conjectures, due to R. Langlands,which unify several apparently very different fields: number theory,group representation theory, and automorphic forms. Each of these fieldswould then benefit from being able to import techniques, problems, and results from the other fields. Progress on these conjectures (which were at one time considered almost hopeless)has been spectacular: many of them are completely solved orverified and all of them have been checked in many special cases. Eventhese partial results are extremely important: for example, A. Wiles'recent and celebrated proof of the Fermat conjecture makesessential use of some of these results from "Langlands' program."Researchers are now optimistic that this program may be successfullycompleted during the next decade.