Geometric Methods in Automorphic Forms

[9900324]戈尔斯基在朗兰兹的纲领中，出现了一个顽固的问题，迄今为止，它使所有试图作一般证明的努力都失败了：朗兰兹和谢尔斯塔德的猜想，这个猜想通常被称为“基本引理”。在其原始公式中，基本引理是两个不同代数群上的数列积分之间的等式。在这个项目中，基本引理被翻译成一个关于两个不同的前代数变体上同调的陈述。然后发展出计算这些上同群的技术，使它们可以相互比较。这一过程将导致对“非分支情况”的基本引理的证明，并可能最终导致对一般情况的证明。在过去的二十年里，数学家们花费了巨大的努力来证明朗兰兹提出的一系列猜想，这些猜想统一了几个显然非常不同的领域：数论、群表示论和自同态形式。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。这些猜想（曾经一度被认为几乎是无望的）取得了惊人的进展：其中许多猜想已被完全解决或证实，所有猜想都在许多特殊情况下得到了检验。甚至这些部分结果也是极其重要的：例如，A.怀尔斯最近对费马猜想的著名证明，就必不可少地利用了“朗兰兹纲领”中的一些结果。研究人员现在乐观地认为，这个项目可能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.