Local and Global Study of Automorphic Forms and Galois Representations

这个研究项目涉及数论的工作，数论是数学的一个分支，在密码学和编码理论中有应用，并且与物理学有联系。数论和其他数学领域之间有许多卓有成效的相互作用，如朗兰兹纲领和费马大定理的证明。费马大定理的证明包括验证朗兰兹程序的一小部分，这是一个庞大的猜想网络，涉及不同的数学领域，在这个例子中，它将伽罗瓦表示、自同构形式和椭圆曲线的l函数联系起来。研究者将追求研究方向，涉及强大的全球技术来研究自同构形式和伽罗瓦表示。这些方法可能包括微量公式、志村变异、p-完全上同源或Taylor-Wiles-Kisin补片构造。本研究计划将涉及以下主题。(1)用Langlands-Kottwitz方法研究了具有充分一般性的良好约简的阿贝尔型Shimura变种的上同调；(2)自同构表示族及其l -函数的算术统计；(3)非拟分裂的局部酉群和全局酉群表示的内窥镜分类；(4)超越GL（2,Qp）的p进Langlands程序。研究者研究的共同主题是通过对当地和全球理论之间相互作用的清晰理解，从这些技术中获得强有力的结果。Langlands- kottwitz方法是Langlands规划的一个基本组成部分，它对阿贝尔类型的Shimura变体的完成不仅是一个里程碑，而且将导致进一步的算术应用。l -函数族的研究与随机矩阵理论有密切的联系，对代数变种族的研究有重要的启示作用。关于酉群的项目解决了Langlands泛函的一个有趣的情况，以及局部Langlands分类和几个期望在算术中的应用，这些主要依赖于使用（通常是非拟分裂的）酉群。针对一般群体的p-adic Langlands计划的持续努力将产生越来越大的影响并开辟新的研究方向。

This research project concerns work in number theory, a branch of mathematics that has seen applications in cryptography and in coding theory and that has connections with physics. There have been many fruitful interactions between number theory and other areas of mathematics, as exemplified by the Langlands program and the proof of Fermat's Last Theorem. The proof of Fermat's last theorem included verification of a small part of the Langlands program, a vast web of conjectures involving disparate areas of mathematics, which, in this case, connected Galois representations and automorphic forms and the L-functions of elliptic curves. The investigator will pursue research directions that involve strong global techniques to study automorphic forms and Galois representations. These may include methods such as the trace formula, Shimura varieties, p-adically completed cohomology, or the Taylor-Wiles-Kisin patching construction. This research project will address the following topics. (1) The Langlands-Kottwitz approach to the cohomology of Shimura varieties of abelian type with good reduction in full generality; (2) Arithmetic statistics for families of automorphic representations and their L-functions; (3) The endoscopic classification for representations of local and global unitary groups that are not quasi-split; and (4) p-adic Langlands program beyond GL(2,Qp). The common theme of the investigator's research is to obtain strong consequences from these techniques via a clear understanding of interactions between local and global theories. The Langlands-Kottwitz method is a fundamental component of the Langlands program, and its completion for Shimura varieties of abelian type will not only be a milestone but also lead to further arithmetic applications. The study of families of L-functions makes a connection with random matrix theory and would shed light on families of algebraic varieties. The project on unitary groups settles an interesting case of Langlands functoriality as well as the local Langlands classification with several expected applications in arithmetic, which rely essentially on the use of (often non-quasi-split) unitary groups. The continued effort on the p-adic Langlands program aiming at general groups would have growing impact and open up new research directions.