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，并与物理学有联系。数论和其他数学领域之间有许多富有成效的相互作用，例如朗兰兹纲领和费马大定理的证明。费马最后定理的证明包括朗兰兹纲领的一小部分的验证，这是一个涉及不同数学领域的庞大网络，在这种情况下，连接了伽罗瓦表示和自守形式以及椭圆曲线的L函数。研究人员将追求研究方向，涉及强大的全球技术来研究自守形式和伽罗瓦表示。这些方法可能包括迹公式、志村变种、p基完全上同调或泰勒-怀尔斯-基辛修补构造。该研究项目将解决以下主题。(1)交换型Shimura簇的上同调的Langlands-Kottwitz方法，具有良好的全一般性约化;（2）自守表示族及其L-函数的算术统计;（3）非拟分裂的局部和整体酉群表示的内窥镜分类;（4）GL（2，Qp）之外的p-adic Langlands程序。研究者研究的共同主题是通过对局部和全局理论之间的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计划的一个基本组成部分，它对交换型Shimura簇的完成不仅是一个里程碑，而且会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联系，并将揭示代数簇族。关于酉群的项目解决了一个有趣的朗兰兹函性的案例，以及局部朗兰兹分类在算术中的几个预期应用，这些应用基本上依赖于酉群的使用（通常是非准分裂的）。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.