Langlands Correspondences and Motivic L-Functions

对数字的抽象性质及其关系的研究在每一个文明历史的早期阶段就出现了，对数论问题的思考一直是当代生活中大多数思想的根源，从计时到现代物理学的对称概念，再到计算机的逻辑。数字可以用两种几乎独立的方式来研究：它们可以用来测量，也可以用来做算术。这两种性质之间的相互作用一直是数论的基础。20世纪下半叶，人们制定了几个雄心勃勃的研究计划，旨在通过研究数字及其在对称性方面的关系，对这些相互作用获得系统的理解。与它们的几何对称性有关的数学分支叫做算术几何；与它们的动态对称性有关的分支称为自同构形式。朗兰兹计划的目标是通过展示每种对称如何编码另一种对称来统一这两个分支。当代理论物理学在新的空间概念的基础上引入了高阶对称性的概念，这些概念本身在很大程度上要归功于数论的早期发展；最近，高阶对称性在朗兰兹程序中变得越来越重要。本项目探讨了高阶对称性在朗兰兹程序中几个具体问题中的作用，最终目的是促进对整数方程解的理解。该项目是对自同构形式的算术理论的贡献，在朗兰兹纲领的背景下，特别关注动机的算术及其相关的伽罗瓦表示，直接或通过应用同余方法。该项目的具体目标是研究一般群的局部朗兰兹参数化，使用迹公式方法；Deligne关于l函数，特别是张量积l函数特殊值的猜想的证明；利用与p进l函数的非预期关系，验证了权值为1的模形式的推导Hecke代数的Venkatesh猜想；以及p进群的模p表示的特征理论的发展。本项目所涉及的方法结合了算术几何和自同构形式的标准技术，基于微分几何和表示理论的上同构自同构形式的方法，范畴表示理论，以及新的方法。

The study of the abstract properties of numbers and their relations has appeared at an early stage in the history of every civilization, and reflection on the problems of number theory is consistently found at the root of most of the ideas that characterize contemporary life, from timekeeping, to the symmetry concepts of modern physics, to the logic of computers. Numbers can be studied in two different ways that are nearly independent: they can be used for measurement and they can be used to do arithmetic. The interaction between these two properties has always been the basis of number theory. The second half of the twentieth century saw the formulation of several ambitious research programs that aimed at obtaining a systematic understanding of these interactions by studying numbers and their relations with the help of symmetry. The branch of mathematics concerned with their geometric symmetries is called arithmetic geometry; the branch concerned with their dynamical symmetries is called automorphic forms. The Langlands program aims to unify these two branches by showing how each kind of symmetry encodes the other. Contemporary theoretical physics has introduced the concept of higher order symmetries, based on new notions of space that themselves owe a great deal to earlier developments in number theory; more recently, higher order symmetries have been of increasing importance in the Langlands program. This project explores the role of higher order symmetries in connection with several specific questions in the Langlands program, with the ultimate aim of contributing to the understanding of solutions of equations in whole numbers.The project is a contribution to the arithmetic theory of automorphic forms, in the setting of the Langlands program, with special attention to the arithmetic of motives and their associated Galois representations, directly or by application of congruence methods. The specific goals of the project are the study of the local Langlands parametrizations for general groups, using trace formula methods; the proof of Deligne's conjecture on special values of L-functions, especially tensor product L-functions; the verification of Venkatesh's conjecture on derived Hecke algebras for modular forms of weight 1, using an unexpected relation with p-adic L-functions; and the development of a character theory for mod p representations of p-adic groups. The methods involved in the present project combine standard techniques from arithmetic geometry and automorphic forms, an approach to cohomological automorphic forms based on differential geometry and representation theory, categorical representation theory, as well as new methods.