Langlands Correspondences and Motivic L-Functions

对数的抽象性质及其关系的研究出现在每一个文明历史的早期阶段，对数论问题的反思始终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不同方式来研究：它们可以用于测量，也可以用于算术。 这两个性质之间的相互作用一直是数论的基础。 世纪后半叶，几个雄心勃勃的研究项目应运而生，旨在借助对称性研究数字及其关系，从而系统地理解这些相互作用。 数学中有关它们的几何对称性的分支叫做算术几何;有关它们的动力学对称性的分支叫做自守形式。 朗兰兹计划旨在通过展示每种对称性如何编码另一种对称性来统一这两个分支。 当代理论物理学引入了高阶对称性的概念，基于新的空间概念，这些空间概念本身在很大程度上归功于数论的早期发展;最近，高阶对称性在朗兰兹纲领中越来越重要。 本项目探讨高阶对称性在朗兰兹纲领中的作用，并与几个具体问题联系起来，最终目的是有助于理解整数方程的解。本项目是对自守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：利用迹公式方法研究一般群的局部Langlands参数化;证明Deligne关于L-函数特别是张量积L-函数的特殊值的猜想;利用与p-adic L-函数的意外关系，验证Venkatesh关于权为1的模形式的导出Hecke代数的猜想;以及发展了p进群的模p表示的特征标理论。 在本项目中所涉及的方法结合联合收割机标准技术从算术几何和自守形式，一种方法上同调自守形式的基础上微分几何和表示理论，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.