Automorphic Galois Representations and Automorphic L-functions

数论起源于对整数方程解的研究。它是数学中最古老的分支之一，几千年来，它的方法一直是基于整数的可整除性和它们的大小之间的相互作用。二十世纪的数论以两种不同的方式将这两个性质形式化。可整除性和尺寸的度量可以被看作是几何性质和动力学性质，后者根植于数学物理方程。研究它们的几何关系的数学分支叫做算术几何；与它们的动态关系有关的分支称为自同构形式。对称在算术几何和自同构形式中都起着中心作用；假设的朗兰兹对应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。PI建议使用这种编码来根据另一端对应对象的属性来理解通信一端的对象。一个特别的焦点是将自同构形式的可整除性质转移到算术几何，这通常会导致关于方程解的惊人精确信息。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，特别关注在志村变量的上同调中产生的动机及其相关的伽罗瓦表示的算术，直接或通过应用同余方法。长期目标是识别所有这样的动机和所有这样的伽罗瓦表示（模块化问题），并证明关于动机算术的杰出猜想，特别是Deligne关于l函数的特殊值的猜想，以及格林伯格、科茨、佩林-里奥等人关于p进l函数存在性的猜想，以这种方式获得动机。特别注意张量积l函数的特殊值。这个项目符合一个国际计划，使用所有可用的技术来扩展到所有建立椭圆模形式的l函数的动机结果；PI正在与法国、奥地利、以色列、日本、香港以及美国和加拿大的同事积极合作。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，PI多年来研究的上同构自同构形式的微分几何方法，p进分析和表示理论。

Number theory originates from the study of solutions to equations in whole numbers. It is one of the oldest branches of mathematics, and its methods have for millenia been based on the interaction between the divisibility properties of whole numbers and their size. Twentieth-century number theory formalized these two properties in two different ways. Divisibility and the measure of size can be seen both as geometric and as dynamical properties, the latter rooted in the equations of mathematical physics. The branch of mathematics concerned with their geometric relations is called arithmetic geometry; the branch concerned with their dynamical relations is called automorphic forms. Symmetry plays a central role in both arithmetic geometry and automorphic forms; the hypothetical Langlands correspondence unifies these two branches by showing how each kind of symmetry encodes the other. The PI proposes to use this coding to understand objects on one side of the correspondence in terms of properties of the corresponding object on the other side. A particular focus is the transfer of divisibility properties of automorphic forms to arithmetic geometry, which often leads to surprisingly precise information about solutions of equations.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 arising in the cohomology of Shimura varieties, directly or by application of congruence methods. The long-term goals are the identification of all such motives and all such Galois representations (the modularity problem) and the proof of outstanding conjectures on the arithmetic of motives, notably Deligne's conjecture on special values of L-functions, and the conjectures of Greenberg, Coates, Perrin-Riou, and others on the existence of p-adic L-functions, for the motives obtained in this way. Special attention is given to special values of tensor product L-functions. This project fits into an international program to use the full range of available techniques to extend to all such motives results established for L-functions of elliptic modular forms; the PI is actively collaborating with colleagues in France, Austria, Israel, Japan, and Hong Kong, as well as in the United States and Canada. The methods involved in the present project combine standard techniques from arithmetic geometry and automorphic forms, the differential-geometric approach to cohomological automorphic forms on which the PI has worked for many years, with p-adic analysis, and representation theory.