Algebraic Cycles and Motivic Cohomology in the Context of the Langlands Program

这个研究项目处于代数几何和数论的交界处。代数几何学是研究由多项式方程组切割出的几何形状的学科，而在数论中，人们通常对用有理数或整数来寻找这种系统的显式解感兴趣。寻找这些解的问题是数学中的一个基础问题，已经研究了两千多年，也是这个提议的中心主题。这个项目将调查这个问题的高维版本在特定的背景下所产生的模和自守函数。模函数和自守函数编码了深层的算术信息，并在其他领域（包括理论物理）中发挥着越来越重要的作用。该项目旨在拓宽和深化数学这一基础领域的知识。更具体地说，该奖项涉及朗兰兹计划背景下的五个不同的代数循环项目：（i）希尔伯特模形式及其四元数志村簇的类似物的整周期关系;（ii）关于Rankin-Selberg L-函数的Bloch-Beilinson猜想，特别是构造对应于中心值为零的圈;（iii）构造对应于Langlands函性情形的绝对Hodge类;（iv）数域上曲面上零圈的Abel-Jacobi映射的内射性研究;（v）动机上同调与自守形式的上同调之间的关系。在许多项目的主题是明确的代数圈或更一般的元素motivic上同调，其存在往往是预测深刻的一般代数圈的建设。

This research project lies at the interface of algebraic geometry and number theory. Algebraic geometry is the study of the geometry of shapes cut out by systems of polynomial equations, while in number theory one is often interested in finding explicit solutions to such systems in rational numbers or integers. The problem of finding such solutions is a foundational question in mathematics, having been studied for more than two thousand years, and is a central theme of this proposal. This project will investigate higher dimensional versions of this question in specific contexts arising from modular and automorphic functions. Modular and automorphic functions encode deep arithmetic information and are increasingly playing important roles in other fields, including theoretical physics. This project aims to broaden and deepen knowledge in this fundamental area of mathematics.More specifically, the award involves work on five distinct projects on algebraic cycles in the context of the Langlands program: (i) integral period relations for Hilbert modular forms and their analogs on quaternionic Shimura varieties; (ii) the Bloch-Beilinson conjecture for Rankin-Selberg L-functions and in particular constructing cycles corresponding to the vanishing of the central value; (iii) the construction of absolute Hodge classes corresponding to cases of Langlands functoriality; (iv) the study of the injectivity of the Abel-Jacobi map for zero cycles on surfaces over number fields; (v) the relation between motivic cohomology and the cohomology of automorphic forms. A theme in many of the projects is the construction of explicit algebraic cycles or more generally elements in motivic cohomology, whose existence is often predicted by deep general conjectures on algebraic cycles.