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

本研究项目是代数几何与数论的交叉领域。代数几何是研究由多项式方程系统切割出的形状的几何，而在数论中，人们通常对在有理数或整数中找到这种系统的显式解感兴趣。找到这样的解是数学中的一个基本问题，已经被研究了两千多年，也是这个提议的中心主题。本项目将在由模函数和自同构函数引起的特定情况下研究这个问题的高维版本。模函数和自同构函数编码了深层次的算术信息，在包括理论物理在内的其他领域发挥着越来越重要的作用。该项目旨在拓宽和深化这一数学基础领域的知识。更具体地说，该奖项涉及在Langlands计划背景下对代数循环的五个不同项目的工作：(i) Hilbert模形式及其在四元数Shimura变体上的类似物的积分周期关系；（ii） Rankin-Selberg l -函数的Bloch-Beilinson猜想，特别是与中心值消失相对应的构造循环；（iii）构造与朗兰泛函性相对应的绝对霍奇类；（iv）研究了数域表面上零环的Abel-Jacobi映射的注入性；(5)动机上同调与自同构形式上同调的关系。许多项目的主题是构造显式代数环或更一般的动机上同调元素，其存在通常是通过对代数环的深刻一般猜想来预测的。

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.