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

朗兰兹纲领背景下的代数环和动机上同调

• 批准号: 1600494
• 负责人: Kartik Prasanna
• 金额: $ 17.66万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-15 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600494&HistoricalAwards=false
• 关键词: Algebraic; Cycles; Motivic; Cohomology; Context

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.

该研究项目位于代数几何学和数理论的界面。代数几何形状是对多项式方程系统所切除的形状几何形状的研究，而在数字理论中，通常有兴趣在有理数或整数中找到对此类系统的明确解决方案。找到这种解决方案的问题是数学中的基本问题，已经进行了超过两千年的研究，这是该提议的核心主题。该项目将在模块化和自动形态功能引起的特定情况下研究该问题的较高维度版本。模块化和自身形态功能编码深层算术信息，并越来越多地在其他领域（包括理论物理学）中扮演重要角色。该项目的目的是在数学的这个基本领域扩大和加深知识。更具体地说，该奖项涉及在Langlands计划的背景下在代数周期上进行五个不同的项目：（i）Hilbert Modular形式的整体关系及其类似物的整体关系关于Quaternionic Shimura品种； （ii）Bloch-beil​​inson猜想Rankin-Selberg L功能，特别是构建与中央值消失的周期； （iii）与Langlands功能案例相对应的绝对Hodge类的构建； （iv）研究ABEL-JACOBI图的注射率在数字字段上表面上的零周期； （v）动机同胞学与自动形式的共同体之间的关系。许多项目中的一个主题是建造明确的代数周期或动机共同体中的更普遍的要素，其存在通常是通过对代数周期的深刻猜想来预测的。