Arithmetic Geometry, Modularity, and L-Functions of Motives

算术几何、模块化和动机的 L 函数

• 批准号: 1702019
• 负责人: Yifeng Liu
• 金额: $ 23万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702019&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Modularity; Functions; Motives

Number theory has a long history in mathematics, beginning with the ancient Greeks. One of the most fundamental and important problems in number theory is to solve Diophantine equations, that is, to find integer solutions to polynomial equations with integer coefficients. For example, the famous Fermat's Last Theorem concerns one class of Diophantine equations. In modern mathematical language, Diophantine equations can be encoded into a geometric notion, called algebraic varieties. Then the information of the solution will give rise to an arithmetic invariant of the corresponding algebraic variety. However, these algebraic varieties carry other important invariants related to the arithmetic one via the so-called L-function. It is often possible to transfer information from different invariants and achieve better understanding of all of them. This project aims to deepen understanding of fundamental relationships among such mathematical constructs.The main theme of this project is to study L-functions of motives, which are a systematic generalization of algebraic varieties, via studying various aspects of Shimura varieties and the method of Arakelov geometry. The L-functions of motives play an important role in the current research in number theory, automorphic representations, and arithmetic geometry, as they encode crucial information from all these aspects. The Langlands Program predicts that motives, like rational elliptic curves, are linked with automorphic forms through L-functions. This project investigates five topics: (1) the Bloch-Kato conjecture for motives appearing in the Gan-Gross-Prasad conjecture; (2) modularity of Hecke actions on special cycles on Shimura varieties; (3) potential theory on non-Archimedean spaces and its connection to Arakelov geometry; (4) a higher derivative version of the Rallis inner product formula over function fields; and (5) nearby cycles for Artin stacks over general bases.

数论在数学中有着悠久的历史，从古希腊开始。数论中最基本、最重要的问题之一是求解丢番图方程，即求整系数多项式方程的整数解。例如，著名的费马大定理涉及一类丢番图方程。在现代数学语言中，丢番图方程可以被编码成一个几何概念，称为代数簇。然后解的信息将产生相应代数簇的算术不变量。然而，这些代数簇通过所谓的L-函数携带与算术不变量相关的其他重要不变量。通常可以从不同的不变量中转移信息，并更好地理解所有这些不变量。该项目旨在加深对此类数学结构之间基本关系的理解。该项目的主题是通过研究志村簇的各个方面和阿拉克洛夫几何方法来研究动机的L-函数，这是代数簇的系统推广。动机的L-函数在数论、自守表示和算术几何的当前研究中起着重要作用，因为它们编码了来自所有这些方面的关键信息。朗兰兹纲领预言，像有理椭圆曲线一样，动机通过L-函数与自守形式联系在一起。本项目研究了五个问题：（1）Gan-Gross-Prasad猜想中动机的Bloch-Kato猜想;（2）Shimura簇上特殊圈上Hecke作用的模性;（3）非Archimedean空间上的势理论及其与Arakelov几何的联系;（4）函数域上Rallis内积公式的高阶导数形式;（5）一般基上Artin堆的邻近圈。

项目成果

Monodromy map for tropical Dolbeault cohomology

热带 Dolbeault 上同调的单峰图

• DOI: 10.14231/ag-2019-018
• 发表时间: 2019
• 期刊: Algebraic Geometry
• 影响因子: 1.5
• 作者: Liu, Yifeng