Arithmetic Geometry, Modularity, and L-Functions of Motives

数论在数学中有着悠久的历史，始于古希腊。数论中最基本、最重要的问题之一是求丢芬图方程，即求系数为整数的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，称为代数变量。然后，解的信息将产生相应代数变量的算术不变量。然而，这些代数变量通过所谓的l函数携带了与算术变量相关的其他重要不变量。通常可以从不同的不变量中传递信息，从而更好地理解所有不变量。该项目旨在加深对这些数学结构之间基本关系的理解。本课题主要研究动机的l函数，动机函数是代数变量的系统推广，通过研究志村变量的各个方面和Arakelov几何的方法。动机的l函数在数论、自同构表示和算术几何的研究中发挥着重要的作用，因为它们编码了这些方面的关键信息。朗兰兹纲领预言，动机就像有理椭圆曲线一样，通过l函数与自同构形式联系在一起。本项目研究五个主题：(1)针对Gan-Gross-Prasad猜想中出现的动机的Bloch-Kato猜想；(2) Shimura变种上特殊循环上Hecke作用的模块化；(3)非阿基米德空间的势理论及其与Arakelov几何的联系；(4)函数域上Rallis内积公式的高阶导数形式；(5)在一般基地上的马丁堆栈附近循环。

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.