Distribution of the Hodge and the Tate locus

Hodge 和 Tate 轨迹的分布

• 批准号: 2302388
• 负责人: Salim Tayou
• 金额: $ 16.5万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302388&HistoricalAwards=false
• 关键词: Distribution; Hodge; Tate; locus

Geometry and arithmetic were first studied by the Greeks, while algebra first emerged centuries later in the hands of Persian scholars as the art of solving equations. The interplay between these three disciplines has been at the center of mathematical research over the past century. Mathematicians have uncovered deep connections between them, leading to many spectacular results in mathematics (e.g., Fermat’s Last Theorem) and other fields, including applications in cryptography, quantum field theory, and string theory in physics. The main object of study at the intersection of these disciplines is a set of algebraic equations. While geometry helps understand the shape of the set of solutions with complex entries (also called algebraic varieties), the goal of arithmetic is to understand the set of solutions with integer entries. There are natural linear structures attached to algebraic varieties called Hodge structures, which in some cases capture faithfully the set of algebraic equations we started with. The study of Hodge structures, their symmetries, and their variations is the main object of investigation of this proposal. It is a topic at the crossroads of several areas of research such as complex algebraic geometry, number theory, and representation theory, with many long-standing conjectures. The PI will involve graduate students in this project and will organize a conference on recent advances in Hodge theory.This project aims to answer several questions regarding the distribution of the exceptional Hodge locus in the theory of variations of Hodge structures and their arithmetic counterpart, the Tate locus. These questions will be addressed using tools from Arakelov intersection theory, ergodic theory, Hodge theory, Ax-Schanuel theorem for Shimura varieties, and Diophantine geometry. The first goal is to study the atypical Hodge locus in some families of algebraic varieties. The second goal is to study the Tate locus and give a concrete application to exceptional algebraicity under specializations of Brauer classes on K3 surfaces. The third goal is to study the modularity behavior of the closure of special cycles in moduli spaces of K3 surfaces, or more generally in orthogonal Shimura varieties. These generating series exhibit a quasi-modularity behavior as well as a mixed mock modularity behavior, depending on the type of degeneration of the family of K3 surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

几何和算术是最初是由希腊人研究的，而代数首先在波斯学者的手中出现了几个世纪，作为解决方程式的艺术。在过去的一个世纪中，这三个学科之间的相互作用一直是数学研究的中心。数学家发现了它们之间的深厚联系，从而导致了许多壮观的数学结果（例如，Fermat的最后一个定理）和其他领域，包括密码学，量子场理论和物理学中的弦理论中的应用。这些学科交集的研究的主要对象是一组代数方程。尽管几何形状有助于了解具有复杂条目（也称为代数品种）的一组解决方案的形状，但算术的目标是了解具有整数条目的解决方案。有天然的线性结构附着在代数品种上称为Hodge结构，在某些情况下，它们忠实地捕获了我们开始使用的代数方程。对霍奇结构，对称性及其变化的研究是对该提案进行调查的主要对象。它是一个研究领域的十字路口的主题，例如复杂的代数几何，数字理论和代表理论，并具有许多长期的猜想。 PI将与研究生参与该项目，并将组织一次有关Hodge理论进展的最新会议。此项目旨在回答有关Hodge结构及其算术差异理论中特殊Hodge基因座的分布的几个问题。这些问题将使用Arakelov交叉点理论，厄运理论，霍奇理论，轴链曲霉定理的工具和二磷的几何形状来解决。第一个目标是研究一些代数品种家族中的非典型霍奇基因座。第二个目标是研究TATE基因座，并在K3表面的Brauer类专业课程下为特殊代数提供具体的代数。第三个目标是研究在K3表面的Modulli空间或正交Shimura品种中闭合特殊周期的模块化行为。这些生成的系列表现出准模块性行为以及混合模拟模块性行为，具体取决于K3表面家族的变性类型。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来通过评估来诚实地支持。