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.

几何和算术最早是由希腊人研究的，而代数在几个世纪后作为解方程的艺术出现在波斯学者手中。在过去的世纪，这三个学科之间的相互作用一直是数学研究的中心。数学家们已经发现了它们之间的深层联系，导致了数学中许多惊人的结果（例如，费马大定理）和其他领域，包括密码学、量子场论和物理学中的弦理论的应用。在这些学科的交叉点的主要研究对象是一组代数方程。虽然几何学有助于理解具有复杂项的解的集合的形状（也称为代数簇），但算术的目标是理解具有整数项的解的集合。有一些自然的线性结构附着在代数簇上，称为霍奇结构，在某些情况下，霍奇结构忠实地捕捉了我们开始时的代数方程组。霍奇结构的研究，它们的对称性，以及它们的变化是这个建议的主要调查对象。这是一个在几个研究领域的十字路口的话题，如复杂的代数几何，数论和表示论，有许多长期的成果。PI将邀请研究生参与该项目，并将组织一次关于霍奇理论最新进展的会议。该项目旨在回答有关霍奇结构变异理论中例外霍奇轨迹及其算术对应物泰特轨迹的分布的几个问题。这些问题将得到解决，使用工具从Arakelov相交理论，遍历理论，霍奇理论，Ax-Schanuel定理志村品种，丢番图几何。第一个目标是研究代数簇族中的非典型Hodge轨迹。第二个目标是研究泰特轨迹，并给出一个具体的应用程序，特殊的代数下专门的Brauer类K3表面。第三个目标是研究K3曲面的模空间中特殊圈的闭包的模性行为，或者更一般地说，研究正交Shimura簇中特殊圈的闭包的模性行为。根据K3曲面家族退化的类型，这些生成系列表现出准模块化行为以及混合模拟模块化行为。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。