Distribution of the Hodge and the Tate locus

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首次出现在波斯学者手中。在过去的一个世纪里，这三个学科之间的相互作用一直是数学研究的中心。数学家们发现了它们之间的深刻联系，在数学(如费马大定理)和其他领域，包括在密码学、量子场论和物理学中的弦理论中的应用，导致了许多壮观的结果。这些学科交叉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数项的解集的形状(也称为代数族)，但算术的目标是理解具有整数项的解集。有一些自然的线性结构依附于代数变体，称为Hodge结构，在某些情况下，它忠实地捕捉到了我们开始时的一组代数方程。霍奇结构、对称性及其变化的研究是这一提议的主要研究对象。它是一个处于复代数几何、数论和表示论等几个研究领域的十字路口的课题，有许多长期存在的猜想。PI将邀请研究生参与这个项目，并将组织一次关于Hodge理论最新进展的会议。本项目旨在回答关于Hodge结构变分理论中例外Hodge轨迹及其算术对应的Tate轨迹的分布的几个问题。这些问题将使用Arakelov交集理论、遍历理论、Hodge理论、Shimura簇的Ax-Schanuel定理和丢番图几何的工具来解决。第一个目标是研究某些代数族中的非典型Hodge轨迹。第二个目标是研究Tate轨迹，并给出K3曲面上Brauer类特殊化下例外代数性的一个具体应用。第三个目标是研究K3曲面的模空间中的特殊圈的闭包的模性行为，或者更一般地，在正交的Shimura簇中。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，这取决于K3表面家族的退化类型。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.