Shimura Varieties and Abelian Varieties

该奖项支持首席研究员在算术几何方面的研究，算术几何是数学的一个分支，研究多项式方程的整数解，也被称为“有理点”。算术几何在解决数论中的许多突出问题中发挥了核心作用，如费马最后定理和关于曲线上有理点的个数的莫德尔猜想。本研究项目的主要研究对象是“阿贝尔族”和“下村族”，它们的研究涉及代数几何、数论和表象理论的交界处，对一些长期存在的猜想有着广泛的应用。该项目为研究生提供了培训机会。本课题是关于阿贝尔簇和Shimura簇的算法及其应用的问题，后者是阿贝尔簇的模空间的推广。该项目的第一个目标是在数域上证明交换变种的同源类的一种新的Northcott性质。也就是说，在同构之前，同源类中只有有限多个有界高度的阿贝尔簇。该项目的第二个目标是研究下村品种上同源的结构，以及它们的mod p点的结构。具体地说，有一个猜想，在某些情况下被主要研究者证明了，每个mod p的同源类包含一个特殊点的约化。这些结果可以用来对下村品种的Hasse-Weil Zeta函数进行频谱解释，遵循Langland的计划。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。

The award supports the principal investigator's research in arithmetic geometry, a branch of mathematics that studies integer solutions of polynomial equations, also called “rational points.” Arithmetic geometry has played a central role in solving many outstanding problems in number theory, such as Fermat's Last Theorem and the Mordell conjecture concerning the number of rational points on a curve. The main objects of study in this research project are called "Abelian varieties" and "Shimura varieties,” the study of which is at the interface of algebraic geometry, number theory, and representation theory and has broad applications to a number of long-standing conjectures. The project provides training opportunities for graduate students. This project concerns problems in and applications of the arithmetic of Abelian varieties and Shimura varieties, the latter being generalizations of the moduli space of abelian varieties. The first goal of the project is to show a new kind of Northcott property for the isogeny class of an abelian variety over a number field. Namely that, up to isomorphism, there are only finitely many abelian varieties of bounded height in the isogeny class. The second goal of the project is to study the structure of the cohomology of Shimura varieties, and the structure of their mod p points. Specifically, there is a conjecture, proved by the principal investigator in some cases, that the isogeny class of every mod p contains the reduction of a special point. These results can be used to give a spectral interpretation of the Hasse-Weil zeta function of a Shimura variety, following a program of Langlands.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.