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，例如费马大定理和关于曲线上有理点的数量的莫德尔猜想。本研究项目的主要研究对象被称为“阿贝尔簇”和“志村簇”，其研究处于代数几何、数论和表示论的界面，并在许多长期存在的结构中具有广泛的应用。该项目为研究生提供培训机会。本计画系关于阿贝尔簇与志村簇算术之问题与应用，志村簇是阿贝尔簇之模空间之推广。该项目的第一个目标是显示一种新的Northcott性质的同构类的阿贝尔品种在数域上。也就是说，直到同构，在同构类中只有100多个有界高度的阿贝尔簇。本项目的第二个目标是研究Shimura簇的上同调结构，以及它们模点的结构。具体来说，有一个猜想，证明了主要调查员在某些情况下，isobutch类的每一个模p包含减少一个特殊的点。这些结果可以用来给出一个光谱解释的Hasse-Weil zeta函数的志村品种，以下程序的Langlands.这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.