Abelian Varieties, Hecke Orbits, and Specialization

椭圆曲线（及其被称为阿贝尔变体的推广）是基本的数学对象，在密码学和纠错码等其他领域也非常重要。有自然发生的几何空间，称为志村变种，其点分类不同的椭圆曲线（和阿贝尔变种）。在这些空间里是轨道，叫做赫克轨道。这些轨道不像行星围绕太阳的定期轨道，而是高度不可预测和混乱的。事实上，每个轨道都被推测均匀地分布在志村星系中。首席研究员和他的合作者将使用不同数学领域的技术，包括数论、代数几何和表示理论来研究这些赫克轨道的几个方面。作为该奖项的一部分，PI计划引导本科生进行数学研究，并就与该项目相关的主题培训研究生。该项目的具体目标是了解等同性和Hecke轨道的特征0和特征p的相互作用，记住在寻找与雅可比矩阵不等同性的阿贝尔变体的长期问题中的应用。PI还计划在Hecke轨道的背景下研究Shimura变体中的刚可能和不可能的交集，并最终在Hecke轨道猜想的背景下，在理解mod p Shimura变体上的Hecke算子的动力学方面取得进展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Elliptic curves (and their generalizations called abelian varieties) are fundamental mathematical objects that are also of great importance in other fields such as cryptography and error correcting codes. There are naturally occurring geometric spaces, called Shimura varieties, whose points classify different elliptic curves (and abelian varieties). Inside these spaces are orbits, called Hecke orbits. These orbits are not like the regular periodic orbits of the planets around the sun, but are highly unpredictable and chaotic. Indeed, each orbit is conjectured to be distributed equally throughout the Shimura variety. The principal investigator and his collaborators will use techniques from various areas of mathematics, including number theory, algebraic geometry and representation theory to study several aspects of these Hecke orbits. As part of this award the PI plans to introduce undergraduates to research in mathematics and to train graduate students on topics related to the project.The specific goals of this project are to understand the characteristic zero and characteristic p interplay of isogenies and Hecke orbits, keeping in mind applications to the long standing question of finding abelian varieties not isogenous to Jacobians. The PI also plans to study just-likely and unlikely intersections in Shimura varieties within the context of Hecke orbits, and to finally make progress towards understanding the dynamics of Hecke operators on mod p Shimura varieties, in the context of the Hecke Orbit conjecture.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.