Abelian Varieties, Hecke Orbits, and Specialization

椭圆曲线(及其推广的阿贝尔变体)是基本的数学对象，在密码学和纠错码等其他领域中也非常重要。存在自然产生的几何空间，称为下村簇，其点将不同的椭圆曲线(和阿贝尔簇)分类。在这些空间中有轨道，称为黑克轨道。这些轨道不像行星围绕太阳运行的规则周期轨道，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下村变种中。首席研究人员和他的合作者将使用不同数学领域的技术，包括数论、代数几何和表示论来研究这些Hecke轨道的几个方面。作为这一奖项的一部分，PI计划向本科生介绍数学研究，并就与该项目相关的主题对研究生进行培训。该项目的具体目标是了解同源和黑克轨道的特征零和特征p的相互作用，同时牢记寻找不同于雅可比的阿贝尔变种的长期问题的应用。PI还计划在Hecke轨道的背景下研究Shimura品种中恰好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品种上的Hecke算子的动态方面取得进展。这一奖项反映了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。

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.