Abelian Varieties, Hecke Orbits, and Specialization

椭圆曲线（和它们的推广称为阿贝尔簇）是基本的数学对象，在其他领域也非常重要，如密码学和纠错码。有自然发生的几何空间，称为志村簇，其点分类不同的椭圆曲线（和阿贝尔簇）。在这些空间内是轨道，称为赫克轨道。这些轨道不像行星围绕太阳的规则周期轨道，而是高度不可预测和混乱的。事实上，每一个轨道都是均匀分布在志村品种。首席研究员和他的合作者将使用数学各个领域的技术，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。作为该奖项的一部分，PI计划介绍本科生在数学研究和培训研究生的主题相关的项目。该项目的具体目标是了解特征零和特征p的相互作用的同源性和赫克轨道，牢记应用程序的长期存在的问题，寻找阿贝尔品种不同源的雅可比矩阵。PI还计划在Hecke轨道的背景下研究Shimura品种的可能性和不可能的交叉点，并最终在Hecke轨道猜想的背景下，朝着理解mod p Shimura品种的Hecke运营商的动态取得进展。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.