Abelian Varieties, Hecke Orbits, and Specialization

阿贝尔簇、赫克轨道和特化

• 批准号: 2337467
• 负责人: Ananth Shankar
• 金额: $ 29.75万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2337467&HistoricalAwards=false
• 关键词: Abelian; Varieties; Hecke; Orbits; Specialization

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.

椭圆曲线（及其称为Abelian品种的概括）是基本的数学对象，在其他领域（例如加密和错误校正代码）中也非常重要。有天然存在的几何空间，称为Shimura品种，其点对不同的椭圆曲线（和Abelian品种）进行了分类。在这些空间内部是轨道，称为Hecke Orbits。这些轨道不像太阳周围行星的常规周期轨道，而是高度不可预测和混乱的。实际上，每个轨道都被认为是在整个Shimura品种中平均分布的。首席研究者及其合作者将使用来自数学各个领域的技术，包括数字理论，代数几何和代表理论来研究这些Hecke Orbits的几个方面。作为该奖项的一部分，PI计划将大学生介绍到数学研究并培训与该项目相关的主题的研究生。该项目的具体目标是了解ISEGENIS和HECKE ORBITS的特征性零和特征性P相互作用，并保持考虑到漫长的问题，即发现雅各布人并非同源性的阿伯利亚品种。 PI还计划在Hecke Orbits的背景下显然研究Shimura品种中的交集，并最终在理解Hecke Operators在Mod P Shimura品种上的动态方面取得了进步，在Hecke Orbit的背景下，奖项反映了NSF的法定任务，并通过使用基金会的智力优点和更广泛的影响审查标准评估值得支持。