Motivic Class Groups and Iwasawa Theory of Function Fields

动机类群和岩泽函数域理论

• 批准号: 1902005
• 负责人: Bryden Cais
• 金额: $ 18万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902005&HistoricalAwards=false
• 关键词: Motivic; Class; Groups; Iwasawa; Theory

The study of the rational numbers and their generalizations (number fields) has occupied mathematicians for thousands of years, and today is the principal object of algebraic number theory. In the past century, giant leaps of understanding have been made by exploring the deep analogy between number fields and and other number systems (called "function fields in positive characteristic") closely related to geometry. The deep relationship between arithmetic and geometry allows the techniques, tools, and intuitions from one to be applied with great effect to the other, leading to new and important insights, results, and conjectures. The research of the PI will continue this tradition and push the analogy in new directions.The project will develop a new analogue of Iwasawa theory for function fields by investigating the structure and growth of the p-divisible groups of the Jacobians associated to a pro-p branched Galois tower of curves over a finite field of characteristic p. These p-divisible groups break up into three pieces: an etale part, a multiplicative part, and a local-local part. Using Dieudonne theory and its relation to p-adic cohomology, the PI expects to prove under very general hypotheses that the etale and multiplicative parts exhibit strikingly regular and predictable behavior in any such tower. While the local-local piece can in many ways be as wild as the imagination allows, the PI nonetheless expects to uncover certain kinds of exceptionally regular behavior and to formulate new conjectures in the spirit of classical Iwasawa theory.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.

对有理数及其概括（数域）的研究已经吸引了数学家数千年的时间，今天是代数数论的主要对象。 在过去的一个世纪里，通过探索数域和与几何密切相关的其他数系统（称为“正特征函数域”）之间的深刻类比，人们的理解取得了巨大的飞跃。 算术和几何之间的深厚关系使得一种技术、工具和直觉能够对另一种产生巨大影响，从而产生新的重要见解、结果和猜想。 PI 的研究将延续这一传统，并将类比推向新的方向。该项目将通过研究与特征 p 的有限域上的 pro-p 分支伽罗瓦曲线塔相关的雅克比行列式 p 可整群的结构和增长，开发函数场的岩泽理论的新类比。 这些 p 可整群分为三部分：etale 部分、乘法部分和局部局部部分。使用 Dieudonne 理论及其与 p-adic 上同调的关系，PI 期望在非常一般的假设下证明 etale 和乘法部分在任何此类塔中都表现出惊人的规则和可预测的行为。 虽然本地与本地的作品在很多方面都可以像想象中的那样疯狂，但 PI 尽管如此，仍希望发现某些异常规律的行为，并本着经典岩泽理论的精神提出新的猜想。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。