Motivic Class Groups and Iwasawa Theory of Function Fields

有理数及其推广（数域）的研究已经占据了数学家数千年，今天是代数数论的主要对象。 在过去的世纪里，通过探索数域与其他与几何密切相关的数系（称为“正特征函数域”）之间的深刻类比，人们的认识有了巨大的飞跃。 算术和几何之间的深刻关系允许技术，工具和直觉从一个被应用到另一个产生巨大的影响，导致新的和重要的见解，结果和启示。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联的雅可比行列式的p-可分群的结构和增长，为函数域开发岩泽理论的新类比。这些p-可分群分为三部分：标准部分、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希望在非常一般的假设下证明，在任何这样的塔中，etale和乘法部分都表现出惊人的规则和可预测的行为。 虽然地方-地方作品在许多方面可以像想象的那样狂野，但PI仍然希望发现某些异常规律的行为，并在经典岩泽理论的精神下制定新的见解。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。

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.