Motivic Class Groups and Iwasawa Theory of Function Fields

有理数及其推广(数域)的研究占据了数学家们几千年的时间，今天是代数数论的主要目标。在过去的一个世纪里，通过探索数域和其他与几何密切相关的数系(称为正特征函数域)之间的深层类比，人们在认识上取得了巨大的飞跃。算术和几何之间的深厚关系允许将一方的技术、工具和直觉应用到另一方，产生新的和重要的见解、结果和猜想。PI的研究将继续这一传统，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Galois曲线塔相关的雅可比的p-可分群的结构和增长，来开发函数域的岩泽理论的新的类比。这些p-可分群分为三部分：集合部分、乘法部分和局部-局部部分。利用Dieudonne理论及其与p-进上同调的关系，PI期望在非常一般的假设下证明在任何这样的塔中，Etale和乘法部分表现出惊人的规则和可预测的行为。虽然当地-当地的作品在许多方面可以像想象的那样狂野，但PI仍然希望发现某些类型的异常规则行为，并本着经典岩泽理论的精神提出新的猜测。这一奖项反映了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.