Iwasawa theory of class group schemes in characteristic p

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，对数域与函数域在正特征上的相似性的探索，使人们的认识有了巨大的飞跃。 在后一种情况下，算术变成了几何，反之亦然，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的研究将继续并在新的方向发展这一传统。 此外，PI将利用他作为亚利桑那大学本科课程副主任的行政角色，收集和分析有关学生挑战的数据，并制定策略，以提高学生在数学科学方面的成功。 该项目将继续开发一个新的模拟岩泽理论的功能领域通过调查的结构和增长的p-整除群体的雅可比矩阵相关的亲p分支伽罗瓦塔曲线在有限领域的特征p.这些p-整除群体分为三块：一个etale部分，乘法部分，和一个本地本地的一部分。 在以前的工作中，PI在非常一般的假设下证明了在任何这样的塔中，标准和乘法部分都表现出规则和可预测的行为。虽然局部-局部部分在许多方面都可以像想象力所允许的那样狂野，但PI和他的合作者最近制定了几个具有惊人规律性和可预测性的假设，并通过利用Dwork理论的思想和工具证明了这些假设的一些基本实例。 该项目将继续通过调整这些方法来处理更一般的情况下，旨在证明充分的知识。这个奖项反映了NSF的法定使命，并已被认为是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的支持。

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 analogy between number fields and function fields in positive characteristic. In this latter setting, arithmetic becomes geometry and vice versa, allowing 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 and develop this tradition in new directions. Additionally, the PI will leverage his administrative role as Associate Head for Undergraduate Programs at the University of Arizona to collect and analyze data on student challenges, and devise strategies to increase student success in the mathematical sciences. The project will continue to 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. In previous work, the PI proved under very general hypotheses that the etale and multiplicative parts exhibit regular and predictable behavior in any such tower. While the local-local part can in many ways be as wild as the imagination allows, the PI and his collaborators have recently formulated several conjectures which posit striking regularity and predictability, and have proved some fundamental instances of these conjectures by leveraging ideas and tools from Dwork theory. The project will continue this research by adapting these methods to handle more general situations, aiming to prove the conjectures in full.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.