Iwasawa theory of class group schemes in characteristic p

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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.