Motivic Class Groups and Iwasawa Theory of Function Fields

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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 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.