Iwasawa Dieudonn\'e theory of function fields

TL;DR

This paper investigates the growth of the $p$-primary part of the Jacobian's class group in towers of function fields over a perfect field, revealing regular asymptotic behaviors in unramified and ramified cases.

Contribution

It extends Iwasawa and Dieudonné theories to function fields, providing new asymptotic formulas for $p$-torsion class groups in $ ext{pro-}p$ towers with ramification.

Findings

Regular growth patterns of $p$-torsion class groups in unramified towers.

Asymptotic formulas for ramified towers generalizing Mazur and Wiles.

Extension of Iwasawa theory to function fields with ramification.

Abstract

Let $k$ be a perfect field of characteristic $p$ and $\Gamma$ an infinite, first countable pro-$p$ group. We study the behavior of the $p$-primary part of the "motivic class group", i.e. the full $p$-divisible group of the Jacobian, in any $\Gamma$-tower of function fields over $k$ that is unramified outside a finite (possibly empty) set of places $\Sigma$, and totally ramified at every place of $\Sigma$. When $\Sigma=\emptyset$ and $\Gamma$ is a torsion free $p$-adic Lie group, we obtain asymptotic formulae which show that the $p$-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $\Sigma\neq\emptyset$, we obtain a similar asymptotic formula for the $p$-torsion in "physical class groups", i.e. the $k$-rational points of the Jacobian, which generalizes the work of Mazur and Wiles, who studied the case $\Gamma=\mathbf{Z}_p$.