Iwasawa Dieudonné theory of function fields

Let $k$ be a perfect field of characteristic $p$ and $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$-tower of function fields over $k$ that is unramified outside a finite (possibly empty) set of places $\mathrm{\Sigma }$, and totally ramified at every place of $\mathrm{\Sigma }$. When $\mathrm{\Sigma }=\mathrm{\varnothing }$ and $\mathrm{\Gamma }$ 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 $\mathrm{\Sigma }\mathrm{\ne }\mathrm{\varnothing }$, 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 $\mathrm{\Gamma }={\mathbf{Z}}_p$.