Iwasawa Theory for $p$-torsion Class Group Schemes in Characteristic $p$

We investigate a novel geometric Iwasawa theory for $\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over $k$ associated to a $\mathbf{Z}_p$-extension of function fields totally ramified over a finite non-empty set of places, we investigate the growth of the $p$-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty$. By Dieudonn\'e theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator $V$. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$-module structure of the space $M_n:=H^0(X_n, \Omega^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty.$ For example, for each tower in a basic class of $\mathbf{Z}_p$-towers we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda_r n + c_r(n)$ for all $n$ sufficiently large, where $a_r, \lambda_r$ are rational constants and $c_r : \mathbf{Z}/m_r \mathbf{Z} \to \mathbf{Q}$ is a periodic function, depending on $r$ and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on $\mathbf{Z}_p$-towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$.