IWASAWA THEORY FORp-TORSION CLASS GROUP SCHEMES IN CHARACTERISTICp

Abstract

AbstractWe investigate a novel geometric Iwasawa theory for${\mathbf Z}_p$-extensions of function fields over a perfect fieldkof characteristic$p>0$by replacing the usual study ofp-torsion in class groups with the study ofp-torsion class groupschemes. That is, if$\cdots \to X_2 \to X_1 \to X_0$is the tower of curves overkassociated with a${\mathbf Z}_p$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of thep-torsion group scheme in the Jacobian of$X_n$as$n\rightarrow \infty $. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of$X_n$equipped with natural actions of Frobenius and of the Cartier operatorV. We formulate and test a number of conjectures which predict striking regularity in the$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)$for allnsufficiently large, where$a_r, \lambda _r$are rational constants and$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$is a periodic function, depending onrand the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on${\mathbf Z}_p$-towers of curves, and we prove our conjectures in the case$p=2$and$r=1$.