IWASAWA THEORY FOR p-TORSION CLASS GROUP SCHEMES IN CHARACTERISTIC p

We investigate a novel geometric Iwasawa theory for Zp-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 ··· →X2 → X1 → X0 is the tower of curves over k associated with a Zp-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of Xn as n → ∞. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of Xn 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 {equation presented} of global regular differential forms as n → ∞. For example, for each tower in a basic class of Zp-towers, we conjecture that the dimension of the kernel of V r on Mn is given by {equation presented} for all n sufficiently large, where ar,λr are rational constants and {equation presented} 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 Zp-towers of curves, and we prove our conjectures in the case p = 2 and r = 1.