TY - JOUR

T1 - IWASAWA THEORY FOR p-TORSION CLASS GROUP SCHEMES IN CHARACTERISTIC p

AU - Booher, Jeremy

AU - Cais, Bryden

N1 - Funding Information:
Booher was partially supported by the Marsden Fund Council administered by the Royal Society of New Zealand. Cais is supported by National Science Foundation grant number DMS-1902005.
Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - 11G20 14H40 14G17 11R23 14Q05 11Y40

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U2 - 10.1017/nmj.2022.30

DO - 10.1017/nmj.2022.30

M3 - Article

AN - SCOPUS:85159378047

SN - 0027-7630

VL - 250

SP - 298

EP - 351

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

ER -