Iwasawa theory of class field towers
Abstract
Let p be an odd prime and let K∞/K be a ℤp-extension. For each finite layer Kn, let Gn be the Galois group of its maximal unramified pro-p extension. We study the augmentation and Zassenhaus filtrations of the groups Gn as n tends to infinity. Assume that the classical Iwasawa μ-invariant is positive. If dn denotes the minimal number of generators of Gn, then there is an integer δ>0 such that dn=δpn+O(1), while the minimal number of relations is O(pn). For each fixed m≥2, as n→∞, we show that the m-th augmentation quotient has the same leading asymptotic as that of a free pro-p group on dn generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.
Related
- Construction of Galois modules describing ideal class groups and applications to Iwasawa theory
- The local structure of refined class groups in cyclotomic Iwasawa theory
- Fermat curves and explicit class field theory for real quadratic number fields
- Higher dimensional local class field theory
- Adelic framed form class groups and explicit class field theory
BibTeX
@misc{ray2026iwasawa,
title = {{Iwasawa theory of class field towers}},
author = {Anwesh Ray},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22297},
eprint = {2609.22297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22297},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Iwasawa theory of class field towers
AU - Anwesh Ray
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.22297
UR - https://arxiv.org/abs/2609.22297
AB - Let p be an odd prime and let K_∞/K be a ℤ_p-extension. For each finite layer K_n, let G_n be the Galois group of its maximal unramified pro-p extension. We study the augmentation and Zassenhaus filtrations of the groups G_n as n tends to infinity. Assume that the classical Iwasawa μ-invariant is positive. If d_n denotes the minimal number of generators of G_n, then there is an integer δ>0 such that d_n=δp^n+O(1), while the minimal number of relations is O(p^n). For each fixed m≥2, as n→∞, we show that the m-th augmentation quotient has the same leading asymptotic as that of a free pro-p group on d_n generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.
ER -