The local structure of refined class groups in cyclotomic Iwasawa theory
Abstract
Let p be an odd prime and let Σ be a nonempty finite set of rational primes different from p. In the cyclotomic tower ℚ(μpn+1), we determine the kernels of the natural maps from the minus p-primary ray class groups, with modulus the product of the primes above Σ, to the corresponding ordinary minus p-primary class groups. We give their character components and elementary divisors at every finite layer, including the layers before splitting stabilizes. In the norm inverse limit, we obtain an explicit integral cyclic decomposition of the local module modulo the single diagonal global Tate module. Its coinvariants recover the finite local kernels. Together with ordinary class-group control, this gives exact control for the minus ray class groups. We identify the local Frobenius factors with Bernoulli regularization factors. For nonexceptional characters, the classical main conjecture then yields regularized measure generators of the ray class characteristic ideals. The exceptional Teichmüller component is treated without the main conjecture: the single Tate correction corresponds to the single pole of the p-adic zeta function. At the bottom layer, the local calculation and Herbrand--Ribet yield characterwise nonvanishing criteria involving regularized Bernoulli values. For a single auxiliary prime, the exceptional component detects the Wieferich condition. We also deduce Eisenstein congruence consequences from existing modular-form theorems.
Related
- Construction of Galois modules describing ideal class groups and applications to Iwasawa theory
- Bloch-regulator Principal Parts of Cyclotomic Iwasawa Pseudomeasures
- Notes on Chevalley Groups and Root Category IV: Iwasawa Decomposition and Total Positivity
- Intrinsic Non-Reductive Geometric Invariant Theory and Moduli Spaces of Principal Bundles with Non-Reductive Structure Groups
- Iwasawa theory of (directed) Cayley graphs
BibTeX
@misc{hu2026local,
title = {{The local structure of refined class groups in cyclotomic Iwasawa theory}},
author = {Su Hu and Min-Soo Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17649},
eprint = {2610.17649},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17649},
}
RIS
TY - UNPB
TI - The local structure of refined class groups in cyclotomic Iwasawa theory
AU - Su Hu
AU - Min-Soo Kim
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17649
UR - https://arxiv.org/abs/2610.17649
AB - Let p be an odd prime and let Σ be a nonempty finite set of rational primes different from p. In the cyclotomic tower ℚ(μ_p^n+1), we determine the kernels of the natural maps from the minus p-primary ray class groups, with modulus the product of the primes above Σ, to the corresponding ordinary minus p-primary class groups. We give their character components and elementary divisors at every finite layer, including the layers before splitting stabilizes. In the norm inverse limit, we obtain an explicit integral cyclic decomposition of the local module modulo the single diagonal global Tate module. Its coinvariants recover the finite local kernels. Together with ordinary class-group control, this gives exact control for the minus ray class groups. We identify the local Frobenius factors with Bernoulli regularization factors. For nonexceptional characters, the classical main conjecture then yields regularized measure generators of the ray class characteristic ideals. The exceptional Teichmüller component is treated without the main conjecture: the single Tate correction corresponds to the single pole of the p-adic zeta function. At the bottom layer, the local calculation and Herbrand--Ribet yield characterwise nonvanishing criteria involving regularized Bernoulli values. For a single auxiliary prime, the exceptional component detects the Wieferich condition. We also deduce Eisenstein congruence consequences from existing modular-form theorems.
ER -