Bloch-regulator Principal Parts of Cyclotomic Iwasawa Pseudomeasures
Abstract
Let p be an odd prime and let K/ℚp be a finite unramified extension. From a finite presentation by roots of unity of order prime to p, we construct a localized Iwasawa pseudomeasure on ℤp×. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on the associated Bloch class: it is the Frobenius-depleted Coleman regulator of that class multiplied by a universal half-shifted zeta principal part. Consequently, every nonexceptional weight component is bounded, while the exceptional component has at most a simple pole with explicitly determined residue. For p>3 and ℤp-valued coefficients, vanishing of the principal part is equivalent to vanishing of the corresponding class in K3(K;ℤp). Cyclotomic refinements preserve the Bloch class and act on the associated pseudomeasures by explicit Iwasawa multipliers. Normalized finite linear combinations of refinements interpolate arbitrary finite jets of the bounded weight-space data, subject only to the normalization at the exceptional point. We also establish a half-shifted complex Mellin factorization, compare the construction with the GSWZ germ family, and derive, at simple degree-one places above primes p>3, a finite-polylogarithm criterion for the local K3-class of the knot 52.
Related
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BibTeX
@misc{fang2026blochregulator,
title = {{Bloch-regulator Principal Parts of Cyclotomic Iwasawa Pseudomeasures}},
author = {Honghuai Fang and Zekun Chen},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29587},
eprint = {2608.29587},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29587},
}
RIS
TY - UNPB
TI - Bloch-regulator Principal Parts of Cyclotomic Iwasawa Pseudomeasures
AU - Honghuai Fang
AU - Zekun Chen
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29587
UR - https://arxiv.org/abs/2608.29587
AB - Let p be an odd prime and let K/ℚ_p be a finite unramified extension. From a finite presentation by roots of unity of order prime to p, we construct a localized Iwasawa pseudomeasure on ℤ_p^×. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on the associated Bloch class: it is the Frobenius-depleted Coleman regulator of that class multiplied by a universal half-shifted zeta principal part. Consequently, every nonexceptional weight component is bounded, while the exceptional component has at most a simple pole with explicitly determined residue. For p>3 and ℤ_p-valued coefficients, vanishing of the principal part is equivalent to vanishing of the corresponding class in K_3(K;ℤ_p). Cyclotomic refinements preserve the Bloch class and act on the associated pseudomeasures by explicit Iwasawa multipliers. Normalized finite linear combinations of refinements interpolate arbitrary finite jets of the bounded weight-space data, subject only to the normalization at the exceptional point. We also establish a half-shifted complex Mellin factorization, compare the construction with the GSWZ germ family, and derive, at simple degree-one places above primes p>3, a finite-polylogarithm criterion for the local K_3-class of the knot 5_2.
ER -