Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups
Abstract
For an odd prime p, let Aj be the ωj-eigenspace of the p-primary class group of ℚ(ζp). Fix an even integer d≥4, put N=(p−1)/d, and let Ud=(ℤ/dℤ)×. For a relative density-one set of primes p≡ d+1pmod2d, we prove that the odd block ⨁a∈ UdAaN has order at most pφ(d)/2−1, and reflection shows that the even block ⨁a∈ UdAp−aN has p-rank at most φ(d)/2−1. For each d∈{4,6}, both blocks vanish for a relative density-one set of primes p≡ d+1pmod2d; in particular, the even components vanish in accordance with Vandiver's conjecture. For each d∈{8,10,12}, the odd block has order at most p for a relative density-one set of primes in the same progression, so every summand AaN, a∈ Ud, is cyclic, in accordance with Iwasawa's cyclicity conjecture. The proof uses an atomless limiting law for products of Dirichlet L-values, integrality of generalized Bernoulli norms, the relative class-number formula and reflection. A separate exact computation proves A34=0 for every odd prime p; a single-file PARI/GP program reproduces the calculation.
Related
- On the class number of certain cyclotomic fields
- A nilpotent quotient algorithm for certain infinitely presented groups and its applications
- Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials
- Permutation automorphism groups of cyclic codes I: cyclotomic association schemes
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
BibTeX
@misc{guo2026density,
title = {{Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups}},
author = {Xuejun Guo and Zhengyu Tao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13932},
eprint = {2609.13932},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13932},
}
RIS
TY - UNPB
TI - Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups
AU - Xuejun Guo
AU - Zhengyu Tao
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13932
UR - https://arxiv.org/abs/2609.13932
AB - For an odd prime p, let A_j be the ω^j-eigenspace of the p-primary class group of ℚ(ζ_p). Fix an even integer d≥4, put N=(p−1)/d, and let U_d=(ℤ/dℤ)^×. For a relative density-one set of primes p≡ d+1pmod2d, we prove that the odd block ⨁_a∈ U_dA_aN has order at most p^φ(d)/2−1, and reflection shows that the even block ⨁_a∈ U_dA_p−aN has p-rank at most φ(d)/2−1. For each d∈{4,6}, both blocks vanish for a relative density-one set of primes p≡ d+1pmod2d; in particular, the even components vanish in accordance with Vandiver's conjecture. For each d∈{8,10,12}, the odd block has order at most p for a relative density-one set of primes in the same progression, so every summand A_aN, a∈ U_d, is cyclic, in accordance with Iwasawa's cyclicity conjecture. The proof uses an atomless limiting law for products of Dirichlet L-values, integrality of generalized Bernoulli norms, the relative class-number formula and reflection. A separate exact computation proves A_34=0 for every odd prime p; a single-file PARI/GP program reproduces the calculation.
ER -