Mild p-Class Tower Groups of Imaginary Quadratic Fields
Abstract
Let K be an imaginary quadratic number field, let p be an odd prime, and let G=G∅(K)(p) be the Galois group of the maximal everywhere unramified pro-p extension of K. To each mod-p character x of G we associate a linear map Dx from Cl(K)[p] to Cl(K)/p; a formula of Ahlqvist and Carlson expresses it through the class of a norm ideal in the unramified cyclic degree-p extension attached to x. These maps determine all triple Massey products on H1(G,𝔽p), and with them the cubic initial relations of G. Suppose that the p-class rank d=dim𝔽pCl(K)/p is at least three. The (d−1)×(d−1) minors of the family x↦ Dx define a subscheme ΣD of ℙd−1𝔽p, an invariant of K, the norm-degeneracy scheme. We prove: if the rank condition rk Dx=d−2 holds transversally at a point of ΣD, over some finite extension of 𝔽p, then G is mild, and hence of cohomological dimension 2. For p>3 transversality means that ΣD is smooth of dimension d−3 at the point; at p=3 the kernel of the Bockstein map enters as an additional constraint. We treat every imaginary quadratic field of p-class rank at least three with |DK|<230. Only p=3, 5, and 7 occur. The criterion decides 206 of the 207 fields at p=5 and 7, and 505 of the 12 750 fields at p=3, where the Bockstein condition restricts its reach. A direct computation with the cubic initial relations settles the remaining field at p=5 and a further 11 765 at p=3, 26 of them not mild, while 480 remain undecided. In all, the p-class tower group is proved mild for 12 451 of the 12 957 fields. These appear to be the first number fields for which the full maximal everywhere unramified pro-p Galois group is proved to be mild, and hence of cohomological dimension 2.
Related
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
- Bowen--Franks groups and minus class groups of imaginary abelian fields
- Imaginary quadratic fields of class number at most 2000
- Admissible sets of integers in imaginary quadratic fields of class number 1
- Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials
BibTeX
@misc{vogel2026mild,
title = {{Mild p-Class Tower Groups of Imaginary Quadratic Fields}},
author = {Denis Vogel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.17072},
eprint = {2608.17072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17072},
}
RIS
TY - UNPB
TI - Mild p-Class Tower Groups of Imaginary Quadratic Fields
AU - Denis Vogel
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2608.17072
UR - https://arxiv.org/abs/2608.17072
AB - Let K be an imaginary quadratic number field, let p be an odd prime, and let G=G_∅(K)(p) be the Galois group of the maximal everywhere unramified pro-p extension of K. To each mod-p character x of G we associate a linear map D_x from Cl(K)[p] to Cl(K)/p; a formula of Ahlqvist and Carlson expresses it through the class of a norm ideal in the unramified cyclic degree-p extension attached to x. These maps determine all triple Massey products on H^1(G,𝔽_p), and with them the cubic initial relations of G. Suppose that the p-class rank d=dim_𝔽_pCl(K)/p is at least three. The (d−1)×(d−1) minors of the family x↦ D_x define a subscheme Σ_D of ℙ^d−1_𝔽_p, an invariant of K, the norm-degeneracy scheme. We prove: if the rank condition rk D_x=d−2 holds transversally at a point of Σ_D, over some finite extension of 𝔽_p, then G is mild, and hence of cohomological dimension 2. For p>3 transversality means that Σ_D is smooth of dimension d−3 at the point; at p=3 the kernel of the Bockstein map enters as an additional constraint. We treat every imaginary quadratic field of p-class rank at least three with |D_K|<2^30. Only p=3, 5, and 7 occur. The criterion decides 206 of the 207 fields at p=5 and 7, and 505 of the 12 750 fields at p=3, where the Bockstein condition restricts its reach. A direct computation with the cubic initial relations settles the remaining field at p=5 and a further 11 765 at p=3, 26 of them not mild, while 480 remain undecided. In all, the p-class tower group is proved mild for 12 451 of the 12 957 fields. These appear to be the first number fields for which the full maximal everywhere unramified pro-p Galois group is proved to be mild, and hence of cohomological dimension 2.
ER -