Invariants for \(A_{4}\) fields and the Cohen-Lenstra heuristics
type: Item
instance of: scholarly article
zbMATH Open document ID: 1300.11106
DOI: 10.1142/S1793042114500250
author: Simon Rubinstein-Salzedo
published in: International Journal of Number Theory
publication date: 2014-08-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.2773
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11Y40
zbMATH DE Number: 6332003
zbMATH Keywords: Cohen-Lenstra heuristics
zbMATH Keywords: class group
zbMATH Keywords: cyclic cubic fields
zbMATH Keywords: \(A_{4}\) fields
zbMATH Keywords: class field theory
MaRDI profile type: Publication
arXiv ID: 1210.2773
cites work: On the Davenport-Heilbronn theorems and second order terms
cites work: Cohen-Lenstra heuristic and roots of unity
cites work: Secondary terms in counting functions for cubic fields
OpenAlex ID: W2011399311
title: Invariants for \(A_{4}\) fields and the Cohen-Lenstra heuristics
review text: The Cohen-Lenstra heuristics fail in some cases, for instance if there is a contribution from genus theory or in the case where the ground field contains \(p\)th roots of unity. In this paper the author discusses a deviation from the Cohen-Lenstra heuristics when roots of unity are present. In particular, he proposes an explanation for the discrepancy between the observed number of cyclic cubic fields whose 2-class group is \(C_{2}\times C_{2}\) and the number predicted by the Cohen-Lenstra heuristics, in term of an invariant, for an \(A_{4}\) field lying in a quotient of the Schur multiplier group. The author also shows that, in some cases, the definition of the invariant can be greatly simplified and he provides a computation of the invariant when the cubic field is ramified at exactly one prime, up to \(10^{8}\).
reviewed by: Abdelmalek Azizi
description: scientific article; zbMATH DE number 6332003