The distribution of number fields with wreath products as Galois groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1270.11116
DOI: 10.1142/S1793042112500492
author: Jürgen Klüners
published in: International Journal of Number Theory
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.5597
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R47
zbMATH DE Number: 6023790
zbMATH Keywords: asymptotics
zbMATH Keywords: number fields
zbMATH Keywords: wreath products
MaRDI profile type: Publication
arXiv ID: 1108.5597
cites work: The density of discriminants of quartic rings and fields
cites work: The density of discriminants of quintic rings and fields
cites work: Q4550589
cites work: On the density of discriminants of cubic fields. II
cites work: The number of extensions of a number field with fixed degree and bounded discriminant
cites work: Computing the multiplicative group of residue class rings
cites work: A counter example to Malle's conjecture on the asymptotics of discriminants
cites work: On the distribution of Galois groups
cites work: On the Distribution of Galois Groups, II
OpenAlex ID: W2964188768
title: The distribution of number fields with wreath products as Galois groups
review text: Let \(K/k\) be an extension of number fields of degree \(n\), then, by an abuse of notation, \(\mathrm{Gal}(K/k)\) is set to be the Galois group of the normal closure of \(K/k\), which is a subgroup of \(S_n\). For a given subgroup \(G\) of \(S_n\) and a number field \(k\), the author considers the number NEWLINE\[NEWLINEZ(k,G;x)=\#\{K/k:\mathrm{Gal}(K/k)=G,\mathcal N(d_{K/k})\leq x\}NEWLINE\]NEWLINE of all the extensions of \(k\) of degree \(n\) with Galois group \(G\) and such that the norm over \(\mathbb Q\) of the discriminant is bounded by \(x\). \textit{G. Malle} conjectured in [J. Number Theory 92, No. 2, 315--329 (2002; Zbl 1022.11058); Exp. Math. 13, No. 2, 129--135 (2004; Zbl 1099.11065)] that for some explicitly defined \(a(G), b(k,G)\in\mathbb R\) there exists a constant \(c(k,G)>0\) such that NEWLINE\[NEWLINEZ(k,G;x)\sim c(k,G)x^{a(G)}\log(x)^{b(k,G)-1}.NEWLINE\]NEWLINE This conjecture is known to hold for abelian groups, for \(S_3\leq S_3\) and for \(D_4\leq S_4\), but in [C. R., Math., Acad. Sci. Paris 340, No. 6, 411--414 (2005; Zbl 1083.11069)], the author showed that the conjecture does not hold for \(k=\mathbb Q\) and \(G=C_3\wr C_2\). Nevertheless the kind of problem which occurs in the counterexample seems not to be very deep, so that some refinement of the conjecture is still expected to be true.NEWLINENEWLINEIn the paper under review the author proves the original conjecture for groups of the form \(C_2\wr H\), under the assumptions that there exists at least one extension of \(k\) with Galois group \(H\) and that \(Z(k,H;x)=O_{k,H,\epsilon}(x^{1+\epsilon})\) for all \(\epsilon>0\). The techniques used in the proof are inspired by the methods described in [\textit{H. Cohen, F. Diaz y Diaz} and \textit{M. Olivier}, Compos. Math. 133, No. 1, 65--93 (2002; Zbl 1050.11104)].
reviewed by: Alessandro Cobbe
description: scientific article; zbMATH DE number 6023790