On \(\ell\)-torsion in class groups of number fields
type: Item
instance of: scholarly article
title: On \(\ell\)-torsion in class groups of number fields
zbMATH Open document ID: 1398.11136
author: Melanie Matchett Wood
author: Jordan S. Ellenberg
author: Lillian Beatrix Pierce
published in: Algebra \& Number Theory
publication date: 2017-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1606.06103
review text: Let \(K\) be an algebraic number field of degree \(d\) over \(\mathbb Q\), \(D_K = |\text{disc}\,K/\mathbb Q|\) the absolute value of the discriminant of \(K\), \(\ell\) a natural number and \(\text{Cl}_K[\ell]\) the \(\ell\)-torsion subgroup of the class group of \(K\). Put \(\delta = \frac 1{2\ell (d-1)}\). Assuming GRH, the first author and \textit{A. Venkatesh} [Int. Math. Res. Not. 2007, No. 1, Article ID rnm002, 18 p. (2007; Zbl 1130.11060)] showed that for any \(\varepsilon >0\) one has \[ |\text{Cl}_K[\ell]| \ll_{d, \ell, \varepsilon} D_K^{\frac 12 - \delta + \varepsilon}. \] In the present paper, the authors prove that this result holds without assuming GRH for \(d \in \{2,3,4,5\}\) for all fields \(K\) with \(D_K \leq X\) up to some small set of exceptions (which is of density \(0\)). For \(d=4\) one must suppose that the Galois group of the normal closure of \(K\) is not \(D_4\) and that \(\ell \geq8\); and for \(d=5\) that \(\ell \geq 25\). From this result one can deduce that under the same suppositions one has \[ \sum_{\overset{\deg(K)=4}{0 < D_K \leq X}} |\text{Cl}_K[\ell]| \ll_{d,\varepsilon} D_K^{\frac {3}{2} - \delta + \varepsilon}. \] Following the ideas of Ellenberg and Venkatesh [loc. cit.], the exceptions are those fields \(K\) which have only ``few'' ``small'' primes which split completely. Therefore the authors develop a new sieve, called ``Chebyshev'' sieve (Section 3 of the paper), and asymptotically count the number of the fields under consideration with \(D_K \leq X\), using results which go back to \textit{M. Bhargava} [Ann. Math. (2) 162, No. 2, 1031--1063 (2005; Zbl 1159.11045)] and [Ann. Math. (2) 172, No. 3, 1559--1591 (2010; Zbl 1220.11139)].
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 6806361
zbMATH Keywords: Cohen-Lenstra heuristics
zbMATH Keywords: Chebotarev's density theorem
zbMATH Keywords: sieve
MaRDI profile type: Publication
OpenAlex ID: W3104171656
arXiv ID: 1606.06103
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DOI: 10.2140/ANT.2017.11.1739
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description: scientific article