Maximal class numbers of CM number fields
type: Item
instance of: scholarly article
title: Maximal class numbers of CM number fields
zbMATH Open document ID: 1185.11065
author: Ryan C. Daileda
author: Raju Krishnamoorthy
author: A. V. Malyshev
published in: Journal of Number Theory
publication date: 2010-04-08T00:00:00Z
review text: The problem of finding bounds for the class number \(h\) of a number field goes back at least to the time of Gauß. Under the assumptions of the generalized Riemann hypothesis (GRH) and Artin's conjecture on the entirety of Artin \(L\)-functions, \textit{W. Duke} [Compos. Math. 136, No. 1, 103--115 (2003; Zbl 1013.11072)] has shown that for families of number fields in which the degree, signature and Galois group are fixed one can deduce bounds of the form \[ h\leq C|d|^{1/2}\frac{(\log\log |d|)^{n-1}}{(\log |d|)^b}, \] where \(d\) is the discriminant, \(n\) the degree over \(\mathbb Q\) and \(C\), \(b\geq 0\) depend on the family under consideration. A natural question to ask is whether such bounds are actually sharp or not. In the paper under review it is shown that for CM number fields with a fixed maximal real subfield \(F\), of degree at least \(2\), the answer to the above question is negative (again under the assumption of GRH). Actually, the authors obtain a stronger bound in the considered situation, namely \[ h\ll_F|d|^{1/2}(\log\log|d|)^{1+o(1)}, \] and show that this bound is the best possible.
reviewed by: Alessandro Cobbe
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R21
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 5690729
zbMATH Keywords: class number
zbMATH Keywords: CM number field
zbMATH Keywords: Artin \(L\)-function
zbMATH Keywords: generalized Riemann hypothesis
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.09.013
OpenAlex ID: W2035964868
cites work: Non-abelian number fields with very large class numbers
cites work: Q4805276
cites work: Q4664217
cites work: Real quadratic fields with large class number
cites work: On the Distribution of Quadratic Residues and Nonresidues Modulo a Prime Number
cites work: An inequality relating the regulator and the discriminant of a number field
DOI: 10.1016/J.JNT.2009.09.013
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description: scientific article; zbMATH DE number 5690729