Heuristics and related results on class groups of real quadratic fields
type: Item
instance of: scholarly article
title: Heuristics and related results on class groups of real quadratic fields
zbMATH Open document ID: 0961.11039
DOI: 10.1007/BF02879027
author: Lawrence C. Washington
author: Xianke Zhang
published in: Science in China. Series A
publication date: 2000-05-08T00:00:00Z
review text: The authors define a probability model for a finite group to have a property \(\pi\) which is a modification of the Cohen-Lenstra heuristics [\textit{H. Cohen} and \textit{H. W. Lenstra}, Lect. Notes Math. 1068, 33-62 (1984; Zbl 0558.12002)]. This model is applied to the class group of a real quadratic field \(K\) to give two heuristic results concerning the divisibility of the class group by an odd prime \(p\). The class group, \(H\), of \(K\) is said to have a potential element, \(P\), of order \(p\) provided \(P^p=1\) but it is not known if \(P=1\). The probability that \(p\) divides the class number of \(K\) provided that \(H\) has a potential element of order \(p\) is \(1-\eta_\infty (p)\) where \(\eta_\infty(p)= (1-p^{-1}) (1-p^{-2})(1-p^{-3}) \dots\). It is interesting to note that this is exactly the probability that the Cohen-Lenstra heuristics give for the probability of an imaginary quadratic field to have class number divisible by \(p\). For \(p=3\) this probability is 0.43987... which is consistent with known numerical data. In a second calculation the authors use their model to compute the probability that a potential element \(P\) in \(H\) of order \(p\) really has order \(p\). This probability is \(1/p\). For \(p=3\) this does not agree well with early numerical data, but agrees quite well for larger primes \(p\).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1329985
zbMATH Keywords: class number
zbMATH Keywords: Cohen-Lenstra heuristics
zbMATH Keywords: class group
zbMATH Keywords: real quadratic field
zbMATH Keywords: divisibility
zbMATH Keywords: potential element
reviewed by: Charles J. Parry
MaRDI profile type: Publication
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cites work: Modification of Cohen-Lenstra heuristics for ideal class groups and numbers of certain real quadratic fields
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description: scientific article; zbMATH DE number 1329985