Heuristics and related results on class groups of real quadratic fields
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 \(p\).
scientific article; zbMATH DE number 1329985
scientific article