On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves
Given a prime number \(l\), a classical problem is to find infinitely many quadratic extensions \(K\) of \(\mathbb{Q}\) such that \(l\) divides the class number of \(K\). In [\textit{T. Nagel}, Hamb. Abh. 1, 140--150 (1922; JFM 48.0170.03)] this has been achieved for imaginary quadratic extensions, while the corresponding result for real quadratic fields has been proved in [\textit{Y. Yamamoto}, Osaka J. Math. 7, 57--76 (1970; Zbl 0222.12003)]. Following [\textit{A. Sato}, Osaka J. Math. 45, No. 2, 375--390 (2008; Zbl 1197.11148)], the authors of the paper under review consider the cases \(l=3\), \(5\) and \(7\) and define some cubic polynomials \(F(a,b,X)\), depending on \(l\) and on the integer parameters \(a\) and \(b\). Under some technical assumptions, Sato proved that for many \(\xi\in\mathbb{Q}\) the class number of \(\mathbb{Q}(\sqrt{F(a,b,\xi)})\) must be divisible by \(l\). Using this result, in the present paper the authors deduce that for \(l=3\), \(5\) or \(7\) and any given nonzero integer \(m\), there exist infinitely many quadratic fields \(\mathbb{Q}(\sqrt{D})\) such that the class numbers of \(\mathbb{Q}(\sqrt{D})\) and \(\mathbb{Q}(\sqrt{mD})\) are both divisible by \(l\). Furthermore, an infinite family of quadratic fields as above can be parametrized by rational points on a specified elliptic curve. A different approach to the case \(l=3\) can be found in [\textit{T. Komatsu}, Acta Arith. 104, No. 2, 129--136 (2002; Zbl 1004.11062)].
scientific article; zbMATH DE number 6597362