An application of the arithmetic of elliptic curves to the class number problem for quadratic fields
This paper generalizes the previous work made by the authors in [J. Math. Soc. Japan 68, No. 2, 899--915 (2016; Zbl 1354.11067)], in which they construct an infinite family of pairs of distinct quadratic fields such that both class numbers are divisible by a prime \(l\), for \(l=3\), \(5\), or \(7\). Finding infinite families of quadratic fields whose class numbers are divisible by a given integer has a long history, whose roots are going back to the classic Gauss's class number problems posed in 1801. In 1922, \textit{T. Nagel} [Abh. Math. Semin. Univ. Hamb. 1, 140--150 (1922; JFM 48.0170.03)] proved that there are infinitely many imaginary quadratic number fields whose class numbers are divisible by a given integer. For real quadratic such a result was proved by \textit{Y. Yamamoto} in 1970 [Osaka J. Math. 7, 57--76 (1970; Zbl 0222.12003)] or by \textit{P. J. Weinberger} in 1973 [J. Number Theory 5, 237--241 (1973; Zbl 0287.12007)]. Finding infinite families of pairs of quadratic fields is of more recent concern, and results can be found in [\textit{T. Komatsu}, Acta Arith. 104, No. 2, 129--136 (2002; Zbl 1004.11062)] or [\textit{Y. Iizuka}, J. Number Theory 184, 122--127 (2018; Zbl 1420.11138)] for the divisibility of both class numbers by \(3\). In this paper, for a given prime \(l=3,5,7\), and for any non-zero rational numbers \(m_1\), \(m_2\), \(n_1\) and \(n_2\), the authors construct an infinite parametrized family of distinct quadratic fields \(\mathbb{Q}(\sqrt{m_1D+n_1})\) and \(\mathbb{Q}(\sqrt{m_2D+n_2})\) whose class numbers are divisible by \(l\). The parametrization is based upon the construction of an explicit elliptic curve over \(\mathbb{Q}\) with positive Mordell-Weil rank, and its rational points. As in their previous paper, one of the main tool is a result of \textit{A. Sato} [Osaka J. Math. 45, No. 2, 375--390 (2008; Zbl 1197.11148)] that states the divisibility by \(l\) of the class number of the family of quadratic fields \(\mathbb{Q}(\sqrt{F(\xi)})\), where \(F\) is a specified cubic polynomial, for all \(\xi \in \mathbb{Q}\) satisfying technical conditions. The theoretical improvements when applying this result allow the authors to find a more general family of pairs of real quadratic fields than in their previous paper.
scientific article; zbMATH DE number 7394479