On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves
type: Item
instance of: scholarly article
title: On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves
zbMATH Open document ID: 1354.11067
author: Yoshichika Iizuka
author: Yutaka Konomi
author: Shin-Ichi Nakano
published in: Journal of the Mathematical Society of Japan
publication date: 2016-06-23T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1460727387
review text: 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)].
reviewed by: Alessandro Cobbe
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6597362
zbMATH Keywords: quadratic field
zbMATH Keywords: class number
zbMATH Keywords: elliptic curve
describes a project that uses: Maple
describes a project that uses: PARI/GP
describes a project that uses: ecdata
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W2186673733
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DOI: 10.2969/JMSJ/06820899
description: scientific article; zbMATH DE number 6597362