On the rank of an elliptic curve in elementary 2-extensions
type: Item
instance of: scholarly article
title: On the rank of an elliptic curve in elementary 2-extensions
zbMATH Open document ID: 0824.14027
author: Masanari Kida
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-08-08T00:00:00Z
review text: The author studies the behaviour of the rank of an elliptic curve \(E\) over a number field \(k\) under transition from \(k\) to an elementary abelian 2-extension \(K\) of \(k\). By adjoining to \(k\) the square roots of a suitable set \(\Sigma\) of integers, the rank of \(E\) over a tower of elementary abelian 2-extensions can be shown to tend to infinity. As an application of the corresponding quantative result to the family of elliptic curves \(E_ n:\) \(y^ 2 = x^ 3 - n^ 2x\) \((n \in \mathbb{N}\) squarefree) arising from the congruent number problem, some prime numbers \(n \in \mathbb{N}\) and sets \(\Sigma\) are specified for which the rank of \(E_ n\) becomes arbitrarily large as one adjoins to \(k = \mathbb{Q}\) a sufficient number of square roots of numbers from \(\Sigma\). As another application it is shown that, for prime numbers \(p\), \(q\), and \(r \equiv 3 \pmod 8\), the curves \(E_ r\) of rank 0 over \(\mathbb{Q}\) have rank 0 also over \(K = \mathbb{Q} (\sqrt p, \sqrt q)\). \{Reviewer's remark: In her diploma thesis ``Rangsprünge elliptischer Kurven mit nichttrivialem 2-Teilungspunkt beim Übergang vom rationalen Zahlkörper \(\mathbb{Q}\) zu multiquadratischen Zahlkörpern'', Saarbrücken 1994, \textit{Margit Sens} considered similar questions. She constructed, among other things, an elliptic curve of rank \(\geq 9\) over \(\mathbb{Q}\), whose rank becomes \(\geq 28\) over the composite of 19 quadratic fields\}.
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 597335
zbMATH Keywords: rank of an elliptic curve
zbMATH Keywords: tower of elementary abelian 2-extensions
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
full work available at URL: pjaa.69.422
OpenAlex ID: W2010754352
cites work: Q5631241
cites work: Q3346401
cites work: All congruent numbers less than 2000
cites work: ELLIPTIC CURVES OF INFINITE RANK OVER Γ-EXTENSIONS
cites work: ON THE RANK OF ELLIPTIC CURVES OVER Γ-EXTENSIONS
cites work: Mock Heegner points and congruent numbers
cites work: On the relative Modell-Weil rank of elliptic quartic curves
cites work: Q4867187
cites work: Q3973403
cites work: Q3973404
DOI: 10.3792/PJAA.69.422
description: scientific article; zbMATH DE number 597335