Using elliptic curves to produce quadratic number fields of high three-rank
type: Item
instance of: scholarly article
title: Using elliptic curves to produce quadratic number fields of high three-rank
zbMATH Open document ID: 1107.11045
author: Matt DeLong
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-10-01T00:00:00Z
review text: Several previous authors have constructed infinite families of quadratic fields possessing non-trivial lower bounds on their three-ranks. In this paper the author gives new and shorter proofs of the results on some of these families. The method relies on the known connection between the arithmetic of the elliptic curves of the form \(y^2=x^3+d\) and the arithmetic of the fields \(\mathbb Q(\sqrt d)\) and \(\mathbb Q(\sqrt{-3d})\). The method is illustrated in detail on a family of \textit{D. Shanks} [J. Number Theory 4, No. 6, 537--556 (1972; Zbl 0265.12001)]. Other previously discovered polynomials are treated in the new context, and it is shown how one can produce infinitely many such polynomials. Finally, specific examples of quadratic fields with high three-rank are given.
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 2104653
zbMATH Keywords: quadratic field
zbMATH Keywords: elliptic curve
zbMATH Keywords: three-rank
zbMATH Keywords: Mordell-Weil group
zbMATH Keywords: Shanks' polynomial
zbMATH Keywords: Spiegelungssatz
reviewed by: Veikko Ennola
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069870
OpenAlex ID: W1977539495
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DOI: 10.1216/RMJM/1181069870
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description: scientific article; zbMATH DE number 2104653