Elliptic curves with rank 0 over number fields
type: Item
instance of: scholarly article
title: Elliptic curves with rank 0 over number fields
zbMATH Open document ID: 1390.14089
author: Pallab Kanti Dey
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2018-04-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1484794815
review text: For any nonzero integer \(b\) define the elliptic curve \(E_b:y^2=x^3+bx\). The paper studies the group \(E_b(k)\) of \(k\)-rational points of \(E_b\) over number fields \(k\) with \([k:\mathbb{Q}]\not\equiv 0\pmod{4}\). By direct computation, the author provides all torsion subgroups of \(E_b(k)\) (\(k\) as above) showing that all points are of even order \(\leqslant 4\). Finally, using the well known relation between the curve \(E_b\) and the diophantine equation \(X^4+bY^4=Z^4\), one checks that nontrivial solutions of \(X^4+bY^4=Z^4\) cannot correspond to points of order 2 or 4; thus proving that \(E_b(k)\) has rank 0 if and only if \(X^4+bY^4=Z^4\) has no nontrivial solutions in \(k^3\).
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 6864143
zbMATH Keywords: elliptic curve
zbMATH Keywords: number field
zbMATH Keywords: Diophantine equation
MaRDI profile type: Publication
OpenAlex ID: W2576942971
cites work: Q5335427
cites work: On the Equation Y 2 = X(X 2 + p)
cites work: Q3185017
cites work: Q3073865
cites work: Q3377780
cites work: Bounds for the torsion of elliptic curves over number fields
cites work: Q4004187
cites work: Q5447643
cites work: On the group structure of elliptic curves y^2=x^3-2px
cites work: Q4406363
DOI: 10.7169/FACM/1585
description: scientific article; zbMATH DE number 6864143