A note on elliptic curves with a rational 3-torsion point
type: Item
instance of: scholarly article
title: A note on elliptic curves with a rational 3-torsion point
zbMATH Open document ID: 1200.11038
DOI: 10.1216/RMJ-2010-40-4-1227
author: Rintaro Kozuma
published in: Rocky Mountain Journal of Mathematics
publication date: 2010-09-07T00:00:00Z
review text: The paper presents a descent via isogenies on elliptic curves defined over a number field \(F\). The known descriptions for the images of the local Kummer maps and for the orders of the Selmer groups require many informations about the curve \(E\) like, for example, the reduction types, the kernel of the isogeny, the number of connected components and so on. The detailed computations of these data are carried out for a 3-isogeny \(\phi\) with \(F\)-rational kernel and curves of the type \(E_{a,b}\,:\;y^2 +axy+by=x^3\) to provide the orders of the Selmer groups \(\mathrm{Sel}^{(\phi)}(E/F)\). The informations on the Selmer groups are then used to find bounds for the rank of the elliptic curve \(E_{a,b}\,\).
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 5779826
zbMATH Keywords: elliptic curves
zbMATH Keywords: Selmer groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5779826