Three-descent and the Birch and Swinnerton-Dyer conjecture
type: Item
instance of: scholarly article
title: Three-descent and the Birch and Swinnerton-Dyer conjecture
zbMATH Open document ID: 1083.11040
author: Andrea Bandini
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-09-27T00:00:00Z
review text: The first part of this article explains how to perform a simple \(3\)-descent on elliptic curves of the form \(y^2 = x^3 + a\), based on the expositions of \textit{P. Satgé} [J. Number Theory 23, 294--317 (1986; Zbl 0601.14027)] and \textit{J. Top} [The proceedings of the third conference of the Canadian Number Theory Association, held at Queen's University, Kingston, Canada, August 18--24, 1991. Oxford: Clarendon Press, 303--317 (1993; Zbl 0804.11040)]. The second part gives numerical results and verifies the conjecture of Birch and Swinnerton-Dyer for some specific examples.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 2103613
zbMATH Keywords: elliptic curves
zbMATH Keywords: 3-descent
zbMATH Keywords: Tate-Shafarevich group
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
MaRDI profile type: Publication
Wikidata QID: Q123132922
full work available at URL: https://doi.org/10.1216/rmjm/1181069889
OpenAlex ID: W2041431610
cites work: Notes on elliptic curves. II.
cites work: Second descents for elliptic curves
cites work: On the conjecture of Birch and Swinnerton-Dyer
cites work: Q3884205
cites work: Q4742927
cites work: Explicit 4-descents on an elliptic curve
cites work: Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication
cites work: The ``main conjectures'' of Iwasawa theory for imaginary quadratic fields
cites work: Q4702590
cites work: Groupes de Selmer et corps cubiques. (Selmer group and cubic fields)
cites work: 2-descent on the Jacobians of hyperelliptic curves
cites work: Class groups and Selmer groups
cites work: Q3710637
cites work: Q4317891
cites work: The diophantine equation X3 + Y3 = DZ3 and the conjectures of Birch and Swinnerton-Dyer.
cites work: Algorithm for determining the type of a singular fiber in an elliptic pencil
cites work: Q4203367
DOI: 10.1216/RMJM/1181069889
description: scientific article; zbMATH DE number 2103613