On elliptic curves \(y^{2} = x^{3}-n^{2}x\) with rank zero
type: Item
instance of: scholarly article
title: On elliptic curves \(y^{2} = x^{3}-n^{2}x\) with rank zero
zbMATH Open document ID: 1076.11036
author: Maosheng Xiong
author: Keqin Feng
published in: Journal of Number Theory
publication date: 2005-02-22T00:00:00Z
review text: A positive square-free integer \(n\) is called a congruent number (CN) if \(n\) is the area of some rational right triangular. It is a classical problem to determine when \(n\) is CN or not CN. By Tunnell's work, such problem is equivalent to determine the rank of the rational points of the elliptic curvs \(E_n: y^2=x^3-n^2x\) is zero or not. In this paper, all elliptic curves \(E_n\) of such type with trivial 2-Selmer group and 2-dual Selmer group \(\{ \pm 1, \pm n\}\) are determined in terms of graph language. The rank of rational points of such elliptic curves is always zero. As an application, the non CN with arbitarily many prime divisors can be given, which also extends the first author's previous result. The proof of the paper uses the 2-descent method and reduces the problem to study the Legendre symbol between prime divisors of \(n\). The graph language shows its advantage for this complicated combinatoric question.
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 2136916
zbMATH Keywords: elliptic curve
zbMATH Keywords: Selmer group
zbMATH Keywords: congruent number
zbMATH Keywords: odd graph
reviewed by: Fei Xu
MaRDI profile type: Publication
full work available at URL: j.jnt.2003.12.015
OpenAlex ID: W2042553442
cites work: Q4950461
cites work: Q3346401
cites work: Q4093521
cites work: Q3973403
cites work: Q3710637
cites work: A classical Diophantine problem and modular forms of weight \(3/2\)
cites work: A criterion for elliptic curves with lowest 2-power in L(1)
DOI: 10.1016/J.JNT.2003.12.015
description: scientific article; zbMATH DE number 2136916