Elliptic curves coming from Heron triangles
type: Item
instance of: scholarly article
title: Elliptic curves coming from Heron triangles
zbMATH Open document ID: 1305.14017
DOI: 10.1216/RMJ-2014-44-4-1145
author: Andrej Dujella
author: Juan Carlos Peral
published in: Rocky Mountain Journal of Mathematics
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1414760946
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6368758
zbMATH Keywords: rank of elliptic curves
zbMATH Keywords: Heron triangles
describes a project that uses: ecdata
describes a project that uses: mwrank
describes a project that uses: PARI/GP
MaRDI profile type: Publication
Wikidata QID: Q57595824
cites work: On the rank of elliptic curves coming from rational Diophantine triples
cites work: Q3499386
cites work: An Infinite Set of Heron Triangles with Two Rational Medians
cites work: Q4341641
cites work: On Mordell-Weil groups of elliptic curves induced by Diophantine triples
cites work: Diophantine triples and construction of high-rank elliptic curves over \(\mathbb{Q}\) with three nontrivial 2-torsion points
cites work: Diophantine \(m\)-tuples and elliptic curves
cites work: High-rank elliptic curves with torsion induced by Diophantine triples
cites work: On Rational Triangles
cites work: Heron triangles via elliptic curves
cites work: A remark on the injectivity of the specialization homomorphism
cites work: Sums of two biquadrates and elliptic curves of rank $\geq 4$
cites work: Some remarks on Heron triangles
cites work: An elliptic \(K3\) surface associated to Heron triangles
cites work: Q3328666
cites work: An example of elliptic curve over \(\mathbb{Q}\) with rank \(\geq 20\)
cites work: Rational triangles with equal area
cites work: Q4317891
OpenAlex ID: W1973470723
review text: Heron triangles are triangles with rational sides \(a\), \(b\), \(c\) and rational area \(Q\). A quadruple \((a,b,c,Q)\) describing a Heron triangle can be parametrized in the following way NEWLINE\[NEWLINE (*)\qquad\qquad \begin{cases} a=n(k^2+m^2)\;\\ b=m(k^2+n^2)\;\\ c=(m+n)(mn-k^2)\;\\ Q=kmn(m+n)(mn-k^2) \end{cases}NEWLINE\]NEWLINE for some positive integers \(k\), \(m\), \(n\) such that \(k^2< mn\). To a Heron triangle one can associate an elliptic curve NEWLINE\[NEWLINE E_{(a,b,c)}\,:\,y^2=(x+ab)(x+ac)(x+bc) NEWLINE\]NEWLINE whose rank over \(\mathbb{Q}\) is at least 2.NEWLINENEWLINE\noindent The authors transform \(E_{(a,b,c)}\) in a curve of type \(y^2=x^3+Ax^2+Bx\) and write \(A\) and \(B\) in terms of the parametrization \((*)\). Then they are able to find explicit families of curves (and the corresponding triangles) of rank at least 3, 4 or 5 by imposing some conditions on the parameters: for example, to get \(-mn(1+m^2)(-2+mn)(1+n^2)\) as the \(x\)-coordinate of a point on \(y^2=x^3+Ax^2+Bx\), they find the substitution \(m=\frac{2}{n(1+\omega^2)}\) (for a new parameter \(\omega\) which ends up describing the family of rank at least 3). The authors then prove the independence of the newly found rational points and (using the sieving method arising from Mestre-Nagao sums and computations with \texttt{mwrank}) provide explicit examples of curves of rank 9 and 10 by specializing the parameters in the family of rank 5.
description: scientific article; zbMATH DE number 6368758