Elliptic curves coming from Heron triangles
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.
scientific article; zbMATH DE number 6368758