Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples (Q2845647)

Let \(a\), \(b\), \(c\) be a nontrivial Pythagorean triple and let \(E_{a,b,c}\) be the elliptic curve \(y^2=x(x-a^2)(x-b^2)\). The paper investigates the Mordell-Weil groups of curves \(E_{a,b,c}\) showing that it has rank at least 2 for infinitely many triples and providing two explicit independent points of infinite order. The problem is strictly related with the study of elliptic surfaces whose generic fiber is equal to (or closely related to) \(E_{a,b,c}\,\). In particular the author studies three such surfaces, namely: NEWLINE\[NEWLINE\begin{aligned} \mathcal{E}_1&: y^2=x(x-(t-1)^2)(x-4t)\;; \\ \mathcal{E}_2&: y^2=x(x-(t^2-1)^2)(x-4t^2) \;\text{i.e.,\;}\mathcal{E}_1\;\text{with}\;t\mapsto t^2\;;\\ \mathcal{E}_3&: y^2=x(x-(u^2-1)^2)(x-4u^2) \;\text{i.e.,\;}\mathcal{E}_2\;\text{with}\;t\mapsto u:=\frac{2t}{5+t^2}\;,\end{aligned}NEWLINE\]NEWLINE and computes their rank over \(\overline{\mathbb{Q}}(t)\) using the Shioda-Tate formula, the formula for the rank of Artin twists and some direct computations (which also use the MAGMA program). In particular one obtains that \(E_3(\overline{\mathbb{Q}}(t))\) (with \(E_3\) the generic fiber of \(\mathcal{E}_3\,\)) is isomorphic to \(\mathbb{Z}^3\otimes\mathbb{Z}/2\otimes\mathbb{Z}/4\) and generated by: NEWLINE\[NEWLINE\begin{aligned} P_1&:=\left(2(1+\sqrt{2})(u-1)^2u,2\sqrt{-1}(1+\sqrt{2})((\sqrt{2}-u)^2-1)(u-1)^2u\right)\;;\\ P_2&:=\left(2(u-1)^2,2(u-1)^2(u^2+2u-1)\right)\;;\\ P_3&:=\left( 1-u^2,\frac{(t^2-5)u(u^2-1)}{5+t^2}\right)\;,\end{aligned}NEWLINE\]NEWLINE for the free part and NEWLINE\[NEWLINE\begin{aligned} T_1&:=( -4u^2,0)\;;\\ T_2&:=\left(2(u^3-u),2\sqrt{-1}(u^2-1)u(u^2-2u-1)\right)\;,\end{aligned}NEWLINE\]NEWLINE for the torsion part. The result on \(E_{a,b,c}\) mentioned above is obtained by specializing \(u\mapsto\frac{2pq}{p^2+5q^2}\) for \(\frac{p}{q}\in \mathbb{Q}^*\,\).