On the rank of elliptic curves arising from Pythagorean quadruplets, II
Volume 163 / 2021
Colloquium Mathematicum 163 (2021), 189-196
MSC: Primary 14H52; Secondary 11G05, 11D25.
DOI: 10.4064/cm8101-3-2020
Published online: 15 June 2020
Abstract
By a Pythagorean quadruplet $(a,b,c,d)$, we mean an integer solution to the quadratic equation $a^2+b^2=c^2+d^2$. We use Pythagorean quadruplets to construct infinite families of elliptic curves with rank as high as possible, including one family with rank at least $4$. We also find a few examples of these curves with rank 8. The families of curves we construct have trivial and $\mathbb Z/2\mathbb Z$ torsion subgroups in general. Previous work of the second named author has studied similar Pythagorean quadruplet elliptic curve families with torsion subgroup $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$.
