Pairing Pythagorean Pairs

TL;DR

This paper explores special classes of Pythagorean pairs and their connection to elliptic curves, establishing criteria for positive rank and classifying elliptic curves with specific torsion groups related to these pairs.

Contribution

It introduces double-pythapotent and quadratic pythapotent pairs, linking them to elliptic curves with particular torsion groups and positive rank conditions, and classifies all such curves with torsion group Z/2Z×Z/8Z.

Findings

Positive rank of associated elliptic curves characterizes pythagorean pairs as double- or quadratic-pythapotent.

Every elliptic curve with torsion Z/2Z×Z/8Z is isomorphic to one derived from a pythagorean pair.

Existence of infinitely many pythagorean pairs related to double- and quadratic-pythapotent pairs.

Abstract

A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2 = \Box$ (i.e., $a^2 + b^2$ is a square). A pythagorean pair $(a, b)$ is called a double-pythapotent pair if there is another pythagorean pair $(k,l)$ such that $(ak,bl)$ is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair $(k,l)$ which is not a multiple of $(a,b)$, such that $(a^2k,b^2l)$ is a pythagorean pair. To each pythagorean pair $(a, b)$ we assign an elliptic curve $\Gamma_{a,b}$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/4\mathbb Z$, such that $\Gamma_{a,b}$ has positive rank if and only if $(a, b)$ is a double-pythapotent pair. Similarly, to each pythagorean pair $(a, b)$ we assign an elliptic curve $\Gamma_{a^2 ,b^2}$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$, such that $\Gamma_{a^2,b^2}$ has positive rank if and only if $(a,b)$ is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve $\Gamma$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$ is isomorphic to a curve of the form $\Gamma_{a^2 ,b^2}$ , where $(a,b)$ is a pythagorean pair. As a side-result we get that if $(a,b)$ is a double-pythapotent pair, then there are infinitely many pythagorean pairs $(k, l)$, not multiples of each other, such that $(ak, bl)$ is a pythagorean pair; the analogous result holds for quadratic pythapotent pairs.