Two infinite families of elliptic curves with rank greater than one

TL;DR

This paper demonstrates that two infinite families of elliptic curves have ranks exceeding one, with specific lower bounds, using elementary and advanced methods, advancing understanding of elliptic curve ranks.

Contribution

It introduces two infinite families of elliptic curves with proven lower bounds on their ranks, employing both elementary and more complex techniques.

Findings

Each $E_m$ has rank at least 2.

Each $E_m'$ has rank at least 3.

Stronger results are obtained with advanced methods.

Abstract

We prove, using elementary methods, that each member of the infinite families of elliptic curves given by $E_m \colon y^2=x^3 - x + m^6$ and $E_m' \colon y^2=x^3 + x - m^6$ have rank at least $2$ and 3, respectively, under mild restrictions on $m$. We also prove stronger results for $E_m$ and $E_m'$ using more technical machinery.