A method for construction of rational points over elliptic curves

TL;DR

This paper introduces a systematic method for constructing rational points on Legendre elliptic curves over number fields, demonstrating that these points can generate subgroups of rank equal to any odd integer n ≥ 3, and extends the approach to all elliptic curves over number fields after finite extensions.

Contribution

It provides a new explicit construction method for rational points on Legendre elliptic curves and shows their potential to generate high-rank subgroups, also applicable to all elliptic curves over number fields.

Findings

Constructs n points on Legendre elliptic curves for any odd n ≥ 3.

Shows the subgroup generated by these points has rank n for n ≥ 7.

Extends the construction to all elliptic curves over number fields after finite base extension.

Abstract

I provide a systematic construction of points (defined over number fields) on Legendre elliptic curves over $\mathbb{Q}$: for any odd integer $n\geq 3$ my method constructs $n$ points on the Legendre curve and I show that rank of the subgroup of the Mordell-Weil group they generate is $n$ if $n\geq 7$. I also show that every elliptic curve over any number field admits similar type of points after a finite base extension.