A method for construction of rational points over elliptic curves II: Points over solvable extensions

TL;DR

This paper presents a systematic method to construct rational points on elliptic curves over finite radical extensions, generalizing previous work and demonstrating the existence of many such points over solvable extensions.

Contribution

It introduces a new systematic construction of rational points over radical extensions on any Legendre elliptic curve, extending Ulmer's work and showing infinite order points over number fields.

Findings

Any elliptic curve over a field with characteristic not two has at least 2n rational points over a finite solvable extension for even n≥4.

Under certain conditions, these points are of infinite order when the base field is a number field.

Ulmer's points can be lifted to characteristic zero, including the canonical lifting.

Abstract

I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over a rational function field in characteristic $p>0$. In particular I show that if $n\geq 4$ is any even integer and not divisible by the characteristic of the field then any elliptic curve $E$ over this field has at least $2n$ rational points over a finite solvable field extension. Under additional hypothesis, when the ground field is a number field, I show that these are of infinite order. I also show that Ulmer's points lift to characteristic zero and in particular to the canonical lifting.