Elliptic Curves with positive rank and no integral points

TL;DR

This paper studies specific families of elliptic curves related to fundamental discriminants, proving they lack integral points under certain conditions and exploring their rank properties, with some results unconditional and others conditional on conjectures.

Contribution

It establishes unconditionally that certain elliptic curves with specific rank relations have no integral points and links their rank parity to discriminant properties, advancing understanding of integral points on elliptic curves.

Findings

Curves with rank difference conditions have no integral points.

Curves with certain discriminants have odd or even rank, conditional on Tate-Shafarevich group finiteness.

Existence of infinite order rational points on parametrized curves with no integral points.

Abstract

We consider all \emph{odd} fundamental discriminants $D \equiv 2 \bmod 3$ and their mirror discriminants $D' = -3D$, and we study the family of elliptic curves $E_{D'}: y^{2} = x^{3} + 16D'$. We denote by $r_{3}(D)$ and $r_{3}(D')$ the rank of the $3$-part of the ideal class group of $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D'})$ respectively. We show that every curve in the subfamily of elliptic curves $E_{D'}$ with $r_{3}(D) = r_{3}(D') + 1$ for $D < 0$ (respectively, with $r_{3}(D) = r_{3}(D')$ for $D > 0$) cannot have any integral points, and this is proved unconditionally. By employing results of Satg\'e and by assuming finiteness of the $3$-primary part of their Tate-Shafarevich group, we show that the curves $E_{D'}$ must have odd rank when $D < 0$ and even rank when $D > 0$. This result is particularly interesting for the case of $D < 0$ since every curve $E_{D'}$ with $r_{3}(D) = r_{3}(D') + 1$ has infinitely many rational points - assuming finiteness of the $3$-primary part of their Tate-Shafarevich group - yet no integral points. We obtain an unconditional result on the existence of elliptic curves with non-trivial rank and no integral points, by defining a parametrised family of such curves with no integral points but with a parametrised rational point, which we prove that it is of infinite order.