Elliptic Curves with positive rank and 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.