Elliptic Curves with positive rank and no integral points
Eleni Agathocleous
TL;DR
The paper investigates elliptic curves $E_{D'}: y^2 = x^3 + 16D'$ attached to odd fundamental discriminants $D \equiv 2 \bmod 3$ and their mirrors $D' = -3D$, tying their rank and integral-point properties to the 3-part of class groups via $3$-descent and the Selmer groups. It provides unconditional no-integral-points results for escalatory/non-escalatory cases and conditional parity results for the rank based on the finiteness of the $3$-primary Tate–Shafarevich group, using Satgé’s Selmer-rank computations and the Fundamental $3$-Descent Map. Moreover, it constructs an explicit parametrised family of curves with positive rank but no integral points, yielding infinitely many examples, and complements this with a concrete infinite subfamily defined by $D(w)$ producing rational points of infinite order. The work highlights a deep link between cubic fields, Scholz reflection, and integral-point behavior on a natural family of $j$-invariant-zero curves, providing both conditional and unconditional existence results for curves with nontrivial rank and no integral points. This has implications for understanding the distribution of integral points on elliptic curves and illustrates how arithmetic of quadratic fields controls integral-point phenomena on associated cubic twists.
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 Satgé 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.