Counting rational points on elliptic curves with a rational 2-torsion point

TL;DR

This paper extends bounds on the number of rational points of elliptic curves with a rational 2-torsion point, using descent methods and refining previous results to achieve tighter bounds without deep transcendence theory.

Contribution

It generalizes existing bounds from elliptic curves with full rational 2-torsion to those with only one rational 2-torsion point, improving the upper bounds on rational points.

Findings

Extended bounds to elliptic curves with one rational 2-torsion point

Derived a stronger upper bound on the number of rational points

Removed deep transcendence theory from the proof

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is $\exp\left(O\left(\frac{\log B}{\sqrt{\log\log B}}\right)\right)$. In this paper we exploit the method of descent via $2$-isogeny to extend this result to elliptic curves with just one nontrivial rational $2$-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound $N_{E}(B) = \exp\left(O\left(\frac{\log B}{\log\log B}\right)\right)$ for these curves and to remove a deep transcendence theory ingredient from the proof.