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

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 $\exp \left(O\right(\frac{\log B}{\sqrt{\log \log B}}\left)\right)$. 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 $N_E(B)=\exp \left(O\right(\frac{\log B}{\sqrt{\log \log B}}\left)\right)$ for these curves and to remove a deep transcendence theory ingredient from the proof.