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