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

Let E=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 NE (B) of rational points with Weil height bounded by B is exp. In this paper we exploit the method of descent log B 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 for these curves and to remove a deep transcendence theory ingredient from the proof. (Formula Presented) © 2021 European Mathematical Society Publishing House. All rights reserved.