Let $E/\\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by work of Bombieri and Zannier, that if $E$ has full 2-torsion, number $N_E(B)$ points with Weil height bounded $B$ $\\operatorname{exp}\\big(O \\big(\\frac{\\operatorname{log}B}{\\sqrt{\\operatorname{log}\\operatorname{log} B}}\\big)\\big)$. In this paper we exploit method descent via 2-isogeny to extend result c...

In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta’s Main Conj...

We show that the number of rational points of height ≤ H on a non-rational plane curve of degree d is Od(H 2/d−δ), for some δ > 0 depending only on d. The implicit constant depends only on d. This improves a result of Heath-Brown, who proved the bound O (H2/d+ ). We also show that one can take δ = 1/450 in the case d = 3.