Diophantine analysis and torsion on elliptic curves

Abstract

In a recent paper of Bennett and the author, it was shown that the elliptic curve defined by y2 = x3 + Ax + B, where A and B are integers, has no rational points of finite order if A is sufficiently large relative to B (at least if one assumes the abc Conjecture of Masser and Oesterlé). In the present article we show, perhaps surprisingly, that the rational torsion on the above curve is also quite restricted if B is sufficiently large relative to A. In particular, we demonstrate that for any ε > 0 there is a constant cε such that if A and B are integers satisfying |B| > cε |A|6+ε, then the elliptic curve defined above has no rational torsion points, other than a possible point of order 2 (again making use of the abc Conjecture in some cases). We then extend this by proving similar results for elliptic curves admitting non-trivial ℚ-isogenies, elliptic curves written in other forms, and elliptic curves over certain number fields. Curiously, the results on isogenies lead to two unexpected irrationality measures for certain algebraic numbers.