Diophantine analysis and torsion on elliptic curves

In a recent paper of Bennett and the author, it was shown thatthe elliptic curve defined by y² = x³ + Ax + B, where A andB are integers, has no rational points of finite order if Ais sufficiently large relative to B (at least if one assumesthe abc Conjecture of Masser and Oesterlé). In the presentarticle we show, perhaps surprisingly, that the rational torsionon the above curve is also quite restricted if B is sufficientlylarge relative to A. In particular, we demonstrate that forany > 0 there is a constant c such that if A and B are integerssatisfying |B| > c |A|⁶⁺, then the elliptic curve definedabove has no rational torsion points, other than a possiblepoint of order 2 (again making use of the abc Conjecture insome cases). We then extend this by proving similar resultsfor elliptic curves admitting non-trivial -isogenies, ellipticcurves written in other forms, and elliptic curves over certainnumber fields. Curiously, the results on isogenies lead to twounexpected irrationality measures for certain algebraic numbers.