Resumen de Diophantine analysis and torsion on elliptic curves
Patrick Ingram
In a recent paper of Bennett and the author, it was shown that the elliptic curve defined by $y^2 = x^3 + Ax + B$, 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 $\epsilon > 0$ there is a constant $c_\epsilon$ such that if $A$ and $B$ are integers satisfying $|B| > c_\epsilon |A|^{6 + \epsilon}$, 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 $\mathbb{Q}$-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.
