Torsion subgroups of elliptic curves in short Weierstrass form

Abstract:

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves $E/\mathbb {Q}$ in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any $\varepsilon >0$, all but finitely many curves \[ E_{A,B} \; : \; y^2 = x^3 + A x + B, \] where $A$ and $B$ are integers satisfying $A>|B|^{1+\varepsilon }>0$, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to $|A|>|B|^{2+\varepsilon }>0$, then the $E_{A,B}$ now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.