Torsion Subgroups of Elliptic Curves in Short Weierstrass Form

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves E/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 ε > 0, all but finitely many curves EA,B : y 2 = x +Ax+B, where A and B are integers satisfying A > |B|1+ε > 0, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to |A| > |B|2+ε > 0, then the EA,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.