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 / Q 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 ε > 0 \varepsilon >0 , all but finitely many curves \[ E A , B : y 2 = x 3 + A x + B , E_{A,B} \; : \; y^2 = x^3 + A x + B, \] where A A and B B are integers satisfying A > | B | 1 + ε > 0 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 + ε > 0 |A|>|B|^{2+\varepsilon }>0 , then the E A , B 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.