Torsion points on elliptic curves in Weierstrass form (Q2864589)

\textit{D. W. Masser} and \textit{U. Zannier} [Am. J. Math. 132, No. 6, 1677--1691 (2010; Zbl 1225.11078)] proved that there are only finitely many complex \(\lambda\neq0,1\) such that \((2,\sqrt{2(2-\lambda})\) and \((3,\sqrt{6(3-\lambda)})\) are torsion points on the elliptic curve given in Legendre form by \(y^2=x(x-1)(x-\lambda)\). The main purpose of the present paper is to generalize this result to the one for elliptic curves in Weierstrass form. More precisely, let \(E_{(a,b)}\) be the elliptic curve in Weierstrass form \(y^2=x^3+ax+b\) with \(a,b\in\mathbb{C}\). The main result of this article shows that there are only finitely many complex pairs \((a,b)\) with \(4a^3+27b^2\neq 0\) such that \((1,\sqrt{1+a+b})\), \((2,\sqrt{8+2a+b})\), and \((3,\sqrt{27+3a+b})\) are torsion points on \(E_{(a,b)}\). In contrast to this result the author remarks that there are infinitely many complex pairs \((a,b)\) with \(4a^3+27b^2\neq 0\) such that \((0,\sqrt{b})\), \((1,\sqrt{1+a+b)})\), and \((-1,\sqrt{-1-a+b})\) are torsion points on \(E_{(a,b)}\). In an unpublished manuscript he describes a necessary condition on the triple of the \(x\)-coordinates to ensure a finiteness statement as above. For example, the first three primes 2, 3, 5 also yield a finiteness result.