Torsion anomalous points and families of elliptic curves

Abstract

We prove that there are at most finitely many complex λ  = 0, 1 such that two points on the Legendre elliptic curve Y2 = X(X − 1)(X − λ) with coordinates X = 2, 3 both have finite order. This is a very special case of some conjectures on unlikely intersections in semiabelian schemes