We improve a result due to Masser and Zannier, who showed that the set {λ∈C∖{0,1}:(2,2(2−λ)),(3,6(3−λ))∈(Eλ)tors} is finite, where Eλ:y2=x(x−1)(x−λ) is the Legendre family of elliptic curves. More generally,
denote by T(α,β), for α,β∈C∖{0,1}, α=β, the set of λ∈C∖{0,1} such that all points with x-coordinate α or β are
torsion on Eλ. By further results of Masser and Zannier, all these
sets are finite. We present a fairly elementary argument showing that the set
T(2,3) in question is actually empty. More generally, we obtain an explicit
description of the set of parameters λ such that the points with
x-coordinate α and β are simultaneously torsion, in the case
that α and β are algebraic numbers that not 2-adically close.
We also improve another result due to Masser and Zannier dealing with the
case that Q(α,β) has transcendence degree 1. In this case
we show that #T(α,β)≤1 and that we can decide whether the set
is empty or not, if we know the irreducible polynomial relating α and
β. This leads to a more precise description of T(α,β) also in
the case when both α and β are algebraic. We performed extensive
computations that support several conjectures, for example that there should be
only finitely many pairs (α,β) such that #T(α,β)≥3.Comment: 24 pages. v2: Improved 2-adic results, leading to more cases that can
be treated explicitly. Used this to solve a problem considered in
arXiv:1509.06573. Added some reference