TORSION POINTS AND THE LATTÈS FAMILY

We give a dynamical proof of a result of Masser and Zannier: for any $a \neq b \in \overline{\mathbb{Q}} \setminus \left \{ 0, 1 \right \}$, there are only finitely many parameters t ∈ ℂ for which points $P_{a} = (a, \sqrt{a(a-1)(a-t)})$ and $P_{b} = (b, \sqrt{b(b-1)(b-t)})$ are both torsion on the Legendre elliptic curve Et = {y2 = x(x − 1)(x − t)}. Our method also gives the finiteness of parameters t where both Pa and Pb have small Néron-Tate height. A key ingredient in the proof is an arithmetic equidistribution theorem on ℙ1. For this, we prove two statements about the degree-4 Lattès family ft on ℙ1: (1) for each c ∈ ℂ(t), the bifurcation measure μc for the pair (ft,c) has continuous potential across the singular parameters t = 0,1,∞; and (2) for distinct points a,b ∈ ℂ \ {0,1}, the bifurcation measures μa and μb cannot coincide. Combining our methods with the result of Masser and Zannier, we extend their conclusion to points t of small height also for a,b ∈ ℂ(t).