Torsion points and the Lattes family

Abstract

We give a dynamical proof of a result of Masser and Zannier [MZ2, MZ3] about torsion points on the Legendre family of elliptic curves. Our methods also treat points of small height. A key ingredient is the arithmetic equidistribution theorem on $\mathbb{P}^1$ of Baker-Rumely, Chambert-Loir, and Favre-Rivera-Letelier. Torsion points on the elliptic curve coincide with preperiodic points for the degree-4 Lattes family of rational functions. Our main new results concern properties of the bifurcation measures for this Lattes family associated to marked points.Comment: Theorem 1.3 now states the strongest form of the main theorem, the result of combining our methods with the conclusions of Masser-Zannier, for rational points with complex coefficients. To appear, American Journal of Mat