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.

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 Math

References in corpus (2)

• Preperiodic points for families of rational map
• Growth of balls of holomorphic sections and energy at equilibrium