Complex dynamics and elliptic curves

In holomorphic dynamical systems, one studies maps on the Riemann sphere (or other complex manifolds) with focus on their Julia sets and invariant measures. From this point of view, the Lattès maps -- those that are quotients of maps on elliptic curves -- are rather uninteresting; their dynamical features are well understood. But viewed algebraically, there are still many unanswered questions. I'll begin the talk with some history of these maps. Then I'll describe a recent question about the geometry of torsion points on elliptic curves and how it has led to interesting complex-dynamical questions about other families of maps and, in turn, new perspectives on the arithmetic side.