Abstract:
After introducing some background in the field of arithmetic dynamics (iteration of rational maps on projective varieties), the speaker will describe a fundamental open problem in the field: the question of a uniform bound for rational preperiodic points. There is a natural analogy between rational maps on the projective line and multiplication-by-m maps on an elliptic curve, in which torsion points on elliptic curves correspond in a natural way to preperiodic points for a rational map. Given Mazur’s and Merel’s theorems on uniform bounds for torsion points on elliptic curves over number fields, the dynamical uniform boundedness conjecture seems reasonable. However, the proof techniques rely heavily on the group structure for elliptic curves, so there is no hope of generalizing them to a dynamics setting.

An elliptic curve with complex multiplication (CM) is one in which the endomorphism ring of the curve is strictly larger than the ring of integers. For these maps, there is a more elementary proof of a uniform bound for torsion points due to Olson. The speaker will briefly outline Olson’s proof, and explain the ways in which she hopes to generalize his work in developing a theory of dynamical complex multiplication.