Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the nth Gleason polynomial G₍Qc comprise the 0-dimensional moduli space of quadratic polynomials with an n-periodic critical point. Per₍ (0) is the 1-dimensional moduli space of quadratic rational maps on P^1 with an n-periodic critical point. We show that if G₍ is irreducible over Q, then Per₍ (0) is irreducible over C. To do this, we exhibit a Q-rational smooth point on a projective completion of Per₍ (0), using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large n, Per₍ (0) itself has no Q-rational points.