Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

We establish an implication between two long-standing open problems in complex dynamics. The roots of the nth Gleason polynomial Gₙ∈Q[c] comprise the $0$-dimensional moduli space of quadratic polynomials with an n-periodic critical point. Perₙ(0) is the $1$-dimensional moduli space of quadratic rational maps on P¹ with an n-periodic critical point. We show that if Gₙ is irreducible over Q, then Perₙ(0) is irreducible over C. To do this, we exhibit a Q-rational smooth point on a projective completion of Perₙ(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, Perₙ(0) itself has no Q-rational points.