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