  Motivated by the dynamical uniform boundedness conjecture of Morton and
Silverman, specifically in the case of quadratic polynomials, we give a formal
construction of a certain class of dynamical analogues of classical modular
curves. The preperiodic points for a quadratic polynomial map may be endowed
with the structure of a directed graph satisfying certain strict conditions; we
call such a graph admissible. Given an admissible graph $G$, we construct a
curve $X_1(G)$ whose points parametrize quadratic polynomial maps -- which, up
to equivalence, form a one-parameter family -- together with a collection of
marked preperiodic points that form a graph isomorphic to $G$. Building on work
of Bousch and Morton, we show that these curves are irreducible in
characteristic zero, and we give an application of irreducibility in the
setting of number fields. We end with a discussion of the Galois theory
associated to the preperiodic points of quadratic polynomials, including a
certain Galois representation that arises naturally in this setting.
