  We show how rational points on certain varieties parametrize phenomena
arising in the Galois theory of iterates of quadratic polynomials. As an
example, we characterize completely the set of quadratic polynomials $x^2+c$
whose third iterate has a "small" Galois group by determining the rational
points on some elliptic curves. It follows as a corollary that the only such
integer value with this property is $c=3$, answering a question of Rafe Jones.
Furthermore, using a result of Granville's on the rational points on quadratic
twists of a hyperelliptic curve, we indicate how the ABC conjecture implies a
finite index result, suggesting a geometric interpretation of this problem.
