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 x²+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 integer value with this property is c=3, answering a question of Rafe Jones. Furthermore, using a result of Granville's on the rational...