Classifying Galois groups of small iterates via rational points

We establish several surjectivity theorems regarding the Galois groups of small iterates of ?(c)(x) = x(2) + c for c ? Q. To do this, we use explicit techniques from the theory of rational points on curves, including the method of Chabauty-Coleman and the Mordell- Weil sieve. For example, we succeed in finding all rational points on a hyperelliptic curve of genus 7, with rank 5 Jacobian, whose points parametrize quadratic polynomials with a "newly small" Galois group at the fifth stage of iteration.