Classifying Galois groups of small iterates via rational points.

We establish several surjectivity theorems regarding the Galois groups of small iterates of ϕc(x)=x2+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. [ABSTRACT FROM AUTHOR]