Classifying Galois groups of small iterates via rational points

Abstract

We establish several surjectivity theorems regarding the Galois groups of small iterates of ${\varphi }_c(x)=x^2+c$ for $c\in \mathbb{Q}$. 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.

References in corpus (1)

• The density of prime divisors in the arithmetic dynamics of quadratic polynomials