## Points on elliptic curves parametrizing dynamical Galois groups

### Volume 159 / 2013

Acta Arithmetica 159 (2013), 149-167
MSC: Primary 14G05; Secondary 37P55.
DOI: 10.4064/aa159-2-5

#### Abstract

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 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.