Points on Elliptic Curves Parametrizing Dynamical Galois Groups

TL;DR

This paper explores how rational points on specific varieties, especially elliptic curves, can parametrize Galois groups of iterated quadratic polynomials, leading to classification results and connections to deep conjectures.

Contribution

It characterizes quadratic polynomials with small Galois groups for their third iterate using elliptic curves and links the problem to the ABC conjecture for a broader finite index result.

Findings

Only c=3 yields a small Galois group for x^2+c with third iterate

Rational points on elliptic curves parametrize Galois phenomena in polynomial iterations

ABC conjecture implies a finite index result related to the problem

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 such 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.