Classifying Galois groups of small iterates via rational points

TL;DR

This paper investigates the Galois groups of small iterates of quadratic polynomials over rationals, using advanced techniques from the theory of rational points on curves to establish surjectivity results.

Contribution

It applies explicit methods like Chabauty-Coleman and Mordell-Weil sieve to determine rational points on complex curves, advancing understanding of Galois groups in polynomial iteration.

Findings

All rational points on a genus 7 hyperelliptic curve identified.

Surjectivity theorems for Galois groups of small polynomial iterates proved.

Explicit techniques successfully applied to high-genus curves with high rank.

Abstract

We establish several surjectivity theorems regarding the Galois groups of small iterates of $\phi_c(x)=x^2+c$ 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.