The arithmetic basilica: a quadratic PCF arboreal Galois group

TL;DR

This paper investigates the arboreal Galois group of a specific quadratic polynomial, revealing a new group called the arithmetic basilica group, which relates to complex dynamics and holds for many rational points.

Contribution

It explicitly computes the arboreal Galois group for the polynomial z^2 - 1 under certain conditions, introducing the novel arithmetic basilica group concept.

Findings

The arithmetic basilica group is explicitly determined for the polynomial z^2 - 1.

The condition for the computation holds for infinitely many rational points.

The work connects Galois groups with complex dynamics through the basilica group.

Abstract

The arboreal Galois group of a polynomial $f$ over a field $K$ encodes the action of Galois on the iterated preimages of a root point $x_0\in K$, analogous to the action of Galois on the $\ell$-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 - 1$ when the field $K$ and root point $x_0$ satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of $f$. For $K=\mathbb{Q}$, our condition holds for infinitely many choices of $x_0$.