The arithmetic basilica: a quadratic PCF arboreal Galois 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 $x_0$.

23 pages, 4 figures. Version 4 includes only minor revisions from version 3. To appear in the Journal of Number Theory

References in corpus (4)

• Odoni's conjecture for number fields
• Large arboreal Galois representations
• Polynomials with Surjective Arboreal Galois Representations Exist in Every Degree
• The inverse problem for arboreal Galois representations of index two