Finite index theorems for iterated Galois groups of unicritical polynomials

TL;DR

This paper investigates the Galois groups of iterated preimages of unicritical polynomials over function fields, establishing finite index results and disjointness of fields for distinct polynomials, with implications for arithmetic dynamics.

Contribution

It proves finite index theorems for Galois groups of iterated unicritical polynomials over function fields and shows disjointness of their preimage fields for different polynomials.

Findings

Galois groups embed into iterated wreath products of cyclic groups.

Finite index of Galois groups unless the point is postcritical or periodic.

Preimage fields of distinct polynomials are disjoint over a finite extension.

Abstract

Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $\beta\in \overline{K}$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(\beta)=\mathop{\rm{Gal}}(K(f^{-n}(\beta))/K(\beta))$ embed into $[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q]^\infty:G_\infty(\beta)]<\infty$ unless $\beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields $\bigcup_{n=1}^\infty K(f_1^{-n}(\beta))$ and $\bigcup_{n=1}^\infty K(f_2^{-n}(\beta))$ are disjoint over a finite extension of $K$.