Finite index theorems for iterated Galois groups of unicritical polynomials

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 $G_n(\beta )=\mathrm{Gal}(K(f^{-n}(\beta ))/K(\beta ))$ 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 $K$.

arXiv comment20 pages