Finite index theorems for iterated Galois groups of unicritical polynomials

Abstract:

Let $K$ be the function field of a smooth irreducible curve defined over $\overline {Q}$. 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 $K$.