# zbMATH — the first resource for mathematics

Arboreal Galois representations and uniformization of polynomial dynamics. (English) Zbl 1280.37058
As an analog of Tate’s uniformization for elliptic curves, the author constructs an (analytic) coordinate-change near infinity for a polynomial $$f$$ so that $$f$$ looks like $$z^{\deg f}$$ (Theorem 2). More precisely, given a monic $$f(z)\in K[z]$$ over a non-Archimedean complete field such that $$\deg(f)$$ is not divisible by the residue characteristic of $$K$$, there exists a Galois-equivariant invertible power series $$\Omega$$ convergent in a neighborhood $$D$$ near infinity such that $$\Omega(f(z)) = \Omega(z)^{\deg f}$$.
As a consequence, the author derives a dynamical analog (Theorem 1) over local fields of the open-image theorem of Serre. Assume $$[K:\mathbb Q_p] < \infty$$, $$f\in K[z]$$ monic of degree $$d \not \equiv 0 \pmod p$$, and $$P\in K$$ not lying in an orbit of any critical point of $$f$$. Let $$T_{f,P}$$ be the rooted $$d$$-ary tree whose nodes at level $$n$$ are preimages of $$P$$ by the $$n$$-th iterate $$f^n$$, with an edge between $$\alpha$$ at level $$n$$ and $$\beta$$ at level $$n-1$$ if and only if $$f(\alpha) = \beta$$. The absolute Galois group $$G_K$$ acts on $$T_{f,P}$$. If $$f$$ has good reduction and if $$P$$ is outside the filled Julia set, the author shows that the entire $$T_{f,P}$$ is inside $$D$$ in terms of Theorem 2, and thus $$T_{f,P}$$ is isomorphic to $$T_{z^d, \Omega(P)}$$ as $$G_K$$-trees. As a result, the image of this arboreal representation is shown to have finite index inside a Kummer subgroup of $$\mathrm{Aut}(T_{f,P})$$. It follows (Corollary 3) that for polynomials defined over number fields satisfying the conditions of Theorem 1 locally for some $$p$$, the fields of definition of preimages eventually grow by $$d$$ at each iterate.
There are also other consequences of Theorem 2. Corollary 4 proves for certain polynomials a conjecture of V. Sookdeo [J. Number Theory 131, No. 7, 1229–1239 (2011; Zbl 1246.37102)], demonstrating that there are only finitely many points in any backward orbit which are integral with respect to a given nonpreperiodic point. Corollaries 5 and 6 give some uniform boundedness results for rational preimages in one-parameter families, cf. [X. Faber et al., Math. Res. Lett. 16, No. 1, 87–101 (2009; Zbl 1222.11086); A. Levin, Monatsh. Math. 168, No. 3–4, 473–501 (2012; Zbl 1302.37060)] for related results.

##### MSC:
 37P20 Dynamical systems over non-Archimedean local ground fields 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 11S20 Galois theory
Full Text: