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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.

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