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Galois theory of quadratic rational functions. (English) Zbl 1316.11104
Let \(K\) be a number field, \(\bar{K}\) its algebraic closure and \(G_{K} :=\mathrm{Gal}(\bar{K}/K)\) its absolute Galois group. Let \(\phi\in K(x)\) be a rational function of degree \(2\). For each \(\alpha\in K\) we consider the tree \(T_{\alpha}\) whose vertex set is the disjoint union \({\textstyle\bigsqcup_{n\geq1}\phi^{-n}}(\alpha)\) of the iterated preimages with edges between \(\phi ^{-n}(\alpha)\) and \(\phi^{-n+1}(\alpha)\) defined by the action of \(\phi\). The elements of \(G_{K}\) commute with \(\phi\) and so we have a homomorphism \(\rho:G_{K}\rightarrow \mathrm{Aut}(T_{\alpha})\) called the arboreal Galois representation attached to \((\phi,\alpha)\). The object of the paper is to study the image \(G_{\infty}\) of \(\rho\) for particular functions \(\phi\).
If \(\phi\) commutes with some \(f\in \mathrm{PGL}_{2}(K)\) and \(f(\alpha)=\alpha\), then the Galois action on \(T_{\alpha}\) commutes with the action of \(f\). Define \(A_{\phi}:=\left\{ f\in \mathrm{PGL}_{2}(\bar{K})\, |\, \phi\circ f=f\circ\phi\right\} \) and let \(A_{\phi,\alpha}\) be the stabilizer of \(\alpha\) in \(A_{\phi}\). Let \(C_{\infty}\leq G_{\infty}\) be the centralizer of action of \(A_{\phi,\alpha}\) on \(\mathrm{Aut}(T_{\alpha})\). The authors are interested in the following conjecture. (Conjecture 1.1): If \(\phi\) is not post-critically finite (that is, the orbit of at least one critical point under \(\phi\) is infinite), then \(\left| C_{\infty}:G_{\infty}\right| <\infty\). In an earlier paper [J. Lond. Math. Soc., II. Ser. 78, No. 2, 523–544 (2008; Zbl 1193.37144)] the first author proved that this conjecture holds for two special polynomials. In the present paper it is proved in some other cases. For example, suppose that \(K=\mathbb{Q}\), \(\phi(x)=k(x^{2}+1)/x\) and \(\alpha=0\). Then it is shown that there exists an effectively computable set \(\Sigma\) of primes of natural density \(0\) in \(\mathbb{Z}\) such that \(G_{\infty}\cong C_{\infty}\) provided the valuations \(v_{p}(k)\) are \(0\) for all \(p\in\Sigma\). More concretely the authors show that if \(k\in\mathbb{Z}\) then Conjecture 1.1 holds whenever \(\left| k\right| <10000\).

11R32 Galois theory
37P15 Dynamical systems over global ground fields
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