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Galois representations from pre-image trees: an arboreal survey. (English) Zbl 1307.11069
Actes de la conférence “Théorie des nombres et applications”. Besançon: Presses Universitaires de Franche-Comté (ISBN 978-2-84867-472-8/pbk). Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres 2013, 107-136 (2013).
Author’s abstract: Given a global field $$K$$ and a rational function $$\phi \in K(x)$$, one may take pre-images of $$0$$ under successive iterates of $$\phi$$, and thus obtain an infinite rooted tree $$T_\infty$$ by assigning edges according to the action of $$\phi$$. The absolute Galois group of $$K$$ acts on $$T_\infty$$ by tree automorphisms, giving a subgroup $$G_\infty (\phi)$$ of the group $$\mathrm{Aut}(T_\infty )$$ of all tree automorphisms. Beginning in the 1980s with work of R. W. K. Odoni [Proc. Lond. Math. Soc. (3) 51, 385–414 (1985; Zbl 0622.12011); J. Lond. Math. Soc., II. Ser. 32, 1–11 (1985; Zbl 0574.10020); Mathematika 35, No. 1, 101–113 (1988; Zbl 0662.12010)], and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. These inquiries arose in part because knowledge of $$G_\infty (\phi)$$ allows one to prove density results on the set of primes of $$K$$ that divide at least one element of a given orbit of $$\phi$$.
Following an overview of the history of the subject and two of its fundamental questions, we survey in Section 2 cases where $$G_\infty (\phi)$$ is known to have finite index in $$\mathrm{Aut}(T_\infty )$$. While it is tempting to conjecture that such behavior should hold in general, we exhibit in Section 3 four classes of rational functions where it does not, illustrating the difficulties in formulating the proper conjecture. Fortunately, one can achieve the aforementioned density results with comparatively little information about $$G_\infty (\phi)$$, thanks in part to a surprising application of probability theory, as we discuss in Section 4. Underlying all of this analysis are results on the factorization into irreducibles of the numerators of iterates of $$\phi$$, which we survey briefly in Section 5. We find that for each of these matters, the arithmetic of the forward orbits of the critical points of $$\phi$$ proves decisive, just as the topology of these orbits is decisive in complex dynamics.
For the entire collection see [Zbl 1282.11004].

##### MSC:
 11F80 Galois representations 11R32 Galois theory 37P15 Dynamical systems over global ground fields 11-02 Research exposition (monographs, survey articles) pertaining to number theory
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