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Arboreal Galois representation for a certain type of quadratic polynomials. (English) Zbl 1435.37108
The results of this paper are nicely described by the author’s introduction:
“Consider the quadratic polynomial $$f(x)=x^2+a\in \mathbb Z[x]$$ and consider the Galois groups of polynomials of the form $$f^n(x)$$, where $$f^n=f\circ\cdots\circ f$$ is the $$n$$-th iterate of $$f$$. Let $$K_n$$ be the splitting field of $$f^n$$ over $$\mathbb Q$$ and denote by $$G_n=\mathrm{Gal}(K_n/\mathbb Q)$$. $$G_n$$ has been called arboreal Galois group because it can be made to act on trees. In fact, let $$T_n$$ be the graph whose vertex set is $$\cup_{0\le i\le n}f^{-i}(0)$$ and where we draw an edge from $$\alpha$$ to $$\beta$$ if $$f(\alpha)=\beta$$. Then clearly $$G_n$$ acts faithfully on $$T_n$$, so that $$G_n$$ can be considered as a subgroup of $$\operatorname{Aut}(T_n)$$, the automorphism group of $$T_n$$. We write $$G_\infty$$ as the inverse limit of the Galois groups $$G_n$$ and write $$T_\infty$$ as the increasing union of $$T_1\subset T_2\subset\cdots$$. Therefore, the injections $$G_n\hookrightarrow \operatorname{Aut}(T_n)$$ naturally extend to the injection $$G_\infty\hookrightarrow \operatorname{Aut}(T_\infty)$$.
In this article, we suppose that $$-a$$ is not a square in $$\mathbb Z$$. This is equivalent to saying that all iterates of $$f$$ are irreducible over $$\mathbb Q$$, and hence ensures that $$T_n$$ is a complete binary rooted tree of height $$n$$ (so there are $$2n$$ leaves at the top). In this case, we can describe $$\operatorname{Aut}(T_n)$$ as the $$n$$-fold iterated wreath product of the cyclic group $$C_2$$ of two elements, as shown by R. W. K. Odoni [Mathematika 35, No. 1, 101–113 (1988; Zbl 0622.12010)]. We also neglect the case that $$a=-2$$. This is the only case that $$f(x)=x^2+a$$ is so called post-critically finite, which means that the forward orbit of its critical points is finite. In fact, when $$f(x)=x^2-2$$, it is well known that $$G_n\cong C_{2^n}$$ and hence $$[\operatorname{Aut}(T_\infty):G_\infty]=\infty$$. Under these assumptions (that is $$-a$$ is not a square in $$\mathbb Z$$ and $$a\ne -2)$$, many evidences indicate that one should expect $$[\operatorname{Aut}(T_\infty):G_\infty]=\infty$$. For example, C. Gratton et al. [Bull. Lond. Math. Soc. 45, No. 6, 1194–1208 (2013; Zbl 1291.37121)] proved that this is the case assuming the ABC conjecture.
The study of the case $$a=1$$ which is a problem originally posed by J. McKay, began by Odoni [loc. cit.] and J. E. Cremona [Mathematika 36, No. 2, 259–261 (1989; Zbl 0699.12018)]. Later, by extending Odoni’s theory and techniques for $$x^2+1$$ to the general case, and by adding an ingenious pair of new ideas of his own, M. Stoll [Arch. Math. 59, No. 3, 239–244 (1992; Zbl 0758.11045)] proved as his main result that if $$a\in\mathbb Z$$ has one of the following properties: (a) $$a>0$$ and $$a\equiv 1\pmod 4$$ or $$a\equiv 2\pmod 4$$; (b) $$a<0$$, $$a\in 4\mathbb Z$$ and $$-a$$ is not a square in $$\mathbb Z$$, then $$G_n=\operatorname{Aut}(T_n)$$ for all $$n\ge 1$$. Not much seems to be known for all other cases where $$-a$$ is not a square.”
In this paper, we solve another piece of this puzzle. We prove that if $$a<0$$ and $$a\equiv -1\pmod 4$$, then $$[\operatorname{Aut}(T_\infty):G_\infty]\le 2$$. More precisely, we have $$G_n=\operatorname{Aut}(T_n)$$ for all $$n\ge1$$ if $$-a$$ is not of the form $$4k^2+1$$ and $$[\operatorname{Aut}(T_n):G_n]=2$$ for all $$n\ge 2$$, otherwise.
The author’s approach follows the methods used by Odoni and Stoll.

##### MSC:
 11R32 Galois theory 11A07 Congruences; primitive roots; residue systems 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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##### References:
  Cremona, Je, On the Galois groups of the iterates of $$x^2+1$$, Mathematika, 36, 259-261 (1989) · Zbl 0699.12018  Gratton, C.; Nguyen, K.; Tucker, Tj, ABC implies primitive prime divisors in arithmetic dynamic, Bull. Lond. Math. Soc., 45, 6, 1194-1208 (2013) · Zbl 1291.37121  Jones, R., The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc., 78, 2, 523-544 (2008) · Zbl 1193.37144  Odoni, Rwk, The Galois theory of iterates and composites of polynomials, Proc. Lond. Math. Soc., 51, 3, 385-414 (1985) · Zbl 0622.12011  Odoni, Rwk, Realising wreath products of cyclic groups as Galois groups, Mathematika, 35, 101-113 (1988) · Zbl 0662.12010  Stoll, M., Galois groups over $${\mathbb{Q}}$$ of some iterated polynomials, Arch. Math., 59, 239-244 (1992) · Zbl 0758.11045
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