×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cremona, Je, On the Galois groups of the iterates of \(x^2+1\), Mathematika, 36, 259-261 (1989) · Zbl 0699.12018
[2] 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
[3] 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
[4] Odoni, Rwk, The Galois theory of iterates and composites of polynomials, Proc. Lond. Math. Soc., 51, 3, 385-414 (1985) · Zbl 0622.12011
[5] Odoni, Rwk, Realising wreath products of cyclic groups as Galois groups, Mathematika, 35, 101-113 (1988) · Zbl 0662.12010
[6] Stoll, M., Galois groups over \({\mathbb{Q}}\) of some iterated polynomials, Arch. Math., 59, 239-244 (1992) · Zbl 0758.11045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.