Arboreal Galois representation for a certain type of quadratic polynomials.

*(English)*Zbl 1435.37108The 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.

“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.

Reviewer: Olaf Ninnemann (Uffing am Staffelsee)

##### 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:

[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 |

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