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Classifying Galois groups of small iterates via rational points. (English) Zbl 1441.11281
Let \(\phi(x)\) be a monic quadratic polynomial over \(Z\) and put \(\phi^0(x)=x\) and \(\phi^n(x)=\phi(\phi^{n-1}(x))\) for \(n\ge1\). The paper deals with Galois group \(G_n(\phi,b)\) of the polynomial \(\phi^n(x)-b\), where \(b\in Z\) is generic for \(\phi\), i.e. for all \(n\) the equation \(\phi^n(x)= b\) has \(2^n\) distinct solutions. Moreover let \(T_{2,n}(\phi)\) be the graph whose set of vertices equals \(\bigcup_{m=0}^n\{z:\ \phi^m(z)=b\}\), and two elements \(z_1,z_2\) are joined by an edge if \(z_2=\phi(z_1)\). If \(T_{2,n}\) is the binary rooted tree with \(n\) levels, then the graphs \(T_{2,n}(\phi)\) and \(T_{2,n}\) are isomorphic. Since \(G_n(\phi,b)\) acts on \(T_{2,n}(\phi)\), it is a subgroup of \(\operatorname{Aut}(T_{2,n})\). Therefore the inverse limit \(G(\phi,b)=\varprojlim G_n(\phi,b)\) is a subgroup of the group of automorphisms \(\operatorname{Aut}(T_2)\) of the full binary rooted tree \(T_2\).
It has been conjectured (see [N. Boston and R. Jones, Pure Appl. Math. Q. 5, No. 1, 213–225 (2009; Zbl 1167.11011)]) that if \(\phi(x) = x^2+c\in Z[x]\), all its iterates are irreducible and \(c\ne-2\), then the index of \(G(\phi,0)\) in \(\operatorname{Aut}(T_2)\) is finite, and this has been shown to be true for certain large families of polynomials (see [M. Stoll, Arch. Math. 59, No. 3, 239–244 (1992; Zbl 0758.11045)] and [H.-C. Li, Arch. Math. 114, No. 3, 265–269 (2020; Zbl 1435.37108)]). C. Gratton et al. [Bull. Lond. Math. Soc. 45, No. 6, 1194–1208 (2013; Zbl 1291.37121)] and the author [Acta Arith. 159, 149–197 (2013; Zbl.1296.14017)] showed that the conjecture follows from the ABC conjecture.
The author established earlier [Proc. Amer. Math.Soc. 144, 1931–1939 (2016; Zbl.1338.14026)] that if the Vojta conjecture holds [P. Vojta, Lect. Notes Math. 2009, 111–224 (2011; Zbl 1258.11076)], then there exist an integer \(n=n(\phi)\) such that if \(G_n(\phi,0) = \mathrm{Aut}(T_n(\phi))\), then \(G(\phi,0) = \mathrm{Aut}(T_2)\). He showed also (J. Number Th. 148, 372–383 (2015); Zbl.1391.37090) that for a large class of quadratic polynomials over the field of rational functions over a field of zero characteristics the analogous assertion holds with \(n=17\) without any unproved assumptions.
In this paper the implications
\[G_3(\phi,0)=\operatorname{Aut}(T_{2,3}) \longrightarrow G_5(\phi,0)=\operatorname{Aut}(T_{2,5})\]
and, if \(c\ne3\) also
\[ G_2(\phi,0)=\operatorname{Aut}(T_{2,2}) \longrightarrow G_5(\phi,0)=\operatorname{Aut}(T_{2,5})\]
are established (Theorem 1.3), and this implies that if \(c\ne3\) and neither \(-c\) nor \(-(c+1)\) is a square, then one has \(G_5(\phi,0)=\operatorname{Aut}(T_{2,5})\). Theorem 1.6 gives similar implications in case \(b=1\). The proofs are based on the determination of all rational points on hyperelliptic curves \[C_\varepsilon: \ y^2 = -x^{\varepsilon_0}\phi^1(x)^{\varepsilon_1}\cdots\phi^n(x)^{\varepsilon_n},\] with \(\varepsilon_i\in\{0,1\}\), which is performed using the Chabauty-Coleman method (see e.g. [W. McCallum and B. Poonen, Panor. Synth. 36, 99–117 (2012; Zbl 1377.11077)]) and the Mordell-Weil sieve (see [N. Bruin and M. Stoll, LMS J. Comput. Math. 13, 272–306 (2010; Zbl 1278.11069)].

11R32 Galois theory
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05 Rational points
37P15 Dynamical systems over global ground fields
Magma; SageMath
Full Text: DOI arXiv
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