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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)].

MSC:
 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:
References:
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