×

zbMATH — the first resource for mathematics

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
Software:
SageMath; Magma
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Balakrishnan, J.; Bradshaw, R.; Kedlaya, K., Explicit Coleman integration for hyperelliptic curves, Int. Algorithmic Number Theory Symp., 16-31, (2010), Springer, Berlin · Zbl 1261.14011
[2] Bosma, W.; Cannon, J.; Playoust, C., The magma algebra system. I. the user language, J. Symbolic Comput., 24, 235-265, (1997) · Zbl 0898.68039
[3] Bruin, N.; Stoll, M., The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math., 13, 272-306, (2010) · Zbl 1278.11069
[4] de la Harpe, P., Topics in Geometric Group Theory, (2000), University of Chicago Press, Chicago, IL · Zbl 0965.20025
[5] Hamblen, S.; Jones, R.; Madhu, K., The density of primes in orbits of \(z^d + c\), Int. Math. Res. Not., 7, 1924-1958, (2015) · Zbl 1395.11128
[6] Hindes, W., The arithmetic of curves defined by iteration, Acta Arith., 169, 1-27, (2015) · Zbl 1330.14032
[7] Hindes, W., Galois uniformity in quadratic dynamics over \(k(t)\), J. Number Theory, 148, 372-383, (2015) · Zbl 1391.37090
[8] Hindes, W., The vojta conjecture implies Galois rigidity in dynamical families, Proc. Amer. Math Soc., 144, 1931-1939, (2016) · Zbl 1338.14026
[9] Jones, R., The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. London Math. Soc. (2), 78, 2, 523-544, (2008) · Zbl 1193.37144
[10] Jones, R., Galois representations from pre-image trees: an arboreal survey, Publ. Math. UFS Sci. Tech. Besançon, 2013, 107-136, (2013) · Zbl 1307.11069
[11] Katz, N., Galois properties of torsion points on abelian varieties, Invent. Math., 62, 3, 481-502, (1980) · Zbl 0471.14023
[12] McCallum, W.; Poonen, B., Explicit Methods in Number Theory: Rational Points and Diophantine Equations, 36, The method of Chabauty and Coleman, 99-117, (2012), Société Mathématiques de France · Zbl 1377.11077
[13] Mumford, D., Tata Lectures on Theta II, (1984), Birkhäuser, Boston
[14] SageMath, the Sage Mathematics Software System (Version 7.2), The Sage Developers (2016), http://www.sagemath.org; see specifically the implementations for “Hyperelliptic curves over \(p\)-adic fields”.
[15] Serre, J.-P., Abelian \(\ell\)-Adic Representations and Elliptic Curves, (1968), W. A. Benjamin, New York
[16] Silverman, J., The Arithmetic of Elliptic Curves, 106, (2009), Springer Science & Business Media · Zbl 1194.11005
[17] Stoll, M., Galois groups over \(\Bbb Q\) of some iterated polynomials, Arch. Math. (Basel), 59, 3, 239-244, (1992) · Zbl 0758.11045
[18] Stoll, M., Rational points on curves, J. Théor. Nombres Bordeaux, 23, 1, 257-277, (2011) · Zbl 1270.11030
[19] Vasiu, A., Surjectivity criteria for \(p\)-adic representations, part I, Manuscripta Math., 112, 3, 325-355, (2003) · Zbl 1117.11064
[20] Vojta, P., Arithmetic Geometry, Diophantine approximation and Nevanlinna theory, 111-224, (2010), Springer, Berlin · Zbl 1258.11076
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.