×

zbMATH — the first resource for mathematics

The Vojta conjecture implies Galois rigidity in dynamical families. (English) Zbl 1338.14026
The author proves that the Hall-Lang conjecture (or Vojta’s conjecture) implies Galois rigidity in a dynamical arboreal setting. More specifically, the author considers the one-parameter family \(\phi_a(x) = (x-\gamma(a))^2 + c(a)\) of quadratic polynomials, where \(\gamma, c\in \mathbb{Z}[t]\) and \(a\in \mathbb{Z}\). Let \(G_n(\phi_a)\) be the Galois group of the splitting field \(K_n(\phi_a)\) of the \(n\)-th iterate \(\phi_a^n\) of \(\phi_a\); this acts on a binary rooted tree \(T_n\). Taking inverse limit, we obtain an arboreal representation of \(G_\infty(\phi_a)\), serving as an analog of the \(\ell\)-adic Galois representations for elliptic curves. The main result is the following:
Theorem 1. If \(\phi_t\) is not isotrivial and \(\phi_t(\gamma(t)) \cdot \phi_t^2(\gamma(t)) \neq 0\), then Hall-Lang (or Vojta) conjecture shows that there exist an integer \(n\) and an effectively computable finite set \(F\) such that for all \(a\in \mathbb Z \backslash F\), \(G_n(\phi_a) \cong \mathrm{Aut}(T_n)\) implies \(G_\infty(\phi_a) \cong \mathrm{Aut}(T_\infty)\). Moreover, there exists a uniform bound on \([\mathrm{Aut}(T_\infty): G_\infty(\phi_a)]\) for \(a\in \mathbb Z\backslash F\) for which all iterates of \(\phi_a\) are irreducible.
As a corollary, when \(G_\infty(\phi_a)\) is maximal, the density of primes dividing the orbit of \(b\in \mathbb Z\) under \(\phi_a\) is shown to be \(0\). The general strategy of the proof is similar to [W. Hindes, Acta Arith. 169, No. 1, 1–27 (2015; Zbl 1330.14032)]: the bounds on integral/rational points on genus 1 or 2 curve (coming from Hall-Lang or Vojta) forces a square-free primitive prime divisor in the critical orbit, but if \(K_n(\phi_a)/K_{n-1}(\phi_a)\) is not maximal, \(\phi_a^n(\gamma(a))\) must be a square in \(K_{n-1}(\phi_a)\), so such a primitive divisor ramifies, contradicting a discriminant formula.

MSC:
14G05 Rational points
37P45 Families and moduli spaces in arithmetic and non-Archimedean dynamical systems
11R32 Galois theory
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.)
37P55 Arithmetic dynamics on general algebraic varieties
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Boston, Nigel; Jones, Rafe, The image of an arboreal Galois representation, Pure Appl. Math. Q., 5, 1, 213-225 (2009) · Zbl 1167.11011
[2] Gratton, C.; Nguyen, K.; Tucker, T. J., \(ABC\) implies primitive prime divisors in arithmetic dynamics, Bull. Lond. Math. Soc., 45, 6, 1194-1208 (2013) · Zbl 1291.37121
[3] Hindes, Wade, Galois uniformity in quadratic dynamics over \(k(t)\), J. Number Theory, 148, 372-383 (2015) · Zbl 1391.37090
[4] [Me] W. Hindes, The arithmetic of curves defined by iteration. Acta Arith., in press, arXiv:1305.0222, (2014). · Zbl 1330.14032
[5] Ih, Su-Ion, Height uniformity for algebraic points on curves, Compositio Math., 134, 1, 35-57 (2002) · Zbl 1031.11041
[6] Ingram, Patrick, Lower bounds on the canonical height associated to the morphism \(\phi (z)=z^d+c\), Monatsh. Math., 157, 1, 69-89 (2009) · Zbl 1239.11071
[7] Jones, Rafe, An iterative construction of irreducible polynomials reducible modulo every prime, J. Algebra, 369, 114-128 (2012) · Zbl 1302.11086
[8] Jones, Rafe, Galois representations from pre-image trees: an arboreal survey. Actes de la Conf\'erence “Th\'”eorie des Nombres et Applications”, Publ. Math. Besan\c con Alg\`“ebre Th\'”eorie Nr., 107-136 (2013), Presses Univ. Franche-Comt\'e, Besan\c con · Zbl 1307.11069
[9] Jones, Rafe, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2), 78, 2, 523-544 (2008) · Zbl 1193.37144
[10] K{\v{r}}{\'{\i }}{\v{z}}ek, Michal; Luca, Florian; Somer, Lawrence, 17 lectures on Fermat numbers. From number theory to geometry; With a foreword by Alena \v Solcov\'a, CMS Books in Mathematics/Ouvrages de Math\'ematiques de la SMC, 9, xxiv+257 pp. (2001), Springer-Verlag, New York · Zbl 1010.11002
[11] Narkiewicz, W{\l }adys{\l }aw, Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics, xii+708 pp. (2004), Springer-Verlag, Berlin · Zbl 1159.11039
[12] Serre, Jean-Pierre, Topics in Galois theory, Research Notes in Mathematics. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author 1, xvi+117 pp. (1992), Jones and Bartlett Publishers, Boston, MA · Zbl 0746.12001
[13] Silverman, Joseph H., Primitive divisors, dynamical Zsigmondy sets, and Vojta’s conjecture, J. Number Theory, 133, 9, 2948-2963 (2013) · Zbl 1297.37046
[14] Silverman, Joseph H., The arithmetic of dynamical systems, Graduate Texts in Mathematics 241, x+511 pp. (2007), Springer, New York · Zbl 1130.37001
[15] Silverman, Joseph H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, xx+513 pp. (2009), Springer, Dordrecht · Zbl 1194.11005
[16] Stoll, Michael, Rational points on curves, J. Th\'eor. Nombres Bordeaux, 23, 1, 257-277 (2011) · Zbl 1270.11030
[17] Vasiu, Adrian, Surjectivity criteria for \(p\)-adic representations. I, Manuscripta Math., 112, 3, 325-355 (2003) · Zbl 1117.11064
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.