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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
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