The Vojta conjecture implies Galois rigidity in dynamical families (Q2790911)

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:NEWLINENEWLINETheorem 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.NEWLINENEWLINEAs 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 [\textit{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.