The Vojta conjecture implies Galois rigidity in dynamical families

Hindes, Wade

doi:10.1090/proc/12877

The Vojta conjecture implies Galois rigidity in dynamical families
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Proc. Amer. Math. Soc. 144 (2016), 1931-1938 Request permission

Abstract:

We show that the Vojta (or Hall-Lang) conjecture implies that the arboreal Galois representations in a 1-parameter family of quadratic polynomials are surjective if and only if they surject onto some finite and uniform quotient. As an application, we use the Vojta conjecture, our uniformity theorem over $\mathbb {Q}(t)$, and Hilbert’s irreducibility theorem to prove that the prime divisors of many quadratic orbits have density zero.

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