The Vojta conjecture implies Galois rigidity in dynamical families

TL;DR

This paper demonstrates that assuming the Vojta conjecture leads to Galois rigidity in quadratic polynomial families, with implications for the distribution of prime divisors in orbits, connecting deep conjectures to dynamical Galois properties.

Contribution

It establishes a link between the Vojta conjecture and Galois rigidity in quadratic dynamical families, providing new insights into arboreal Galois representations and prime divisor distributions.

Findings

Vojta conjecture implies Galois surjectivity conditions

Proves prime divisors of orbits have density zero under assumptions

Connects uniformity over $\

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.