Newly reducible iterates in families of quadratic polynomials

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We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, fⁿ(x) irreducible over K but fⁿ⁺¹(x) reducible over K, where fⁿ denotes the nth iterate of f. For each choice of critical point γof f(x), we consider the family gᵧ,ₘ(x)= (x - γ)² + m + γ, m ∈ K. For fixed n ≥ 3 and nearly all values of γ, we show that there are only finitely many m such that gᵧ,ₘ has a newly reducible nth iterate. For n = 2 we show a similar result for a much more restricted set of γ. These results complement those obtained by Danielson and Fein in the higher-degree case. Our method involves translating the problem to one of finding rational points on certain hyperelliptic curves, determining the genus of these curves, and applying Faltings' theorem.