Specializations of one-parameter families of polynomials

Let K be a number field, and suppose λ(x,t)∈K[x,t] is irreducible over K(t). Using algebraic geometry and group theory, we describe conditions under which the K-exceptional set of λ, i.e. the set of α∈K for which the specialized polynomial λ(x,α) is K-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed n≥10, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L n (t) (x) are K-irreducible and have Galois group S n . Second, we study specializations of the modular polynomial Φ n (x,t) (which vanishes on the j-invariants of pairs of elliptic curves related by a cyclic n-isogeny), and show that for any n≥53, all but finitely many of the K-specializations of Φ n (x,t) are K-irreducible and have Galois group containing SL 2 (ℤ/n)/{±I}. Third, for a simple branched cover π:Y→ℙ K 1 of degree n≥7 and of genus at least 2, all but finitely many K-specializations are K-irreducible and have Galois group S n