Specializations of one-parameter families of polynomials. (English) Zbl 1160.12004

Summary: Let \(K\) be a number field, and suppose \(\lambda(x,t)\in K[x,t]\) is irreducible over \(K(t)\). Using algebraic geometry and group theory, we describe conditions under which the \(K\)-exceptional set of \(\lambda\), i.e. the set of \(\alpha\in K\) for which the specialized polynomial \(\lambda (x,\alpha)\) is \(K\)-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed \(n\geq 10\), 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 \(\Phi_n(x,t)\) (which vanishes on the \(j\)-invariants of pairs of elliptic curves related by a cyclic \(n\)-isogeny), and show that for any \(n\geq 53\), all but finitely many of the \(K\)-specializations of \(\Phi_n(x,t)\) are \(K\)-irreducible and have Galois group containing \(\text{SL}_2 (\mathbb Z/n)/\{\pm I\}\). Third, for a simple branched cover \(\pi: Y\to\mathbb P_K^1\) of degree \(n\geq 7\) and of genus at least 2, all but finitely many \(K\)-specializations are \(K\)-irreducible and have Galois group \(S_n\).


12H25 \(p\)-adic differential equations
11C08 Polynomials in number theory
11G15 Complex multiplication and moduli of abelian varieties
11R09 Polynomials (irreducibility, etc.)
14H25 Arithmetic ground fields for curves
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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