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Let K be a number field, and let λ(x, t) ∈ K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of α ∈ K for which the specialized polynomial λ(x, α) is K-reducible. We apply this to show that for any fixed n ≥ 10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L (t) n (x) are K-irreducible and have Galois group S n . In conjunction with the theory of complex multiplication, we also show that for any K and for any n ≥ 53, all but finitely many of the K-specializations of the modular equation Φ n (x, t) are K-irreducible and have Galois group containing P SL 2 (Z/n).

Abstract. Let N (d, G, X) be the number of degree d number fields K with Galois group G and whose discriminant D K satisfies |D K | ≤ X. Under standard conjectures in diophantine geometry, we show that N (4, A 4 , X) X 2/3+ , and that there are N 3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S 4 , confirming a question of Gallagher. Unconditionally we have N (4, A 4 , X) X 5/6+ , and that the 2-class groups of almost all Abelian cubic fields k have size D 1/3+ k . The proofs depend on counting integral points on elliptic fibrations.

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We show that two surjective *-adic Galois representations which are *-adically close near the supersingular primes are equivalent up to a twist and a standard automorphism of GL n . In particular, two elliptic curves over a number field which are locally twist of each other in fact differ by a global twist. The proof depends on determining the automorphisms of PGL n over a complete local ring. 1999Academic Press