Let \(K\) be a field, \(S\) a finite set of independent, non-trivial discrete valuations of \(K\) and \(G\) a finite group. Given a family \((E_v/K_v)_{v \in S}\) of Galois extensions of the \(v\)-completions \(K_v\) of \(K\), a solution to the Grunwald problem is a Galois extension \(L/K\) such that for all \(v \in S\) its local extensions are isomorphic to \(E_v/K_v\). The main result of this paper (Theorem 1.2) considers the case that \(K\) is an algebraic number field. It shows that a Galois cover \(f: X \to \mathbb P^1\) with group \(G\) and some good-reduction condition imply that each unramified Grunwald problem has solutions which are specializations of \(f\) at points in \(\mathbb A^1(K)\) outside the branch divisor.
The authors deduce this result from a more general one (Theorem 3.2), which considers an arbitrary field \(K\) and a \(G\)-cover over a smooth projective \(K\)-variety with some further integrality conditions.
For the proof, the authors employ a “Twisting Lemma” (2.1) to obtain the result in the local situation (Proposition 2.2), and then globalize (Chapter 3).
The results have connections to Hilbert’s Irreducibility Theorem and the Regular Inverse Galois problem.