The Grunwald problem and specialization of families of regular Galois extensions

TL;DR

This paper explores how specializations of regular Galois extensions over number fields can be controlled to solve Grunwald problems, extending previous work on local behavior prescriptions.

Contribution

It generalizes prior results by providing a partial solution to Grunwald problems through specializations of families of regular Galois extensions.

Findings

Extended the understanding of local behavior in specializations

Provided a partial solution to Grunwald problems

Connected specialization techniques with Galois extension constructions

Abstract

We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular Galois extension can be prescribed has been investigated by D\`ebes and Ghazi in the unramified case, and by Legrand, Neftin and the author in general. Here, we generalize these results and give a partial solution to Grunwald problems using Galois extensions arising as specializations of a family of regular Galois extensions.