Given a field \(K\), the regular inverse Galois problem over \(K\) consists of showing that each finite group \(G\) is the Galois group of a regular extension of \(K(T)\), or, equivalently, in finding a \(G\)-cover of \(\mathbb{P}^1\) of group \(G\) defined over \(K\). The first result of the paper is a solution of the regular inverse Galois problem over the field \(\mathbb{Q}^{tp}\) of totally \(p\)-adic numbers for each prime \(p\). This generalizes a previous result of M. Fried and the author where the prime \(p\) was the prime \(p = \infty\), in which case “\(p\)-adic” should be understood as “real” and \(\mathbb{Q}^{tp}\) is denoted by \(\mathbb{Q}^{tr}\). Similar results have been independently obtained by F. Pop. Both our approach and his rely on patching and glueing techniques introduced by D. Harbater for formal analytic covers and revisited by Q. Liu from the rigid point of view. The Hurwitz space theory is the other important tool of our proof. The second result of the paper is a criterion for a \(G\)-cover a priori defined over \(\mathbb{Q}^{tr}\) and with \(\mathbb{Q}\)-rational branch points to be defined over \(\mathbb{Q}\). The extra condition “with \(\mathbb{Q} \)-rational branch points” forbids at the moment the combination of this result with the first one.
The second half of the paper is devoted to “local-global” questions. In his thesis, E. Dew conjectures that a \(G\)-cover defined over all completions of a number field \(K\) is necessarily defined over \(K\). It is proved here that Dew’s conjecture holds except possibly in a very special case coming from Grunwald’s theorem. This special case cannot occur if \(K = \mathbb{Q}\). If a \(G\)-cover (or a mere cover) is only defined over all but finitely many completions of \(K\), then only the field of moduli has to be equal to \(K\). The last result asserts that the converse holds as well: only for finitely many places \(v\) may the completion \(K_v\) of the field of moduli \(K\) not be a field of definition. This result also originated in a question of E. Dew. These two results are related to the classical obstruction to the field of moduli being a field of definition.
For the entire collection see [Zbl 0823.00012].