Finitely generated pro-\(p\)-groups as Galois groups of maximal \(p\)-extensions of function fields over \(\mathbb{Q}_ q\) (Q1922565)

Let \(p\) and \(q\) be two different rational primes. This article completely characterizes those (topologically) finitely generated abelian pro-\(p\)-groups which occur as Galois groups of the maximal abelian \(p\)-extension of some field of transcendence degree one over the field \(\mathbb{Q}_q\) of \(q\)-adic numbers. It is shown that the structure of these groups depends only on the algebraic properties of \(\mathbb{Q}_q\), i.e. the result is the same if \(\mathbb{Q}_q\) is replaced by any \(q\)-adically closed field. The article concludes with explicit constructions of fields realizing the groups in question.