Embedding problems with bounded ramification over function fields of positive characteristic (Q2662936)

Assume that \(K/K_0\) is a finite Galois extension, where \(K_0\) is an algebraic function field in one variable over a field \(F\) of characteristic \(p > 0\) (i.e. \(K_0/F\) is a finitely-generated regular extension of transcendence degree \(1\)). The paper under review deals with the study of the finite embedding problem \[ 1 \to H \to G \to\mathrm{Gal}(K/K_0) \to 1\tag{1} \] with a solvable kernel \(H\) along the lines drawn by the authors in [Proc. Lond. Math. Soc. (3) 117, No. 1, 149--191 (2018; Zbl 1451.11119); J. Lond. Math. Soc., II. Ser. 100, No. 1, 323--340 (2019; Zbl 1473.11206)], in the case where \(K_0\) is a global field (possibly, a number field in the former paper). For convenience of the reader, the authors state in the paper under review (as Theorem A) a combination of the main results of the cited earlier papers. The main result of the present paper is stated as Theorem B. It shows that if \(F\) is a Hilbertian field, then the embedding problem (1) is properly solvable whenever \(H\) is a \(p\)-group. Moreover, the solution can be chosen to locally coincide with finitely many, given in advance, weak local solutions. Finally, and this is the main point of the present article, the number of prime divisors of \(K_0 = F\) that ramify in the solution field is bounded by the number of prime divisors of \(K_0\) trivial on \(F\) that ramify in \(K\) plus the length of the maximal \(G\)-invariant sequence of subgroups of \(H\). The proof of Theorem B is a modification of that part of the authors' proof of Theorem A (in their joint paper of 2019 (loc. cit.)) in which \(H\) is a simple \(p\)-module over the absolute Galois group of \(K_0\). The authors point out the main changes made in the proof of Theorem B and explain the reasons for these changes.