Given a field \(F\), a finite group \(G\) is called \(F\)-admissible if there is a \(G\)-Galois extension \(E/F\) such that \(E\) is a maximal subfield of some central division \(F\)-algebra [see M. M. Schacher, J. Algebra 9, 451–477 (1968; Zbl 0174.34103)]. There has been a vast literature on this subject; see, for example, the references of the paper under review. In fact, many papers are devoted to the situation when \(F\) is a global field. Among these results, Schacher proved that, if \(G\) is \(Q\)-admissible, then every \(p\)-Sylow subgroup of \(G\) is meta-cyclic, i.e., a \(p\)-group generated by two cyclic subgroups such that one of them is a normal subgroup.
This paper considers the case when \(F\) is an algebraic function field of one variable over a field \(K\), which is complete discrete-valued with algebraically closed residue field \(k\). In this case, if \(G\) is a finite group with \(\gcd \{\text{char}\,k, |G| \}=1\), the main theorem of this paper is that, \(G\) is \(F\)-admissible if and only every \(p\)-Sylow subgroup of \(G\) is bicyclic, i.e., abelian and meta-cyclic.
The proof of the forward direction (“\(F\)-admissible” \(\Rightarrow\) “bicyclic”) is to show first that every Sylow subgroup of \(G\) is meta-cyclic (Theorem 3.3), which is analogous to Schacher’s proof for \(Q\)-admissibility. Once the base field \(F\) has enough roots of unity, then the bicyclic property will follow (see Lemma 3.1). But the authors prove more; the interested reader should consult Theorem 3.3 and its various consequences, e.g. Corollaries 3.4, 3.6, 3.7, besides the main application Proposition 3.5.
In the proof of Theorem 3.3, it is important to have a central division \(F\)-algebra \(D\) which realizes the \(F\)-admissibility for \(G\) and satisfies the index-period property for \(D\), an assumption reminiscent of the situations of global or local fields.
To prove the converse direction, take a suitable integral model of \(F\). More precisely, let \(T\) be a complete DVR with quotient field \(K\), \(\hat{X}\) be a regular projective \(T\)-curve with function field \(F\), \(X\) the closed fiber of \(\hat{X}\) such that the reduced irreducible components of \(X\) are regular. It is possible to find a finite morphism \(f:\hat{X} \rightarrow P^1_T\) so that \(f^{-1}(\infty)\) contains all the singular points of \(X\) (and possibly some other points) where \(\infty \in P^1_k\).
Suppose that all Sylow subgroups of \(G\) are bicyclic. Consider first \(G\) is a \(p\)-group of the form \(C_q \times C_{q'}\). Take a regular point \(Q \in X\), \(R_Q\) the local ring of \(\hat{X}\) at \(Q\), \(\hat{R_Q}\) its completion. Choose suitable generators \(f, t\) for the maximal ideal of \(\hat{R_Q}\). With the aid of Lemma 4.3 of this paper, it is not difficult to construct a central division \(F\)-algebra with a maximal subfield which is Galois over \(F\) with group isomorphic to \(C_q \times C_{q'}\).
In general, if \(p_1, \dots, p_r\) are all the prime numbers dividing \(|G|\) and \(P_1, \dots, P_r\) the corresponding Sylow subgroups, choose distinct regular points \(Q_1, \dots, Q_r\) as above. The difficulty is how to globalize these data. The key idea is the patching method [see D. Harbater and J. Hartmann, Isr. J. Math. 176, 61–107 (2010; Zbl 1213.14052), D. Harbater, Number theory, Semin. New York 1984/85, Lect. Notes Math. 1240, 165–195 (1987; Zbl 0627.12015)]. The patching method is summarized in Theorem 4.1. Lemma 4.2 is the version which is used in the present situation. The converse direction is proved in Proposition 4.4 and Theorem 4.5.