The group of automorphisms of cyclic extensions of rational function fields
Let \(K = k(t)\) be a rational function field over an algebraically closed field of characteristic \(p \geq 0\). We have that the group of \(k\)-automorphisms of \(K\), \(\Aut_k K\) is isomorphic to \(PGL(2,k)\) the projective linear group. In his Ph.D. dissertation, \textit{R. Brandt} [Essen: Univ. (Gesamthochschule) Essen, Fachbereich Mathematik, Diss. (1988; Zbl 0716.11058)] studied the group of automorphisms of function fields \(F\) which are cyclic extensions of \(K\) with \(\text{Gal}(F/K) \cong C _q\) the cyclic group of prime order \(q \neq p\). In the paper under review the author studies function fields \(F\) with \(k\)-automorphism groups \(\Aut_k F \cong G\) which are cyclic extensions of order \(n\) of \(K\). It is assumed that \(p\) does not divide \(n\), that all the ramified places in \(F/K\) are fully ramified and that \(\text{Gal}(F/K)\) is normal in \(G\). This last condition is met when the genus of \(F\), \(g_F\), is larger than \((n-1) ^2\) or equivalently, when the number of ramified places, \(s\), in \(F/K\) satisfies \(s > 2 n\). The finite subgroups of \(PGL(2,k)\) are described by \textit{R. Valentini} and \textit{M. Madan} in [J. Reine Angew. Math. 318, 156-177 (1980; Zbl 0426.12016)]. In that paper, Valentini and Madan found the ramification type in each possible case. Using this description and the ramification type, the author determines the structure of \(G\) in term of generators and relations when \(G/C _n\) is isomorphic to \(C _n, D _n, A _4, A _5, S _4\) or a semidirect product of an elementary abelian group by a cyclic one, and in terms of cohomology class of the group extension in all other cases. It turns out that in most cases the ramification type in \(F/K\) and in \(K/F ^G\) determines the group structure of the extension \(G\). However there are cases of function fields with the same ramification type but different automorphism groups. It is also proved that for every finite subgroup of \(PGL(2, k)\) there is a cyclic Kummer extension \(F\) which realizes every possible ramification type. It is also provided a method to write down an equation \(y ^n = f(x)\) realizing every possible automorphism group.
scientific article; zbMATH DE number 1340596