Unramified Galois coverings of algebraic curves
Let \(X\) be a complete nonsingular irreducible algebraic curve of genus \(g \) defined over an algebraically closed field \(k\) of characteristic \(p > 0\). Let \(\pi_1 (X)\) be its algebraic fundamental group, and \(\Gamma_g\) the topological fundamental group of (any) compact Riemann surface of genus \(g\). It is well known since Grothendieck, that a finite group \(G\) of order prime to \(p\) can be realized as the Galois group of an unramified cover of \(X\) if and only if it is a quotient of \(\Gamma_g\). Other information on the structure of \(\pi_1 (X)\) comes from the Hasse-Witt invariant of \(X\), i.e. the \(\mathbb{F}_p\)-dimension of the \(p\)-torsion subgroup of the Jacobian variety of \(X\). In this paper the author counts the number of unramified Galois coverings of \(X\) whose Galois group is isomorphic to an extension of a group \(G\) of order prime to \(p\) by a finite group \(H\), which is an irreducible \(\mathbb{F}_p [G]\)-module. This counting depends on some invariants attached to a Galois cover \(Y \to X\) of group \(G\), called generalized Hasse-Witt invariants, arising in the canonical decomposition of the \(p\)-torsion space of the Jacobian variety \(J_Y\) of \(Y\). -- Some particular cases had already been treated by \textit{S. Nakajima} [in: Galois groups and their representations, Proc. Symp., Nagoya 1981, Adv. Stud. Pure Math. 2, 69-88 (1983; Zbl 0529.14016)] and \textit{H. Katsurada} [J. Math. Soc. Japan 31, 101-125 (1979; Zbl 0401.14004)], whose results are generalized in the present paper.
scientific article; zbMATH DE number 798698
scientific article