Equivariant form of the Deuring-Šafarevič formula for Hasse-Witt invariants
Let \(\pi: X\to Y\) be a finite Galois covering of complete non-singular curves over an algebraically closed field k of characteristic p, and let \(\tau_ X\) and \(\tau_ Y\) be the Hasse-Witt invariants of X and Y respectively. We assume \(G=Gal(X/Y)\) is a p-group. The Deuring- Šafarevič formula expresses \(\tau_ X\) by using \(\tau_ Y\), \(| G|\) and the ramification indices of \(\pi\). Recall \(\tau_ X=_{k} \Omega_ X(O)^ s,\) where \(\Omega_ X(O)\) is the space of holomorphic differentials and \(\Omega_ X(O)^ s\) is its ''semi-simple'' part with respect to the Cartier operator. Equivariant form of the Deuring-Šafarevič formula means determining \(\Omega_ X(O)^ s\) as k[G]-module, not only its dimension over k. In the paper we treat a more general problem. Namely, we consider differentials with poles, and give the k[G]-module structure of \(\Omega_ X(-D)^ s\) for G-invariant effective divisors D on X, where \(\Omega_ X(-D)^ s\) is the ''semi- simple'' part of \(\Omega_ X(-D)=\{\omega | (\omega)\geq -D\}.\) When \(\pi\) is unramified, we can also determine the k[G]-module structure of the ''nilpotent'' part of \(\Omega_ X(-D)\).
Galois covering of complete non-singular curves
scientific article; zbMATH DE number 3891521