A remark on Artin-Schreier curves whose Hasse-Witt maps are the zero maps (Q1177329)

Let \(X\) be a complete smooth curve over an algebraically closed field \(k\). We assume that \(\hbox{char}(k)=p\) and we let \(F:{\mathcal O}_ X\to {\mathcal O}_ X\) be the Frobenius morphism, \(F(\alpha)=\alpha^ p\). There is an induced functorial action (called the ``Haase-Witt map'') of \(F\) on the cohomology group \(H^ 1(X,{\mathcal O}_ X)\); the dimension of the semi-simple part \(H^ 1(X,{\mathcal O}_ X)_ s\subset H^ 1(X,{\mathcal O}_ X)\) is denoted by \(\sigma (X)\). One knows that \(\sigma (X)\) is also equal to the \(p\)-rank of the Jacobian of \(X\). Suppose now that \(X\) is not isomorphic to \(\mathbb{P}^ 1\) but does possess a \(p\)-cyclic covering of \(\mathbb{P}^ 1\). One calls \(X\) an `` Artin-Schreier curve''. Classical results allow one immediately to compute \(\sigma (X)\). In the paper under review the authors discuss the rank of the total Hasse-Witt map and give a criterion for it to vanish.