Powers of the cartier operator on Artin-Schreier covers (Q6594436)

Let \(k\) be an algebraically closed field of positive characteristic \(p\), and let \(\pi:Y\to X\) be a finite morphism of smooth, projective, geometrically connected curves over \(k\) that is generically Galois with Galois group \(G\), branched at a finite set \(S \subset X(k)\). \NIn this paper, the author focuses on the case when \(G\) is cyclic of order \(p\), i.e. when the cover \(\pi\) is an Artin-Schreier cover. Let \(\sigma\) be the Frobenius morphism of \(k\). The Cartier operator \(V_X\) is a \(\sigma^{-1}\)-linear operator on \(\mathrm{H}^0(X,\Omega_X^1)\), and the \(a\)-number \(a_X\) is the dimension of the kernel of \(V_X\). \N\N\textit{J. Booher} and \textit{B. Cais} used a sheaf-theoretic approach in [Algebra Number Theory 14, No. 3, 593--653 (2020; Zbl 1454.14080)], to give bounds on the \(a\)-numbers of Artin-Schreier covers. \NIn the paper under review, the authors generalizes their approach to arbitrary powers of the Cartier operator, yielding bounds for the dimensions of the kernels of these powers. These bounds give new restrictions on the Ekedahl-Oort type of Artin-Schreier covers.