The de Rham cohomology of covers with cyclic $p$-Sylow subgroup
Jędrzej Garnek, Aristides Kontogeorgis
TL;DR
The article extends Chevalley–Weil type results to characteristic p by focusing on curves with group actions whose p-Sylow subgroup is cyclic. By combining modular representation theory for cyclic p-Sylow subgroups with ramification data from the cover X -> X/G and induction techniques, it derives explicit decompositions of H^1_{dR}(X) as a k[G]-module, notably for G ≅ Z/p^n and for p-Sylow subgroups of order p that are normal. It also treats the case of p-subgroups with nontrivial semidirect actions, providing a concrete metacyclic example that yields explicit formulas for the G-structure in terms of ramification invariants. The results enable Chevalley–Weil–type formulas in characteristic p for a broad class of covers and pave the way for effective computations of de Rham cohomology under cyclic p-Sylow actions.
Abstract
Let $X$ be a smooth projective curve over a field $k$ with an action of a finite group $G$. A well-known result of Chevalley and Weil describes the $k[G]$-module structure of cohomologies of $X$ in the case when the characteristic of $k$ does not divide $\# G$. It is unlikely that such a formula can be derived in the general case, since the representation theory of groups with non-cyclic $p$-Sylow subgroups is wild in characteristic $p$. The goal of this article is to show that when $G$ has a cyclic $p$-Sylow subgroup, the $G$-structure of the de Rham cohomology of $X$ is completely determined by the ramification data. In principle, this leads to new formulas in the spirit of Chevalley and Weil for such curves. We provide such an explicit description of the de Rham cohomology in the cases when $G = \mathbb Z/p^n$ and when the $p$-Sylow subgroup of $G$ is normal of order $p$.