The crystalline cohomology of covers with a cyclic $p$-Sylow subgroup

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 $\# G$. In case when $G$ has a cyclic $p$-Sylow subgroup, it is known that the $G$-structures of the module of holomorphic differentials and of de Rham cohomology of $X$ are completely determined by the ramification data of the cover $X \to X/G$. In this article we extend this result to the crystalline cohomology of $X$. Also, we provide an explicit description of the structure of the crystalline cohomology when $G = \mathbb Z/p^n$. The main used tool is the theory of Yakovlev diagrams - algebraic objects that classify the $G$-modules over Witt vectors.