Holomorphic differentials of Klein four covers

Abstract

Let $k$ be an algebraically closed field of characteristic two, and let $G$ be isomorphic to $\mathbb{Z}/2\times \mathbb{Z}/2$. Suppose $X$ is a smooth projective irreducible curve over $k$ with a faithful $G$-action, and assume that the cover $X\to X/G$ is totally ramified, in the sense that it is ramified and every branch point is totally ramified. We study to what extent the lower ramification groups of the closed points of $X$ determine the isomorphism types of the indecomposable $kG$-modules and the multiplicities with which they occur as direct summands of the space ${\mathrm{H}}^0(X,{\Omega }_{X/k})$ of holomorphic differentials of $X$ over $k$. In the case when $X/G={\mathbb{P}}_k^1$, we completely determine the decomposition of ${\mathrm{H}}^0(X,{\Omega }_{X/k})$ into a direct sum of indecomposable $kG$-modules. Moreover, we show that the isomorphism classes of indecomposable $kG$-modules that actually occur as direct summands belong to an infinite list of non-isomorphic indecomposable $kG$-modules that contain modules of arbitrarily large $k$-dimension. In particular, our results show that [14, Theorem 6.4] is incorrect.

25 pages; changed name of covered curve from $U$ to $Y$, fixed statement of Theorem 1.1, fixed typo in Example 2.19; added explanation why [14, Theorem 6.4] is incorrect in general (see Remark 2.21)