Holomorphic differentials of Klein four covers

Abstract

Let k be an algebraically closed field of characteristic two, and let G be isomorphic to Z/2×Z/2. Suppose X is a smooth projective irreducible curve over k with a faithful G-action, and assume that the cover X→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 H0(X,ΩX/k) of holomorphic differentials of X over k. In the case when X/G=Pk1, we completely determine the decomposition of H0(X,ΩX/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.