Holomorphic differentials of certain solvable covers of the projective line over a perfect field

Abstract

We provide a Boseck‐type basis of the space of holomorphic differentials for a large class of solvable covers of the projective line with perfect field of constants of characteristic p>0. Within this class, we also describe the Galois module structure of holomorphic differentials for abelian covers.

$p$-group Galois covers of curves in characteristic $p$

“…For example, connected étale Z/pcovers do not have a global standard form (cf. [17,Subsection 3.1]). We present another example in Subsection 7.4.…”

“…• y p−1 n ∈ k(X) is a magical element for π. The notion of a global standard form of a p-group cover appeared already in [17], where it was used to construct a basis of holomorphic differentials of X in some cases. 8.1.…”

“…[14], [19], [16], [5]) or focus on some special groups (see e.g. [24] for the case of cyclic groups, [17] for abelian groups or [1] for groups with a cyclic Sylow subgroup). Even less is known about the de Rham cohomology of X.…”

Galois structure of the holomorphic differentials of curves

“…In [39], the case when G is an elementary abelian p-group and X/G ∼ = P 1 k was studied. In [33], solvable groups G were considered under some additional assumptions on the ramification of the cover X → X/G. In [23] and, independently, in [37], the case when G is arbitrary and the cover X → X/G is tamely ramified was analyzed.…”

“…We would like to point out that the determination of the kG-module structure of H 0 (X, Ω ⊗m X ) for m ≥ 1 can be divided into two basic approaches. The authors of [24,25,33,44] have used so-called Boseck bases and Artin-Schreier extensions. In contrast, the authors of [5] have taken a more geometric approach, decomposing the cover X → X/G into a wildly ramified cover followed by a tamely ramified cover and treating the wildly ramified cover by working locally and applying the results of [37] to the tamely ramified cover.…”