Automorphisms and holomorphic differentials in characteristic p

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

“…(This is not necessary for the proof of Proposition , which gives the decomposition of in terms of indecomposable ‐modules.) This form was mentioned by Valentini and Madan in , who gave it when the field of constants is algebraically closed; we include the analogous result for when k is only assumed to be perfect. Corrollary Let K be any function field of characteristic with perfect field of constants , and let be an Artin–Schreier extension.…”

“…Our next stated result (Lemma ) is the natural extension of Lemma to generalised Artin–Schreier extensions. One may compare this to [, Lemma 2]; we remark that Lemma differs slightly from this result, as the generators of unramified steps in the tower do not appear, so that we only need to require the weak standard form given by Lemma . This is done in order to employ a version of [, Theorem 1] over a constant field which is only assumed to be perfect.…”

“…Remark For the results of this section, it is not necessary to construct this basis in terms of an action of a generator of a cyclic group , nor is it necessary to assume that the field of constants k is algebraically closed, which is particularly important for situations when the constant field is finite. The basis is defined completely in terms of the ramification data and those valuations arising from global standard form.…”

“…Boseck bases may also be used to understand the ‐module structure of the k ‐vector space of holomorphic differentials of the field L . The description of the ‐module structure in terms of the group structure of G was first done by Valentini and Madan and later generalised by Rzedowski–Calderón et al. .…”

“…§4 gives the k ‐basis of for a global standard function field L . In §5, we give examples of the construction of solvable towers with global standard form, and we employ [, Theorem 7] and [, Theorem 1] to generalise some cases of known results on cyclic groups using the explicit basis of Theorem and a weak standard form (which is stronger than local standard form, but weaker than global standard form). In §6, we give the Galois module structure when L is an abelian cover exhibiting global standard form, following an argument of Rzedowski–Calderón et al.…”