Structure of semisimple differentials and p-class groups in ?p-extensions

Abstract

For a wildly ramified p-extension E/F of algebraic function fields of one variable in an algebraically closed field k of characteristic p with Galois Group G, Nakajima obtained two exact sequences which determined implicitly the structure of the holomorphic semisimple differentials as k[G]-module. In this paper, in many cases, e.g., if there is a fully ramified prime, the structure is determined explicitly. Analogous results are obtained for p-extensions of ℤp-fields of CM-type. In the latter situation, if E/F is unramified, the structure of the minus part of the p-class group of E is determined as ℤp[G]-module.