Exponential power series, Galois module structure and differential modules

TL;DR

This paper introduces new p-adic exponential power series to construct self-dual normal basis generators for specific abelian extensions of p-adic fields, linking Galois module structures with differential modules.

Contribution

It develops a novel method using over-convergent p-adic exponential power series to explicitly construct basis generators and analyze Galois module structures in ramified extensions.

Findings

Constructed self-dual normal basis generators for certain abelian extensions.

Established a one-to-one correspondence between projective space of residue field and extension set.

Extended exponential power series to differential modules with Frobenius structure.

Abstract

We use new over-convergent p-adic exponential power series, inspired by work of Pulita, to build self-dual normal basis generators for the square root of the inverse different of certain abelian weakly ramified extensions of an unramified extension K of Qp. These extensions, whose set we denote by M, are the degree p subextensions over K of Mp,2, the maximal abelian totally, wildly and weakly ramified extension of K, whose norm group contains p. Our construction follows Pickett's, who dealt with the same set M of extensions of K, but does not depend on the choice of a basis of the residue field k of K. Instead it furnishes a one-to-one correspondence, commuting with the action of the Galois group of K/Qp, from the projective space of k onto M. We describe very precisely the norm group of the extensions in M. When K is not equal to Qp, their compositum Mp,2 yields an interesting example of non abelian weakly ramified extension of Qp, with Galois group isomorphic to a wreath product. Finally we show that, with a slight modification, our over-convergent exponential power series endow certain differential modules with a Frobenius structure, generalising a result of Pulita. Unfortunately, they then lose the property we need to build self-dual normal basis generators, hence the desirable link between Galois module structure and differential modules is not yet obtained.