  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.
