On the necessity of new ramification breaks
by
Griff Elder
University of Nebraska at Omaha
Coauthors: Nigel Byott

Let N/K be a Galois, local number field extension with K a finite extension of the p-adic numbers Q_p. Ideals of the ring of integers of N provide interesting, canonical modules over the group ring Z_p[G], where Z_p is the p-adic integers and G=Gal(N/K). When N/K is wildly ramified (which is often the case), it becomes apparently quite difficult to say much about these modules that is of a concrete, explicit nature.

Still there are a few instances where such descriptions are available. Considering these cases, one might be struck by the observation that ramification invariants have been sufficient for these descriptions precisely when there have been a maximal number of breaks in the ramification filtration of G=Gal(N/K).

What happens when the number of breaks is not maximal? Apparently, other information is required. What is the nature of this information? We propose a new ramification filtration involving truncated exponentiation; prove that these new filtrations contain a maximal number of new breaks; show that the new breaks contain the required 'other information' and that more generally these new breaks will be necessary for our understanding of the Galois module structure of ideals.

Date received: April 27, 2004