Jump sets in local fields

TL;DR

This paper introduces a combinatorial approach using jump sets to classify the structure of principal units in local fields, linking ramification theory and automorphism groups through a mass formula.

Contribution

It develops a new parametrization of unit groups via jump sets and establishes a bijection with automorphism orbit spaces, providing explicit invariants and a conceptual proof of wild character jump classifications.

Findings

Parametrization of principal units using jump sets.

A mass formula for unit filtrations.

Explicit invariants linking Eisenstein polynomials and ramification.

Abstract

We show how to use the combinatorial notion of jump sets to parametrize the possible structures of the group of principal units of local fields, viewed as filtered modules. We establish a natural bijection between the set of jump sets and the orbit space of a $p$-adic group of filtered automorphisms acting on a free filtered module. This, together with a Markov process on Eisenstein polynomials, culminates into a mass-formula for unit filtrations. As a bonus the proof leads in many cases to explicit invariants of Eisenstein polynomials, yielding a link between the filtered structure of the unit group and ramification theory. Finally, with the basic theory of filtered modules developed here, we recover, with a more conceptual proof, a classification, due to Miki, of the possible sets of upper jumps of a wild character: these are all jump sets, with a set of exceptions explicitly prescribed by the jump set of the local field and the size of its residue field.