On Galois Extensions of Local Fields with a Single Wild Ramification Jump

TL;DR

This paper classifies and counts Galois extensions of local fields with a single wild ramification jump, revealing bounded degrees and extending classical results from tame to wild ramification cases.

Contribution

It provides a new enumeration method for wild ramification jumps in Galois extensions using Lubin-Tate theory and local class field theory.

Findings

Number of such extensions is finite and bounded for fixed parameters.

Extension degrees are bounded in the wild ramification case.

The method reconstructs norm subgroups from ramification data.

Abstract

For a given positive integer $n$ and $K/\mathbb{Q}_p$ a finite extension of ramification degree $e$, we determine the number of finite Galois extensions $L/K$ with inertia degree $f$ and a single nonnegative ramification jump at $n$ as long as $(p,e)$ is outside of a finite set. This builds upon the tamely ramified case, which is a classical consequence of Serre's Mass Formula, exhibiting a more restrictive behavior than in the tamely ramified case because the degrees of such extensions are bounded. We do this by working in a fixed Lubin-Tate extension and exploiting the surjectivity of a map corresponding to the ramification jump to reconstruct the $U^1$ part of the norm subgroup (coming from local class field theory) from its fibers and then by understanding how the fibers interact by studying them in terms of properties of the formal logarithm and partitions.