On Galois Extensions of Local Fields with a Single Wild Ramification Jump
Samuel Goodman
TL;DR
The article resolves the problem of counting $(-1,n)$ Galois extensions of a local field $K$ with a single wild ramification jump at $n$, for all but finitely many pairs $(p,e)$. By fixing a Lubin-Tate tower and applying Local Class Field Theory, it reduces the global counting to norm-subgroup data and fiberwise compatibilities governed by ramification jumps, formal logarithms, and partitions. It proves an explicit counting framework that yields concrete formulas depending on residue-field data (via $G$-invariant subspaces of the extended residue field and the trace-zero subspace) and shows the counts vanish when $p$ divides $n$, while giving precise positive formulas when $p mid n$ and $e\ge n$. The results include a clean special case for totally ramified bases, a complete treatment of the $n=2$, $p=2$ exceptional scenario, and a finite-set-of-exceptions statement ensuring broad applicability. Overall, the paper advances understanding of wild ramification structure and provides practical, residue-field–driven counts for such extensions.
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.