Upper ramification jumps in abelian extensions of exponent p

TL;DR

This paper classifies the possible upper ramification jumps in elementary abelian p-extensions of p-adic fields, extending previous results by computing ramification filtrations and applying class field theory.

Contribution

It generalizes earlier work by removing the assumption that the base field contains a primitive p-th root of unity, providing a comprehensive classification of ramification jumps.

Findings

Computed ramification filtration for maximal elementary abelian p-extension

Derived necessary and sufficient conditions for upper ramification jumps

Extended classification to broader class of p-adic field extensions

Abstract

In this paper we present a classification of the possible upper ramification jumps for an elementary abelian p-extension of a p-adic field. The fundamental step for the proof of the main result is the computation of the ramification filtration for the maximal elementary abelian p-extension of the base field K. This is a generalization of a previous work of the second author and Dvornicich where the same result is proved under the assumption that K contains a primitive p-th root of unity. Using the class field theory and the explicit relations between the normic group of an extension and its ramification jumps, it is fairly simple to recover necessary and sufficient conditions for the upper ramification jumps of an elementary abelian p-extension of K.