On refined metric and hermitian structures in arithmetic, I: Galois-Gauss sums and weak ramification

TL;DR

This paper develops a unified approach to Galois structures in arithmetic using algebraic K-theory, leading to new results and conjectures about wildly ramified Galois-Gauss sums.

Contribution

It introduces a common refinement of metrized and hermitian Galois structures and applies it to derive new arithmetic properties and conjectures related to Galois-Gauss sums.

Findings

New results on wildly ramified Galois-Gauss sums

Framework for future conjectures in arithmetic Galois theory

Refinement of existing Galois structure theories

Abstract

We use techniques of relative algebraic K-theory to develop a common refinement of the existing theories of metrized and hermitian Galois structures in arithmetic. As a first application of this very general approach, we then use it to prove several new results, and to formulate a framework of new conjectures, concerning the detailed arithmetic properties of wildly ramified Galois-Gauss sums.