Mass formulas for local Galois representations and quotient singularities II: dualities and resolution of singularities

TL;DR

This paper investigates dualities in total mass formulas for local Galois representations, exploring their connection to singularity resolutions and the wild McKay correspondence through examples and theoretical analysis.

Contribution

It extends the understanding of dualities in total mass formulas and links them to simultaneous resolution of singularities using advanced geometric and representation-theoretic tools.

Findings

Duality of total masses observed in specific cases

Relation between dualities and simultaneous resolution of singularities

Examples illustrating the theoretical connections

Abstract

A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper, the authors observed that in a particular example, two total masses coming from two different weightings are dual to each other. We discuss the problem how general such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincar\'e duality for stringy invariants. We also exhibit several examples.