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Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions

Melanie Machett Wood, Takehiko Yasuda

Abstract

We study a relation between the Artin conductor and the weight coming from the motivic integration over wild Deligne-Mumford stacks. As an application, we prove some version of the McKay correspondence, which relates Bhargava's mass formula for extensions of a local field and the Hilbert scheme of points.

Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions

Abstract

We study a relation between the Artin conductor and the weight coming from the motivic integration over wild Deligne-Mumford stacks. As an application, we prove some version of the McKay correspondence, which relates Bhargava's mass formula for extensions of a local field and the Hilbert scheme of points.

Paper Structure

This paper contains 17 sections, 24 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Convention
  3. Acknowledgements
  4. Total masses and the Artin conductor
  5. Total masses and counting functions
  6. The Artin conductor
  7. The Artin conductor as a counting system
  8. Bhargava's mass formula
  9. Weights
  10. A comparison of the Artin conductor and the weight
  11. The tame case
  12. Permutation representations
  13. The McKay correspondence
  14. The Grothendieck ring of varieties and the point counting realization
  15. The tame case
  16. ...and 2 more sections

Key Result

Theorem 1.2

The conjecture above holds for $Y=H_{\mathcal{O}_{K}}$, $X=S^{n}\mathbb{A}_{\mathcal{O}_{K}}^{2}$ and $\Gamma=S_{n}$.

Theorems & Definitions (64)

  • Conjecture 1.1: Conjecture \ref{['conj: McKay points']}
  • Theorem 1.2: Theorem \ref{['thm: Hilb']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 54 more