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Nonarchimedean and motivic stationary phase formulas

Téofil Adamski

TL;DR

This work analyzes stationary phase phenomena over nonarchimedean local fields, establishing a detailed p-adic stationary phase formula for nondegenerate critical points. It then constructs a motivic analogue via Cluckers–Loeser motivic integration, including a motivic Morse lemma and an explicit motivic stationary phase formula. The results unify real and p-adic oscillatory analysis within the motivic framework, enabling uniform treatments across fields and potential applications to motivic wavefront sets. The approach leverages a nonarchimedean Morse lemma, Fourier-analytic tools, and the formalism of motivic exponential functions to express oscillatory integrals in a way that is definable and transferable between settings.

Abstract

In this article, for a non degenerate singular phase, we reconsider a stationary phase formula of Heifetz in the nonarchimedean local field setting and give a motivic analogue using Cluckers-Loeser's motivic integration.

Nonarchimedean and motivic stationary phase formulas

TL;DR

This work analyzes stationary phase phenomena over nonarchimedean local fields, establishing a detailed p-adic stationary phase formula for nondegenerate critical points. It then constructs a motivic analogue via Cluckers–Loeser motivic integration, including a motivic Morse lemma and an explicit motivic stationary phase formula. The results unify real and p-adic oscillatory analysis within the motivic framework, enabling uniform treatments across fields and potential applications to motivic wavefront sets. The approach leverages a nonarchimedean Morse lemma, Fourier-analytic tools, and the formalism of motivic exponential functions to express oscillatory integrals in a way that is definable and transferable between settings.

Abstract

In this article, for a non degenerate singular phase, we reconsider a stationary phase formula of Heifetz in the nonarchimedean local field setting and give a motivic analogue using Cluckers-Loeser's motivic integration.

Paper Structure

This paper contains 17 sections, 21 theorems, 108 equations.

Table of Contents

  1. The nonarchimedean local field case
  2. Integration and Fourier transform
  3. Setting and integration
  4. Fourier transform
  5. A Morse lemma
  6. The nonarchimedean stationary phase formula
  7. The motivic case
  8. A quick reminder about motivic integration
  9. Cluckers--Loeser's motivic integration
  10. Cluckers--Halupczok's evaluation
  11. Definable constants and open definable subassignments
  12. Definable constants
  13. Open definable subassignments
  14. Globally analytic functions and critical points
  15. A motivic Morse lemma
  16. ...and 2 more sections

Key Result

Theorem 1.1

The Fourier transform $\varphi \longmapsto \hat{\varphi}$ induces a $\mathbf{C}$-linear automorphism on the space $\mathscr{S}(\mathbf{K}^n)$. More precisely, for all Schwartz-Bruhat function $\varphi$ on $\mathbf{K}^n$ and for all element $x \in \mathbf{K}^n$, we have the equality

Theorems & Definitions (55)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • proof
  • Lemma 1.6: CluckersHerremans
  • Lemma 1.7
  • proof
  • Corollary 1.8
  • ...and 45 more