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A motivic Poisson formula for split algebraic tori with an application to motivic height zeta functions

Margaret Bilu, Loïs Faisant

Abstract

We prove a motivic version of the Poisson formula on the adelic points of a split algebraic torus and apply it to the study of the motivic height zeta function of split projective toric varieties, in the context of the motivic Manin-Peyre principle.

A motivic Poisson formula for split algebraic tori with an application to motivic height zeta functions

Abstract

We prove a motivic version of the Poisson formula on the adelic points of a split algebraic torus and apply it to the study of the motivic height zeta function of split projective toric varieties, in the context of the motivic Manin-Peyre principle.

Paper Structure

This paper contains 50 sections, 30 theorems, 344 equations.

Table of Contents

  1. Grothendieck rings
  2. Rings of varieties
  3. Biduality for constant commutative group schemes
  4. Ring of varieties with multiplicative characters
  5. A local integration operator
  6. Dimensional filtration and convergence of series
  7. Motivic stabilisation
  8. Symmetric products and motivic Euler products
  9. Symmetric products
  10. Mixed symmetric products
  11. Motivic Euler products
  12. Symmetric products of varieties with characters
  13. Higher order Grothendieck rings
  14. Motivic Euler products and characters
  15. Symmetric products and Hilbert 90
  16. ...and 35 more sections

Key Result

Theorem 1

There exists an $\eta >0$ and an integer $a_{X_\Sigma}$ such that the formal series converges for $| T | < \mathbf{L}_k^{-1 + \eta }$, with its value at $\mathbf{L}^{-1}$ being a non-zero element of $\widehat{\mathscr{M}_k}^{\dim}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (100)

  • Definition
  • Theorem 1: Meromorphic continuation
  • Theorem 2: Multi-height motivic stabilisation
  • Definition 1.1
  • Lemma 1.2: chambert-loir-loeser2016motivic or cluckers-halupczok2022evaluation
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5: SGA3-II
  • Proposition 1.6
  • Remark 1.7
  • ...and 90 more