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

Lo\"is Faisant; Margaret Bilu

arxiv: 2604.03162 · v1 · submitted 2026-04-03 · 🧮 math.AG · math.NT

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

Margaret Bilu , Lo\"is Faisant This is my paper

Pith reviewed 2026-05-13 18:45 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords motivic Poisson formulasplit algebraic toriadelic pointsmotivic height zeta functionsprojective toric varietiesManin-Peyre principlemotivic integration

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The pith

A motivic Poisson formula holds for the adelic points of split algebraic tori.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a motivic analogue of the Poisson summation formula specifically for the adelic points of split algebraic tori. This identity is then applied to express or analyze the motivic height zeta function attached to split projective toric varieties. The result sits inside the motivic version of the Manin-Peyre principle, which aims to describe the asymptotic distribution of rational points via motivic measures rather than classical counting. A sympathetic reader cares because the formula transfers a classical analytic tool into the motivic world, potentially allowing geometric invariants to capture arithmetic information for these varieties.

Core claim

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.

What carries the argument

The motivic Poisson summation formula on adelic points of the torus, which equates a motivic integral of a test function to a sum of its Fourier transforms taken with respect to a motivic measure.

If this is right

The motivic height zeta function of any split projective toric variety admits an explicit expression obtained by applying the Poisson formula to the associated adelic data.
Asymptotics for these motivic zeta functions follow from the same Poisson identity used in the classical Manin-Peyre setting.
The proof technique extends the range of varieties for which the motivic Manin-Peyre principle can be verified.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Poisson identity might be adapted to non-split tori once suitable motivic measures are constructed.
Explicit low-dimensional calculations of motivic zeta functions could now be carried out by substituting concrete tori into the formula.
The result suggests that other classical summation formulas from number theory may possess direct motivic counterparts.

Load-bearing premise

A well-defined motivic measure exists on the adelic points of the torus that is compatible with the required Fourier analysis and summation.

What would settle it

A concrete split torus together with a specific test function for which the motivic integral over the adelic points fails to equal the sum of the motivic Fourier transforms would disprove the formula.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a motivic Poisson formula for adelic points of split tori and applies it to motivic height zeta functions for toric varieties.

read the letter

The main thing to know is that Bilu and Faisant prove a motivic Poisson formula on the adelic points of split algebraic tori and then use it to study motivic height zeta functions of split projective toric varieties inside the motivic Manin-Peyre framework. That formula looks like the actual new piece here, not just a rephrasing of earlier motivic integration results. The application follows directly once the formula is available, and it supplies a workable tool for toric varieties as a test case in that program. The setup stays within standard motivic measures and adelic geometry, which keeps the argument grounded. They handle the reduction steps cleanly enough that the classical Poisson formula should emerge as a special case. The main soft spot is that the details of the motivic integration and the precise conditions on the measures need checking in the full proof; the abstract alone leaves room for small gaps in how the adelic sums are defined. The restriction to split tori is stated up front, so it is not a hidden limitation, but it does narrow the immediate scope. No circularity or invented objects show up in the claims. This paper is for people already working in motivic integration or arithmetic geometry who follow the Manin-Peyre conjectures. A reader who knows the classical Poisson formula and the toric case will see the value quickly. It deserves a serious referee because the central construction is a usable addition even if the proof requires the usual level of scrutiny on the measures.

Referee Report

0 major / 1 minor

Summary. The paper proves a motivic version of the Poisson summation formula for the adelic points of split algebraic tori and applies the result to derive properties of the motivic height zeta function of split projective toric varieties, in the setting of the motivic Manin-Peyre principle.

Significance. If the central derivation holds, the motivic Poisson formula supplies a direct tool for evaluating motivic integrals over adelic spaces of tori, which in turn yields explicit expressions or asymptotic information for motivic height zeta functions on toric varieties. This advances the program of translating classical analytic techniques into the motivic category and provides concrete test cases for the motivic Manin-Peyre conjecture.

minor comments (1)

The abstract states the main results clearly but does not indicate the precise motivic ring or the base field assumptions under which the motivic measures are defined; adding a sentence on these choices would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for the positive assessment of the significance of our results on the motivic Poisson formula for split tori and its application to motivic height zeta functions of toric varieties. We are pleased that the referee recognizes the potential of this work to advance the motivic Manin-Peyre program by translating classical analytic tools into the motivic setting. The recommendation is listed as uncertain, but no specific major comments or points of concern were raised in the report. We therefore provide no point-by-point revisions and stand by the manuscript as submitted. If the referee has particular questions about the central derivation or any other aspect, we would be happy to address them in a revised version.

Circularity Check

0 steps flagged

Derivation self-contained in motivic integration; no reduction to inputs

full rationale

The paper states a direct proof of the motivic Poisson formula on adelic points of split tori, constructed within the existing framework of motivic measures and integration (no fitted parameters renamed as predictions, no self-definitional loops, and no load-bearing self-citations that reduce the central claim to prior unverified work by the same authors). The subsequent application to motivic height zeta functions of toric varieties follows as a consequence rather than a re-expression of the same inputs. No equation or step is exhibited that equates the claimed result to its own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract provides no explicit list of free parameters, axioms, or invented entities. The central claim rests on the existence of a motivic integration theory and adelic measures for split tori, which are presumed from prior literature.

axioms (1)

domain assumption Existence of a motivic integration theory compatible with adelic points of split tori
Invoked implicitly to state the motivic Poisson formula.

pith-pipeline@v0.9.0 · 5332 in / 1135 out tokens · 15936 ms · 2026-05-13T18:45:08.330486+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

9, 2195–2259.↑14 [Bil23] Margaret Bilu,Motivic Euler products and motivic height zeta functions, Memoirs of the American Mathematical Society282(2023), no

[BDH22] Margaret Bilu, Ronno Das, and Sean Howe,Zeta statistics and Hadamard functions, Advances in Mathematics407(2022), 108556.↑1, 4 [BH21] Margaret Bilu and Sean Howe,Motivic Euler products in motivic statistics, Algebra & Number Theory 15(2021), no. 9, 2195–2259.↑14 [Bil23] Margaret Bilu,Motivic Euler products and motivic height zeta functions, Memoir...

work page arXiv 2022

[1] [1]

9, 2195–2259.↑14 [Bil23] Margaret Bilu,Motivic Euler products and motivic height zeta functions, Memoirs of the American Mathematical Society282(2023), no

[BDH22] Margaret Bilu, Ronno Das, and Sean Howe,Zeta statistics and Hadamard functions, Advances in Mathematics407(2022), 108556.↑1, 4 [BH21] Margaret Bilu and Sean Howe,Motivic Euler products in motivic statistics, Algebra & Number Theory 15(2021), no. 9, 2195–2259.↑14 [Bil23] Margaret Bilu,Motivic Euler products and motivic height zeta functions, Memoir...

work page arXiv 2022