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.

• The paper establishes a motivic Poisson formula for split algebraic tori by developing a motivic Fourier analysis framework in the Grothendieck ring.
• It applies the formula to compute meromorphic continuation and residue evaluations for motivic height zeta functions on toric varieties, aligning with Batyrev-Manin-Peyre predictions.
• The approach innovatively utilizes Grothendieck rings with character data, motivic Euler products, and stabilization techniques to advance arithmetic geometry.

Motivic Poisson Formula and Height Zeta Functions for Split Algebraic Tori

Context and Motivation

The Batyrev-Manin-Peyre conjectures have defined expectations for the distribution of rational points on Fano varieties over global fields, with harmonic analysis and the Poisson summation formula as crucial tools in the classical number-theoretic setting. In the motivic framework, analogous counting problems arise in the Grothendieck ring of varieties, demanding categorial and geometric generalizations of these analytic techniques. Previous motivic Poisson formulae, notably in the additive group setting [hrushovski-kazhdan], have facilitated the study of moduli spaces through motivic Fourier analysis. However, the extension to the multiplicative setting, specifically algebraic tori, remained structurally more involved due to the lack of self-duality and the necessity to encode character data within motivic functions.

This paper establishes a motivic Poisson formula for split algebraic tori, using a Grothendieck ring of varieties equipped with character structures, and applies the theory to motivic height zeta functions for split projective toric varieties over function fields of curves. The methodology diverges fundamentally from earlier universal torsor approaches, instead developing a motivic Fourier theory for tori.

Construction of the Motivic Poisson Formula

The classical Poisson summation formula operates on locally compact commutative groups and their duals, relating sums over discrete subgroups to sums over their annihilators via Fourier transforms. Adapting this to the motivic context, which lacks a topology and conventional integration, necessitates the construction of motivic analogues of Schwartz-Bruhat functions, motivic Fourier transforms, and integration over characters, all within the framework of the Grothendieck ring of varieties with exponentials and characters.

Let $U \cong \mathbf{G}_m^n$ be a split algebraic torus over a base field $k$, $C$ a smooth projective curve over $k$, and $F = k(C)$ its function field. The discrete group structure emerging from $U(F)$ and its adelic embedding is encoded geometrically by moduli spaces of morphisms from $C$ to $X$, a split projective toric compactification of $U$. The Grothendieck ring $K_0 k$ and its localization $k$0 serve as receptacles for motivic counting functions, with virtual dimension filtrations yielding a completion to accommodate convergent series.

The motivic Poisson formula constructed in this paper employs:

• A Grothendieck ring of varieties with character data, $k$1, encoding motivic functions with values in diagonalizable (Cartier dual) group schemes,
• Motivic Fourier transforms defined via pushforward in this ring, associating motivic functions on the constant group scheme to functions on its dual,
• Integration operators over character schemes, respecting motivic orthogonality and compatible with quotient structures,
• Motivic Euler products realized as symmetric products indexed by divisors on the curve, with precise control over convergence and virtual dimension.

The construction identifies the group of divisors $k$2 and principal divisors $k$3 as analogues of idèles and their reciprocals, respectively. The motivic Poisson formula relates summation over principal divisors to integration over characters trivial on principal divisors, precisely mirroring the classical analytic correspondence.

Application: Motivic Height Zeta Functions for Toric Varieties

The motivic height zeta function encodes counting data for morphisms from $k$4 to $k$5 of prescribed multidegree, parametrized by the effective cone of the Picard group. Using the geometric-motivic Poisson formula, the motivic height zeta function admits a representation as an integral over the character domain, with local factors determined by explicit polynomial expressions ($k$6) associated to the fan $k$7 describing the toric variety.

The paper establishes, for the motivic height zeta function $k$8 indexed by $k$9:

• Meromorphic continuation: There exist $C$0 and $C$1 such that the normalized series

$C$2

converges for $C$3, and its value at $C$4 is non-zero in $C$5 (2604.03162).

• Multi-height motivic stabilization: As $C$6 tends to infinity within the movable cone, the normalized class

$C$7

stabilizes to a motivic Euler product determined by the curve genus $C$8, the class of $C$9, the dimension and Picard rank of $k$0, and local contributions from closed points.

These results precisely recover and extend prior stabilizations and residue-type calculations achieved by torsor and Chow-motive methods [bourqui2009produit, bilu-das-howe2022zeta, faisant2025motivic-distribution].

Technical Innovations

Grothendieck Rings with Characters

The introduction of $k$1 allows the motivic encoding of multiplicative character actions, crucial for realizing motivic analogues of harmonic analysis on tori. Integration and Fourier transformation are realized as algebraic pushforwards, respecting cut-and-paste relations.

Motivic Euler Products and Symmetric Products

Motivic Euler products constructed via symmetric products of varieties with character structure enable not only the analytic continuation of motivic height zeta functions but also explicit calculation of residues and stabilization limits. This is achieved through detailed analysis of cones, combinatorics of fans, and decomposition lemmas controlling polynomial coefficients and convergence domains.

Formal Motivic Fourier Theory for Tori

By departing from the additive self-dual structure and developing formal motivic Fourier theory adapted to the multiplicative case, the paper enables applications to toric varieties, capturing both the local and global arithmetic structure within the motivic framework.

Numerical and Structural Results

Meromorphic Continuation and Residue Computation:

• The normalized motivic height zeta function exhibits convergence and non-vanishing residues controlled by combinatorial invariants of the toric variety and the genus of the base curve.
• Explicit combinatorial expressions, particularly the polynomial $k$2, control the analytic continuation and pole structure, paralleling the role of Tamagawa measures and Peyre constants in the classical theory.

Motivic Stabilization:

• The stabilization of normalized motivic classes, as multidegrees recede in the movable cone, validates motivic analogues of Batyrev-Manin-Peyre predictions for distribution of rational curves on toric varieties.
• The limiting motivic Euler product expression includes contributions from the Picard variety and affine line, reflecting geometric and arithmetic invariants.

Compatibility with Prior Work:

• The global motivic Poisson formula and its applications generalize and subsume prior universal torsor and residue analyses, aligning with expectations from motivic counting statistics and Hadamard convergence [faisant2025motivic-distribution, bilu-das-howe2022zeta].

Implications and Speculative Directions

The formalism developed in this paper paves the way for motivic harmonic analysis on broader classes of algebraic groups and varieties, potentially extending to non-split tori, spherical varieties, and higher-dimensional bases. The motivic stabilization and residue computations strengthen conjectural frameworks for Manin-type motivations in the Grothendieck ring, facilitating explicit geometric interpretations in terms of principal value integrals and motivic measure theory.

The motivic Poisson formula for tori enables further development of motivic automorphic forms, the study of motivic L-functions, and the analysis of motivic height zeta functions for general types of varieties. Potential generalizations include:

• Extending motivic Fourier theory to torus torsors and quasi-toric varieties,
• Refinements in the structure of Grothendieck rings with additional symmetries or exponentials,
• Relations to motivic Mellin transforms and motivic sheaf-theoretic structures [cluckers-motivic-mellin], possibly connecting to advanced applications in arithmetic geometry and motivic integration.

Conclusion

The paper rigorously constructs a motivic Poisson formula for split algebraic tori and demonstrates its application to motivic height zeta functions, verifying motivic Batyrev-Manin-Peyre stabilization in the Grothendieck ring setting. The theory encompasses a formal motivic Fourier analysis tailored for the multiplicative case, provides explicit, convergent motivic Euler product expressions, and resolves residue computations and stabilization phenomena with precision. Further exploration of this motivic harmonic framework has the potential to deepen connections between arithmetic geometry, motivic integration, and categorical counting invariants.