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Arithmetic Siegel-Weil for the spherical Hecke algebra

This paper formulates and verifies in a low-dimensional case a conjectural generalization of the arithmetic Siegel-Weil formula that relates intersection multiplicities of cycles on unitary Rapoport-Zink spaces under Hecke correspondences to central derivatives of local Whittaker functions, supported by a novel generalization of classical representation densities for local Hermitian spaces.

Original authors: Benjamin Howard

Published 2026-09-15

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CC BY 4.0

Original authors: Benjamin Howard

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of number theory, mathematicians often study the hidden patterns within numbers by looking at how they relate to one another through geometry. Imagine a world where numbers are not just abstract symbols but points on a map, and the rules of arithmetic create shapes and landscapes that can be explored. Two powerful ideas have long guided explorers in this realm: the study of "cycles," which are like closed loops or paths drawn on these geometric maps, and the study of "Whittaker functions," which are complex mathematical tools used to measure the density of numbers in specific regions, much like a weather map measures atmospheric pressure. For decades, a profound connection has been suspected between these two worlds: that the way these geometric loops intersect with one another is directly linked to the precise values of these number-density measurements. This connection, known as the arithmetic Siegel-Weil formula, has been a cornerstone of modern research, helping mathematicians understand deep questions about the distribution of prime numbers and the structure of algebraic equations.

Building on this established foundation, Benjamin Howard has formulated a new, broader version of this formula that incorporates the influence of specific symmetries, known as Hecke correspondences. These symmetries act like a set of rules that can transform one geometric shape into another, or shift a collection of numbers into a new arrangement, without changing their fundamental nature. The central question Howard addresses is whether the deep link between geometry and number density holds true even when these shapes and numbers are being actively transformed by these symmetry rules. While the original formula described a static relationship, this new conjecture asks if the relationship remains stable and predictable as the geometric objects are moved and reshaped by these algebraic forces.

In this work, Howard proposes a specific conjecture that extends the known formula to include these dynamic transformations. He suggests that if you take a geometric cycle on a special type of space called a Rapoport-Zink space and apply a symmetry operation to it, the resulting change in its intersection properties can be precisely calculated using a modified version of the number-density measurements. To test this idea, he focuses on a specific, manageable case involving two-dimensional spaces. In this scenario, the geometric space is relatively simple, and the symmetry operations are well-understood. Howard does not merely guess at the answer; he performs a direct, rigorous calculation of both sides of the proposed equation. On one side, he computes the geometric intersection numbers after the symmetry operations are applied. On the other side, he calculates the corresponding values of the number-density functions, which requires developing a new, more flexible way to measure how numbers are distributed in these transformed settings.

The result of this detailed calculation is a confirmation of the conjecture in this specific low-dimensional case. Howard demonstrates that the geometric side and the analytic side match perfectly, proving that the relationship between the transformed shapes and the number densities is indeed consistent with his proposed formula. A significant portion of the paper is dedicated to the development of the tools needed for this calculation, specifically a generalized method for computing representation densities. These are the mathematical quantities that measure how often a particular pattern of numbers appears within a given region. The standard methods for these calculations were insufficient for the complex, transformed regions Howard needed to study, so he constructed a new theoretical framework to handle them. This new framework allows for the explicit computation of these densities in situations where the underlying structures have been altered by the symmetry operations.

The paper concludes by verifying that for the specific case of two-dimensional spaces, the arithmetic Siegel-Weil formula successfully incorporates the action of these symmetry correspondences. The equality holds true, meaning the geometric intersection numbers of the transformed cycles are exactly equal to the derivatives of the corresponding Whittaker functions. This finding provides strong evidence that the broader conjecture is correct and that the deep connection between geometry and number theory is robust enough to withstand these complex transformations. While the proof is currently limited to this specific dimension, the methods developed and the conjecture formulated offer a clear path forward for exploring these relationships in higher dimensions and more complex settings, potentially unlocking new insights into the fundamental structure of numbers.

Problem Statement

The paper addresses the formulation and verification of a conjectural generalization of the arithmetic Siegel-Weil formula for the spherical Hecke algebra. The classical arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, establishes a relation between local arithmetic intersection numbers of Kudla-Rapoport cycles on unitary Rapoport-Zink spaces and the central derivatives of local Whittaker functions.

The specific problem tackled here is to incorporate the action of the spherical Hecke algebra into this framework. Inspired by the conjectural arithmetic fundamental lemma for the spherical Hecke algebra formulated by Li-Rapoport-Zhang, the author seeks to relate the intersection multiplicities of cycles acted upon by Hecke correspondences to the central derivatives of Whittaker functions associated with Hecke operators acting on Schwartz functions. A significant technical hurdle identified is that the standard theory of representation densities for local Hermitian spaces is insufficient for the explicit computation of the relevant Whittaker functions in this generalized setting.

Methodology

The paper employs a combination of arithmetic geometry, representation theory, and explicit local calculations.

Generalized Representation Densities: A substantial portion of the work is dedicated to developing a generalized theory of representation densities. The author defines a representation density $\text{Den}_T(C)$ for a nonsingular Hermitian matrix $T$ and an arbitrary compact open subset $C \subset V^d$ . This generalizes the classical density $\alpha(M, T)$ defined for lattices. The definition utilizes algebro-geometric tools, specifically the moment morphism $\tau: V^d \to \text{Herm}_d(F')$ , and measures induced by differential forms on smooth schemes.
Density Polynomials: The author constructs density polynomials $\text{Den}_T(s, C) \in \mathbb{Q}[q^{-s}]$ such that $\text{Den}_T(0, C) = \text{Den}_T(C)$ . It is shown that these polynomials essentially coincide with local Whittaker functions $W_T(s, \mathbf{1}_C)$ for characteristic functions of compact open sets.
Computational Tools: To compute these densities explicitly, the paper develops two technical tools:
- Integral Models: A method to compute representation densities by counting points on the special fibers of smooth integral models of the relevant schemes.
- Reduction Formulas: A Fubini-style reduction formula that expresses the representation density of a block-diagonal matrix in terms of densities of its diagonal blocks, allowing for recursive computation.
Verification in Low Dimensions: The conjecture is verified in the specific case where the dimension $n=2$ $n = 2$ . This involves:
- Explicitly calculating the arithmetic intersection numbers on the geometric side using the action of Hecke correspondences on Kudla-Rapoport cycles.
- Explicitly calculating the derivatives of Whittaker functions on the analytic side using the newly developed density polynomials and combinatorial counting of lattice paths in the Bruhat-Tits tree.

Key Contributions and Results

Conjectural Formulation: The paper formulates Conjecture 5.3.3, a generalization of the arithmetic Siegel-Weil formula. It posits that for a tuple $x$ with nonsingular moment matrix $T$ and Hecke functions $f_1, \dots, f_r$ , the Euler-Poincaré characteristic of the derived intersection of Hecke-acted cycles equals a normalized central derivative of a Whittaker function involving the corresponding Hecke actions on Schwartz functions:
$\chi(f_1 Z^L(x_1) \otimes_L \dots \otimes_L f_r Z^L(x_r)) \stackrel{?}{=} \frac{W'_T(0, f_1 \mathbf{1}_{\Xi}^{d_1} \otimes \dots \otimes f_r \mathbf{1}_{\Xi}^{d_r})}{c(n) \log(q^2)}$
Theorem A (Verification for $n=2$ ): The author proves the conjecture for the case $n=2$ . Specifically, for any Hecke functions $f_1, f_2 \in H(V)$ and a basis $x_1, x_2$ with specific norm conditions, the equality holds:
$\chi(f_1 Z^L(x_1) \otimes_L f_2 Z^L(x_2)) = \frac{W'_T(0, f_1 \mathbf{1}_{\Xi} \otimes f_2 \mathbf{1}_{\Xi})}{c(2) \log(q^2)}$
The proof involves a direct calculation of both sides. The geometric side reduces to known formulas for Kudla-Rapoport cycles and their Hecke transforms, while the analytic side requires the new machinery of generalized representation densities.
Generalized Representation Density Theory: The paper provides a systematic development of representation densities for arbitrary compact open subsets, not just lattices. This includes the construction of density polynomials and the derivation of explicit formulas for $2 \times 2$ matrices involving self-dual lattices in relative positions.

Significance

The paper claims significance primarily in two areas:

Advancement of the Arithmetic Siegel-Weil Program: It extends the scope of the arithmetic Siegel-Weil formula to include the action of the spherical Hecke algebra, aligning it with the arithmetic fundamental lemma for the spherical Hecke algebra. This is a necessary step for potential applications to the Beilinson-Bloch conjecture for unitary Shimura varieties in the presence of Hecke operators.
Independent Interest of Generalized Densities: The author explicitly states that the generalization of the classical theory of representation densities is expected to be of independent interest. The developed tools for computing densities of arbitrary compact open sets and the associated density polynomials provide a framework for handling Whittaker functions that are linear combinations of standard ones, a situation that arises naturally when incorporating Hecke actions.

The work is presented as a verification of a nontrivial special case ( $n=2$ ) of a broader conjecture, with the explicit calculations serving as evidence for the general validity of the proposed formula.

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