PreprintCC BY 4.0arXiv
Arithmetic Siegel-Weil for the spherical Hecke algebra
Abstract
We formulate a conjectural generalization of the arithmetic Siegel-Weil formula, relating intersection multiplicities of cycles on unitary Rapoport-Zink spaces to central derivatives of local Whittaker functions. The new aspect of the conjecture is that it incorporates the action of certain Hecke correspondences on the Rapoport-Zink space. We verify the conjecture in a low-dimensional case. A significant portion of the paper is devoted to generalizing the classical theory of representation densities for local Hermitian spaces, as this is needed for explicit computation of the relevant Whittaker functions. We expect this generalization to be of independent interest.
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BibTeX
@misc{howard2026arithmetic,
title = {{Arithmetic Siegel-Weil for the spherical Hecke algebra}},
author = {Benjamin Howard},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13435},
eprint = {2609.13435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13435},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Arithmetic Siegel-Weil for the spherical Hecke algebra
AU - Benjamin Howard
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13435
UR - https://arxiv.org/abs/2609.13435
AB - We formulate a conjectural generalization of the arithmetic Siegel-Weil formula, relating intersection multiplicities of cycles on unitary Rapoport-Zink spaces to central derivatives of local Whittaker functions. The new aspect of the conjecture is that it incorporates the action of certain Hecke correspondences on the Rapoport-Zink space. We verify the conjecture in a low-dimensional case. A significant portion of the paper is devoted to generalizing the classical theory of representation densities for local Hermitian spaces, as this is needed for explicit computation of the relevant Whittaker functions. We expect this generalization to be of independent interest.
ER -