Zero-dimensional Shimura varieties and central derivatives of Eisenstein series
Number Theory
2022-10-31 v1
Abstract
We formulate and prove a version of the arithmetic Siegel-Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of ``special" divisors in terms of Green functions at archimedean and non-archimedean places, and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel-Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature.
Keywords
Cite
@article{arxiv.2210.16218,
title = {Zero-dimensional Shimura varieties and central derivatives of Eisenstein series},
author = {Siddarth Sankaran},
journal= {arXiv preprint arXiv:2210.16218},
year = {2022}
}