Zero-dimensional Shimura varieties and central derivatives of Eisenstein series

TL;DR

This paper establishes a version of the arithmetic Siegel-Weil formula for zero-dimensional Shimura varieties linked to tori, connecting special divisors with derivatives of Eisenstein series.

Contribution

It formulates and proves a new arithmetic Siegel-Weil formula for zero-dimensional Shimura varieties, providing a conceptual framework and extending prior results.

Findings

Degrees of special divisors match Fourier coefficients of Eisenstein series derivatives

Provides a new conceptual proof using the classical Siegel-Weil formula

Extends the understanding of arithmetic intersections on Shimura varieties

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.