Zero-Dimensional Shimura Varieties and Central Derivatives of Eisenstein Series

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.