On the arithmetic Siegel--Weil formula for GSpin Shimura varieties

TL;DR

This paper establishes a local arithmetic Siegel--Weil formula for GSpin Rapoport--Zink spaces, linking intersection numbers of special cycles to derivatives of local representation densities, and applies it to prove a semi-global formula related to Kudla's conjecture.

Contribution

It formulates and proves a local arithmetic Siegel--Weil formula for GSpin Rapoport--Zink spaces, and verifies Kudla's semi-global conjecture for GSpin Shimura varieties.

Findings

Proved a local arithmetic Siegel--Weil formula for GSpin Rapoport--Zink spaces.

Established a semi-global arithmetic Siegel--Weil formula at good reduction places.

Connected intersection numbers of special cycles with derivatives of Eisenstein series.

Abstract

We formulate and prove a local arithmetic Siegel--Weil formula for GSpin Rapoport--Zink spaces, which is a precise identity between the arithmetic intersection numbers of special cycles on GSpin Rapoport--Zink spaces and the derivatives of local representation densities of quadratic forms. As a first application, we prove a semi-global arithmetic Siegel--Weil formula as conjectured by Kudla, which relates the arithmetic intersection numbers of special cycles on GSpin Shimura varieties at a place of good reduction and the central derivatives of nonsingular Fourier coefficients of incoherent Siegel Eisenstein series.