Co-rank 1 Arithmetic Siegel--Weil

Abstract

We prove the arithmetic Siegel–Weil formula in co-rank 1, for Kudla–Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. We also propose a construction for arithmetic special cycle classes associated to possibly singular matrices of arbitrary co-rank. Our arithmetic Siegel–Weil formula implies that degrees of Kudla–Rapoport arithmetic special 1-cycles are encoded in near-central first derivatives of unitary Eisenstein series Fourier coefficients. The key input is a new limiting method at all places. On the analytic side, the limit relates local Whittaker functions on different groups. On the geometric side at nonsplit non-Archimedean places, the limit relates degrees of 0-cycles on Rapoport–Zink spaces and local contributions to heights of 1-cycles in mixed characteristic.