A proof of the Kudla-Rapoport conjecture for Kr\"amer models

TL;DR

This paper proves the Kudla-Rapoport conjecture for Kr"amer models of unitary Rapoport--Zink spaces at ramified places, establishing a key identity linking intersection numbers and local densities, with applications to the arithmetic Siegel--Weil formula.

Contribution

It provides the first proof of the Kudla-Rapoport conjecture for Kr"amer models at ramified places, extending the understanding of arithmetic intersections in unitary Shimura varieties.

Findings

Established the identity between intersection numbers and local densities for Kr"amer models.

Relaxed local assumptions in the arithmetic Siegel--Weil formula at ramified places.

Applicable to unitary Shimura varieties over imaginary quadratic fields.

Abstract

We prove the Kudla--Rapoport conjecture for Kr\"amer models of unitary Rapoport--Zink spaces at ramified places. It is a precise identity between arithmetic intersection numbers of special cycles on Kr\"amer models and modified derived local densities of hermitian forms. As an application, we relax the local assumptions at ramified places in the arithmetic Siegel--Weil formula for unitary Shimura varieties, which is in particular applicable to unitary Shimura vartieties associated to unimodular hermitian lattices over imaginary quadratic fields.