Kudla-Rapoport conjecture for Kr\"amer models

TL;DR

This paper proposes a modified Kudla-Rapoport conjecture for Kr"amer models in unitary Rapoport-Zink spaces at ramified primes, relating intersection numbers to derivatives of local density polynomials, and proves it for dimension three.

Contribution

It introduces a new conjecture for Kr"amer models, defines special difference cycles, and proves the conjecture for the case n=3, advancing understanding of arithmetic intersection theory.

Findings

Proposed a modified Kudla-Rapoport conjecture for Kr"amer models.

Proved the conjecture for n=3 using induction formulas.

Connected the conjecture to the arithmetic Siegel-Weil formula.

Abstract

In this paper, we propose a modified Kudla-Rapoport conjecture for the Kr\"amer model of unitary Rapoport-Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla-Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies arithmetic Siegel-Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice.