Improper Intersections of Kudla-Rapoport divisors and Eisenstein series

TL;DR

This paper links the arithmetic degrees of certain Kudla-Rapoport cycles on a Shimura variety to Fourier coefficients of a derivative of an Eisenstein series, confirming conjectures relating cycle intersections and automorphic forms in a degenerate, dimension 2 setting.

Contribution

It establishes a novel connection between degenerate Kudla-Rapoport cycle degrees and Eisenstein series derivatives on a specific integral model of a Shimura variety.

Findings

Arithmetic degrees match Fourier coefficients of Eisenstein series derivatives.

Degenerate cycles can contain positive-dimensional components.

Supports Kudla-Kudla-Rapoport conjectures in a new setting.

Abstract

We consider a certain family of Kudla-Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1,1), and prove that the arithmetic degrees of these cycles can be identified with the Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parametrizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirmation, in the degenerate setting and for dimension 2, of conjectures of Kudla and Kudla-Rapoport that predict relations between the intersection numbers of special cycles and the Fourier coefficients of automorphic forms.