Chow groups andL-derivatives of automorphic motives for unitary groups, II.

Chao Li,Yifeng Liu

doi:10.1017/fmp.2022.2

Abstract

AbstractIn this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension$E/F$at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures.

Highlights

For the higher dimensional case, in a series of papers starting from the late 1990s, Kudla and Rapoport developed the theory of special cycles on integral models of Shimura varieties for GSpin groups in lower rank cases and for unitary groups of arbitrary ranks [KR11, KR14]
In [LL21], we proved that for certain cuspidal automorphic representations π of U(r, r), if the central derivative L (1/2, π) is nonvanishing, the π-nearly isotypic localisation of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish
We improve the main results from [LL21] in two directions: First, we allow ramified places in the CM extension E/F at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zink spaces

Summary

Introduction

In 1986, Gross and Zagier [GZ86] proved a remarkable formula that relates the Néron–Tate heights of Heegner points on a rational elliptic curve to the central derivative of the corresponding Rankin– Selberg L-function. A decade later, Kudla [Kud97] revealed another striking relation between Gillet– Soulé heights of special cycles on Shimura curves and derivatives of Siegel Eisenstein series of genus 2, suggesting an arithmetic version of theta lifting and the Siegel–Weil formula (see, for example, [Kud[02], Kud03]) This was later further developed in his joint work with Rapoport and Yang [KRY06]. For the higher dimensional case, in a series of papers starting from the late 1990s, Kudla and Rapoport developed the theory of special cycles on integral models of Shimura varieties for GSpin groups in lower rank cases and for unitary groups of arbitrary ranks [KR11, KR14] They studied special cycles on the relevant Rapoport–Zink spaces over non-Archimedean local fields.

Main results

Two new ingredients

Notation and conventions

Intersection of special cycles at ramified places

A Kudla–Rapoport type formula We fix an embedding φ0 : E → Cp and let

Fourier transform of analytic side

Bruhat–Tits stratification

Linear invariance of intersection numbers

Fourier transform of geometric side

Comparison with absolute Rapoport–Zink spaces

Local theta lifting at ramified places

Weil representation and spherical module