Chow groups and $L$-derivatives of automorphic motives for unitary groups


In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $\pi $ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions on the ramification of $\pi $, if the central derivative $L'(1/2,\pi )$ is nonvanishing, then the $\pi $-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson–Bloch conjecture for Chow groups and $L$-functions, which generalizes the Birch and Swinnerton-Dyer conjecture. Moreover, assuming the modularity of Kudla’s generating functions of special cycles, we explicitly construct elements in a certain $\pi $-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi )$ and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by one of us, which generalizes the Gross–Zagier formula to higher dimensional motives.


Chao Li

Department of Mathematics, Columbia University, New York NY, USA

Yifeng Liu

Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China