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

Abstract

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.