Professor Liu Yifeng has online published a research paper by Annals of Mathematics

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 LL-packet ππ of unitary groups of even rank with respect to a CM extension, whose global root number is −1−1. We show that, under some restrictions on the ramification of ππ, if the central derivative L′(1/2,π)L′(1/2,π) is nonvanishing, then the ππ-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 LL-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 ππ-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative L′(1/2,π)L′(1/2,π) 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.