November 2021 Chow groups and L-derivatives of automorphic motives for unitary groups
Chao Li, Yifeng Liu
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Ann. of Math. (2) 194(3): 817-901 (November 2021). DOI: 10.4007/annals.2021.194.3.6


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.


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Chao Li. Yifeng Liu. "Chow groups and L-derivatives of automorphic motives for unitary groups." Ann. of Math. (2) 194 (3) 817 - 901, November 2021.


Published: November 2021
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2021.194.3.6

Primary: 11G18 , 11G40 , 11G50 , 14C15

Keywords: arithmetic inner product formula , Beilinson--Bloch conjecture , Chow groups, $L$-derivatives , height pairing

Rights: Copyright © 2021 Department of Mathematics, Princeton University


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Vol.194 • No. 3 • November 2021
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