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.

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    @BOOK{YZZ,
      author = {Yuan, Xinyi and Zhang, Shou-Wu and Zhang, Wei},
      title = {The {G}ross-{Z}agier Formula on {S}himura curves},
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      year = {2009},
      pages = {48},
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    @INCOLLECTION{Zha12,
      author = {Zhang, Wei},
      title = {Gross-{Z}agier formula and arithmetic fundamental lemma},
      booktitle = {Fifth {I}nternational {C}ongress of {C}hinese {M}athematicians. {P}art 1, 2},
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      volume = {2},
      pages = {447--459},
      publisher = {Amer. Math. Soc., Providence, RI},
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      mrclass = {11G18 (11F72 11G50 22E55)},
      mrnumber = {2908086},
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      author = {Zhang, Wei},
      title = {Weil representation and arithmetic fundamental lemma},
      journal = {Ann. of Math. (2)},
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      volume = {193},
      year = {2021},
      number = {3},
      pages = {863--978},
      issn = {0003-486X},
      mrclass = {11F27 (11F67 11G40 14C25 14G35)},
      mrnumber = {4250392},
      doi = {10.4007/annals.2021.193.3.5},
      url = {https://doi.org/10.4007/annals.2021.193.3.5},
      zblnumber = {07353244},
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Authors

Chao Li

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

Yifeng Liu

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