Algebra & Number Theory

Arithmetic theta lifting and $L$-derivatives for unitary groups, II

Yifeng Liu

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We prove the arithmetic inner product formula conjectured in the first paper of this series for n=1, that is, for the group U(1,1)F unconditionally. The formula relates central L-derivatives of weight-2 holomorphic cuspidal automorphic representations of U(1,1)F with ϵ-factor 1 with the Néron–Tate height pairing of special cycles on Shimura curves of unitary groups. In particular, we treat all kinds of ramification in a uniform way. This generalizes the arithmetic inner product formula obtained by Kudla, Rapoport, and Yang, which holds for certain cusp eigenforms of PGL(2) of square-free level.

Article information

Algebra Number Theory, Volume 5, Number 7 (2011), 923-1000.

Received: 2 April 2010
Revised: 20 October 2010
Accepted: 21 October 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 20G05: Representation theory 11G50: Heights [See also 14G40, 37P30] 11F27: Theta series; Weil representation; theta correspondences

arithmetic inner product formula arithmetic theta lifting L-derivatives unitary Shimura curves


Liu, Yifeng. Arithmetic theta lifting and $L$-derivatives for unitary groups, II. Algebra Number Theory 5 (2011), no. 7, 923--1000. doi:10.2140/ant.2011.5.923.

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