Vol. 5, No. 7, 2011

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

Yifeng Liu

Vol. 5 (2011), No. 7, 923–1000

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.

arithmetic inner product formula, arithmetic theta lifting, L-derivatives, unitary Shimura curves
Mathematical Subject Classification 2000
Primary: 11G18
Secondary: 20G05, 11G50, 11F27
Received: 2 April 2010
Revised: 20 October 2010
Accepted: 21 October 2010
Published: 11 April 2012

Proposed: Richard Taylor
Seconded: Brian Conrad, Hendrik W. Lenstra
Yifeng Liu
Department of Mathematics
Columbia University
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New York, NY 10027
United States