#### 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
##### Abstract

We prove the arithmetic inner product formula conjectured in the first paper of this series for $n=1$, that is, for the group $U{\left(1,1\right)}_{F}$ unconditionally. The formula relates central $L$-derivatives of weight-$2$ holomorphic cuspidal automorphic representations of $U{\left(1,1\right)}_{F}$ 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 $PGL{\left(2\right)}_{ℚ}$ of square-free level.

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

Proposed: Richard Taylor
Seconded: Brian Conrad, Hendrik W. Lenstra
##### Authors
 Yifeng Liu Department of Mathematics Columbia University 2990 Broadway New York, NY 10027 United States