## Algebra & Number Theory

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

Yifeng Liu

#### Abstract

We prove the arithmetic inner product formula conjectured in the first paper of this series for $n=1$, that is, for the group $U(1,1)F$ 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 $PGL(2)ℚ$ of square-free level.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729714

Digital Object Identifier
doi:10.2140/ant.2011.5.923

Mathematical Reviews number (MathSciNet)
MR2928564

Zentralblatt MATH identifier
1258.11061

#### Citation

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. https://projecteuclid.org/euclid.ant/1513729714

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