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

Liu, Yifeng

doi:10.2140/ant.2011.5.923

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

Yifeng Liu

Vol. 5 (2011), No. 7, 923–1000

DOI: 10.2140/ant.2011.5.923

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 $P G L {(2)}_{ℚ}$ 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

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

Authors

Yifeng Liu
Department of Mathematics Columbia University 2990 Broadway New York, NY 10027 United States

Vol. 5, No. 7, 2011