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

Liu, Yifeng

doi:10.2140/ant.2011.5.849

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

Yifeng Liu

Vol. 5 (2011), No. 7, 849–921

DOI: 10.2140/ant.2011.5.849

Abstract

We study cuspidal automorphic representations of unitary groups of $2 n$ variables with $ϵ$ -factor $- 1$ and their central $L$ -derivatives by constructing their arithmetic theta liftings, which are Chow cycles of codimension $n$ on Shimura varieties of dimension $2 n - 1$ of certain unitary groups. We give a precise conjecture for the arithmetic inner product formula, originated by Kudla, which relates the height pairing of these arithmetic theta liftings and the central $L$ -derivatives of certain automorphic representations. We also prove an identity relating the archimedean local height pairing and derivatives of archimedean Whittaker functions of certain Eisenstein series, which we call an arithmetic local Siegel–Weil formula for archimedean places. This provides some evidence toward the conjectural arithmetic inner product formula.

Keywords

arithmetic inner product formula, arithmetic theta lifting, L-derivatives, special cycles

Mathematical Subject Classification 2000

Primary: 11G18

Secondary: 11F27, 11G50, 20G05

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