Vol. 5, No. 7, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Arithmetic theta lifting and $L$-derivatives for unitary groups, I

Yifeng Liu

Vol. 5 (2011), No. 7, 849–921

We study cuspidal automorphic representations of unitary groups of 2n 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 2n 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.

arithmetic inner product formula, arithmetic theta lifting, L-derivatives, special cycles
Mathematical Subject Classification 2000
Primary: 11G18
Secondary: 11F27, 11G50, 20G05
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
2990 Broadway
New York, NY 10027
United States