Arithmetic theta lifting andL-derivatives for unitary groups, I

Abstract

We study cuspidal automorphic representations of unitary groups of 2n variables with -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.

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“…We want to give another interpretation for the formula (2-6) when (π, χ ) = 1, which is crucial for our proof in [Liu 2011]. For this purpose, let us assume the following conjecture raised by Kudla and Rallis (see …”

“…We also prove some partial results toward the general arithmetic inner product formula, namely the modularity theorem on the (noncompactified) generating series and the arithmetic analogue of the local Siegel-Weil formula at archimedean places. In the second part of this paper [Liu 2011], we will give a full proof of the arithmetic inner product formula for n = 1.…”

“…As we do in [Liu 2011], to prove the arithmetic inner product formula, we introduce analytic kernel functions and geometric kernel functions which carry over all cusp forms simultaneously. The former is the derivative of certain Eisenstein series on the doubling group which deals with derivatives of L-functions, while the latter is the height pairing of the generating series which deals with that of the arithmetic theta lifting.…”

“…The Green's function, originally constructed in , is related to the Kudla-Millson form [1986] and very closely related to derivatives of Whittaker functions. But this is not the admissible Green's function used in the Beilinson-Bloch height pairing (at an archimedean place); instead, they have a certain relation which will be elaborated in [Liu 2011] when n = 1 and we expect that they relate for general n. For any x = (x 1 , . .…”

“…He proposed the project of the arithmetic Siegel-Weil formula and proved a special form of the arithmetic inner product formula with Rapoport and Yang. Our work follows the second approach, establishing an explicit form of the arithmetic inner product formula and, together with [Liu 2011], proving the complete version of the arithmetic inner product formula in the case of unitary groups of two variables over totally real fields.…”