On the Siegel–Weil formula for unitary groups

Ichino, Atsushi

doi:10.1007/s00209-006-0045-8

Cited by 34 publications

(29 citation statements)

References 17 publications

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“…This is a special case of the regularized Siegel-Weil formula, proved in the case of unitary groups by Ichino [17,18]. There is a second type of irreducible constituent of I.0; Á/ -note that this representation is unitarizable and hence completely reducible.…”

Section: Introductionmentioning

confidence: 87%

“…It then remains to calculate the Fourier coefficient corresponding to T of the derivative of the incoherent Eisenstein series. For this we use the Siegel-Weil formula established by Ichino [17,18] in this case. We must ultimately calculate some representation densities for hermitian forms, which is in general a very difficult task, even though a general formula due to Hironaka [13,14] exists.…”

Section: Introductionmentioning

confidence: 99%

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Special cycles on unitary Shimura varieties II: Global theory

Kudla

Rapoport

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

145

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We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature .n 1; 1/. We define arithmetic cycles on these models and study their intersection behavior. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s D 0 of a certain incoherent Eisenstein series for the group U.n; n/. This is done by relating the arithmetic cycles to their formal counterpart from part I [33] via nonarchimedean uniformization, and by relating the Fourier coefficients to the derivatives of representation densities of hermitian forms. The result then follows from the main theorem of part I and a counting argument.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Special cycles on unitary Shimura varieties II: Global theory

Kudla

Rapoport

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

145

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“…In fact, using the nonvanishing result of these L-values, Li [1992] was able to prove some nonvanishing results for the cohomology of certain arithmetic quotients, which is an important and well-known application of the inner product formula. Kudla and Rallis [1994] extended the Siegel-Weil formula with great generality for symplectic-orthogonal pairs and Ichino [2004;2007] did that for unitary pairs using the similar idea of Kudla and Rallis. Now we can extend Rallis' original inner product formula "below the convergence line" (after regularization if necessary) which enables us to say some words about the global L-values at other points, especially the central point 1 2 .…”

Section: Introductionmentioning

confidence: 99%

Arithmetic theta lifting andL-derivatives for unitary groups, I

Liu

2011

ANT

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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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“…See [27] for a survey of recent work in this direction. For similar extensions in the unitary group case see Theorems 4.1 and 4.2 of [22] and Theorem 1.1 of [23], and a version for Hermitian orthogonal groups is Theorem 2 of [42].…”

Section: Automorphic Forms Eisenstein Series and Weil Representationsmentioning

confidence: 93%