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Local models for ramified unitary groups

TL;DR: In this paper, local models associated to certain Shimura varieties are studied and a resoultion of their singularities is presented, which is able to determine the alternating semisimple trace of the geometric Frobenius on the sheaf of nearby cycles.
Abstract: In this article, we study local models associated to certain Shimura varieties. In particular, we present a resoultion of their singularities. As a consequence, we are able to determine the alternating semisimple trace of the geometric Frobenius on the sheaf of nearby cycles.
Citations
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Journal ArticleDOI
TL;DR: In this article, a transfer homomorphism t : HK∗ (G∗) → HK(G) where G∗ is the quasi-split inner form of G is defined.
Abstract: Let G denote a connected reductive group over a nonarchimedean local field F . Let K denote a special maximal parahoric subgroup of G(F ). We establish a Satake isomorphism for the Hecke algebra HK of K-bi-invariant compactly supported functions on G(F ). The key ingredient is a Cartan decomposition describing the double coset space K\G(F )/K. As an application we define a transfer homomorphism t : HK∗ (G∗) → HK(G) where G∗ is the quasi-split inner form of G. We also describe how our results relate to the treatment of Cartier [Car], where K is replaced by a special maximal compact open subgroup K ⊂ G(F ) and where a Satake isomorphism is established for the Hecke algebra H K .

64 citations

Posted Content
TL;DR: The theory of local models of Shimura varieties has been surveyed in this paper, where the authors give an overview of the results on their geometry and combinatorics obtained in the last 15 years.
Abstract: We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.

58 citations


Cites methods from "Local models for ramified unitary g..."

  • ...The blowup M̃ locG,{µ},L occurring in (ii) is described explicitly by Krämer in [Kr] in terms of a moduli problem analogous to the Demazure resolution of a Schubert variety in the Grassmannian....

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  • ...Finally, we mention the papers by Gaitsgory [30], Haines and Ngô [42], Görtz [36], Haines [40, 41, 44], Krämer [57], Pappas and Zhu [82], Rostami [88], and Zhu [99] addressing the problem of determining the complex of nearby cycles for local models (Kottwitz conjecture)....

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  • ...Finally, we mention the papers by Gaitsgory [Ga], Haines and Ngô [HN1], Görtz [Gö3], Haines [H1, H2, HP], and Krämer [Kr], addressing the problem of determining the complex of nearby cycles for local models (Kottwitz conjecture)....

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  • ...The blowup M̃loc G,{μ},L occurring in (ii) is described explicitly by Krämer in [57] in terms of a moduli problem analogous to the Demazure resolution of a Schubert variety in the Grassmannian....

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Journal ArticleDOI
TL;DR: In this article, the authors constructed an arithmetic theta lift from harmonic Maass forms of weight 2 n to the arithmetic Chow group of the integral model of a unitary Shimura variety associated with unitary similitude groups of signature (n-1, 1, 1) by associating to a linear combination of Kudla-Rapoport divisors.
Abstract: We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature \((n-1,1)\). We construct an arithmetic theta lift from harmonic Maass forms of weight \(2-n\) to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form \(f\) a linear combination of Kudla–Rapoport divisors, equipped with the Green function given by the regularized theta lift of \(f\). Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of \(f\) against a CM cycle, and (2) the central derivative of the convolution \(L\)-function of a weight \(n\) cusp form (depending on \(f\)) and the theta function of a positive definite hermitian lattice of rank \(n-1\). When specialized to the case \(n=2\), this result can be viewed as a variant of the Gross–Zagier formula for Shimura curves associated to unitary groups of signature \((1,1)\). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

46 citations

Posted Content
TL;DR: In this article, the geometrical Satake isomorphism for a reductive group defined over F = k((t)), and split over a tamely ramified extension is proved.
Abstract: We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension As an application, we give a description of the nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models

45 citations

Journal ArticleDOI
Benjamin Howard1
TL;DR: In this paper, the intersections of special cycles on a unitary Shimura variety of signature (n-1,1) were studied and it was shown that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series.
Abstract: We study the intersections of special cycles on a unitary Shimura variety of signature (n-1,1) and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla and suggest a Gross-Zagier theorem for unitary Shimura varieties.

44 citations


Cites background from "Local models for ramified unitary g..."

  • ...The stack M is only flat and regular after inverting disc(K0), but Pappas [37] and Krämer [19] have modified the moduli problem defining M in order to obtain a flat and regular moduli stack which agrees with M over OK0 [disc(K0) −1]....

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  • ...It seems likely that the results of Lan’s thesis can be extended to give a compactification of the integral model of Pappas and Krämer, but the details have not been written down....

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  • ...Thus one expects to obtain a class in the generalized arithmetic Chow group of Burgos-Gil–Kramer–Kühn [4, 5]....

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  • ...stack M is only flat and regular after inverting disc(K0), but Pappas [37] and Krämer [19] have modified the moduli problem definingM in order to obtain a flat and regular moduli stack that agrees withM over OK0 [disc(K0)]....

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References
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Book
22 Dec 1995
TL;DR: In this paper, the relation of "p"-adic period domains to moduli space of arbitrary reductive groups is investigated, and nonarchimedean uniformization theorems for general Shimura varieties are established.
Abstract: In this monograph "p"-adic period domains are associated to arbitrary reductive groups. Using the concept of rigid-analytic period maps the relation of "p"-adic period domains to moduli space of "p"-divisible groups is investigated. In addition, non-archimedean uniformization theorems for general Shimura varieties are established.The exposition includes background material on Grothendieck's "mysterious functor" (Fontaine theory), on moduli problems of "p"-divisible groups, on rigid analytic spaces, and on the theory of Shimura varieties, as well as an exposition of some aspects of Drinfelds' original construction. In addition, the material is illustrated throughout the book with numerous examples.

555 citations


"Local models for ramified unitary g..." refers background in this paper

  • ...el strucure where K⊂G(Af) is an open and compact subgroup. If we fix a prime p, the investigation of a model S′ at pcan be reduced to the investigation of the associated “local model” M, introduced in [9] – at least if K= Kp.Kp where Kp ⊂G(Ap f) and Kp ⊂G(Qp) is a parahoric subgroup. In this work we will consider the case that Kp is a maximal parahoric subgroup of G(Qp). We further fix a prime p of OE ...

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  • ...model is and both schemes have the same singularities. We are going to study local models Mr,s of Shimura varieties where Gis a 1 1 INTRODUCTION 2 reductive group over Q such that G⊗Q R = GU(r,s). In [9] it is conjectured that Mr,s is flat over the base R. This is true if pis unramified by the work of Go¨rtz [4], but an argument of Pappas shows that this is not true in general: Using the facts that the...

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Book
01 Jan 1993
TL;DR: In this paper, Kitaoka provides an introduction to quadratic forms that builds from basics up to the most recent results, including lattice theory, Siegel's formula, and some results involving tensor products.
Abstract: The aim of this book is to provide an introduction to quadratic forms that builds from basics up to the most recent results. Professor Kitaoka is well known for his work in this area, and in this book he covers many aspects of the subject, including lattice theory, Siegel's formula, and some results involving tensor products of positive definite quadratic forms. The reader is required to have only a knowledge of algebraic number fields, making this book ideal for graduate students and researchers wishing for an insight into quadratic forms.

243 citations


Additional excerpts

  • ... point ythe alternating semisimple trace of Frobenius equals Trss(Frob q;RΨt(Ql)y) = z·(1 −q)+(|P(ΠΛ)(Fq)|−z) ·1 = z·(1 −q)+ qn −1 q−1 −z = qn −1 q−1 −z·q. There is an explicit formula for z(see e.g. [6], lemma 1.3.1): z= qn−1 −χqn 2 −1 +χqn 2 −1 q−1 ; with χ= 0 if nis odd and χ= 1 or −1 if nis even, depending on whether Fn q is hyperbolic with respect to v→ P v2 i or not. Combining these two results...

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Journal Article

94 citations


"Local models for ramified unitary g..." refers background or methods in this paper

  • ...While the naive local model M cannot be flat over Spec(R) if |r − s| > 1 , Pappas conjectures in [7] that M loc is flat for every pair (r, s) ....

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  • ...In [7], this was done by blowing up the singular locus....

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  • ...In fact, Pappas is able to show the flatness of M loc r,s if (r, s) = (n− 1, 1) (see [7], theorem 4....

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  • ...This proof is more or less a replication of Pappas’ proof in [7]....

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  • ...Pappas conjectures in [7] that M loc = M if |r − s| ≤ 1 and is able to prove this in the case (r, s) = (2, 1) (see [7] 4....

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