Arithmetic intersection on GSpin Rapoport–Zink spaces

Li, Chao; Zhu, Yihang

doi:10.1112/s0010437x18007108

Cited by 14 publications

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“…The AFL is proven in the case of dimension n 3; see [23]. In the subsequent work [18], Rapoport, Terstiege and Zhang verify the AFL for arbitrary n and the so-called minuscule group elements g. Their proof was later simplified by Li and Zhu [12,13] and He, Li and Zhu [8].…”

Section: Introductionmentioning

confidence: 95%

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Relative Unitary Rz-Spaces and the Arithmetic Fundamental Lemma

Andreas¹

2020

J. Inst. Math. Jussieu

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Definition 1.1. Let S be a scheme over Spf OȆ. 2 A hermitian O E -module over S is a triple (X, ι, λ) where X/S is a supersingular strict O E0 -module, ι : O E −→ End(X) an action and λ : X ∼ −→ X ∨ a principal polarization that is compatible with the Rosati involution, see Definition 3.1. The hermitian O E -module (X, ι, λ) over S is of signature (r, s) if, for all a ∈ O E , charpol(ι(a) | Lie(X))(T ) = (T − a) r (T − σ(a)) s ∈ O S [T ].Up to quasi-isogeny, there is a unique hermitian O E -module (X E0, (r,s) , ι, λ) of signature (r, s) over F.Definition 1.2. For an OȆ-scheme S, we denote by S := S ⊗ F its special fiber. Let N E0,(r,s) be the following set-valued functor on the category of schemes over Spf OȆ.To any S, we associate the set of isomorphism classes of quadruples (X, ι, λ, ρ), where (X, ι, λ) is a hermitian O E -module of signature (r, s) and whereProposition 1.3. The functor N E0,(r,s) is representable by a formal scheme which is locally formally of finite type and formally smooth of dimension rs over Spf OȆ.Two remarks are in order. First, if E 0 = Q p , then this moduli problem is of PEL-type in the sense of Rapoport and Zink, see [16]. By contrast, if E 0 = Q p , then this moduli problem is not covered by their book. This is due to the polarization λ, which is a polarization as a strict O E0 -module. We call N E0,(r,s) a relative Rapoport-Zink space since the underlying moduli problem is formulated in strict O E0 -modules as opposed to p-divisible groups.Second, the formal scheme N Qp,(1,n−1) uniformizes the supersingular locus in a certain unitary Shimura variety, see [18, Section 5]. Essentially, this follows directly from the moduli description of the Shimura variety in terms of abelian varieties. An analogous result is not known for the formal schemes N E0,(1,n−1) a priori.These two remarks motivate our main result from Part I, which we now state in a rather informal way. See Theorem 4.1 for the precise statement.Theorem 1.4. There exists an RZ-space N E0/Qp,(r,s) of PEL-type in the sense of [16] together with an isomorphismThis isomorphism is equivariant with respect to the unitary group acting on both sides.In particular, the RZ-space N E0/Qp,(r,s) is smooth over Spf OȆ. This is remarkable since we do not impose any conditions on the ramification behavior of E 0 /Q p . Instead, we impose a very specific Kottwitz condition for the moduli problem N E0/Qp, (r,s) . Namely, the Kottwitz condition has to be induced from the maximal unramified intermediate field Q p ⊂ E u 0 ⊂ E 0 at all but possibly one place ψ 0 : E u 0 ֒→Ȇ, see Definition 2.8. Our definition bears some similarity with the situation in [17, Equation (2.1)]. But note that the unramified intermediate field does not play a role in loc. cit. Instead, the authors impose the Eisenstein condition to get a regular moduli problem. A similar definition is made in [11]. Int(g) := len OȆ O Im(δ)∩Z(g) .Actually, Wei Zhang defines an intersection product for all regular semi-simple elements g ∈ U (J 1 ) rs . Then the schematic intersection ...

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Section: Introductionmentioning

confidence: 95%

“…Note that in these cases, g is also artinian. Their proof was subsequently simplified by Li and Zhu [12,13] and He, Li and Zhu [8].…”

Section: Orbital Integrals As Lattice Countsmentioning

confidence: 99%

Relative Unitary Rz-Spaces and the Arithmetic Fundamental Lemma

Andreas¹

2020

J. Inst. Math. Jussieu

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“…Here we recall the theory of GGP cycles on Rapoport-Zink spaces ("GGP" means Gan, Gross and Prasad). In [LZhu18], Li and Zhu defined the GGP cycles for codimension 1 embeddings of basic Rapoport-Zink spaces for GSpin(d, 2) with hyperspecial level. Moreover, they computed the intersection multiplicities of the GGP cycles in a special case.…”

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confidence: 99%

Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure

Oki¹

2020

Preprint

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In this article, we give a concrete description of the underlying reduced subscheme of the Rapoport-Zink spaces for spinor similitude groups with special maximal parahoric (and nonhyperspecial) level structure. Moreover, we give two applications of the above result. One of which is describing the structure of the basic loci of mod p reductions of Kisin-Pappas integral models of Shimura varieties for spinor similitude groups with special maximal parahoric level structure at p. The other is constructing a variant of the result of He, Li and Zhu, which gives a formula on the intersection multiplicity of the GGP cycles associated codimension 1 embeddings of Rapoport-Zink spaces for spinor similitude groups.

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“…Our new proofs are largely inspired by our previous work on arithmetic intersections on GSpin Rapoport-Zink spaces [LZ17]. The GSpin Rapoport-Zink spaces considered in [LZ17] are not of PEL type, which makes them technically more complicated. So the unitary case treated here can serve as a guide to [LZ17].…”

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confidence: 99%

“…The GSpin Rapoport-Zink spaces considered in [LZ17] are not of PEL type, which makes them technically more complicated. So the unitary case treated here can serve as a guide to [LZ17]. We have tried to indicate similarities between certain statements and proofs, for both clarity and the convenience of the readers.…”

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confidence: 99%

Remarks on the arithmetic fundamental lemma

Zhu

2017

Alg. Number Th.

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W. Zhang's arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport-Zink space. In the minuscule case, Rapoport-Terstiege-Zhang have verified the AFL conjecture via explicit evaluation of both sides of the identity. We present a simpler way for evaluating the arithmetic intersection number, thereby providing a new proof of the AFL conjecture in the minuscule case.

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Arithmetic intersection on GSpin Rapoport–Zink spaces

Cited by 14 publications

References 17 publications

Relative Unitary Rz-Spaces and the Arithmetic Fundamental Lemma

Relative Unitary Rz-Spaces and the Arithmetic Fundamental Lemma

Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure

Remarks on the arithmetic fundamental lemma

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