Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

Liu, Yifeng; Zhu, Yihang; Li, Chao

doi:10.4310/cjm.2021.v9.n1.a1

Cited by 10 publications

(7 citation statements)

References 35 publications

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“…For (4.9), by a similar argument for [Liu21, Theorem 5.22], the identity holds with replaced by . Then it follows by Corollary 2.66.…”

Section: Arithmetic Inner Product Formulamentioning

confidence: 80%

Chow groups andL-derivatives of automorphic motives for unitary groups, II.

Liu

2022

Forum of Mathematics, Pi

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In this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures.

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“…For (4.9), by a similar argument for [Liu21, Theorem 5.22], the identity holds with replaced by . Then it follows by Corollary 2.66.…”

Section: Arithmetic Inner Product Formulamentioning

confidence: 80%

Chow groups andL-derivatives of automorphic motives for unitary groups, II.

Liu

2022

Forum of Mathematics, Pi

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“…Now we formulate two semi-Lie version arithmetic transfer conjectures for any vertex lattice L using Kudla-Rapoport cycles Z(u) and Y(u) (u ∈ V) [5,27] on N = N U(L) . The semi-Lie version arithmetic transfer conjecture generalizes the arithmetic fundamental lemma conjecture in the set up of [30] to maximal parahoric levels. For g ∈ U(V)(F 0 ), the naive fixed pointed locus Fix(g) → N is poorly behaved.…”

Section: Introductionmentioning

confidence: 90%

“…sending a locally noetherian O E [∆ −1 ]-scheme S to the groupoid M 0 (S) of tuples (A 0 , ι 0 , λ 0 , η 0 ) where [30,Section C.3]. An isomorphism between two tuples is a quasi-isogeny preserving the polarization and…”

Section: Integral Models and Balloon-ground Stratificationmentioning

confidence: 99%

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Maximal parahoric arithmetic transfers, resolutions and modularity

Zhang¹

2021

Preprint

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For any unramified quadratic extension of p-adic local fields F/F 0 (p > 2), we formulate several arithmetic transfer conjectures at any maximal parahoric level, in the context of Zhang's relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture. The singularity is resolved via an arithmetic Atiyah flop using formal Balloon-Ground stratification. Then we do intersections and modify derived fixed points on the resolution.By a local-global method and double induction, we prove these conjectures for F 0 unramified over Qp, including the arithmetic fundamental lemma for p > 2. We introduce the relative Cayley map and also establish Jacquet-Rallis transfers of vertex lattices. We also prove new modularity results for arithmetic theta series at maximal parahoric levels via modifications over Fq and C. Along the way, we study the complex and mod p geometry of Shimura varieties and special cycles through natural stratifications.

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“…Remark 1.2.4. Theorem 1.2.3 is also used to prove the minuscule case of Liu's arithmetic fundamental lemma for Fourier-Jacobi cycles, see [Liu18,Appendix E].…”

Section: Introductionmentioning

confidence: 99%

Fine Deligne–lusztig Varieties and Arithmetic Fundamental Lemmas

Zhu

2019

Forum of Mathematics, Sigma

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We prove a character formula for some closed fine Deligne-Lusztig varieties. We apply it to compute fixed points for fine Deligne-Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport-Zink spaces arising from the arithmetic Gan-Gross-Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.Here the product is over pairs {Q, Q * } of monic irreducible non-self-reciprocal polynomials in F q 2 [λ] with non-zero constant terms. Theorem 1.2.1 then follows immediately from Theorem 1.2.3 and the explicit formula for the analytic side given in [RTZ13, Proposition 8.2]. Remark 1.2.4. Theorem 1.2.3 is also used to prove the minuscule case of Liu's arithmetic fundamental lemma for Fourier-Jacobi cycles, see [Liu18, Appendix E].Remark 1.2.5. In Theorem 5.2.4 we also establish an analogous arithmetic intersection formula for GSpin Rapoport-Zink spaces arising from the AGGP conjectures for orthogonal groups. This provides a new proof of the main result of [LZ18], and also removes the assumption that p n+1 2 in loc. cit. 1.3. Computing the arithmetic intersection. The starting point of the proof of Theorem 1.2.3 is the observation made in [LZ17, Proposition 4.1.2] that, in the minuscule case, the formal scheme (1.1.2) can be identified with the fixed point scheme Vḡ of an explicitly given smooth projective variety V over k, under a finite-order automorphismḡ. It also turns out that Vḡ is an Artinian scheme. Hence Int(g) is given by the k-length of Vḡ.In order to compute the k-length of Vḡ, there are two apparent approaches. One approach, taken in [LZ17], is to explicitly study all the local equations. The other approach, which we take in the current paper, is to compute it using the Lefschetz trace formula. Thus we obtainProof. This follows immediately from Proposition 2.5.1.2.6. Review of regular elements. We recall the definition of regular elements and some standard facts. Let G be a reductive group over k.Definition 2.6.1. An element g ∈ G is called regular, if the centralizer G g of g in G has dimension equal to the rank of G. The set of regular elements is denoted by G reg .If G is semi-simple, the above definition is the same as [Ste65]. In general, one easily checks that g ∈ G is regular in the above sense if and only if the image of g in G ad is regular. Thus we can easily transport the results from [Ste65], which only discusses semi-simple groups, to reductive groups.Theorem 2.6.2. An element g ∈ G is regular if and only if there are only finitely many Borel subgroups of G that contain g.Proof. This follows from [Ste65, Theorem 1.1] applied to G ad .Proposition 2.6.3. Assume G ′ is a reductive group over k that contains G as a closed subgroup. ThenProof. Fix a Borel subgroup B ′ ⊂ G ′ that contains B. By Theorem 2.6.2, it suffices to show that the natural map between flag varie...

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Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

Cited by 10 publications

References 35 publications

Chow groups andL-derivatives of automorphic motives for unitary groups, II.

Chow groups andL-derivatives of automorphic motives for unitary groups, II.

Maximal parahoric arithmetic transfers, resolutions and modularity

Fine Deligne–lusztig Varieties and Arithmetic Fundamental Lemmas

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