We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport-Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan-Gross-Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic-geometric side of the arithmetic fundamental lemma proved by Rapoport-Terstiege-Zhang in the minuscule case.

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.

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and qanalogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the σ-centralizer group to the Mirković-Vilonen basis of a certain weight space of a representation of the Langlands dual group.Understanding of basic geometric properties of affine Deligne-Lusztig varieties has been fruitful for arithmetic applications. For instance an understanding of the connected components [CKV15] was applied to the proof of a version of Langlands-Rapoport conjecture by Kisin [Kis17]. The geometry of the supersingular locus of Hilbert modular varieties, which is also a question closely related to affine Deligne-Lusztig varieties via p-adic uniformizatoin, was applied to arithmetic level raising in the recent work of Liu-Tian [LT17].In this paper, we concern the problem of parameterizing the irreducible components of affine Deligne-Lusztig varieties. This problem was initiated in the work of Xiao-Zhu [XZ17]. These authors studied this problem in some special cases, as an essential ingredient in their proof of a version of Tate's conjecture for special fibers of Shimura varieties. After that, Miaofen Chen and Xinwen Zhu formulated a general conjecture, relating the set of top dimensional irreducible components of general affine Deligne-Lusztig varieties to the Mirković-Vilonen cycles in the affine Grassmannian, and thus to the representation theory of the Langlands dual group via the geometric Satake. Partial results towards this conjecture have been obtained by Xiao-Zhu [XZ17], Hamacher-Viehmann [HV17], and Nie [Nie18a], based on a common idea of reduction to the superbasic case (which goes back to [GHKR06]).In this paper we present a new method and prove:Theorem. The Chen-Zhu Conjecture (see Conjecture 1.2.1) holds in full generality.Our proof is based on an approach completely different from the previous works. The problem is to compute the number of top dimensional irreducible components of an affine Deligne-Lusztig variety (modulo a certain symmetry group). We use the Lang-Weil estimate to relate this number to the asymptotic behavior of the number of points on the affine Deligne-Lusztig variety over a finite field, as the finite field grows. We show that the number of points over a finite field is computed by a twisted orbital integral, and thus we reduce the problem to the asymptotic behavior of twisted orbital integrals. We study the latter using explicit methods from local harmonic analysis and representation theory, including the Base Change Fundamental Lemma and the Kato-Lusztig formula.An interesting point in our proof is that we apply the Base Change Fundamental Lemma, which is only available in general for mixed characteristic local fields as the current known proofs of it r...

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...

In this article, we develop an arithmetic analogue of Fourier-Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier-Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan-Gross-Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin-Selberg L-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case. YIFENG LIU B.2. Main theorem and consequences 58 B.3. Proof of Theorem B.4 60 Appendix C. Shimura varieties for hermitian spaces 64 C.1. Case of isometry 64 C.2. Case of similitude 66 C.3. Their connection 68 C.4. Integral models and uniformization 70 Appendix D. Cohomology of unitary Shimura curves 75 D.1. Oscillator representations of local unitary groups 75 D.2. Setup for cohomology of Shimura varieties 77 D.3. Statements for cohomology of unitary Shimura curves 78 D.4. Proof of Theorem D.6 81 References 85