THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SIMPLE SHIMURA VARIETIES

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1 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SIMPLE SHIMURA VARIETIES PETER SCHOLZE Abstract. We show how the Langlands-Kottwitz method can be used to determine the semisimple local factors of the Hasse-Weil zeta-function of certain Shimura varieties. On the way, we prove a conjecture of Haines and Kottwitz in this special case. Contents 1. Introduction 2 2. A base change identity 5 3. The Geometry of some simple Shimura Varieties 7 4. Description of isogeny classes Calculation of the nearby cycles Orbital integrals for GL n Comparison with the Arthur-Selberg trace formula 20 References 23 Date: October 7,

2 2 PETER SCHOLZE 1. Introduction The aim of this paper is to extend the method used in [19] for determining the zetafunction of the modular curve to the case of the unitary group Shimura varieties with signature (1, n 1), considered e.g. in the book of Harris-Taylor, [12]. These are proper Shimura varieties for which endoscopy does not occur and for which particularly nice integral models exist by the theory of Drinfeld level structures. Our first main result is of a general geometric nature. Let O be the ring of integers in a local field K. Assume X/O is a separated scheme of finite type. Then we denote by X η ur the base-change of X to the maximal unramified extension K ur of K and by X O ur the base-change to the ring of integers in K ur. Also let X s be the geometric special fibre. We get maps ι : X s X O ur and j : X η ur X O ur. Now assume that X is regular and flat of relative dimension n over O and that the special fibre X s possesses a stratification X s = Z i i into locally closed strata Z i whose closures Z i are regular. We call the Z i special fibre components. Let c(z) be the codimension of Z in X, for any special fibre component Z in X. To any special fibre component Z, we associate a Q l -vector space W Z together with maps W Z W Z whenever c(z) = c(z ) + 1, by induction on c(z). If c(z) = 1, we simply set W Z = Q l. In general, we define W Z to be the kernel of the map Z Z c(z )=c(z) 1 W Z Z Z c(z )=c(z) 2 W Z, where in the case c(z) = 2, the target has to be replaced by Q l. We make the following assumption: ( ) The sequence 0 W Z... Z Z W Z Z Z c(z )=i c(z )=i 1 is exact for all special fibre components Z. W Z... Q l 0 Theorem A. Assume that the condition ( ) holds. Then for any x X s (F q ) with geometric point x over x, there are canonical isomorphisms (ι R k j Q l ) x = W Z ( k) for all k. Z,c(Z)=k x Z This generalizes Theorem A of [19], and again makes use of Thomason s purity theorem, [21], a special case of Grothendieck s purity conjecture. It is worthwhile noting that this theorem implies the (nontrivial) fact that the groups (ι R k j Q l ) x are pure (of weight 2k). Also, the description is analogous to results of Brieskorn, [3], Lemma 5, on the cohomology of complements of hyperplane arrangements. Indeed, one may think of the special fibre components of maximal dimension as hyperplanes, and the cohomology groups can be thought of as the cohomology groups of a small neighborhood of x in the complement of the geometric special fibre X s in X O ur. In order to formulate our other results, we need to introduce certain Shimura varieties. Fix an imaginary quadratic field k and a central division algebra D over k of dimension n 2, with an involution of the second kind. Further, let an R-algebra homomorphism h 0 : C D R

3 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 3 be given such that h 0 (z) = h 0 (z) and such that the involution x h 0 (i) 1 x h 0 (i) is positive. These data give rise to an algebraic group G over Q whose R-valued points are G(R) = {g (D Q R) g g R }, and a homomorphism h : R C/R G m G R. Taking any small compact open subgroup K G(A f ), one gets a smooth and proper Shimura variety Sh K associated to (G, h 1, K), canonically defined over k, although the reflex field may be even smaller, i.e. Q. We make the assumption that G = GU(1, n 1). It is well-known that under these assumptions the associated Shimura varieties Sh K have good integral models at primes p above primes p which split in k and such that D splits at p, if the level is given by a principal congruence subgroup. In fact, these integral models satisfy the hypotheses of Theorem A, as proved in Theorem 3.4 and Lemma 5.5. For any r 1 and any h Cc (GL n (Z p )), we define a function φ h = φ r,h in the Hecke algebra of GL n (Q p r), cf. Definition 5.8. Its values encode the semisimple trace of Frobenius on the vanishing cycles, cf. Theorem 5.9. Finally, let f n,p be the function of the Bernstein center of GL n (Q p ) such that for all irreducible admissible smooth representations π of GL n (Q p ), f n,p acts by the scalar p n 1 2 r tr ss (Φ r p σ π ), where σ π is the representation of the Weil group W Qp of Q p with values in Q l associated to π by the Local Langlands Correspondence, cf. [12]. Assume that the Haar measures give maximal compact subgroups measure 1. Theorem B. For all h Cc (GL n (Z p )), the functions φ h and f n,p h have matching (twisted) orbital integrals. This theorem furnishes a comparison between a purely geometric expression, φ h, and a purely automorphic expression, f n,p h. It is analogous to Theorem B in [19] for the case of modular curves, with some notable differences. Usually, one first compares φ h with a function in the Bernstein center of GL n (Q p r) and then invokes a base-change identity as in the case of a hyperspecial maximal compact subgroup. Our Theorem B does both steps at once. To get the comparison with the twisted orbital integrals of a function in the Bernstein center of GL n (Q p r), we prove the following base-change identity. In Section 2, we define a certain class of smooth group schemes G = G M,I over Z p, whose generic fibre is isomorphic to GL n. They include the case of principal congruence subgroups and parahoric subgroups. As a general piece of notation, we let Z(G, K) be the center of the Hecke algebra H(G, K) of compactly supported K-biinvariant functions on G, for any compact open subgroup K of G = GL n (Q p ) or G = GL n (Q p r). Theorem C. Let f Z(GL n (Q p ), G(Z p )) and φ Z(GL n (Q p r), G(Z p r)) be given. For every tempered irreducible smooth representation π of GL n (Q p ) with base-change lift Π, the set of invariants π G(Zp) is nonzero only if Π G(Z p r ) is nonzero. Assume that for all π with π G(Zp) 0, the scalars c f,π resp. c φ,π through which f resp. φ act on π G(Zp) resp. Π G(Z p r ), agree: c f,π = c φ,π. Then f and φ have matching (twisted) orbital integrals. Remark 1.1. In fact, we prove this theorem for any local field of characteristic 0 in place of Q p. This theorem generalizes, for the special case of the group GL n, the classical basechange fundamental lemma as well as the results of Haines about parahoric subgroups

4 4 PETER SCHOLZE G in [11]. The proof of Theorem C makes strong use of (unramified) base change for GL n and hence does not generalize to arbitrary groups. Using Theorem C, one arrives at an expression for the semisimple Lefschetz number as a sum of a volume factor times an orbital integral away from p times a twisted orbital integral of a central function in a certain Hecke algebra at p, cf. Theorem 7.4, procing a general conjecture of Haines and Kottwitz in this special case. Our last main theorem calculates the semisimple local factor of the Hasse-Weil zetafunction of Sh K /k in terms of automorphic L-functions. Recall that p is a prime of k lying above a rational prime p that splits in k and such that D splits at p. In particular, G Qp = GLn G m. Theorem D. Let K G(A f ) be any small compact open subgroup of the form K = K 0 p Z p K p GL n (Q p ) Q p G(A p f ) = G(A f ). As in [15], p.656, there is a representation r of the local L-group L G kp. semisimple local factor of Sh K at p is given by ζ ss p (Sh K, s) = π f L ss (s n 1 2, π p, r) a(π f )dimπ K f, Then the where π f runs over irreducible admissible representations of G(A f ) and where a(π f ) Z is given by a(π f ) = π m(π f π ) tr(f π ). Here m(π f π ) is the multiplicity of π f π in L 2 (G(Q)A G (R) 0 \G(A)), where A G is the split component of the center of G. Finally, f is the function defined by Kottwitz in [15]. It is up to sign a pseudo-character of a discrete series representation with trivial central and infinitesimal character. Remark 1.2. For a discussion of the semisimple local factor, we refer e.g. to [8], cf. also [19]. Note that under the assumptions, the local group G Qp = GLn G m. Therefore the local Langlands Correspondence for G Qp is known and hence one can define the semisimple L-factor L ss (s, π p, r) as the semisimple L-factor L ss (s, r σ πp ) of the Weil group representation σ πp associated to π p through the Local Langlands Correspondence, see e.g. [12]. However, unraveling all the definitions, one sees that one could define this L-factor directly, without appealing to the work of Harris-Taylor, cf. Remark 6.2. Note that Theorem D is essentially known by the work of Harris-Taylor, [12]. The emphasis of this paper lies in the method of proof. It is an extension of the method used by Kottwitz in [15] for a hyperspecial maximal compact level structure at p, and more classically by Langlands in [17] to prove a local-global compatibility statement for the cohomology of modular curves in the case that the local representation is not supercuspidal. In a similar spirit is the proof of Theorem 11.7 in [10] in the case that K p is an Iwahori subgroup (but without restriction on the signature of G) and the article of Haines and Rapoport, [9], for the case that the level structure at p is given by the pro-unipotent radical of an Iwahori subgroup. Here, as in [19], which discusses the case of GL 2, we show that this method can be applied for arbitrarily small level structures at p. In the paper [20], we use the methods and results of this paper to give a new proof of the Local Langlands Correspondence for p-adic local fields, avoiding the use of the numerical Local Langlands Correspondence of Henniart, [13]. We now describe the content of the different sections. In Section 2, we prove the basechange identity, Theorem C. It makes strong use of base-change for GL n. Afterwards, we

5 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 5 review in Section 3 the geometric results concerning the Shimura varieties, particularly their interpretation as moduli spaces of abelian varieties and a description of the special fibre. In Section 4, we recall some of the aspects of Kottwitz work on the description of isogeny classes in the special fibre of these Shimura varieties, for a hyperspecial maximal compact level structure at p. The main geometric result, Theorem A, is then formulated and proved in Section 5 along with the application to the Shimura varieties in question. We remark that the result is closely related to the cohomology of Drinfeld s upper half plane, as determined by Schneider-Stuhler, [18]. These results are then reinterpreted as orbital integrals in Section 6 to prove Theorem B, again making use of base-change for GL n. Finally, Section 7 concludes with the proof of Theorem D. Remark 1.3. Throughout the paper we fix a rational prime l different from p and an isomorphism Q l = C. Acknowledgments. I thank my advisor M. Rapoport for introducing me to this area, his constant encouragement during the process of writing this paper, and his interest in these ideas. 2. A base change identity Let K be a local field of characteristic 0 with ring of integers O K and uniformiser ϖ and let L be an unramified extension of K with ring of integers O L. Consider the group G = GL n over K. We give a construction of compact open subgroups of G(K) and G(L). Start with an O K -subalgebra M of M n (O K ) such that M K = M n (K). Further, take a (two-sided) ideal I M such that M/I is finite. Then one gets the smooth group scheme G over O K defined by G M,I (R) = {x I OK R g = 1 + x (M OK R) }. In particular, we get compact open subgroups G M,I (O L ), resp. G M,I (O K ), of G(L), resp. G(K). We write G M = G M,M. Let us give two examples. First, taking M = M n (O K ) and I = ϖ m M n (O K ), we get the principal congruence subgroups Second, one may fix a lattice chain and consider the subalgebra G M,I (O K ) = ker(gl n (O K ) GL n (O K /ϖ m O K )). Λ 0 = O n K Λ 1 Λ k = ϖ 1 O n K M = {m M n (O K ) mλ i Λ i for all i} M n (O K ). Then G M is the parahoric group scheme given by the lattice chain (Λ i ) i. We begin by proving the following comparison of conjugacy and σ-conjugacy classes. Proposition 2.1. There is a unique map N from the set of σ-conjugacy classes in G M (O L )/G M,I (O L ) to the set of conjugacy classes in G M (O K )/G M,I (O K ), satisfying the requirement that if δ G M (O L ) has the property δδ σ δ σr 1 G M (O K ), then the σ-conjugacy class of δ is sent to the conjugacy class of δδ σ δ σr 1. Moreover, the map N is a bijection, and the size of the σ-centralizer of some element δ G M (O L )/G M,I (O L ) equals the size of the centralizer of Nδ G M (O K )/G M,I (O K ). Proof. Let γ G M (O K ). Consider the subalgebra R γ = O K [γ] M. Note that this is a finite free O K -algebra, so that the functor Z γ (R) = (R γ OK R) describes a smooth commutative group scheme over O K.

6 6 PETER SCHOLZE Lemma 2.2. The norm homomorphism N : Z γ (O L ) Z γ (O K ) is surjective. δ δδ σ δ σr 1 Proof. The proof is identical to the proof of Lemma 3.4 of [19]. Hence, given γ G M (O K )/G M,I (O K ), choose some lift γ G M (O K ) and δ Z γ (O L ) with Nδ = γ, with reduction δ G M (O L )/G M,I (O L ). We claim that {x G M (O L )/G M,I (O L ) x 1 δx σ = δ} = {x G M (O K )/G M,I (O K ) x 1 γx = γ}. Take x G M (O L )/G M,I (O L ) with x 1 δx σ = δ. Then x σi δ σi x σi+1 = δ σi for all i = 0,..., r 1 and multiplying these equations gives x 1 Nδx = Nδ, hence x commutes with γ, i.e. x commutes with γ modulo I. But then x commutes with δ modulo I, because δ O L [γ], i.e. x commutes with δ. Therefore x 1 δx = δ = x 1 δx σ, and hence x = x σ, whence x G M (O K )/G M,I (O K ). The other direction is clear. This proves that the size of centralizers equals the size of σ-centralizers. Now a counting argument finishes the proof, cf. proof of Proposition 3.3 of [19]. We use this proposition to prove the following identity. Corollary 2.3. Let f be a conjugation-invariant locally integrable function on G M (O K ). Then the function φ on G M (O L ) defined by φ(δ) = f(nδ) is locally integrable. Furthermore, (e GM,I (O L ) φ)(δ) = (e GM,I (O K ) f)(nδ) for all δ G M (O L ). Proof. Assume first that f is locally constant, say invariant by G M,I (O K ) for some ideal I. Of course, φ is then invariant under G M,I (O L ) as well and in particular locally integrable. The desired identity follows on combining the above Proposition for the ideals I and I : Using it for I, we see that we may average over the σ-conjugacy class of δ. Then we get the sum over all δ G M (O L )/G M,I (O L ) which are congruent to δ modulo I of the averages of φ, resp. f, over the σ-conjugacy class of δ, resp. the conjugacy class of Nδ, which agree by the Proposition for I. The corollary now follows by approximating f by locally constant functions. Let tempered representations π, Π of G(K), G(L), resp., be given. σ Gal(L/K) be the lift of Frobenius. Further, let Definition 2.4. In this situation, Π is called a base-change lift of π if Π is invariant under Gal(L/K) and for some extension of Π to a representation of G(L) Gal(L/K), the identity tr(ng π) = tr((g, σ) Π) holds for all g G(L) such that the conjugacy class of Ng is regular semisimple. It is known that there exist unique base-change lifts by the work of Arthur-Clozel, cf. [1]. Write Z(G 1, G 2 ) for the center of the Hecke algebra of compactly supported G 2 -biinvariant functions on G 1, for any parameters G 1, G 2 occuring in the sequel.

7 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 7 Theorem 2.5. Let f Z(G(K), G M,I (O K )) and φ Z(G(L), G M,I (O L )) be given. For every tempered irreducible smooth representation π of G with base-change lift Π, the set of invariants π G M,I(O K ) is nonzero only if Π G M,I(O L ) is nonzero. Assume that for all π with π G M,I(O K) 0, the scalars c f,π resp. c φ,π through which f resp. φ act on π G M,I(O K ) resp. Π G M,I(O L ), agree: c f,π = c φ,π. Then f and φ have matching (twisted) orbital integrals. Proof. First, because (twisted) characters are locally integrable, cf. [1], Proposition 2.2, we find that Corollary 2.3 implies tr(e GM,I (O K ) π) = tr((e GM,I (O L ), σ) Π), taking f to be the character of π and δ = 1. This implies that π G M,I(O K) 0 only if Π G M,I(O L) 0. Further, we see that tr(f π) = c f,π tr(e GM,I (O K ) π) = c φ,π tr((e GM,I (O L ), σ) Π) = tr((φ, σ) Π). We may find a function f Cc (G(K)) that has matching (twisted) orbital integrals with φ, cf. [1], Proposition 3.1. This implies that tr((φ, σ) Π) = tr(f π) by the Weyl integration formula, cf. [1], p. 36, for the twisted version. Hence tr(f f π) = 0 for all tempered irreducible smooth representations π of G(K). By Kazhdan s density theorem, Theorem 1 in [14], all regular orbital integrals of f f vanish. Hence f and φ have matching regular (twisted) orbital integrals. By [5], Prop. 7.2, all semi-simple (twisted) orbital integrals of f and φ match. 3. The Geometry of some simple Shimura Varieties We briefly recall the construction of integral models for the Shimura varieties considered, as explained in [10], p As in the introduction, fix an imaginary quadratic field k and a central division algebra D over k of dimension n 2 and an involution on D of the second kind. Let G be the algebraic group over Q representing the following functor on Q-algebras R: G(R) = {g (D Q R) g g R }. Then the kernel G 0 of the map G G m sending g to g g is an inner form of the unitary groups associated to the extension k of Q. We assume that an R-algebra homomorphism h 0 : C D R is given such that h 0 (z) = h 0 (z) and the involution x h 0 (i) 1 x h 0 (i) is positive. By restriction to C, it gives rise to h : R C/R G m G R. One can describe the Shimura variety associated to (G, h 1 ) as a moduli space of abelian varieties, of the type considered by Kottwitz in [16], cf. also [10]. In the notation of that paper, let B = D op and V = D as a left B-module through right multiplication. Then one gets a natural identification C = End B (V ) = D. We are given h 0 : C D R = C R. Further, there is some ξ D with ξ = ξ such that the involution B on B = D op given by x B = ξx ξ 1 is positive: Simply take ξ close to h 0 (i) in D R to achieve the second condition. Then, defining the hermitian form (, ) : V V Q by (x, y) = tr k/q tr D/k (xξy ), where tr D/k is the reduced trace, we have (bx, y) = (x, b By) for all b B. One can further assume that the form (, h(i) ) is positive definite, where we recall that h(i) D acts by left multiplication on V.

8 8 PETER SCHOLZE To summarize, we have a simple Q-algebra B equipped with a positive involution B, a left B-module V equipped with a B -hermitian form (, ), and a homomorphism h 0 : C C R = End B (V ) R such that the form (, h 0 (i) ) is positive-definite. This is precisely the basic set-up of [16]. Using the homomorphism h 0, we get a decomposition of D C = V 1 V 2 under the action of C R C = C C. This decomposition is stable under right multiplication of D C = D k,v C D k,v C, where v, v : k C are the two embeddings. We make the assumption that as a right D C -module, V 1 is isomorphic to W (W ) n 1, where W and W are the simple right modules for D k,v C resp. D k,v C. Up to exchanging v and v, this is equivalent to asking G R to be isomorphic to GU(1, n 1). In section I.7 of the book [12], it is discussed how to achieve this situation. These assumptions imply that the cocharacter associated to h is given by µ : G m D C = D k,v C D k,v C = M n (C) M n (C) Then µ is already defined as a cocharacter µ(z) = diag(z, 1,..., 1) diag(1, z,..., z). µ : G m G k. Let p be a prime which splits in k as p = pp and such that D splits at p (and p). We fix isomorphisms k p = kp = Qp and D p = Mn (Q p ). This induces an isomorphism Under the induced isomorphism the cocharacter µ takes the form G Qp = GLn G m. G kp = GQp = GLn G m, µ(z) = diag(z, 1,..., 1) z. Lemma 3.1. The representation r of the local L-group L G kp = GLn (C) C W kp defined by Kottwitz in [15], p. 656, is given by r : GL n (C) C W kp GL n (C) (g, x, σ) (g 1 ) t x 1. Proof. It clearly satisfies both conditions (a) and (b) in [15], p It remains to fix the integral data. Note that the isomorphism D p = Mn (Q p ) gives a natural lattice M n (Z p ) M n (Q p ), which extends to a unique self-dual lattice Λ V Q p. The stabilizers of Λ give rise to Z (p) -orders O B, resp. O C, in B, resp. C. Locally at p, we have an isomorphism O B,p = Mn (Z p ) op. Using the involution B, this gives rise to an isomorphism O B Z p = Mn (Z p ) op M n (Z p ) op, such that B is given by (X, Y ) (Y t, X t ). Fix a compact open subgroup K p of G(A p f ). We consider the following moduli scheme.

9 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 9 Proposition 3.2. Let M be the functor from schemes over O kp = Zp to sets, which maps a locally noetherian scheme S to the set of equivalence classes of abelian varieties (A, λ, ι, η p ), where (1) A is projective abelian scheme over S up to prime-to-p-isogeny; (2) λ is a polarization of A of degree prime to p; (3) ι : O B End S (A) such that the Rosati involution restricts to B on O B and the Kottwitz condition det OS (T b LieA) = det D/k (T b V 1 ) holds for all b O B, where det D/k is the reduced norm for D/k. Note that this equation makes sense because O k is mapped to O S via O k O kp = Zp O S ; (4) η p is a level-k p -structure on A. Here, (A, λ, ι, η p ) and (A, λ, ι, η p ) are said to be equivalent if there exists an O B -linear isogeny α : A A of degree prime to p such that α (λ ) = cλ for some c Z (p) and α (η p ) = η p. Then M is representable by a smooth projective scheme M over Z p of relative dimension n 1, if K p is sufficiently small. Furthermore, if K = K p K p for the hyperspecial maximal compact subgroup then there is an isomorphism K p = GL n (Z p ) Z p GL n (Q p ) Q p = G(Q p ), Sh K k k p = M Ok p k p. Proof. This follows from [16], once one checks that ker 1 (Q, G) = 1. By the discussion of Section 7 of [16], this is automatic if n is even; for n odd, one has ker 1 (Q, Z(G)) ker 1 (Q, G), where Z(G) is the center of G. But, as explained there, for n odd, the center of Z(G) is isomorphic to Res k/q G m, which has trivial cohomology. Associated to A we get a p-divisible group X A, which decomposes under the action of O B Z p = Mn (Z p ) op M n (Z p ) op as X A = X n p X n p. Furthermore, X p is a p-divisible group of dimension 1 and height n because of the Kottwitz condition and the assumption of signature (1, n 1). Hence the notion of a Drinfeld level structure on X p makes sense. Proposition 3.3. Fix m 1. Let M Γ(p m ) be the Galois covering of M which parametrizes Drinfeld level-p m -structures on X p, i.e. sections P 1,..., P n : S X p such that X p [p m ] = [i 1 P i n P n ] (i 1,...,i n) (Z/p m Z) n as relative Cartier divisors. Then M Γ(p m ) is representable by a projective scheme M Γ(p m ), if K p is sufficiently small. Further, if K = K p K p, where K p is the compact open subgroup (1 + p m M n (Z p )) Z p GL n (Q p ) Q p = G(Q p ), then Sh K k k p = MΓ(p m ) Ok p k p. Proof. We refer to the discussion in the book of Harris-Taylor, [12], sections III.1 and III.4.

10 10 PETER SCHOLZE For any direct summand H of (Z/p m Z) n, let M H Γ(p m ) be the reduced subscheme of the closed subscheme of M Γ(p m ) where [i 1 P i n P n ] = H [e]. (i 1,...,i n) H (Z/p m Z) n The following theorem is known to the experts, and is easy to deduce from the explicit description of the completed local rings of M Γ(p m ) also used by Yoshida in [22]. Theorem 3.4. (i) For all H, the scheme M H Γ(p m ) is regular. (ii) The special fibre of M Γ(p m ) is the union of M H Γ(p m ) over all H 0. (iii) M H 1 Γ(p m ) MH 2 Γ(p m ) if and only if H 2 H 1. (iv) This induces a stratification M Γ(p m ) = H M H Γ(p m ) into locally closed strata M H Γ(p m ) = MH Γ(p m ) \ M H Γ(p m ). H H Proof. We may pass to the formal complection at a geometric point x of the special fibre. Then we get a deformation space of the p-divisible group X p of dimension 1 and height n: Deforming (A, ι, λ, η p, P 1,..., P n ) is the same as deforming (A[p ], ι, λ, P 1,..., P n ) by the Serre-Tate theorem and rigidity of level structures away from p. As λ : A[p ] = A[p ], this is the same as deforming (A[p ], ι Mn(Zp) op, P 1,..., P n ). But this reduces to deforming (X p, P 1,..., P n ), as A[p ] = Xp n through the action of ι Mn(Zp) op. Let X p = X inf X et be the decomposition into étale and infinitesimal part, where X inf has height k. Applying some element of GL n (Z/p m Z) if necessary, we can assume that P 1 =... = P k = e and P k+1,..., P n generate X et. Then the deformation space of (X p, P 1,..., P n ) maps to the deformation space of (X inf, P 1,..., P k ) and this map is formally smooth, as shown in [12], p.80. If x lies in M H Γ(p n ), then necessarily H (Z/p m Z) k 0 n k. Further, these strata M H Γ(p m ) are pullbacks of the corresponding strata for the deformation space of (X inf, P 1,..., P k ), as their equations only involve the deformation of X inf and P 1,..., P k. We are reduced to the case of the deformation of a 1-dimensional formal group X inf with Drinfeld-level-structure P 1,..., P k. By the results of Drinfeld, [6], Proposition 4.3, the deformation ring R of (X inf, P 1,..., P k ) is a complete regular local ring with parameters X 1,..., X k. There is a group structure + Σ on the maximal ideal m of R given by the universal deformation Σ of the formal group law of X inf, after the choice of a formal parameter on the universal deformation of X inf. We may assume that H = (Z/p m Z) j 0 n j (Z/p m Z) n for some j k. The condition H [e] = [i 1 P i j P j ] is equivalent to T pmj = (unit) i 1,...,i j Z/p m Z i 1,...,i j Z/p m Z (T ([i 1 ](X 1 ) + Σ... + Σ [i j ](X j ))) as power series in the formal variable T. It follows that all symmetric polynomials in the variables x i1,...,i j = [i 1 ](X 1 ) + Σ... + Σ [i j ](X j ) vanish, hence on the reduced subscheme all x i1,...,i j vanish. This implies X 1 =... = X j = 0 on the reduced closed subscheme, and conversely if X 1 =... = X j = 0, then H [e] = [i 1 P i j P j ]. Because

11 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 11 X 1,..., X k form a regular sequence, the closed subscheme given by X 1 =... = X j = 0 is regular and in particular reduced. This describes the closed subscheme corresponding to H locally and claims (i)-(iii) are clear. For claim (iv), it is enough to check that for any point x M Γ(p m ), there is a unique maximal H such that x M H Γ(p m ). This is also clear from our description. 4. Description of isogeny classes We recall some results from [16] that we will need. In that paper, Kottwitz associates to any point x M(F p r) a triple (γ 0, γ, δ) consisting of (1) a semisimple element γ 0 G(Q), elliptic in G(R) and well-defined up to G(Q)- conjugation; (2) an element γ G(A p f ), well-defined up to conjugacy; (3) an element δ G(Q p r), well-defined up to σ-conjugacy; satisfying certain compatibilities, e.g. the image of γ 0 in G(Q l ) is stably conjugate to the l-adic component of γ and the image of γ 0 in G(Q p ) is stably conjugate to Nδ. Roughly, these parametrize F p r-isogeny classes in M(F p r), cf. [16] for precise statements. Let us briefly recall how δ is defined, as this is what is used below, cf. [16], p Let x correspond to (A, ι, λ, η p ). The dual of the rational crystalline cohomology H p = Hom(H 1 cris(a/z p r), Q p r) has an action of B, a semilinear Frobenius F and a principal polarization λ. As shown in [16], p.430, there is an isomorphism of B-modules H p = V Q Q p r, preserving the hermitian forms up to a scalar. Choosing such an isomorphism, F is mapped to an endomorphism δσ for some skew-hermitian B-linear automorphism δ of V Q Q p r, i.e. δ G(Q p r), which is well-defined up to σ-conjugacy. In the following lemma, this is computed more directly. Lemma 4.1. Let x M(F p r). One has the associated p-divisible group X p,x and hence a (contravariant) rational Dieudonné module N x, abstractly isomorphic to Q n p r. Fixing such an isomorphism, the Frobenius operator F takes the form δ 0 σ, and under the isomorphism G Qp = GLn G m, the element δ is sent to ((δ 1 0 )t, p 1 ). Moreover, δ 0, resp. δ, is uniquely determined as an element of GL n (Q p r) M n (Z p r), resp. GL n (Q p r) Q p r, up to GL n(z p r)-σ-conjugation, resp. G(Z p r)-σ-conjugation. Remark 4.2. This stronger normalization of δ will be essential. Proof. Fix an isomorphism ι : N x Q n p r. Recall that the p-divisible group X A of the universal abelian variety A decomposes as X A = X n p X n p, equivariant for action of O B Z p = Mn (Z p ) op M n (Z p ) op, and the polarization λ gives an isomorphism λ : X n p = (X n p ). In particular, H p = Hom(N n x (N x ) n, Q p r), compatible with O B -action and polarization. Hence H p = Hom(N n x (N x ) n, Q p r) = M n (Q p r) M n (Q p r) = V Q Q p r

12 12 PETER SCHOLZE is compatible with the action of O B and the hermitian forms, up to the scalar p. The scalar p enters, because by the definition of N x, there is a perfect pairing of Dieudonné modules N x N x Q p r(1). This implies the first part of the lemma. One gets the stronger normalization of δ 0 by picking an isomorphism of the integral Dieudonné module of X p,x with Z n p r. 5. Calculation of the nearby cycles In this section we consider the following general situation. Let O be the ring of integers in a local field K. Let X/O be a separated scheme of finite type. Let X η ur be the base-change of X to the maximal unramified extension K ur of K and let X O ur be the base-change to the ring of integers in K ur. Also let X s be the geometric special fibre of X. Then we have ι : X s X O ur and j : X η ur X O ur. Let R IK be the derived functor of taking invariants under the inertia group I K. Lemma 5.1. For any sheaf F on the generic fibre X η, R IK (RψF) = ι Rj F η ur. Proof. Both sides are the derived functors of the same functor, cf. [19], Lemma 8.1. We now make further assumptions. We assume that X is regular and flat of relative dimension n over O and that the special fibre X s possesses a stratification X s = Z i i into locally closed strata Z i whose closures Z i are regular. We call the Z i special fibre components. Let c(z) be the codimension of Z in X, for any special fibre component Z in X. To any special fibre component Z, we associate a Q l -vector space W Z together with maps W Z W Z whenever c(z) = c(z ) + 1, by induction on c(z). If c(z) = 1, we simply set W Z = Q l. In general, we define W Z to be the kernel of the map Z Z c(z )=c(z) 1 W Z Z Z c(z )=c(z) 2 W Z, where in the case c(z) = 2, the target has to be replaced by Q l. We make the following assumption: ( ) The sequence 0 W Z... Z Z W Z Z Z c(z )=i c(z )=i 1 is exact for all special fibre components Z. W Z... Q l 0 Remark 5.2. By definition of W Z, this sequence is exact at the first step. It is not clear to the author whether the condition ( ) is automatic in general. Using a combinatorial result of Folkman on the homology of geometric lattices, [7], one can check that it is fulfilled if the reduced intersection of any two special fibre components is again a special fibre component. Note that in the general case, this intersection will only be a union of special fibre components. Finally, let x X s (F q ) be a point of the special fibre, with geometric point x X s (F q ) over x.

13 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 13 Theorem 5.3. Assume that ( ) holds. Then for all k, there are canonical isomorphisms (ι R k j Q l ) x = W Z ( k). Z,c(Z)=k x Z Proof. Let b Z : Z X be the closed embeddings, for all special fibre components Z. Let I be an injective resolution of the constant sheaf Q l on X. Consider the following complex 0... W Z b Z b! ZI... Z,c(Z)=i... Z,c(Z)=1 W Z b Z b! ZI ι I ι j j I 0. Proposition 5.4. The hypercohomology of this complex vanishes. Proof. We prove the proposition for any complex of injective sheaves I. This reduces the problem to doing it for a single injective sheaf I. Recall that for any injective sheaf I on a scheme X = U Z with j : U X an open and i : Z X a closed embedding, one gets a decomposition I = j I U i I Z for certain injective sheaves I U and I Z on U, resp. Z. In our situation, we get a decomposition of I as I = Z f Z I Z, where I Z is an injective sheaf on Z and where f Z : Z X is the natural locally closed embedding. But that the complex is exact for I = f Z I Z is a direct consequence of the condition ( ) on the vector spaces W Z. As in [19], proof of Theorem 8.2, this implies the Theorem, using cohomological purity, as proved by Thomason, [21]. As a corollary of Theorem 5.3, we can compute the semisimple trace of Frobenius on the nearby cycles in our situation. We start by analyzing the combinatorics of the situation that will arise. Recall from Theorem 3.4 that the special fibre components are parametrized by (nontrivial) direct summands of (Z/p m Z) n. The next lemma shows that the condition ( ) holds in this case. Note that if a group G acts on X preserving the stratification of the special fibre, then all vector spaces W Z acquire an action of the stabilizer of Z in G. In our case M Γ(p m ) F p = H 0 M H Γ(p m ), and the stabilizer of MH Γ(p m ) is GL(H) = GL k (Z/p m Z), if H = (Z/p m Z) k. Lemma 5.5. Let the rank of H be k. Then the representation W H of GL(H) on the Q l -vector space of functions f : {0 = H 0 H 1 H k = H} Q l on complete flags of direct summands of H satisfying f(h 0 H i H k ) = 0 H i direct summand H 0 H i 1 H i H i+1 H k =H

14 14 PETER SCHOLZE for all i = 1,..., k 1 is isomorphic to the Steinberg representation St H of GL(H) given by St H = ker Ind GL(H) B 1 Ind GL(H) B P P 1 where B GL(H) is a Borel subgroup, P GL(H) runs through parabolic subgroups and 1 denotes the trivial representation. Moreover, if H H is a direct summand of rank k 1, then there are natural transition maps W H W H defined by, f (H 0 H k 1 = H ) = f(h 0 H k 1 H k = H). Finally, the complex 0 W H... is exact. H H rank H =i W H H H rank H =i 1 W H... Q l 0 Proof. Under the standard identifications of the homogeneous spaces with spaces of flags, the first assertion is immediate. For the last assertion, assume that H = (Z/p m ) k, so that GL(H) = GL k (Z/p m ). Let P i GL k be the standard parabolic with Levi GL i GL k i for i = 1,..., k 1. Let St i be the Steinberg representation of GL i (Z/p m ). Then the complex gets identified with 0 St k Ind GL k(z/p m ) P 1 (Z/p m ) 1 St k 1 Ind GL k(z/p m ) P i (Z/p m ) 1 St i Ind GL k(z/p m ) P k 1 (Z/p m ) This is the complex of 1 + p m M k (Z p )-invariants in a corresponding complex 0 St k Ind GL k(q p) P 1 (Q p) 1 St k 1 Ind GL k(q p) P i (Q p) 1 St i Ind GL k(q p) P k 1 (Q p) of GL k (Q p )-representations. But note that Ind GL k(q p) P i (Q p) 1 St i is an extension of two irreducible representations π i 1 and π i of GL n (Q p ), cf. e.g. Lemma I.3.2 of [12]. Let d i denote the i-th differential of this complex. As all differentials of the complex are nonzero, one sees by induction that im d i+1 = π i = ker d i, whence the complex is exact. Corollary 5.6. The condition ( ) holds for the scheme M Γ(p m ), and W Z = W H if Z = M H Γ(p m ). In particular, let x M Γ(p m )(F q ) be a point such that the infinitesimal part of the associated p-divisible group X p,x has height k, i.e. x M H Γ(p m ) (F q) for some direct summand H (Z/p m Z) n of rank k. Fix an isomorphism H = (Z/p m Z) k. Then ( ) (R j I F Rψ MΓ(p m ) Ql ) x = Ind GL k(z/p m Z) P j,k (Z/p m Z) St j 1 ( j), where P j,k GL k is the standard parabolic with Levi GL j GL k j. Fix a positive integer r and let 1 k n. We define certain (virtual) representations. First, we define the virtual representation I 0 k = 1 1 p r k j=0 ( 1) j p rj Ind GL k(z p) P j,k (Z p) St j 1

15 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 15 of GL k (Z p ), where St j is the Steinberg representation of GL j (Z p ). Note that Ik 0 is selfdual. In the situation of the corollary, it follows from Lemma 7.7 of [19] that for all g GL k (Z/p m Z) we have tr ss (Φ p r g (Rψ MΓ(p m ) Ql ) x ) = tr(ge Γ(p m ) I 0 k ), where e Γ(p m ) is the idempotent associated to Γ(p m ) = 1+p m M n (Z p ). Moreover, the long exact sequence in Lemma 5.5 shows that Ik 0 can be rewritten as a Z-linear combination of representations: k 1 Ik 0 = j p rj ( 1) i Ind GL k(z p) P i,k (Z St p) i 1. j=0 We remark without proof that the virtual representation j ( 1) i Ind GL k(q p) P i,k (Q St p) i 1 i=0 i=0 is up to the sign ( 1) j the irreducible subquotient of Ind GL k(q p) B(Q p) 1 corresponding to {1,..., j + 1} {1,..., k} in the standard enumeration of the irreducible subquotients. These representations also show up in the cohomology of Drinfeld s upper half space, cf. [18]. Consider the standard parabolic P k GL n with Levi subgroup GL k GL n k and define the virtual representation I k = Ind GLn(Zp) P k (Z p) I 0 k C c (GL n k (Z p )) which carries a left action of GL n (Z p ) and a left action of g GL n k (Z p ) through multiplication by g 1 on the right. Next, we use these representations to define the test functions φ h whose twisted orbital integrals will appear in the formula for the Lefschetz number. We need to introduce some terminology. Definition 5.7. An element δ 0 GL n (Q p r) M n (Z p r) is said to be of height k if v p (δ 0 ) = 1 and δ 0 is GL n (Z p r)-σ-conjugate to an element (δ 1, δ 2 ) (GL k (Q p r) M k (Z p r)) GL n k (Z p r) such that Nδ 1 is elliptic. It is clear that if δ 0 is of height k, then δ 0 GL n (Z p r)diag(p, 1,..., 1)GL n (Z p r). On the other hand, it is easy to see that one-dimensional p-divisible groups X of height n over F p r are in bijection with GL n (Z p r)-σ-conjugacy classes δ 0 GL n (Z p r)diag(p, 1,..., 1)GL n (Z p r), and under this bijection, δ 0 is of height k if and only if the infinitesimal part of X has height k. In particular, this shows that any δ 0 GL n (Z p r)diag(p, 1,..., 1)GL n (Z p r) is of height k for a unique k = 1,..., n. Definition 5.8. Let h C c (GL n (Z p )). Let φ h C c (GL n (Q p r)) be the unique function with support in GL n (Z p r)diag(p, 1,..., 1)GL n (Z p r) and invariant under GL n (Z p r)-σconjugation, such that if δ 0 = (δ 1, δ 2 ) (GL k (Q p r) M k (Z p r)) GL n k (Z p r) is of height k, then φ h (δ 0 ) = tr(h Nδ 2 I k ). This function will take the place of the characteristic function, often called φ r, of GL n (Z p r)diag(p, 1,..., 1)GL n (Z p r)

16 16 PETER SCHOLZE whose twisted orbital integrals intervene in the proof of the Theorem D for the case that K p is a maximal compact subgroup. Recall that we associated an element δ 0 GL n (Q p r) M n (Z p r) to any point x M(F p r), well-defined up to GL n (Z p r)-σ-conjugation. Fix a geometric point x over x. We have the covering π m : M Γ(p m ) M and the sheaf F m = π mη Q l on the generic fibre of M Γ(p m ). Define the vector space (RψF ) x = lim (RψF m ) x. It carries a natural smooth admissible action of GL n (Z p ) and a commuting continuous action of Gal(Q p r/q p r). Finally, fix h C c (GL n (Z p )) and define a new function h C c (GL n (Z p )) by h (g) = h((g 1 ) t ). Theorem 5.9. In this situation, tr ss (Φ p r h (RψF ) x ) = φ h (δ 0 ). Proof. We can assume that h equals ge Γ(p m ) for some g GL n (Z/p m Z). Let us describe the fiber πm 1 (x) over x M(F p r). The Drinfeld level-p m -structures are parametrized by n-tuples (P 1,..., P n ) X et,x [p m ](F p r) generating X et,x [p m ](F p r). The action of Φ p r on πm 1 (x) is given by the action of Φ p r on X et,x [p m ](F p r), which is given by right multiplication with (Nδ 2 ) 1. Further, Lemma 7.7 of [19] says that tr ss (Φ p r (g 1 ) t (RψF m ) x ) = 1 1 p r tr(φ p r (g 1 ) t R IQp (RψF m ) x ). But one can rewrite R IQp Rψ M F m = R IQp Rψ M π mη Q l = π ms R IQp Rψ MΓ(p m ) Q l = π ms ι M Γ(p m ) Rj MΓ(p m ) Q l, because π m is finite. Here subscripts for Rψ indicate with respect to which scheme the nearby cycles are taken, and ι MΓ(p m ) and j MΓ(p m ) are as defined before Lemma 5.1, for the scheme M Γ(p m ). Now the theorem follows from Corollary 5.6 and the definition of I k. 6. Orbital integrals for GL n Fix an integer r 1 throughout this section. First, we construct the function f n,p which will turn out to have the correct orbital integrals. Lemma 6.1. There is a function f n,p of the Bernstein center for GL n (Q p ) such that for all irreducible smooth representations π of GL n (Q p ), f n,p acts by the scalar p n 1 2 r tr ss (Φ r p σ π ), where σ π is the representation of the Weil group W Qp of Q p with values in Q l associated to π by the Local Langlands Correspondence. Proof. The proof is identical to the proof of Lemma 9.1 in [19]. Remark 6.2. The definition as given needs the existence of the Local Langlands Correspondence. However, one can give a direct definition of f n,p, because it is easy to evaluate tr ss (Φ r p σ π ): If π is a subquotient of the normalized induction of a supercuspidal representation π 1 π t, then take the sum of π i (p r ) over all π i which are unramified characters. This is the definition that we are going to use.

17 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 17 Theorem 6.3. Let h Cc (GL n (Z p )). Then f n,p h and φ h have matching (twisted) orbital integrals. Remark 6.4. All Haar measures are normalized by giving a hyperspecial maximal compact subgroup measure 1. For the case h = e GLn(Zp), this is exactly the usual base change identity, used e.g. in [15]. Proof. Arguing as in the proof of Theorem 2.5, it is enough to check that for any tempered irreducible representation π of GL n (Q p ) with base-change lift Π, the equality tr(f n,p h π) = tr((φ h, σ) Π) holds. We begin with a computation of the right hand side. Let P k be the standard parabolic with Levi GL k GL n k and let N k be its unipotent radical. For any admissible representation π of GL n (Q p ) of finite length, let π Nk be its (unnormalized) Jacquet module with respect to N k. Assume that t π,k π Nk = πn 1 k,i πn 2 k,i i=1 as elements of the Grothendieck group of representations of GL k (Q p ) GL n k (Q p ). Let Θ Π be the distribution on GL n (Q p r) given by Θ Π (δ) = Θ π (Nδ), where Θ π is the character of π. Define Θ Π 1 in the same way. For any f C Nk,i c (GL n (Q p r)), we will write Θ Π (f) as tr((f, σ) Π), thinking of Π as the base-change lift of π. Lemma 6.5. In this situation, t n π,k tr((φ h, σ) Π) = p (n k)r tr((χ Bk, σ) Π 1 N k,i) tr(h Ind GLn(Zp) P k (Z p) Ik 0 π2 N k,i), k=1 i=1 where χ Bk is the characteristic function of the set B k of all elements δ of GL k (Q p r) M k (Z p r) with v p (δ) = 1 such that Nδ is elliptic. Proof. Let V k n be the set of δ GL n (Q p r) M n (Z p r) which are of height k. It is clearly enough to prove that t π,k tr((φ h χ V k n, σ) Π) = p (n k)r tr((χ Bk, σ) Π 1 N k,i) tr(h Ind GLn(Zp) P k (Z p) Ik 0 π2 N k,i). i=1 But by the twisted Weyl integration formula, cf. [1], p. 36, tr((φ h χ V k n, σ) Π) = W (T k T n k, GL k GL n k ) 1 T k GL k,t n k GL n k T k anisotropic 2 GL (Nt)T O n(q p) tσ(φ h )Θ π (Nt). T k (Q p r ) 1 σ T n k (Q p r ) 1 σ \T k (Q p r ) 1 T n k (Z p r ) Here T k (Q p r) 1 denotes the set of elements t T k (Q p r) with v p (det t) = 1. Also recall that 2 GL n(q p) (t) = det(1 Ad t Lie gl n) p. We can make several simplifications. First of all, writing t = (t 1, t 2 ) T k (Q p r) T n k (Q p r), t π,k Θ π (Nt) = Θ πnk (Nt) = Θ π 1 (Nt Nk 1)Θ,i π 2 (Nt Nk 2),i i=1 by a result of Casselman, [4]. We may use this result because of the conditions t 1 T k (Q p r) 1 with T k anisotropic and t 2 T n k (Z p r).

18 18 PETER SCHOLZE Second, 2 GL n(q p) (Nt) = p(n k)r 2 GL k (Q p) (Nt 1) 2 GL n k (Q p) (Nt 2), because the action of Ad Nt on the Lie algebra of N k is by multiplication through p, and on the opposite nilpotent Lie algebra by multiplication through p 1. Third, we have the following lemma. Lemma 6.6. Let φ Cc (GL n (Q p r)) have support in the elements which are of height k and assume that φ is invariant under GL n (Z p r)-σ-conjugation. Then for t = (t 1, t 2 ) GL k (Q p r) GL n k (Z p r) with Nt 1 elliptic, v p (det t 1 ) = 1, we have T O tσ (φ) = T O GL k GL n k tσ (φ), where the right-hand side is the twisted orbital integral on the Levi subgroup GL k GL n k. Proof. We apply Proposition 3.12 of [1] with P being the opposite parabolic of P k. We have to check that the constant term φ P of φ along P equals φ. Let N be the nilpotent radical of P. It is enough to see that for any GL k (Q p r) GL n k (Q p r)-σ-conjugate t of t and n N(Q p r), { φ(t φ(t n) = ) n N(Z p r) 0 else. This holds true, because both sides vanish unless t (GL k (Q p r) M k (Z p r)) GL n k (Z p r) and n N(Z p r), in which case, t and t n are easily seen to be GL n (Z p r)-σ-conjugate. If one defines h n k C c (GL n k (Z p r)) by h n k (δ 2 ) = tr(h Nδ 2 I k ), then inserting φ = φ h χ V k n in this lemma implies that for t = (t 1, t 2 ) T k (Q p r) 1 T n k (Z p r) as above, T O tσ (φ h ) = T O t1 σ(χ Bk )T O t2 σ(h n k ). Now, it is clear that the Weyl integration formula factors as a sum over i = 1,..., t π,k of products of the Weyl integration formulas for GL k (Q p r) and GL n k (Q p r). To finish the proof, we only need to show that tr(h n k Π n k ) = tr(h Ind GLn(Zp) P k (Z p) Ik 0 π n k) for any irreducible smooth representation π n k of GL n k (Q p ). If h n k C c (GL n k (Z p )) is defined by h n k (γ 2) = tr(h γ 2 I k ), then tr(h n k Π n k ) = tr(h n k π n k) by Corollary 2.3 and Proposition 2.1, taking M = M n (Z p ) and I small. We need to see that tr(h n k π n k) = tr(h Ind GLn(Zp) P k (Z p) I 0 k π n k). But for this we can replace Z p by Z/p m Z for m large and π n k by an irreducible representation of GL n k (Z/p m Z). Now, the computation is easy, using the orthogonality relations: #GL n k (Z/p m Z) tr(h n k π n k) = γ 2 tr(h γ 2 Ind GLn(Z/pm Z) P k (Z/p m Z) I 0,Γ(pm ) k C[GL n k (Z/p m Z)]) tr(γ 2 π n k ) = tr(h γ 2 Ind GLn(Z/pm Z) P k (Z/p m Z) I 0,Γ(pm ) k ( π n k π n k )) tr(γ 2 π n k ) γ 2 π n k = #GL n k (Z/p m Z) tr(h Ind GLn(Z/pm Z) P k (Z/p m Z) I 0,Γ(pm ) k π n k ),

19 THE LANGLANDS-KOTTWITZ APPROACH FOR SOME SHIMURA VARIETIES 19 where π n k runs through irreducible representations of GL n k(z/p m Z). Lemma 6.7. Let π be an irreducible smooth representation of GL n (Q p ) which is a subquotient of a parabolic induction from a supercuspidal representation π 1 π t with no π i being an unramified character. Then tr((χ Bn, σ) Π) = 0. If π is the trivial representation of GL n (Q p ), then tr((χ Bn, σ) Π) = 1. Proof. Consider the first assertion. Put h = e GLn(Zp) in Lemma 6.5. The left hand side vanishes, because φ h is bi-gl n (Z p r)-invariant and π, hence Π, is not unramified, e.g. by Theorem 2.5 for G = GL n. Further, for all k < n, we have by induction tr((χ Bk, σ) Π 1 N k,i) = 0. Hence also the term for k = n vanishes, which gives exactly the desired identity. The same inductive computation also works if π is the trivial representation, using that the left hand side is now the volume of GL n (Z p r)diag(p, 1,..., 1)GL n (Z p r), which can be determined to be 1+p r +...+p (n 1)r, e.g. by interpreting it as the number of neighbors of the vertex corresponding to GL n (Z p r) in the building of PGL n. It is enough to show that for all tempered irreducible representations π of GL n (Q p ) p n 1 2 r tr ss (Φ r p σ π )π = t n π,k k=1 i=1 p (n k)r tr((χ Bk, σ) Π 1 N k,i)ind GLn(Zp) P k (Z p) I 0 k π2 N k,i (1) as virtual representations of GL n (Z p ). We prove the theorem by induction on n, knowing that the equation (1) holds true for all irreducible smooth representations of GL n (Q p ) with n < n. Lemma 6.8. Assume that π is not necessarily irreducible, but is the parabolic induction of an irreducible smooth representation. Then equation (1) holds true. Proof. We first remark that the equation (1) makes sense for π of this form, because by Remark 6.2, tr ss (Φ r p σ π ) has a definitive meaning. Assume that π = Ind GLn(Qp) P m(q p) π π, where Ind denotes unnormalized induction. We remark that in this case, we have equalities in the Grothendieck group of admissible representations: π = Ind GLn(Qp) P m(q p) π π = n-ind GLn(Qp) P m(q p) π [ m n 2 ] π [ m 2 ] = Ind GLn(Qp) P m(q p) π [m] π [m n], where [x] means twisting by det x p and n-ind denotes normalized induction. In particular, tr ss (Φ r p σ π ) = p m n 2 tr ss (Φ r p σ π ) + p m 2 tr ss (Φ r p σ π ). Further, assume by induction that the equation (1) holds true for π and π. Note that by the restriction induction formula of Bernstein-Zelevinsky, [2], Lemma 2.12, we have π Nk = Ind GL k GL n k (Q p) (P m GL k GL n k )(Q p) π N k π +Ind GL k GL n k (Q p) (P n m GL k GL n k )(Q p) π N k [m] π [m n]+r, where the rest R is a sum of representations of the form π k π n k with π k properly induced. Let π N k = t π,k i=1 π 1 N k,i π 2 N k,i.

20 20 PETER SCHOLZE By induction, we have an equality of virtual representations of GL m (Z p ): = p m 1 2 r tr ss (Φ r p σ π )π m t π,k k=1 i=1 p (m k)r tr((χ Bk, σ) Π 1 N k,i)ind GLm(Zp) (P k GL m)(z p) I0 k π 2 N k,i. Hence, defining P k,m to be the parabolic with breakpoints k m, = = p 2n m 1 2 r tr ss (Φ r p σ π )π m t π,k k=1 i=1 m t π,k k=1 i=1 p (n k)r tr((χ Bk, σ) Π 1 N k,i)ind GLn(Zp) P k,m (Z p) I0 k π 2 N k,i π p (n k)r tr((χ Bk, σ) Π 1 N k,i)ind GLn(Zp) P k (Z p) I 0 k π2 N k,i, as virtual representations of GL n (Z p ), where π 1 N k,i = π 1 N k,i and π2 N k,i = Indπ 2 N k,i π. This definition is useful, since t π Ind GL k GL n k (Q p),k (P m GL k GL n k )(Q p) π N k π = πn 1 k,i πn 2 k,i expresses the first part of π Nk for k m. Note that for k > m, i=1 Ind GL k GL n k (Q p) (P m GL k GL n k )(Q p) π N k π is also of the form π k π n k with π k properly induced. Repeating this with π and π exchanged, we are left to show that any representation of the form π k π n k in π Nk with π k properly induced contributes trivially to (1). This follows from the fact that if π k is properly induced, then tr((χ Bk, σ) Π k ) = 0, as χ Bk is supported in elements whose norm is elliptic, and the character of π k vanishes on elliptic elements. Further, the equation (1) is trivial for representations which are subquotients of the parabolic induction of a supercuspidal representation π 1 π t with no π i being an unramified character, as both sides vanish. Hence we are left to check it for an unramified twist of the Steinberg representation, or equivalently by Lemma 6.8 for an unramified character χ det. The equation (1) reduces to n (1 + p r p (n 1)r )χ(p r )(χ det) = p (n k)r χ(p r )Ind GLn(Zp) P k (Z p) Ik 0 (χ det). k=1 We see that we may assume χ = 1. Then this is a trivial consequence of the definition of Ik 0 and the long exact sequence in Lemma Comparison with the Arthur-Selberg trace formula We now conclude the proof of Theorem D, i.e. the determination of the semisimple local factor of the Shimura varieties Sh K at the place p in terms of L-functions of automorphic forms, cf. the introduction for the precise statement. The main ingredients are the work of Kottwitz on the number of points of the Shimura variety over finite fields, cf. [16], with the refinements in the case at hand in [15], and the Arthur trace formula, which is very simple in our case, because our Shimura varieties are proper. Let H = ( 1) i H i et(sh K k Q p, Q l )