A quotient of the Lubin-Tate tower II

In this article we construct the quotient M_1/P(K) of the infinite-level Lubin-Tate space M_1 by the parabolic subgroup P(K) of GL(n,K) of block form (n-1,1) as a perfectoid space, generalizing results of the second author to arbitrary n and K/Q_p finite. For this we prove some perfectoidness results for certain Harris-Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze's candidate for the mod p Jacquet-Langlands and the mod p local Langlands correspondence.


INTRODUCTION
This article generalises the main results of [Lud17]. Let K/Q p be a finite extension with ring of integers O K , uniformizer ̟ and residue field k. Fix an algebraically closed and complete non-archimedean field C containing K. Let M 1 denote the infinite-level Lubin-Tate space over C. By work of Weinstein, M 1 is a perfectoid space equipped with an action of GL n (K). Let P ⊂ GL n be the parabolic subgroup consisting of upper triangular block matrices of block size (n − 1, 1). In this article we prove the following theorem.
Theorem A. The quotient M P (K) := M 1 /P (K) is a perfectoid space over C of Krull dimension n − 1.
The construction of the perfectoid structure on M P (K) follows the strategy via globalisation from [Lud17], where the quotient was constructed in the case when n = 2 and K = Q p . In that case, modular curves were used to globalise and one could rely on the perfectoidness results of [Sch15b]. For our generalisation we make use of the Shimura varieties studied by Harris-Taylor in their proof of the local Langlands correspondence for GL n [HT01], and this necessitates some new perfectoidness results.
Let us now describe the strategy of [Lud17] and this paper in slightly more detail; the reader may also consult the introduction to [Lud17]. The space M 1 has a GL n (O K )-equivariant decomposition M 1 ∼ = i∈Z M (i) 1 into pairwise isomorphic spaces (coming from the decomposition of the Lubin-Tate space at level 0 into connected components). As in [Lud17] we reduce the construction of M 1 /P (K) to the construction of M as an open subspace of a certain infinite level perfectoid Harris-Taylor Shimura variety X 1 . The image lands inside what we call the "complementary locus" X comp 1 , which is a subspace of X 1 defined in terms of the Hodge-Tate period map. We show that the quotient X comp 1 /P (O K ) exists and is perfectoid, and existence and perfectoidness of M (0) 1 /P (O K ) is then a direct consequence. The main ingredient of the proof is the construction of a perfectoid overconvergent anticanonical tower for our Harris-Taylor Shimura varieties (analogous to [Sch15b,Corollary 3.2.20]), and this forms the technical heart of this paper.
Theorem A has the following application. Let D × be the group of units in the central division algebra D over K with invariant 1/n. In [Sch15a], Scholze constructs a functor that is expected to be simultaneously related to a conjectural mod p local Langlands correspondence for the group GL n (K) and an equally conjectural mod p Jacquet-Langlands transfer between GL n (K) and D × . For any admissible smooth representation π of GL n (K) on a F p -vector space, Scholze constructs anétale sheaf F π on P n−1 using the Gross-Hopkins period morphism M 1 → P n−1 . The cohomology groups S i (π) := H í et (P n−1 , F π ), i ≥ 0, are admissible D × -representations which carry an action of Gal(K/K) and vanish in degree i > 2(n − 1) ([Sch15a, Theorem 1.1]). As an application of the construction of M P (K) we prove the following vanishing result.
Theorem B (Theorem 5.3.1). Let P * ⊂ GL n be a parabolic subgroup contained in P and let σ be a smooth admissible representation of P * (K) with parabolic induction π := Ind GLn(K) P * (K) σ to GL n (K). Then S i (π) = 0 for all i > n − 1.
This theorem generalises [Lud17,Theorem 4.6], which is the special case when n = 2, K = Q p and σ is a character.
Let us now describe the contents of this paper. Sections 2 and 3 are devoted to proving the perfectoidness results for the Harris-Taylor Shimura vartieties that we need. While it might be possible to deduce what we need from [Sch15b], certain technicalities made such an approach seem very cumbersome and unsatisfactory to us. We have therefore elected to construct the anticanonical tower in the Harris-Taylor setting directly, following the approach in [Sch15b] (simplified by the absence of a boundary). Scholze's approach relies on an integral theory of canonical subgroups and on the Hasse invariant, so we need a version of these notions for our Harris-Taylor Shimura varieties (which have empty ordinary locus in general). Section 2 develops a theory of µ-ordinary Hasse invariants and canonical subgroups for one-dimensional compatible Barsotti-Tate O K -modules G/S of height n, where S is a k-scheme. We use a Hasse invariant due to Ito [Ito,Ito06] which turns out to be perfect for adapting Scholze's approach to canonical subgroups based on Illusie's deformation theory for group schemes [Ill85]. We refer to Remark 2.2.4 for further discussion of the Hasse invariants used in this paper.
Equipped with a theory of canonical subgroups, Section 3 proceeds to construct the ǫ-neighbourhoods of the anticanonical tower in our setting. It is a tower of formal schemes ( X(ǫ) m,a ) m≥0 whose generic fibres X (ǫ) m,a embed into the adic Shimura varieties X U0(̟ m ) , where the level at the important prime is U 0 (̟ m ) := {g ∈ GL n (O K ) | g mod ̟ m ∈ P (O K /̟ m )}; we refer to the main body of the paper for precise definitions. In the limit we get a perfectoid space (Theorem 3.1.8).
This then allows us to prove the analogues of the main geometric results of [Sch15b], importantly including the construction of a Hodge-Tate period map π HT : X 1 → P n−1 (see Theorem 3.3.3). For this we have found it convenient to use the language of diamonds [Sch17], but we emphasise that the role the theory of diamonds plays in this paper is secondary. We end Section 3 by using the geometry of the Hodge-Tate period map to show that the quotient X comp 1 /P (O K ) of the complementary locus is perfectoid (Theorem 3.3.6).
Section 4 then uses the results of Section 3 to prove Theorem A and deduce some properties of the space M P (K) . The Gross-Hopkins period map plays a prominent role in the proofs, and it induces a quasicompact map π GH : M P (K) → P n−1 .
Finally, Section 5 proves Theorem B. The calculations follow the same path as Section 4 of [Lud17], the idea being that pushforward along the map π GH : M P (K) → P n−1 is a geometric realisation of the parabolic induction functor, soétale cohomology of F π on P n−1 is equal toétale cohomology of an analogously defined sheaf F σ on M P (K) . For the reader familiar with [Lud17], we mention that our argument deviates somewhat from that of [Lud17]. The most important point is that, by invoking a general result of Scheiderer [Sch92] on the cohomological dimension of spectral spaces, it suffices for us to relate theétale cohomology of F σ on M P (K) to an analytic cohomology group on M P (K) . In [Lud17] it was instead related to an analytic cohomology group on P 1 , which necessitated the study of the fibres of π GH . Moreover, to deal with the fact that σ will typically be infinite-dimensional, we use some additional limit arguments.
The authors wish to thank David Hansen for his interest in our work and for a number of discussions relating to it. The authors would furthermore like to thank Florian Herzig and Vytautas Paškūnas for helpful conversations. C.J. also wishes to thank Daniel Gulotta, Chi-Yun Hsu, Lucia Mocz, Emanuel Reinecke, Sheng-Chi Shih and especially Ana Caraiani for all discussions relating to [CGH + 18], which have had a large influence on this paper. C.J. was supported by the Herchel Smith Foundation for part of the work on this paper. J.L. was supported by the Max Planck Institute for Mathematics and the IWR Heidelberg.

HASSE INVARIANTS AND CANONICAL SUBGROUPS
2.1. Global setup. We start by introducing some notation which will be in place throughout the paper. Fix, once and for all, a prime p and an integer n ≥ 2. We also fix a finite extension K/Q p with ring of integers O K , uniformizer ̟, residue field k, ramification index e, and inertia degree f . Set q = p f . As in [HT01], we choose a totally real field F + of degree d, with primes v = v 1 , v 2 , . . . , v r above p, such that F + v ∼ = K (we fix such an isomorphism and think of it as an equality). We then choose an imaginary quadratic field E in which p splits as p = uu c , where c denotes complex conjugation, and let F = EF + ; this is a CM field. We let w i , i = 1, . . . , r, denote the unique prime in F above u and v i , and put w = w 1 .
Let us now recall the setup of [HT01], to which we refer for more details. Following [HT01, §I.7], we let B/F denote a central division algebra of dimension n 2 such that • The opposite algebra B op is isomorphic to B ⊗ F,c F ; • B is split at w; • at any place x of F which is not split over F + , B x is split; • at any place x of F which is split over F + , B x is either split or a division algebra; • if n is even, then the number of finite places of F + above which B is ramified is congruent to 1 + dn/2 modulo 2.
Choose an involution * of the second kind on B. Let V = B and consider it as a B ⊗ F B op -module. For any β ∈ B with β * = −β, we can define an alternating * -Hermitian where tr B/F denotes the reduced trace. We fix a β ∈ B with β * = −β. We define another involution # of the second kind on V by x # = βx * β −1 . We let G/Q be the reductive group with the functor of points (R any Q-algebra) The map (g, λ) → λ defines a homomorphism ν : G → G m (the similitude factor) and we denote its kernel by G 1 . If x is a prime in Q which splits as x = yy c in E, then y induces an isomorphism In particular, u induces an isomorphism We will assume (see [HT01,Lemma I.7.1] and the discussion following it; we assume that β is chosen so that this applies) that • if x is a prime in Q which does not split in E, then G × Q x is quasi-split; • the pairing (−, −) on V ⊗ Q R has invariants (1, n − 1) at one embedding F + ֒→ R and (0, n) at all other embeddings F + ֒→ R. Next, fix a maximal order Λ i = O B,wi ⊆ B wi for each i = 1, . . . , r. The pairing (−, −) gives a perfect duality between V wi := V ⊗F F wi and V w c i , and we let and (−, −) restricts to a perfect pairing Λ × Λ → Z p . We fix an isomorphism O B,w ∼ = M n (O K ), and we compose it with the transpose map to get an isomorphism O op is the idempotent which has 1 in the (1, 1)-entry and 0 everywhere else; ǫ induces an isomorphism [HT01, for further discussion). Let us now recall the integral models of the Shimura varieties for G; we refer to [HT01,§III.4] for more details. We remark that we will only need integral models in the case m 1 = 0 below, when the models are smooth, but we recall the definitions in the general case. If S is an O K -scheme and A/S is an abelian scheme with an injective homomorphism i : Fix a sufficiently small compact open subgroup U p ⊆ G(A ∞,p ) and a tuple m = (m 1 , . . . , m r ) ∈ Z r ≥0 . The moduli functor X m (we suppress U p from the notation) is defined as follows: If S is a connected locally noetherian O K -scheme and s is a geometric point of S, X m is the set of equivalence classes of (r + 4)-tuples (A, λ, i, η p , α i ) where • A/S is an abelian scheme of dimension dn 2 ; is canonically independent of the choice of s, and we get a functor on all locally noetherian O K -schemes by requiring that This functor is representable by a projective scheme over O K , which is smooth when m 1 = 0. By abuse of notation, we will denote it by X m . If m ′ ≥ m (by which we mean m ′ i ≥ m i for all i), then the natural map X m ′ → X m is finite and flat; moreover it isétale if m ′ 1 = m 1 . See [HT01,. We will denote the special fibre of X m by X m , and the generic fibre by X m . Over X m , we have a universal abelian scheme A m and the associated Barsotti-Tate O K -module G Am , which we will denote by G m or just G if the context is clear. One defines a locally closed subscheme X 2.2. Hasse invariants. In this section, we let S be a scheme over k and we let G/S be a compatible Barsotti-Tate O K -module of dimension 1 and height n (throughout this article, heights are O K -heights unless otherwise specified). We refer to [HT01,p. 59] for the notion of compatibility. The goal of this section is to define a so-called µ-ordinary Hasse invariant for G/S. The topic of generalized Hasse invariants has received a lot of attention recently. In the case of µ-ordinary Hasse invariants we mention the works [GN17,KW18,Her16,BH17]; moreover the works [GK16,Box15] construct generalized Hasse invariants on all Ekedahl-Oort strata (in the cases when they apply). In particular, µ-ordinary Hasse invariants have been defined in large generality (including the cases needed here) by Bijakowski and Hernandez [BH17]. We have nevertheless opted for a direct approach. It should be noted that 'Hasse invariants', as the term exists in the literature, are not unique. The definition given here is chosen because it is very well suited for adapting Scholze's approach to the canonical subgroup to the situation of our Harris-Taylor Shimura varieties, which is the topic of the next subsection. After writing a first draft of this section, we learnt that the definition of a µ-ordinary Hasse invariant we give here was first given by Ito [Ito,Ito06]. Since we are not aware of any detailed account of Ito's construction, we give our construction (it seems very likely that they are the same, judging from the sketch in [Ito]). Ito did not only construct a µ-ordinary Hasse invariant but also 'strata' Hasse invariants on Harris-Taylor Shimura varieties, and the construction below can easily be adapted to produce such Hasse invariants (see Remark 2.2.4).
We start with a description of some Dieudonné modules. Let κ be an algebraically closed field containing k and assume that S = Spec κ. Then we have G ∼ = G et × G 0 (étale and connected parts) and both G 0 and G et are Barsotti-Tate O K -modules. Let h be the height of G et , then 0 ≤ h ≤ n − 1. By the Dieudonné-Manin theorem, G et and G 0 are determined up to isomorphism by their Dieudonné modules. The Dieudonné module of G 0 is isomorphic to a Dieudonné module M n−h , which we now describe. We write W (κ) for the Witt vectors of κ, and σ for the lift of the p-th power Frobenius. M n−h has a Frobenius F and a Verschiebung V , and has a basis ω, F ω, F 2 ω, . . . , To finish the description, we need to describe F , and we know that it is σ-linear and it sends F i ω to F i+1 ω for i = 0, . . . , n − h − 1, so it remains to determine F n−h ω. For this, we write where T = Gal(k/F p ), K 0 is the maximal unramified subextension of K/Q p , and ι : O K0 ֒→ W (κ) is the lift of the inclusion k ⊆ κ. Then where a id = ̟ ⊗ 1 and a τ = 1 ⊗ 1 if τ = id. V is then defined uniquely by the condition with F acting as x ⊗ y → x ⊗ σ(y) on every factor. Taking the direct sum gives us the Dieudonné module of G.
Definition 2.2.1. Let S = Spec κ, where κ ⊇ k algebraically closed. We say that G is µ-ordinary if G et has height n − 1. For a general S/k and G/S, we say that G is µ-ordinary if G x is µ-ordinary for every geometric point x of S.
We now give an axiomatic definition of the µ-ordinary Hasse invariant. Here and elsewhere we use the following piece of notation: For any integer m ≥ 1, the twist G (q m ) is defined as the pullback of G along the absolute q m -th power Frobenius F q m : S → S. The relative q m -power Frobenius will be denoted by F r q m .
Definition 2.2.2. Let S/k be a scheme and let G/S be a one-dimensional compatible Barsotti-Tate O Kmodule of height n.
(1) If ̟ : G → G factors through F r q : G → G (q) , then we denote by V the unique isogeny G (q) → G such that V • F r q = ̟.
(2) In the situation in (1), V induces a pullback map V * : ω G → ω G (q) ∼ = ω ⊗q G on top differentials, which corresponds to an element H ∈ H 0 (S, ω q−1 G ). We define H to be the µ-ordinary Hasse invariant.
The following proposition shows that we have µ-ordinary Hasse invariants whenever S is reduced.
Proposition 2.2.3. Let S be a reduced scheme over k and G/S a one-dimensional compatible Barsotti-Tate O K -module of height n. Then the isogeny ̟ : G → G factors through the q-th power Frobenius isogeny F r q : G → G (q) .
Proof. The proposition is equivalent to showing that Ker F r q ⊆ Ker ̟ = G[̟]. Both are finite locally free subschemes of the finite locally free scheme G[q], so we are in the situation where we have a finite locally free scheme G over a reduced k-scheme S, and two finite locally free subschemes H, K ⊆ G and we want to show that H ⊆ K. We claim that it is enough to check this on geometric points.
To see this we argue as follows. First, it is enough to check it Zariski-locally on S. So without loss of generality S = Spec(A) is affine, and G = Spec(B) where A → B is projective; moreover H = Spec(C) and K = Spec(D) with A → C, D projective and B ։ C, D. Let I = Ker(B → C) and J = Ker(B → D); we want J ⊆ I. J and I are also projective as A-modules, so localising further on S we may assume that I, J, C, D are all free over A (which implies that B is free as well, since B ∼ = I ⊕ C ∼ = J ⊕ D). Choose a basis e 1 , . . . , e r , . . . , e t of B over A such that e 1 , . . . , e r is a basis of I, and choose another basis f 1 , . . . , f s , . . . , f t of B over A such that f 1 , . . . , f s is a basis for J. We can write a ji e i for unique a ji ∈ A. To check that J ⊆ I we need to check that a ji = 0 when 1 ≤ j ≤ s and i > r. But this can be checked at geometric points of S since S is reduced.
So, let us go back to our original situation. Let x : Spec(κ) → S be a geometric point. We need to show that ̟ : x . This follows from a direct calculation on the Dieudonné module. In fact, if h is the height of G et x , then F f (n−h) acts as ̟σ f (n−h) on the Dieudonné module of G 0 x and as σ f (n−h) on the Dieudonné module of G et x by the description of the Dieudonné modules above; this implies what we want.
Remark 2.2.4. The proof above works to give 'strata' Hasse invariants cutting out the Ekedahl-Oort strata, in the sense of [Box15,GK16]. These strata Hasse invariants were already defined by Ito [Ito,Ito06]. More precisely, assume that there are no points s of S where G et x has height > h. Then the proof above shows that there exists an isogeny V h : m is precisely the Ekedahl-Ort stratification in this case, moreover it is also equal to the Newton stratification). This was the main point of Ito's work, and some further properties and applications are stated in [Ito] in the case when F + = Q.
Moving on, we record some basic properties of our Hasse invariants.
Proposition 2.2.5. Let S/k be a scheme and let G/S be a one-dimensional compatible Barsotti-Tate O K -module of height n. Assume that the µ-ordinary Hasse invariant of G exists and denote it by H ∈ H 0 (S, ω q−1 G ).
(1) Let φ : S ′ → S be a k-morphism and let G ′ = G × S S ′ . Then the µ-ordinary Hasse invariant of G ′ exists and is equal to φ * H.
(2) Assume that S = Spec κ, where κ is an algebraically closed field. Then H = 0 if and only if G is µ-ordinary.
Proof. The first part follows from the fact that both F r q and ̟ are functorial, so the factorization For the second part, we note that H = 0 if and only if V isétale. Let h denote the height of G et ; by the calculation in the proof of Proposition 2.2.3 we see that ̟ factors through F r q (n−h) so we must have h = n − 1 for V to beétale. The calculation also shows that if h = n − 1 then V isétale, which is what we wanted.
In particular, we have a µ-ordinary Hasse invariant whenever G/S comes by pullback from some G ′ /S ′ with S ′ reduced, and the non-vanishing locus is precisely the open whose geometric points x are those for which G x is µ-ordinary.
Remark 2.2.6. We note a particular consequence of Proposition 2.2.5(1). Let G/S be a one-dimensional Barsotti-Tate O K -module of height n over a k-scheme S, and assume that the µ-ordinary Hasse invariant H(G) exists. Let m ≥ 1 and consider the q m -power Frobenius twist G (q m ) , which is the pullback of G under the absolute q m -th power Frobenius map F q m : S → S. Then Proposition 2.2.5(1) implies that Let us now return to the setting of our Shimura varieties. Recall X m , which is reduced and has the one-dimensional compatible Barsotti-Tate O K -module G on it, so we have a µ-ordinary Hasse invariant The following proposition is presumably well known to experts. We state it for completeness and sketch the proof, though it is not necessary for the main results of this paper.
Proposition 2.2.7. In the setting above, ω G is ample. As a consequence, X (n−1) m is affine.
Proof. When p is unramified in F and m = (0, . . . , 0) this is a special case of [LS12, Proposition 7.15], but the proof of that result also works when p is ramified in F , using that the models X m are smooth and defined by a Kottwitz condition when m 1 = 0. The case of general m then follows since the natural map X m → X (0,...,0) is finite and surjective.
Our goal in this section is to establish a theory of canonical subgroups for one-dimensional Barsotti-Tate O K -modules of height n, under the assumption that the Hasse invariant exists. We follow the approach of Scholze closely [Sch15b, 3.2.1], which relies on Illusie's deformation theory for group schemes [Ill72].
Let Q cycl p denote the completion of the p-power cyclotomic extension of Q p ; this is a perfectoid field. We let Z cycl p denote the ring of integers of Q cycl contains elements of valuation ǫ for any ǫ ∈ Q ≥0 of the form ae ′ /(p − 1)p n for a, n ∈ Z ≥0 (here we normalise the valuation so that ̟ has valuation 1); we will let ̟ ǫ denote such an element.
Proposition 2.3.1. Let R be a ̟-adically complete flat O cycl K -algebra. Let G be a finite locally free commutative group scheme over R and let C 1 ⊆ G 1 := G ⊗ R R/̟ be a finite locally free subgroup scheme. Assume that multiplication by ̟ ǫ on the Lie complexľ G1/C1 of G 1 /C 1 is homotopic to zero, where 0 ≤ ǫ < 1/2. Then there is a finite locally free subgroup scheme Proof. The proof of [Sch15b, Corollary 3.2.2] goes through verbatim (substituting ̟ for p).
Proposition 2.3.2. Let R be a ̟-adically complete flat O cycl K -algebra and let G be a one-dimensional compatible Barsotti-Tate O K -module of height n over R, with reduction G 1 to R/̟. Assume that the µ- Proof. The proof of [Sch15b, Corollary 3.2.6] goes through with only superficial changes; we sketch it for completeness. Fix m and set H 1 := Ker(V m : G (q m ) 1 → G 1 ) (which makes sense by assumption); then there is an exact sequence by definition. By Lemma 2.3.4 below, the Lie complex of H 1 is isomorphic tǒ We calculate the determinant of Lie V m to be H(G 1 ) q m −1 q−1 using Remark 2.2.6. Multiplication by the determinant Lie V m is then null-homotopic on the complex Lie G Lie V m −→ Lie G 1 (using the adjugate endomorphism of Lie V m as the chain homotopy), so multiplication by ̟ ǫ is null-homotopic using the as- The existence of C m then follows from Proposition 2.3.1. Uniqueness is a consequence of the final statement of the proposition, which is proved in the same way as the analogous part of [Sch15b, Corollary 3.2.6], using Lemma 2.3.3.
We have used the following two lemmas in the proof.
Proof. The proof of [Sch15b, Lemma 3.2.4] goes through, replacing p ǫ and p δ by ̟ ǫ and ̟ δ , respectively. Proof. We may identify Lie G 1 and Lie G we regard modules as complexes concentrated in degree 0). We have exact sequences By the remarks above, we have and hence a distinguished triangle A →ľ ≤0 We may then construct a commutative diagram for all large enough k ′ ≥ k, where the two unmarked vertical arrows are canonical and f then exists for abstract reasons (we remark that we can and do choose f to be independent of k ′ ). We claim that f is an isomorphism; it suffices to check this on cohomology groups in degrees 0 and 1 (all other cohomology groups vanish). Taking long exact exact sequences we get a commutative diagram (with exact rows) Now take the direct limit over k ′ in the bottom row. We have lim Proposition 2.2.1(c)(i)], and the maps Lie G and are both isomorphisms. This implies that H 0 (f ) and H 1 (f ) are both isomorphisms, which finishes the proof.
Remark 2.3.5. Morally, the Lemma above should be proven by taking the homotopy colimit of the trian- However, since homotopy colimits are poorly behaved, such an argument seems to require some work to carry out. The argument above may be viewed as an elementary workaround.
Definition 2.3.6. Let R be a ̟-adically complete flat O cycl K -algebra and let G be a one-dimensional compatible Barsotti-Tate O K -module of height n over R, with reduction G 1 to R/̟. We say that G has a weak canonical subgroup of level m if the µ-ordinary Hasse invariant H(G 1 ) exists and H(G 1 ) q m −1 q−1 divides ̟ ǫ for some ǫ < 1/2, and we then call the subgroup C m ⊆ G[̟ m ] (given by Proposition 2.3.2) the weak canonical subgroup of level m. If in addition H(G 1 ) q m divides ̟ ǫ , we call C m the (strong) canonical subgroup.
One then has the following analogue of [Sch15b, Proposition 3.2.8], which is proved by exactly the same arguments.
Proposition 2.3.7. Let R be a ̟-adically complete flat O cycl K -algebra, and let G and H be one-dimensional compatible Barsotti-Tate O K -modules of height n over R.
(1) If G has a (weak) canonical subgroup of level m, then it has a (weak) canonical subgroup of level Assume that G has a canonical subgroup C m1 of level m 1 , and that H = G/C m1 . Then H has a canonical subgroup D m2 of level m 2 if and only if G has a canonical subgroup C m1+m2 of level m 1 + m 2 . If so, there is a short exact sequence

PERFECTOID SHIMURA VARIETIES
In this section we prove our results about Harris-Taylor Shimura varieties. We first prove an analogue of Scholze's result [Sch15b, Corollary 3.2.19] that the 'anticanonical tower' for Siegel modular varieties is perfectoid at Γ 0 (p ∞ )-level; this is the main result of this section. Using this, we reprove results of Scholze [Sch15b] and Caraiani-Scholze [CS17] that the tower of Harris-Taylor Shimura varieties is perfectoid at full infinite level and admits a Hodge-Tate period map to P n−1 . For this, we follow Scholze's arguments for the Siegel case, but the situation is much simpler in our case due to the absence of a boundary. We also take advantage of the formalism of diamonds, which provide a good setting in which to carry out the arguments.
and U = U p U p (recall that we have fixed a sufficiently small compact open subgroup U p ⊆ G(A p,∞ ) throughout this article). We define a contravariant functor X U from locally noetherian K-schemes to sets as follows. If S is a connected locally Noetherian K-scheme and s is a geometric point of S, we define X U (S, s) to be the set of equivalence classes of (r + 4)-tuples (A, λ, i, η p , η i ) where • A is an abelian scheme over S of dimension dn 2 ; Before defining equivalence, let us define compatibility. The map i induces an action of E on Lie A, and we let Lie + A denote the summand of Lie A where E acts in the same way as via the structure morphism E → O S . We then say that (A, i) is compatible if Lie + A has rank n (over O S ) and the actions of F + on Lie + A via i and via the structure morphism Again the set X U (S, s) is canonically independent of the choice of s, giving X U on connected S, and one extends to disconnected S in the usual way. This functor is representable by a smooth projective K-scheme which we will also denote by X U . If m = (m 1 , . . . , m r ) and U vi = 1 + ̟ mi i O op B,wi , then X U is canonically isomorphic to the generic fibre X m of X m ; see [HT01,.
For the rest of this article, we will fix non-negative integers m 2 , . . . , m r and the corresponding compact open subgroups U vi = 1 + ̟ mi i O op B,wi for i = 2, . . . , r. We drop the levels U p , U vi , i = 2, . . . , r, and Z × p from all notation and only indicate the level at v. In particular, we write X m for what was previously called X (m,m2,...,mr) , etc. Let us now introduce the level subgroups U 0 (̟ m ) ⊆ GL n (K) that we will use to define the anticanonical tower. Let P ⊆ GL n denote the (n − 1, 1)-block upper triangular parabolic. We define, for m ≥ 0, Let us also put U (̟ m ) = 1 + ̟ m M n (O K ). Consider X U0(̟ m ) . It is the quotient of X m by the free action of the finite group U 0 (̟ m )/U (̟ m ) ∼ = P (O K /̟ m ). Since the level structure at w defining X m are isomorphisms α 1 : it follows that the level structure at w defining For the rest of this section we will base change all Shimura varietes X U to K cycl . We will now define some formal schemes whose generic fibres embed in the rigid analytification of X U0(̟ m ) (for suitable m). Set X := X 0 and let X be the formal completion of X ⊗ OK O cycl K along ̟. Recall our conventions about elements ǫ ∈ Q ≥0 and elements ̟ ǫ ∈ O cycl K from §2.3.
Definition 3.1.1. Assume that 0 ≤ ǫ < 1/2. Let X(ǫ) → X be the functor on ̟-adically complete flat O cycl K -algebras sending such an S to the set of equivalence classes of pairs (f, u), where f : Spf S → X is a morphism and and u ∈ H 0 (Spf S, Proposition 3.1.2. X(ǫ) is representable by a flat formal scheme over O cycl K which is affine over X.
Proof. It suffices to work Zariski locally on X, so let Spf R ⊆ X be an affine open over which ω q−1 is trivial. Choose a non-vanishing section η ∈ ω q−1 and choose a lift H ∈ H 0 (Spf R, ω q−1 ) of H. We claim that X(ǫ) × X Spf R is represented by Spf(R T /(T ( Hη −1 ) − ̟ ǫ )). The formal scheme Spf(R T /(T ( Hη −1 ) − ̟ ǫ )) represents pairs (f, u) with f : Spf S → Spf R a morphism and u ∈ H 0 (Spf S, (f * ω) 1−q ) such that u H = ̟ ǫ in S. There is a natural transformation from pairs (f, u) to equivalence classes of pairs (f, u) parametrized by X(ǫ) × X Spf R, and one shows that this is an isomorphism by the same argument as in [Sch15b, Lemma 3.2.13]. This shows that X(ǫ) is representable and is affine over X.
It remains to show that R T /(T ( Hη −1 ) − ̟ ǫ ) is flat over O cycl K , for which it suffices to show that it has no ̟ ǫ -torsion. Set A = R T and g = T ( Hη −1 ) − ̟ ǫ . Taking the long exact sequence of Since g = T (Hη −1 ) in A/̟ ǫ and Hη −1 is not a zero divisor in R/̟ ǫ , the assertion follows.
For any formal scheme whose notation involves X..., we will use X ... to denote its generic fibre, and X... the reduction modulo ̟. We record two corollaries.
Corollary 3.1.3. The reduction X(ǫ) represents the functor on O cycl K /̟-algebras sending such an S to the set of pairs f : Spec S → X and u ∈ H 0 (Spec S, (f * ω) 1−q ) such that uH = ̟ ǫ .
Proof. It suffices to prove this locally on X, so we pick an open affine Spf R ⊆ X and η trivialising ω q−1 as in the proof of Proposition 3.1.2. Then, by the proof, X(ǫ) is represented over then corresponds to a morphism R/̟ → S plus an element t ∈ S such that t(Hη −1 ) = ̟ ǫ ; setting u = tη −1 gives the desired element of H 0 (Spec S, (f * ω) 1−q ). One checks that this is independent of the choice of η, which finishes the proof.
Corollary 3.1.4. Let 0 ≤ ǫ < 1/2. Let S be a ̟-adically complete and flat O cycl K -algebra and let f : Spf S → X be a morphism. Assume that the reduction f : Then there exists a unique map g : Spf S → X(ǫ) lifting g such that the composition Spf S g −→ X(ǫ) → X is f .
Proof. The assertion is local on the target and the source, so we may use the local description of X(ǫ) from the proof of Proposition 3.1.2; we use the notation of that proof. The problem then becomes to prove the following: If h : R → S is an O cycl K -algebra homomorphism and u 0 ∈ S is an element such that u 0 h( Hη −1 ) ≡ ̟ ǫ modulo ̟ 1−ǫ , then there is a unique u ∈ S such that uh( Hη −1 ) = ̟ ǫ and u ≡ u 0 modulo ̟ 1−ǫ . For existence, write u 0 h( Hη −1 ) = ̟ ǫ + ̟ 1−ǫ v for some v ∈ S, then we can K -flat, existence shows that S is h( Hη −1 )-torsionfree, which implies uniqueness.
Remark 3.1.5. Note that the map X(0) → X is an open immersion; it identifies X(0) with the open subset {H = 0} of X. In particular, X(0) is formally smooth over O cycl K . Note also that, for any 0 ≤ ǫ < 1/2, the natural map X(0) → X(ǫ) (given by multiplying the section by ̟ ǫ ) is an open immersion, again identifying X(0) as the subset {H = 0} ⊆ X(ǫ). Similar remarks then apply modulo ̟, in particular X(0) is formally smooth over O cycl K /̟.
Let A be the universal abelian (formal) scheme over X, with pullback A(ǫ) to X(ǫ). We may define canonical subgroups of A(ǫ) whenever they exist for G A(ǫ) , as follows. Recall that we have a decomposition Here − ∨ denotes the Cartier dual. If G A(ǫ) has a (weak) canonical subgroup C m of level m, then we let m is the annihilator of C m with respect to the duality pairing. We say that D m is the (weak) canonical subgroup of A(ǫ). Note that D m modulo ̟ is the kernel of the qth power Frobenius on A(ǫ) (since A(ǫ)[w ∞ i ] isétale for i = 2, . . . , r).
Next, we note that there is a natural isomorphism X (q) ∼ = X over O cycl K /̟ (or any other base), since X comes by base change from k. Let F r = F r X/(O cycl K /̟) : X → X (q) be the relative (qth power) Frobenius map 1 ; note that the composition is the map coming from the abelian scheme A/ Ker F r A/X → X (with extra structures), where F r A/X is the relative Frobenius. We may then pull back this situation to X(ǫ) to obtain the following analogue of [Sch15b, Lemma 3.2.14].
Proof. That X (q) ∼ = X induces an isomorphism X(q −1 ǫ) (q) ∼ = X(ǫ) follows (for example) by explicit calculation in the local coordinates of the proof of Corollary 3.1.3, assuming in addition that the ring R/̟ in that proof as well as the non-vanishing section η comes by base change from k. It then follows that A(ǫ) pulls back to A(q −1 ǫ)/ Ker F r A(q −1 ǫ)/X(q −1 ǫ) via the map X(q −1 ǫ) → X(ǫ) since A pulls back to A/ Ker F r A/X via F r : X → X (with extra structures). Finally, one identifies the pullback of the universal section by explicit calculation in the local coordinates used in the first part of the proof.
We will abuse the terminology and write F r for the map X(q −1 ǫ) → X(ǫ), and refer to it as the relative Frobenius.
For finiteness, first note that the morphism is affine by construction. Finiteness of F then follows from the fact that F is finite modulo ̟ 1−ǫ , since it is the relative Frobenius of a morphism of finite presentation (see e.g. [Sta, Tag 0CCD] for the case q = p). To prove that the generic fibre is finite flat of degree q n−1 , we first do the case ǫ = 0. In this case X(0) is smooth of relative dimension n − 1 (Remark 3.1.5), so the relative Frobenius is finite and locally free of degree q n−1 (see e.g. [Ill96, Proposition 3.2] when q = p), and hence the same is true for F and its generic fibre. For general ǫ, the generic fibre is a finite surjective morphism between smooth rigid spaces, hence flat. To compute the degree, we use that the diagram is cartesian; then the right vertical morphism has the same degree as the left vertical morphism, which we already know has degree q n−1 .
We now turn to part (2). The existence of canonical subgroups C m of level m again follows from Proposition 2.3.2. The formula in the proposition then defines a morphism X (q −m ǫ) → X U0(̟ m ) by Proposition 2.3.7(4). To see that it is an open immersion, we consider the map π 2 : X U0(̟ m ) → X sending a pair (A,

→ X sends an abelian variety A (with extra structures) to A/A[p m ]
(with extra structures) by direct computation. It follows that the composition is equal to the forgetful map X (q −1 ǫ) → X (which is an open immersion) followed by an isomorphism of X (which only changes the level structures away from w), and is hence an open immersion. Since π 2 isétale, it follows that X (q −1 ǫ) → X U0(̟ m ) is an open immersion as desired.
The commutation of the diagram in (2) follows from Proposition 2.3.7. To see that it is cartesian we argue as follows. The horizontal maps are open embeddings, and the right vertical map is finiteétale of degree q n−1 . Since the left vertical map is finite flat of degree q n−1 by part (1), it follows that the induced map X (q −m−1 ǫ) → X (q −m ǫ) × X U 0 (̟ m ) X U0(̟ m+1 ) is a finite surjective morphism of degree 1 between smooth rigid spaces, and hence an isomorphism. In particular, F isétale, and the diagram is cartesian. This finishes the proof of (2).
For (3), we first need to establish that X (q −1 ǫ) → X U0(̟) has image inside X U0(̟) (ǫ) a . This is done as in the last part of the proof of [Sch15b, Theorem 3.2.15]. After this, we look at the diagram As in the proof of part (2), it commutes. We claim that it is cartesian; since the bottom horizontal arrow is the identity this gives the desired conclusion. The left vertical map is finite of degree q n−1 , and one checks that the right vertical map is finiteétale of degree q n−1 . An argument as in the proof of (2) then shows that the diagram is cartesian, and finishes the proof.
For the next result, which is the main result of this subsection, we use the notion X ∼ lim ← −i X i for an adic space X with a collection of compatible maps to a cofiltered inverse system of adic spaces (X i ) from [SW13, Definition 2.4.1].
For m ≥ 1 we define X U0(̟ m ) (ǫ) a as the image of X (q −m ǫ) in X U0(̟ m ) .
Theorem 3.1.8. Fix 0 ≤ ǫ < 1/2. There is a unique (affinoid) perfectoid space X P (OK ) (ǫ) a over K cycl such that Proof. We start by showing the existence of such a perfectoid space X P (OK ) (ǫ) a . By Theorem 3.1.7 we may identify the tower (X U0(̟ m ) (ǫ) a ) m≥0 with (X (q −m ǫ)) m≥0 , with transition maps given by F . This gives us a formal model ( X(q −m ǫ)) m≥0 for this tower, and we may take the inverse limit in the category of ̟-adic formal schemes since the transition maps are affine. We define X P (OK ) (ǫ) a to be the generic fibre of X ∞ in the sense of [SW13, §2.2]. Since the transition maps agree with Frobenius modulo ̟ 1−ǫ , we may argue as in the proof of [Sch15b, Corollary 3.2.19] to conclude that X P (OK ) (ǫ) a is perfectoid and that Finally, to show that X P (OK ) (ǫ) a is affinoid perfectoid, one may argue using tilts as in [Sch15b, Corollary 3.2.19, Corollary 3.2.20]. Since this additional information is not needed for the results of this paper we will not give further details.
3.2. The Hodge-Tate period map. We now introduce some notation for more general 'infinite level Shimura varieties'. These will be defined (a priori) as diamonds, and we refer to [Sch17] for the definitions and terminology concerning diamonds. Let H v ⊆ GL n (O K ) be a closed subgroup. We define where U v ranges through all the open subgroups U v ⊆ GL n (O K ) containing H v , and Y → Y ♦ is the 'diamondification functor' on rigid spaces [Sch17, Definition 15.5]. We remark that each X ♦ Uv is a spatial diamond, and that the inverse limits above exist (as diamonds) and are spatial by [Sch17,Lemma 11.22], which also says that the natural map is a homeomorphism, where |Y | denotes the underlying topological space of an adic space or a diamond [Sch17,Definition 11.14] (and the equality follows from [Sch17, Lemma 15.6]). Note that if H v = U v is open, our definition above is essentially saying that we will conflate X Uv with its corresponding diamond; this abuse of notation is mostly harmless since the diamondification functor is fully faithful on the category of normal rigid spaces (over a fixed nonarchimedean field, remembering the structure morphism). Thus, writing 1 ⊆ GL n (O K ) for the trivial subgroup, we have a diamond X 1 = lim ← − X Uv with an action of GL n (O K ), which extends to an action of GL n (K) by using the maps g : X gUv g −1 → X Uv for g ∈ GL n (K) and any open U v such that U v , gU v g −1 ⊆ GL n (O K ). Our goal is to show that a certain open subset X comp P (OK ) ⊆ X P (OK ) (containing X P (OK ) (ǫ) a for sufficiently small ǫ > 0) is perfectoid. To do this, we proceed from the previous subsection by going further up the tower. By Theorem 3.1.8 and [SW13, Proposition 2.4.5], we have and X P (OK ) (ǫ) a is naturally an open subdiamond of X P (OK ) .
Proposition 3.2.1. Let 0 ≤ ǫ < 1/2 and let H v ⊆ GL n (O K ) be a closed subgroup contained in P (O K ).
Then the spatial diamond X Hv (ǫ) a := X P (OK ) (ǫ) a × X P (O K ) X Hv is an (affinoid) perfectoid space.
Proof. First assume that H v has finite index inside P (O K ). Then X Hv (ǫ) a → X P (OK ) (ǫ) a is finiteétale, and the result then follows. In general X Hv (ǫ) a = lim has finite index, and the result follows.
To continue, we construct the Hodge-Tate period map X 1 → (P n−1 ) ♦ on diamonds; this is the content of the following proposition. We keep the statement vague; the meaning of the name 'Hodge-Tate period map' should be clear from the construction.
Proposition 3.2.2. There exists a GL n (K)-equivariant Hodge-Tate period map π HT : X 1 → (P n−1 ) ♦ over (K cycl , O cycl K ). Proof. By the definitions, we may regard X 1 and (P n−1 ) ♦ as sheaves on the pro-étale site of perfectoid spaces over K cycl , so to construct a map of sheaves it suffices to work with a basis for the topology. Let Spa(R, R + ) be a strictly totally disconnected perfectoid space over (K cycl , O cycl K ). A map Spa(R, R + ) → X 1 is the same as a compatible system of maps Spa(R, R + ) → X U(̟ m ) for all m, and we may assume that the map Spa of finite projective R-modules. By the compatibility of G R and the fact that it has dimension 1, defines an (R, R + )-point of P n−1 . This gives the desired map, and GL n (K)-equivariance is clear from the construction.
We remark that any map between spatial diamonds induces a spectral map of the underlying spectral topological spaces, so π HT is spectral. The next lemma characterises the image of the µ-ordinary locus under the Hodge-Tate period map. For more general results under the assumption that K/Q p is unramified, see [Her16,§11].
Lemma 3.2.3. Let C be a complete algebraically closed extension of K with valuation ring O C and residue field k C . Let G be a compatible Barsotti-Tate O K -module over O C of dimension 1 and height n. Then the special fibre G × OC k C is µ-ordinary if and only if the subspace Lie(G) ⊗ OC C(1) ⊆ T G ⊗ Zp C is K-rational (here T G is the Tate module of G).
Proof. We use the Scholze-Weinstein classification of Barsotti-Tate groups over O C [SW13, §5]. To simplify the notation, we will take the linear algebra data (T, W ) in the Scholze-Weinstein equivalence [SW13, Theorem 5.2.1] to be a finite free Z p -module T together with a C-subspace W ⊆ T ⊗ Zp C rather than a subspace of T ⊗ Zp C(−1) (from the point of view of Barsotti-Tate groups G, we take W to be Lie(G) ⊗ OC C(1) rather than Lie(G) ⊗ OC C). We start by assuming that the special fibre of G is µ-ordinary, and consider the connected-étale sequence and an equality Lie(G 0 ) ⊗ OC C = Lie(G) ⊗ OC C (since Lie(G et ) = 0), so it suffices to show that Lie(G 0 ) ⊗ OC C(1) ⊆ T G 0 is K-rational. Since the special fibre is µ-ordinary, G 0 has height 1 (using that the connected-étale sequence is compatible with reduction). But, by the Scholze-Weinstein classification [SW13, Theorem 5.2.1], there is a unique compatible Barsotti-Tate O K -module LT of dimension 1 and height n over O C , given by the linear algebra datum ( C τ and σ : K → C is the inclusion (recall that C was defined to be an extension of K), and this W is visibly K-rational. Note that LT is the unique lift of the Lubin-Tate O K -module of height 1 over k C .
For the converse, assume that (T = O n K , W ) is the linear algebra datum of G, assume that W is Krational and use the notation established in the previous paragraph. Write W = W K ⊗ K C with W K a K-rational structure on W . We can canonically identify T [1/p] = K n with the K-rational structure on (T ⊗ Zp C) σ and hence think of W K as a subspace of T [1/p]; the intersection W OK = W K ∩ T is then an O K -module direct summand of T of rank 1; let T ′ ⊆ T be a complement. It follows that we can write compatibly with the O K -action. It then follows from the Scholze-Weinstein equivalence that G is isomorphic to LT × (K/O K ) n−1 as a Barsotti-Tate O K -module, and hence has µ-ordinary reduction.
Let us now define P n−1 (K) a := (a 1 : · · · : a n ) ∈ P n−1 (O K ) | a n ∈ O × K . We then get the following corollaries.
Proof. By Lemma 3.2.3 the rank one points of π −1 HT (P n−1 (K)) are precisely the rank one points of the µ-ordinary locus X 1 (0), so it follows that π −1 HT (P n−1 (K)) is precisely the set of specializations of points in X 1 (0). Since X 1 (0) is a quasicompact open subset of X 1 , the set of such specializations is precisely X 1 (0). Moreover X 1 (0) is GL n (O K )-stable and P n−1 (K) is a GL n (O K )-orbit, so by equivariance of π HT the image of X 1 (0) has to be all of P n−1 (K). Finally, to deduce the corollary from this one checks easily that the anticanonical condition on a rank 1 point is equivalent to the image under π HT being in P n−1 (K) a , and then we argue similarly using that X 1 (0) a is also quasicompact and open.
Corollary 3.2.5. For every 0 < ǫ < 1/2 there exists a quasicompact open subset U ⊆ P n−1 containing P n−1 (K) a such that π −1 HT (U ) ⊆ X 1 (ǫ) a . Conversely, for every open subset V ⊆ P n−1 containing P n−1 (K) a , we have X 1 (ǫ) a ⊆ π −1 HT (V ) for all sufficiently small ǫ > 0. Proof. We may write P n−1 (K) a = U , where U runs through the quasicompact open subsets of P n−1 containing P n−1 (K) a . Fix ǫ > 0 small enough. We have X 1 (0) a ⊆ X 1 (ǫ) a , so by Corollary 3.2.4 we have X 1 (ǫ) a ⊇ π −1 HT (U ), and it follows (by a short argument using the constructible topology) that π −1 HT (U ) ⊆ X 1 (ǫ) a for some U since the π −1 HT (U ) are quasicompact opens (since π HT is spectral). This proves the first part, and the converse is proved in exactly the same way using the fact that X 1 (0) a = ǫ>0 X 1 (ǫ) a .
3.3. Perfectoid spaces. In this subsection we will prove the (global) perfectoidness results that we will need in this paper. We start with some remarks on the geometry of P n−1 , to set up notation. We have a cover of P n−1 by open affinoid subsets V i = {(a 1 : · · · : a n ) | |a j | ≤ |a i | j = i}.
Note also that the V i are translates of one another under the action of the Weyl group of GL n (with respect to the diagonal torus). We have a similar 'algebraic' cover by open subsets V i = {(a 1 : · · · : a n ) | |a i | = 0}.
Let γ = diag(̟, . . . , ̟, 1) ∈ GL n (K). We then have the following elementary lemma. Recall that we are using the right action of GL n on P n−1 which is the inverse of the usual left action. Lemma 3.3.1. We have V n = k≥0 V n γ k , and the sets V n γ −k , k ≥ 0, form a basis of quasicompact open neighbourhoods of (0 : . . . 0 : 1) ∈ P n−1 .
Proof. By Corollary 3.2.5 and the second part of Lemma 3.3.1 we can choose a U , and ǫ > 0 and a k ≥ 0 such that π −1 HT (U ) ⊆ X 1 (ǫ) a ⊆ X comp 1 and V n γ −k ⊆ U . The first assertion of this corollary then follows from the first part of Lemma 3.3.1 (using the equivariance of π HT ), and the second part of the corollary is immediate from the first and Proposition 3.2.1.
As an aside, which won't be used in this paper, we note the following theorem.
Theorem 3.3.3. X 1 is a perfectoid space and π HT comes from a unique map X 1 → P n−1 of adic spaces.
Proof. The fact that X 1 is a perfectoid space follows Corollary 3.3.2 and the fact that (which is immediate from equivariance of π HT and P n−1 = V 1 ∪ · · · ∪ V n ). The second part then follows immediately, since any map of diamonds from a perfectoid space S to the diamond Z ♦ of a rigid space Z corresponds to a unique map of adic spaces S → Z, by the definition of the diamondification functor.
We now turn to the main result of this section. The natural map |X 1 | → |X P (OK ) | is open, so we may define X comp P (OK ) ⊆ X P (OK ) to be the open subdiamond given as the image of |X comp 1 |. Note that X comp 1 is P (O K )-stable. From the next lemma, we see that X comp Proof.
. Then X U ′ v,m → X Uv,m is a U v,m /U ′ v,m -torsor, compatibly in m. Diamondification preserves torsors by finite groups, so we have compatible isomorphisms for all m. Taking the inverse limit over m then gives the result.
Let us recall that Huber defined the category of adic spaces as a full subcategory of a category he called V in [Hub94]. This category has quotients by arbitrary group actions, cf. [Lud17, §2.2]. Let us explicitly record the following link between torsors and group quotients in V, in the case of perfectoid spaces. Write U K = Spa(R K , R + K ) and U = Spa(S, S + ). If π is a pseudouniformizer for R we then have (R + K /π m ) H/K = a R + /π m for all m, compatibly in K (= a for almost equal). This implies that S H = R and that (R  Proof. We have an isomorphism we have a natural finiteétale map X γ k P (OK )γ −k (ǫ) a → X P (OK ) (ǫ) a . It follows that X γ k P (OK )γ −k (ǫ) a is (affinoid) perfectoid, and hence that the diamond corresponding to |X 1 (ǫ) a γ k |/P (O K ) is (affinoid) perfectoid. This proves the theorem, except for the 'moreover' part, which then follows from Lemma 3.3.5 since X comp

THE LUBIN-TATE TOWER
In this section we prove our geometric results on the Lubin-Tate tower.
4.1. Preliminaries. We begin by recalling the Lubin-Tate spaces that we will be working with, cf. [GH94,RZ96]. Let G 0 be the unique one-dimensional compatible Barsotti-Tate O K -module of O K -height n and with G et 0 = 0 over k, and setK = K ⊗ W (k) W (k). The Lubin-Tate space M is the formal scheme over OK whose R-points, for R an OK-algebra with ̟ nilpotent, is the set of pairs (G, ρ) where G is a one-dimensional compatible Barsotti-Tate O K -module over R and ρ : H similarly). We have two period maps; the Gross-Hopkins period map π GH : M GLn(OK ) → P n−1 and the Hodge-Tate period map π HT : M 1 → P n−1 . The map π GH isétale, surjective and admits local sections 3 . Moreover, the composite is a GL n (K)-torsor in the sense of [Sch17, Definition 10.12]. The image of the Hodge-Tate period map π HT is the Drinfeld upper halfspace Ω n−1 ⊆ P n−1 .
We now relate our Lubin-Tate spaces to the Shimura varieties from the previous section. We use the notation and conventions of the previous sections freely, except that we will base change all analytic adic spaces to a complete and algebraically closed non-archimedean field extension C of K (e.g. . We also have a compatibility between the local and global Hodge-Tate period maps: Composing the immersion M 1 ֒→ X 1 and the global π HT : X 1 → P n−1 gives the local π HT : M 1 → P n−1 . Since the Drinfeld upper halfspace Ω n−1 is contained in the 'complementary locus' V n ⊆ P n−1 from Subsection 3.3, we obtain M    [Lud17,§3.6] in the case n = 2, F = Q p . To clarify, the quotient above is taken in the category V; this makes sense since M 1 is a perfectoid space (using Proposition 4.1.1 and the GL n (K)-action). Set This is the kernel of the homomorphism GL n (K) → Z given by g → v K (det(g)), where v K is the normalised valuation on K, and this homomorphism is split. Moreover, for g ∈ GL n (K), one has M (0) 1 .g = M (vK (det(g))) 1 by looking at the degree of the quasi-isogeny. From this we see that G ′ is the stabiliser of the component M  Since the Gross-Hopkins period map M 1 → P n−1 is GL n (K)-equivariant for the trivial action on the target, it factors over M 1 → M P (K) ; we write π GH : M P (K) → P n−1 for this factorization. We get the following generalization of [Lud17, Proposition 3.15], by exactly the same proof.
Proof. The proof that π GH is quasicompact is identical to the proof of the special case [Lud17, Proposition 3.15] when n = 2 and K = Q p ; we recall it briefly since the argument also proves that M P (K) is quasiseparated. In short, since π GH has local sections, P n−1 is covered by quasicompact open subsets V for which there exists an open U ⊆ M (0) such that π GH | U is an isomorphism onto V . By the argument in the proof of Theorem 4.2.1, which is quasicompact, so π GH is quasicompact (and hence so is M P (K) since P n−1 is quasicompact). To show that M P (K) is quasiseparated we first show that π −1 GH (V ) is qcqs. To see this, note that pr −1 (U ) is an inverse limit of qcqs spaces, hence qcqs and therefore a spectral space. It then follows that the quotient π −1 GH (V ) ∼ = pr −1 (U )/P (O K ) is a spectral space by [BFH + 18, Lemma 3.2.3], so in particular qcqs. The intersection of two such subsets of M P (K) is also quasicompact (π −1 Thus we have shown that |M P (K) | is a spectral space. We will also need the fact that it has Krull dimension n − 1, i.e., that the supremum of all lengths k of generalizations x 0 ≺ · · · ≺ x k is equal to n − 1. To make the proof transparent, we record a few simple observations on Krull dimensions.
Lemma 4.2.3. Let X and Y be locally spectral spaces.
(1) If X is a cofiltered inverse limit lim ← −i X i of locally spectral spaces, then dim X ≤ sup i dim X i .
(2) If f : X → Y is a surjective and generalizing continuous map, then dim X ≥ dim Y .
Proof. We start with (1). Write q i : X → X i for the natural map. If x 0 ≺ · · · ≺ x n is a chain of distinct generalizations in X, then q i (x 0 ) · · · q i (x n ) is a chain of generalizations in X i for any i, and the q i (x j ) will be distinct for some i. This proves (1).
For (2), let y 0 ≺ · · · ≺ y m be a chain of distinct generalizations in Y . Then we can lift y 0 to a point x 0 ∈ X by surjectivity of f , and then successively lift the y i , i ≥ 2, using that f is generalizing, to obtain a chain x 0 ≺ · · · ≺ x m in X, proving (2).
Proposition 4.2.4. |M P (K) | is a spectral space of Krull dimension n − 1.
Proof. Since M 1 is an inverse limit of rigid analytic varieties of dimension n− 1, it has dimension ≤ n− 1 by Lemma 4.2.3(1). Applying Lemma 4.2.3(2) to the surjective and generalizing 4 maps M 1 → M and M 1 → M P (K) , we see that dim M 1 = n − 1 and that dim M P (K) ≤ n − 1. To prove equality, one may argue exactly as at the end of the proof of [BFH + 18, Lemma 3.2.3], using that M P (K) is the quotient of M 1 by P (K) in the category V.
We will end this section by showing that M 1 is a P (K)-torsor over M P (K) . For this, we first record two lemmas concerning the pushouts defined in [Lud17,§2].
Lemma 4.2.5. Let G be a locally profinite group and let H ⊆ G be a compact open subgroup. Assume that H acts on a perfectoid space X, that G acts on a perfectoid space Y and that we have an H-invariant map of perfectoid spaces X → Y . Then there is a natural G-invariant map X × H G → Y , and if Z → Y is a map of perfectoid spaces then the natural map Proof. The existence of X × H G → Y is [Lud17, Lemma 2.16]. For the compatibility with fibre products we note that there is indeed a natural map (X × Y Z)×G → (X × H G)× Y Z given by (x, z, g) → (x, g, z). It is easily checked to be both H-invariant for the action (x, z, g).h = (xh, z, h −1 g) on (X × Y Z)× G, and G-equivariant for the action given by acting by right translation on the G-factor on the target and source. These actions commute and so induce the natural To see that it is an isomorphism, use the description of the pushout from [Lud17, Proposition 2.15] and the fact that disjoint unions commute with fibre products.
Lemma 4.2.6. Let G be a locally profinite group and let H ⊆ G be a compact open subgroup. If X → Y is an H-torsor of perfectoid spaces, then X × H G → Y is a G-torsor of perfectoid spaces.
Proof. X → Y is a v-cover, so it suffices to show that (X × H G) × Y X ∼ = X × G, G-equivariantly. Using Lemma 4.2.5 and the fact that X → Y is an H-torsor we see that and one checks that these isomorphisms are all G-equivariant.
Using these we can now prove that M 1 is a P (K)-torsor over M P (K) . 5. APPLICATION TO SCHOLZE'S FUNCTOR 5.1. Recollections. We recall some results of [Sch15a]. Let D/K be a central division algebra of invariant 1/n. For a smooth admissible representation π of GL n (K) on a F p -vector space, Scholze defines a sheaf F π on (P n−1 )é t by F π (U ) = Map cont,GLn(K) (|U × P n−1 M 1 |, π) (where U → P n−1 is anétale map) and shows that the cohomology groups S i (π) := H í et (P n−1 , F π ), i ≥ 0, are admissible D × -representations which carry an action of Gal(K/K) and vanish in degree i > 2(n − 1) ([Sch15a, Theorem 1.1]). The main result of this section is Theorem 5.3.1, which shows that in fact S i (π) = 0 for i > n − 1 whenever π is induced from the parabolic P . 5.2. Some cohomological calculations. In preparation for Theorem 5.3.1, we carry out some auxiliary calculations. We begin with some remarks about the geometric fibres of π GH . Let x : Spa(E, E + ) → P n−1 be a geometric point. The fibre (M P (K) ) x may be defined either as the fibre product (M P (K) ) x := M P (K) × (P n−1 ) ♦ Spa(E, E + ) in the category of diamonds or like in [CS17,Lemma 4.4.1]; these notions agree (by the proof of [CS17, Lemma 4.4.1]) and (M P (K) ) x is a perfectoid space. Since M 1 → P n−1 is a GL n (K)-torsor and M 1 → M P (K) is a P (K)-torsor (by Proposition 4.2.7), the geometric fibres of π GH are profinite sets (M P (K) ) x ∼ = x × S, with S = GL n (K)/P (K) = GL n (O K )/P (O K ) (we refer to e.g. [Lud17, Proposition 2.10] for a definition of the notation x × S; see also [Sch17,Example 11.12]).
Lemma 5.2.1. Let F be a sheaf of abelian groups on (M P (K) )é t . Then Proof. This is proved exactly as [Lud17, Proposition 4.4], using Proposition 4.2.2 and the fact that the geometric fibres (M P (K) ) x are profinite sets over x.
Proof. We give a slightly different proof than in [Lud17, Lemma 4.5]. There is a natural map (π GH, * F ) ⊗ O + P n−1 /p → π GH, * (F ⊗ O + M P (K) /p), so we can check the assertion on stalks at geometric points. For that let x = Spa(E, E + ) be a geometric point of P n−1 . On the one hand − →j∈J F j of sheaves F j that arise as the inverse image of a system of sheaves F j i on the spaces (x × S i )é t . The topos (x × S)é t is coherent, so (étale) cohomology commutes with direct limits. As tensor products also commute with direct limits it suffices to prove (5.2.1) for sheaves of the form F ∼ = lim − → q −1 i F i for some sheaves F i on (x × S i )é t . Note that O + x×S /p ∼ = lim − → q −1 i (O + x×Si /p) 5 . Using [SW13, Theorem 2.4.7] we see that we can rewrite (5.2.1) as x×Si /p , and we see this map is indeed an isomorphism as the spaces x×S i are just finite disjoint unions of geometric points with the same underlying affinoid field (E, E + ).
Next, let σ be a smooth admissible representation of P (K). Define a sheaf F σ on (M P (K) )é t by for any smooth admissible P (K)-representation σ and its restriction σ| P (OK ) to P (O K ). third is Lemma 5.2.1. By Proposition 5.2.4, theétale cohomology group H í et (M P (K) , F σ ⊗ O + /p) is almost isomorphic to the analytic cohomology group H i (|M P (K) |, λ * (F σ ⊗ O + /p)).
Proof. This follows from exactness of the functor π → F π , Theorem 5.3.1 and the long exact sequence in cohomology.
Remark 5.3.3. We finish with a few remarks on our results.
(1) The bound on cohomological vanishing in Corollary 5.3.2 (combined with [Sch15a, Theorem 3.2]) is sharp in general, and for general subquotients of representations induced from P (K) the bound from [Sch15a, Theorem 3.2] cannot be improved. To see these two things (simultaneously), consider the trivial representation 1 and the exact sequence 0 → 1 → σ = Ind GLn(K) P (K) 1 → Q → 0, where Q is simply defined to be the quotient. From this we get an exact sequence ofétale sheaves 0 → F 1 → F σ → F Q → 0 on P n−1 . Note that F 1 is the trivial local system on P n−1 , so S 2(n−1) (1) = 0; this shows the second point. The long exact sequence then shows (using Theorem 5.3.1) that S 2n−3 (Q) surjects onto S 2(n−1) (1), so S 2n−3 (Q) = 0 as well, proving the first point. Note also that, as a consequence of Corollary 5.3.2, the trivial representation cannot be written as a quotient of a representation induced from P (K). We thank Florian Herzig for informing us that this is well known, and is easily proved using the adjunction formula between parabolic induction and Emerton's ordinary parts functor.
(2) It is also natural to ask about vanishing below the middle degree, but here things seem to be much more unclear. For S 0 , we have S 0 (π) = S 0 (π SLn(K) ) by [Sch15a, Proposition 4.7], so e.g. when π is irreducible and infinite-dimensional we know that S 0 (π) = 0. When n = 2 the middle degree is 1, so in this case (for arbitrary K), we can say that S i (π) is concentrated in degree 1 for irreducible π = Ind GLn(K) P (K) σ.
(3) We end by remarking that Paškūnas [Paš18] has used the results of [Lud17] to show a nonvanishing result in degree one for (a version of) Scholze's functor for Banach space representations of GL 2 (Q p ) corresponding to reducible two-dimensional representations of Gal(Q p /Q p ) via the p-adic local Langlands correspondence (we refer to [Paš18] for precise statements). It would be interesting to see similar consequences for GL 2 (K), where K/Q p is arbitrary. However, Paškūnas informs us that our results would not be sufficient even assuming a p-adic local Langlands correspondence for GL 2 (K), as it is expected that supersingular representations will contribute to the Banach space representation corresponding to reducible two-dimensional representations of Gal(K/K) when K = Q p . Nevertheless, we hope that our results will be useful for the further study of Scholze's functor.