A QUOTIENT OF THE LUBIN TATE TOWER

Transcription

1 A QUOTIENT OF THE LUBIN TATE TOWER JUDITH LUDWIG Abstract. In this article we show that the quotient M /B(Q p) of the Lubin Tate space at infinite level M by the Borel subgroup of upper triangular matrices B(Q p) GL 2(Q p) exists as a perfectoid space. As an application we show that Scholze s functor H í et(p 1 C p, F π) is concentrated in degree one whenever π is a principal series representation of GL 2(Q p). 1. Introduction In [12], Scholze constructs a candidate for the mod p local Langlands correspondence for the group GL n (F ), where n 1 and F/Q p is a finite extension. His construction is purely local, satisfies some local-global compatibility and also gives a candidate for the mod p Jacquet Langlands transfer from GL n (F ) to D, the group of units in the central division algebra over F with invariant 1/n. Slightly more precisely, for any admissible smooth representation π of GL n (F ) on an F p -vector space, Scholze constructs an étale sheaf F π on PC n 1 p using the infinite level Lubin Tate space M and the Gross Hopkins period morphism M PC n 1 p. The cohomology groups S i (π) := H í et(p n 1 C p, F π ), i 0, are admissible D -representations which carry an action of Gal(F /F ) and vanish in degree i > 2(n 1) ([12, Theorem 1.1]). Let n = 2 and F = Q p, and denote by B(Q p ) GL 2 (Q p ) the Borel subgroup of upper triangular matrices. For two smooth characters χ i : Q p F q, i = 1, 2, let Ind GL 2(Q p) B(Q p) (χ 1, χ 2 ) denote the parabolic induction. In this article we show the following vanishing result. Theorem (4.6). Let χ i : Q p F q, i = 1, 2, be two smooth characters. Then ( S 2 Ind GL ) 2(Q p) B(Q p) (χ 1, χ 2 ) = 0. The main ingredient to prove the theorem is the construction of a quotient M /B(Q p ) of the infinite-level Lubin Tate tower M by the Borel subgroup B(Q p ) as a perfectoid space. In order to explain our results in more detail we introduce some notation. We fix a connected p-divisible group H/F p of dimension one and height two. Then for any compact open subgroup K GL 2 (Q p ) we have the Lubin Tate space M K of level K, which is a rigid analytic variety over C p. Varying K one gets a tower (M K ) K GL2 (Q p) with finite étale transition maps and in the limit one has a perfectoid space M lim K M K. 1

2 2 JUDITH LUDWIG In particular, M is an adic space and an object of Huber s ambient category V. The group GL 2 (Q p ) acts on M. We have a decomposition M = i Z M (i) and for any i Z, there is a non-canonical isomorphism M (i) =. The stabilizer in GL 2 (Q p ) of is given by G := {g GL 2 (Q p ) : det(g) Z p}. As we recall in Section 2.2 we can form a quotient M /B(Q p ) in the category V. The key to proving Theorem 4.6 is the following. Theorem (3.14). The object of V is a perfectoid space. M /B(Q p ) The proof of this theorem is done in three steps, the first step is the most difficult one. There one constructs a candidate perfectoid space M B for the quotient /B(Z p ) using infinite-level modular curves, the geometry of the Hodge-Tate period map and various other results of [10]. Theorem (3.11). There exists a unique (up to unique isomorphism) perfectoid space M B lim m /B(Z/p m Z). In a second step one shows that the candidate space M B agrees with the object /B(Z p ) in V. Finally one gets from /B(Z p ) to a perfectoid space M /B(Q p ) using the geometry of the Gross Hopkins period morphism. The Gross Hopkins period morphism π GH : M P 1 C p factors through the quotient M /B(Q p ) and we get an induced map π GH : M /B(Q p ) P 1 C p with many nice properties (cf. Section 3.6), e.g. it is quasi-compact. From a character χ : B(Q p ) F q one can construct an étale local system F χ on M /B(Q p ) such that F π = πgh, F χ, where π := Ind GL 2(Q p) B(Q p) (χ). We show that all higher direct images R i π GH, F χ, i > 0, vanish (Proposition 4.4) and so ( S i Ind GL 2(Q p) B(Q p) (χ) ) = Het(M í /B(Q p ), F χ ). We see that at the expense of a more complicated space, the sheaf has simplified a great deal. And although the space is more complicated, it is perfectoid. This ( fact and the good properties of π GH make it possible to show vanishing of S 2 Ind GL ) 2(Q p) B(Q p) (χ) in Theorem 4.6. The article is structured as follows. In Section 2 we give some background on inverse limits of adic spaces, quotients of adic spaces and we introduce some auxiliary spaces that we need in the construction of M /B(Q p ). The quotient is constructed in Section 3. In the last section we prove the vanishing result in cohomological degree two.

3 A QUOTIENT OF THE LUBIN TATE TOWER 3 Acknowledgments. The author would like to thank Peter Scholze for suggesting the project underlying this article to her and for sharing his ideas which made this paper possible. She would like to thank him for this constant support, for his patience in answering her many questions and for the careful reading of the manuscript. The author would furthermore like to thank Ana Caraiani for many helpful conversations and Sug Woo Shin for pointing out an error at an earlier stage of the project. The author was supported by the SFB/TR 45 of the DFG Notation. All adic spaces occurring are defined over (K, O K ), where K is a non-archimedean field, i.e., a non-discrete complete topological field K whose topology is induced by a non-archimedean norm : K R 0. Throughout we fix a topologically nilpotent unit ϖ K. In particular all adic spaces considered in this article are analytic. We will often use that if X and Y are affinoid analytic adic spaces and f : X Y is a morphism, then the preimage of a rational subset U Y is a rational subset in X. We use the pro-étale site as defined in [11], i.e., in a tower we want the transition maps to eventually be finite étale surjective. 2. Background on adic spaces and quotients 2.1. Inverse limits of adic spaces. Definition 2.1 (cf. Def in [13]). Let (X m ) m I be a cofiltered inverse system of adic spaces with finite étale transition maps. Let X be an adic space and let p m : X X m be a compatible family of maps. Write X lim X m if the map of underlying topological spaces X lim X m is a homeomorphism and if there is an open cover of X by affinoid subsets Spa(A, A + ) X such that the map lim A m A Spa(A m,a + m) X m has dense image, where the direct limit runs over all open affinoids Spa(A m, A + m) X m over which Spa(A, A + ) X X m factors. Recall the following result. Proposition 2.2. Let (K, O K ) be a perfectoid field, (X m ) m I a cofiltered inverse system of adic spaces with finite étale transition maps. Assume there is a perfectoid space X over Spa(K, O K ) such that X lim X m. Then for any affinoid uniform (K, O K )-algebra (R, R + ), we have X(R, R + ) = lim X m (R, R + ). In particular, if Y is a perfectoid space over Spa(K, O K ) with a compatible system of maps Y X m, then Y factors over X uniquely, making X unique up to unique isomorphism. Proof. For (R, R + ) affinoid perfectoid, this is [13, Prop ]. However the same proof works for any affinoid (K, O K )-algebra (R, R + ) with R uniform or what is equivalent, with R + ϖ-adically complete. Consider also the following stronger notion.

4 4 JUDITH LUDWIG Definition 2.3. Let (K, O K ) be a non-archimedean field. Let (X m = Spa(R m, R + m)) m I be a cofiltered inverse system of affinoid adic spaces over Spa(K, O K ) with finite étale transition maps. Assume X = Spa(R, R + ) is an affinoid adic space over Spa(K, O K ) with a compatible family of maps p m : X X m. We write X lim X m if R + is the ϖ-adic completion of lim R + m m. Obviously if X lim X m then also X lim X m. The reason for introducing this stronger notion is the following. When X lim X m, we do not know whether an arbitrary affinoid open U = Spa(R, R + ) X is the preimage p 1 m (U m ) of an open affinoid subset U m = Spa(R m, R m) + X m for sufficiently large m and if it satisfies R + = (lim R + m). For this reason some of the arguments below become slightly technical. We can however always pass to rational subsets. Lemma 2.4. Let (X m = Spa(R m, R m)) + m I be a cofiltered inverse system of affinoid analytic spaces over Spa(K, O K ). Assume X = Spa(R, R + ) is an affinoid perfectoid space over Spa(K, O K ), with compatible maps p m : X X m and such that X lim X m. Let U X be a rational subset. Then U = p 1 m 0 (U m0 ) for a rational subset U m0 X m0 and m 0 sufficiently large and U lim U m0 Xm0 X m. ( ) Proof. We write U = X f1 g,, fn g for some elements f i, g R such that f 1,..., f n generate R. Then for some small open neighbourhood V of 0 R, the rational subset defined by any choice of functions f i f i + V, g g + V agrees with U. As lim R m is dense in R, we may assume the f i and g are elements of R m0 for some m 0. Define ( f U m0 := X 1 m0 g,, f n ) ( f = g Spa R 1 m0 g,, f n ) g, B m0, where B m0 is the completion of the integral closure B m0 of R m + 0 [ f 1 g,, f n g ] in R m0 [g 1 ]. Then ( U m := U m0 Xm0 X m f = Spa R 1 m g,, f n g with B m the obvious ring and ( U f = X 1 g,, f n ) g, B m ) ( f = Spa R 1 g,, f n ) g, B. As (R, R + ) is perfectoid, B is ϖ-adically complete. We see that ( U f = X 1 g,, f n ) = g U m0 Xm0 X lim U m0 Xm0 X m as lim B m B is dense in the ϖ-adic topology.

5 A QUOTIENT OF THE LUBIN TATE TOWER 5 Recall that an analytic adic space X over a non-archimedean field (K, K + ) is called partially proper if X is quasi-separated and if for all affinoid (K, K + )-algebras (k, k + ) with k a non-archimedean field and k + a valuation ring, X(k, k + ) = X(k, O k ). From Lemma 2.2 above one immediately concludes Proposition 2.5. Let X lim X m and assume X is perfectoid. If there exists m 0 such that X m is partially proper for all m m 0, then X is partially proper Quotients in Huber s category V. The category V is defined in [7] as follows. Its objects are triples X = ( X, O X, { x } x X ), where X is a topological space, O X is a sheaf of complete topological rings on X and x is a valuation on the stalk O X,x := lim O x U X (U). A morphism X Y in V is a pair (f, ϕ) consisting of a continuous map f : X Y and a morphism ϕ : O Y f O X of sheaves of topological rings, such that for every x X, the induced ring homomorphism ϕ x : O Y,f(x) O X,x is compatible with the valuations x and f(x). Let X V be an object with a right action of a group G, i.e., with a homomorphism G op Aut V (X). Consider the natural continuous and open map p : X X /G. Define the triple where the valuations x are defined as X/G := ( X /G, (p O X ) G, { x } x X /G ) x x : O X /G,x O X,x Γ x {0} where x is the valuation of any preimage x p 1 (x). Then this defines an object of V. Indeed (p O X ) G is again a sheaf as taking G-invariants is left exact. Furthermore for any open subset U X /G, (p O X ) G (U) = O X (p 1 (U)) G O X (p 1 (U)) is closed, therefore complete. One checks that X/G is the categorical quotient of X in V Quotients of adic spaces. We specialize to the case we are interested in, which is when X is an analytic adic space over Spa(K, O K ) for a non-archimedean field K, and mostly either a rigid space or a perfectoid space. In this section we also assume that K is of characteristic zero. Let G be a locally profinite group. Below we want our actions to be continuous in the following sense. Definition 2.6 (cf. [12, Def. 2.1]). Let G be a locally profinite group and let X be an analytic adic space. An action of G on X is said to be continuous if there exists a cover of X by affinoid open subspaces Spa(A, A + ) X invariant under the action of open subgroups H A G such that the action morphism H A A A is continuous.

6 6 JUDITH LUDWIG Lemma 2.7. Let X be an analytic adic space and G a finite group with an action of G on X. Assume that X has a cover by G-stable affinoid open subsets Spa(A, A + ). Then X/G is an analytic adic space. The quotient Spa(A, A + )/G is affinoid and given by Spa(A G, A + G ). Proof. This is Theorem 1.1 and Theorem 1.2 in [6]. Lemma 2.8. Let (K, O K ) be an affinoid perfectoid field. Let G be a finite group acting on an affinoid perfectoid space X = Spa(R, R + ) for some affinoid perfectoid (K, O K )-algebra (R, R + ). Then X/G is again affinoid perfectoid and given by Spa(R G, R + G ). Proof. This is Theorem 1.4 in [6]. Lemma 2.9. Let X be a rigid space or a perfectoid space over Spa(K, O K ) and assume a finite group G acts continuously on X. Assume furthermore that X X/G is finite étale Galois with Galois group G. Then X G = X X/G X. Proof. We may assume X = Spa(A, A + ) is affinoid. As X X/G is a finite étale Galois cover with Galois group G the map ϕ : A A G A g G A (g) a b (a(b g)) is an isomorphism. The image of A + (A + ) G A+ is contained in A + and as A + is integrally closed in A, so is the image of the integral closure (A + (A + ) G A+ ) int, in fact ϕ((a + (A + ) G A+ ) int ) = A Some auxiliary spaces. Proposition Let (R, R + ) be a perfectoid (K, O K )-algebra. Assume P is a profinite set. Then (S = Map cont (P, R), S + = Map cont (P, R + )) is again a perfectoid (K, O K )-algebra. Let X = Spa(R, R + ), then we define X P := Spa(S, S + ). If we write P = lim P m m for finite sets P m, then X P lim X P m. Proof. One easily checks that the pair (S, S + ) is perfectoid (K, O K )-algebra. Define (S m, S m) + := (Map(P m, R), Map(P m, R + )). Then and lim S m = Map loc.cst (P, R), lim S+ m = Map loc.cst (P, R + ). We have that lim S + m is dense in S + in the ϖ-adic topology. Now Spa(S, S + ) = lim Spa(S m, S m) + = lim X P m = X P.

7 A QUOTIENT OF THE LUBIN TATE TOWER 7 Proposition Keep the notation from above and let Y be a perfectoid space, then Hom(Y, X P ) = Hom(Y, X) Map cont ( Y, P ). Proof. This follows from the previous Lemma and Proposition 2.2. Namely we have Hom(Y, X P ) = lim Hom(Y, X P m ) = lim(hom(y, X) Map( Y, P m )) = Hom(Y, X) lim Map( Y, P m ) = Hom(Y, X) Map cont ( Y, P ). Lemma We keep ( the notation ) from the Proposition Consider the rational subset U = X f1 g,, fn g for some elements f i, g R such that f 1,..., f n generate R. Then ( f1 (X P ) g,, f ) n = U P, g where f i Map cont (P, R) (resp. g Map cont (P, R)) is the constant map with value f i R (resp. g R). Proof. It is clear that the underlying topological spaces agree. We show the corresponding perfectoid algebras are isomorphic by showing that the Huber pair ( ( (S, S + f1 ) := (Map cont P, R g,, f )) ( ( n, Map g cont P, R + f1 g,, f ))) n g satisfies the same universal property as the Huber pair ( ) S ( f1 g,, f n g Clearly we have a map ( ( f1 S g,, f n g ), S + ( f1 g,, f n g )). (, S + f1 g,, f )) n (S, S + ). g Now let (A, A + ) be perfectoid and assume we have a morphism of Huber pairs (S, S + ) (A, A + ) ( ) such that map ϕ : Spa(A, A + ) Spa(S, S + ) factors over (X P ) f1g,, fn g. Let p : X P X denote the projection map (corresponding to the map (R, R + ) (S, S + ) that sends an element r R to the constant map with value r). Then the morphism p ϕ : Y := Spa(A, A + ) Spa(R, R + ) factors over U. By the universal property of rational subsets we get a corresponding unique morphism of Huber pairs ( ϕ U : R ( f1 g,, f n g By the last proposition, ϕ corresponds to a pair ) (, R + f1 g,, f )) n (A, A + ). g (ϕ X, ϕ P ) Hom(Y, X) Map cont ( Y, P ).

8 8 JUDITH LUDWIG Then (S, S + ) (A, A + ) factors over the unique morphism (S, S + ) (A, A + ) corresponding to (ϕ U, ϕ P ) Hom(Y, U) Map cont ( Y, P ). We see that we can glue the construction X P and locally on X, so we can build a perfectoid space X P from any perfectoid space X. Even more generally: Given a perfectoid space X and a locally profinite set Q there is a unique perfectoid space such that for any perfectoid space Y, X Q Hom(Y, X Q) = Hom(Y, X) Map cont ( Y, Q). If we write Q = i I P i for profinite sets P i, then X Q = i X P i. Lemma Let x = Spa(K, K + ) with K algebraically closed and K + K an open and bounded valuation subring. Let P = lim P m be a profinite set. Then for any sheaf F of abelian groups on (x P )ét for all i > 0. Proof. By Lemma 2.10 we have H í et(x P, F) = 0 x P lim x P m. Any sheaf F on (x P )ét can be written as a filtered colimit lim j J F j of sheaves F j arising as the pullback of a system of sheaves F j m on (x P m )ét [1, VI, ]. The topos (x P )ét is coherent, so for all i 0 H í et(x P, F) = lim H í et(x P, F j ). But by [13, Theorem 2.4.7] Het(x í P, F j ) = lim H í et(x P m, Fm) j and the latter terms are all zero for i > 0 as x P m = p P m x is a finite disjoint union of geometric points. Now let H G be a profinite subgroup of a locally profinite group G and let X be a separated perfectoid space equipped with a continuous right action of H. Then H acts on X G as follows. Firstly H acts on the topological space X G via (x, g) h := (xh, h 1 g) for (x, g) X G and h H and on the structure sheaf via h : O X G (U H ) h O X G (U H ) = O X G (U h 1 hh ) (h f)(g) = f(h 1 g)h for any open U H X G with U = Spa(A, A + ) affinoid perfectoid. (As X is separated, there is a basis B for the topological space X G = X G of open subspaces of this form.)

9 Consider the quotient A QUOTIENT OF THE LUBIN TATE TOWER 9 X H G := (X G)/H V. Note that X H G comes equipped with a right action of G which is given by [(x, g )] g := [(x, g g)] for g G and [(x, g )] ( X G)/H on the topological spaces and in the natural way on the structure sheaf, i.e., on the basis B by right translation of the function. Proposition Using the notation above, there is an isomorphism in V (X H G)/G = X/H. Proof. There is a natural continuous map of topological spaces f : X H G X/H given by [(x, g)] [x]. This is well defined as for (y, g ) [(x, g)] there exists h H such that y = xh, g = h 1 g. This map factors through the quotient (X H G)/G X/H. One immediately checks it is bijective and open, so f is a homeomorphism. Consider the commutative diagram X G X q r q 1 X H G q 2 (X H G)/G f X/H Then for U X /H open, O X/H (U) = O X (r 1 (U)) H. On the other hand O (X H G)/G(f 1 (U)) = O (X H G)(q 1 2 (f 1 (U))) G = = (O X G ((f q) 1 (U)) H ) G = (O X G ((f q) 1 (U)) G ) H p = (O X G ((r 1 (U) G)) G ) H = O X (r 1 (U)) H. Compatibility of the valuations on the stalks is clear. Proposition Assume H G again. Using the notation above, choose a set of representatives g i, i I, of H\G. The natural map X G is an isomorphism. g i,i I X (g i ) f X H G Proof. Define X H\G := g i,i I X (g i ). There is a continuous map on topological spaces f : X H\G X H G, sending x X (gi ) to (x, g i ). One easily checks that it is bijective and open, therefore a homeomorphism. For any i I, there is a morphism in V, ϕ i : X X G, x (x, g i ).

10 10 JUDITH LUDWIG On a basis it is given as follows: If U = V H V g i X H G open, with V = Spa(A, A + ) affinoid perfectoid and H V invariant and such that H V A A is continuous, then the morphism of sheaves ϕ i evaluates to O X G (V H V g i ) O X (V ), f f(g i ). Note that the map has a section given by sending r O X (V ) to the function hg i r h. This gives an isomorphism O X (V ) O X G (V H V g i ) H V. For U = V H X H G with g i / H, the corresponding map between sheaves is the zero map. Denote by p : X G X H G the open projection map. Let V X H G be an open subset. We may assume that V = f(u) and U X (gi ) for some i and that U = Spa(B, B + ) is affinoid open. Then there is H V H normal and open such that H V leaves U X (gi ) invariant and such that H V B B is continuous. Then p 1 (V ) = V h hg i = finite V h k H V h 1 k g i where H/H V = H V h k. Finally O X G (p 1 (V )) H = ( O X G (V h k H V h 1 k g i)) H = O X G (V H V g i ) H V. Let Y be a separated perfectoid space with a continuous right action of a locally profinite group G. We define an action morphism α : Y G Y, (y, g) yg as follows. Let U = Spa(A, A + ) be an affinoid perfectoid open subspace of Y, and H U G an open subgroup leaving U invariant. Choose coset representatives g i, i I, for H U \G. Then the natural map (U H U ) gi U G is an isomorphism. Then i I (y, h) gi (y, hg i ) α (U HU ) gi : (U H U ) gi Y is defined as the composite g i α HU, where α HU : (U H U ) gi U is the map induced by A Map cont (H U, A), a (h a h) and g i : U Y is the morphism from the action of G on Y. Lemma Let G be a locally profinite group, H G an open subgroup. Assume f : X Y is a morphism between separated perfectoid spaces, that H acts continuously on X and G on Y and that f is H-equivariant. Then there exists a unique G-equivariant map f : X H G Y such that f is the composite of f and the natural inclusion X = X H H X H G. If f is an open immersion and f : X H G Y is an injection, then f is also an open immersion.

11 A QUOTIENT OF THE LUBIN TATE TOWER 11 Proof. Consider the morphism (f, id) : X G Y G and the action morphism α : Y G Y, (y, g) yg defined above. The composition is H-equivariant for the trivial action of H on Y, therefore factors through the quotient X H G which gives the morphism f. To show the last claim, it suffices to show that the restriction f to each copy X (gi ) of X is an open immersion. But g i : Y Y is an isomorphism and f : X (gi ) Y agrees with g i f. 3. The quotient M /B(Q p ) 3.1. The Lubin Tate tower. Let k = F p and fix a connected p-divisible group H/k of dimension one and height two. Definition 3.1. Let Nilp W (k) be the category of W (k)-algebras R in which p is nilpotent. A deformation of H to R Nilp W (k) is a pair (G, ρ) where G is a p- divisible group over R and is a quasi-isogeny. Define the functor Def H : Nilp W (k) Sets ρ : H k R/p G R R/p R {(G, ρ) : deformation of H}/ =. Theorem 3.2 ([9]). The functor Def H is representable by a formal scheme M/W (k). We have a decomposition M = M (i) i Z according to the height i of the quasi-isogeny and non-canonically M (i) = Spf(W (k)[[t]]). Let M 0 := M ad η Qp C p be the (base change to C p of the) adic generic fibre of M. For an integer m 0 let K(m) := ker(gl 2 (Z p ) GL 2 (Z/p m Z)). One can introduce level structures to get spaces M m := M K(m) and finite étale Galois covers M m M 0 with Galois group GL 2 (Z/p m Z). More generally one can construct spaces M K for any compact open subgroup K GL 2 (Q p ) (see [9, Section 5.34]). For i Z let M (i) K := M K M0 M (i) 0 so that we have a decomposition M K = i Z M (i) K. For two compact open subgroups K K of GL 2 (Q p ) such that K is normal in K there is a finite étale Galois cover M K M K with Galois group K /K. In particular the spaces (M K ) K GL2 (Q p) form an inverse system. In the limit we get a perfectoid space.

12 12 JUDITH LUDWIG Theorem 3.3 ([13, 6.3.4]). There exists a unique (up to unique isomorphism) perfectoid space M over C p such that M lim K M K. The space M represents the functor from complete affinoid (W (k)[ 1 p ], W (k))- algebras to Sets which sends (R, R + ) to the set of triples (G, ρ, α) where (G, ρ) M 0 (R, R + ) and α : Z 2 p T (G) ad η (R, R + ) is a morphism of Z p -modules such that for all points x = Spa(K, K + ) Spa(R, R + ), the induced map α(x) : Z 2 p T (G) ad η (K, K + ) is an isomorphism, cf. [13, Section 6.3]. Here T (G) ad η denotes the generic fibre of the Tate module of G. We have a continuous right action of GL 2 (Q p ) on M. The action of GL 2 (Z p ) is easy to describe: An element g GL 2 (Z p ) acts as (G, ρ, α) (G, ρ, α g). We have a decomposition into perfectoid spaces M = i Z M (i) at infinite level and for each i Z the stabilizer Stab GL2 (Q p)(m (i) ) is given by the subgroup G := {g GL 2 (Q p ) det(g) Z p}. As the system (K(m)) m 0 is cofinal in the system of compact open subgroups K GL 2 (Q p ) we also have (1) M (i) lim M (i) for any i Z. We denote by p m the map implicit in (1). m. p m : m 3.2. The quotient /B(Z p ). Consider the perfectoid space. Let B GL 2 the Borel subgroup of upper triangular matrices. The group B(Z p ) acts continuously on. In Section 3.4 we prove the following theorem. Theorem 3.4. The object /B(Z p ) V is a perfectoid space. It comes equipped with compatible maps and satisfies /B(Z p ) m /B(Z/p m Z) /B(Z p ) lim m m /B(Z/p m Z). There is a cover (U i = Spa(R i, R i + )) i I of by B(Z p )-invariant affinoid perfectoid spaces such that the image φ(u i ) under the projection map φ : /B(Z p )

13 A QUOTIENT OF THE LUBIN TATE TOWER 13 is affinoid perfectoid and given by Spa(R B(Zp) i, (R + i )B(Zp) ). The strategy to prove the above theorem is the following: Using infinite level modular curves, we will first construct a perfectoid space M B with the property that M B lim m /B(Z/p m Z). By Proposition 2.2 the space M B is then unique up to unique isomorphism. In Section 3.5 we then identify M B with /B(Z p ) Background on infinite level modular curves. The infinite-level modular curve XΓ(p ) was constructed as a perfectoid space over Qcycl p in [10]. In this paper we will work with the base change to C p. Furthermore Γ 0 (p m ) denotes the subgroup of GL 2 (Z p ) consisting of matrices that are upper triangular mod p m, without the determinant condition in [10]. This affects only connected components as we work over C p. In what follows, all other notation is as in [10], but the spaces are always implicitly base changed to C p. The perfectoid space XΓ(p ) comes equipped with a compatible family of maps q m : X Γ(p ) X Γ(p m ) such that X Γ(p ) lim X Γ(p m ). The group GL 2 (Q p ) acts continuously on X Γ(p ) and there is a surjective GL 2(Q p )- equivariant map, the Hodge-Tate period map π HT : X Γ(p ) P1 C p. The space P 1 C p comes equipped with a right action of GL 2 (Q p ), which on C p -points is given by ( ) a b [x 0 : x 1 ] = [dx c d 0 bx 1 : cx 0 + ax 1 ], i.e., it is the usual left action of g 1. Furthermore part of the tower (X Γ 0 (p m ) ) m is perfectoid. More precisely let 0 ɛ < 1/2. Then Scholze constructs open neighbourhoods X Γ 0 (p m ) (ɛ) a X Γ 0 (p m ) of the anticanonical locus. For m large enough these are affinoid spaces. Roughly speaking X Γ 0 (p m ) (ɛ) a is the locus, where the subgroup is anticanonical and the Hasse invariant satisfies Ha p p mɛ. By Corollary III.2.19 of [10], there exists a unique affinoid perfectoid space X Γ 0 (p ) (ɛ) a such that (2) X Γ 0 (p ) (ɛ) a lim X Γ 0 (p m ) (ɛ) a. This is in fact the first step in showing that X Γ(p ) is perfectoid. From Γ 0(p ) one can pass to deeper level structures as follows. Firstly, in [10, Proposition III.2.33], the fibre product X Γ 1 (p m ) Γ 0 (p ) (ɛ) a := X Γ 1 (p m ) X Γ 0 (p m ) X Γ 0 (p ) (ɛ) a is shown to be affinoid perfectoid. Passing from Γ 1 (p m ) to full level is easy: The map X Γ(p m ) (ɛ) a X Γ 1 (p m ) (ɛ) a

14 14 JUDITH LUDWIG is finite étale. Therefore the space X Γ(p m ) Γ 0 (p ) (ɛ) a := X Γ(p m ) (ɛ) a X Γ 1 (p m ) (ɛ)a X Γ 1 (p m ) Γ 0 (p ) (ɛ) a = X Γ(p m ) (ɛ) a X Γ 0 (p m ) (ɛ)a X Γ 0 (p ) (ɛ) a is affinoid perfectoid. In the limit (cf. [10, Theorem III.2.36]) we get an affinoid perfectoid space X Γ(p ) (ɛ) a which satisfies (3) X Γ(p ) (ɛ) a lim X Γ(p m ) (ɛ) a and in particular Proposition 3.5. The natural map is open. X Γ(p ) (ɛ) a lim X Γ(p m ) Γ 0 (p ) (ɛ) a. φ : X Γ(p ) (ɛ) a X Γ 0 (p ) (ɛ) a Proof. Using the above relations (2) and (3) it suffices to show that the map X Γ(p m ) X Γ 0 (p m ) is open. But this map is flat, therefore open. We have a decomposition XΓ(p ) = XΓ(p ord ) X Γ(p ss ) = πht 1 (Q p )) πht 1 ). Here Ω 2 denotes the Drinfeld upper half plane Ω 2 = P 1 \P 1 (Q p ). The supersingular locus XΓ(p ss ) is contained in translates of the affinoid perfectoid ( ) p space XΓ(p ) (ɛ) a. Concretely, let γ := p 1 GL 2 (Q p ). Then X ss Γ(p ) X Γ(p ) \π 1 HT ([1 : 0]) = i=0 X Γ(p ) (ɛ) aγ i. The Lubin Tate tower sits inside the infinite level modular curve. More precisely we have a decomposition of the supersingular locus X ss Γ(p ) = j J into a finite disjoint union of copies of. The embeddings are equivariant for the GL 2 (Z p )-action. We fix once and for all an embedding ι : embeddings ι m : m X ss Γ(p m ) X ss Γ(p ) as well as compatible such that for any m 0 the following diagram

15 A QUOTIENT OF THE LUBIN TATE TOWER 15 commutes 1 (4) p m m ι XΓ(p ). q m ι m X Γ(p m ) 3.4. Construction of M B. Define the underlying topological space of M B to be the quotient space /B(Z p ) := /B(Z p ). Then /B(Z p ) = lim m m /B(Z/p m Z). In analogy with Definition III.3.5 of [10] we make the following definition Definition 3.6. (1) A subset V /B(Z p ) is affinoid perfectoid if it is the preimage of some affinoid open V m = Spa(R m, R + m) m /B(Z/p m Z) for all sufficiently large m, and if (R, R ) + is an affinoid perfectoid C p - algebra, where R + is the p-adic completion of lim R + m m, and R := R [p + 1 ]. (2) A subset V /B(Z p ) is called perfectoid if it can be covered by affinoid perfectoid subsets. Once we show that /B(Z p ) is perfectoid in the sense of the previous definition, we may use the affinoid perfectoid algebras on the affinoid perfectoid pieces to define a structure sheaf on /B(Z p ) turning it into the perfectoid space M B that we seek to construct. In a first step we show that every point x has a nice B(Z p )-stable neighbourhood. For that we recall some explicit B(Z p )-invariant affinoid open subspaces of Ω 2. Let D 1 P 1 be the closed unit disc embedded in P 1 via x [x : 1] in usual homogeneous coordinates. For an integer n 1 let z 1,..., z p n Z p be a set of representatives of Z p /p n Z p and consider the rational subset X n D 1 defined by X n (C p ) = {x D 1 (C p ) : x z i p n for all i = 1,..., p n } = {x D 1 (C p ) : x z p n for all z Z p }. It is well known that (X n ) n N is a cover of Ω 2 D 1.( For) any n 0, the affinoid open X n P 1 is stable under B(Z p ). Indeed let γ = a bd B(Z p ) and x X n (C p ), then x z p n, z Z p and therefore xγ z = (dx b)/a z = dx b az = x (az b)/d p n. 1 Equivalently: Fix a supersingular elliptic curve E/Fp and an isomorphism E[p ] = H.

16 16 JUDITH LUDWIG For i 0 the translate D 1 γ i is the disk of radius p 2i D 1 γ i (C p ) = {[x : 1] P 1 (C p ) : x p 2i }. The translates X n γ i are invariant under the group {( ) γ i B(Z p )γ i a b := B(Q d p ) : a, d Z p, } b p 2i Z p. Note that B(Z p ) γ i B(Z p )γ i. Explicitly we have X n γ i (C p ) := {x D 1 γ i (C p ) : x z p 2i n z Q p D 1 γ i (C p )}. By [10] Theorem III.3.17(i), the preimage π 1 HT (D1 ) X Γ(p ) under the Hodge-Tate period morphism is affinoid perfectoid 2. Furthermore there exists 0 < ɛ < 1 2 such that X Γ(p ) (ɛ) a πht 1 (D1 ). Note this is an inclusion of affinoid perfectoid spaces. We fix such an ɛ and abbreviate Y := X Γ(p ) (ɛ) a. Proposition 3.7. Let x Y γ i be a point. Then there exists an open neighbourhood U of x, such that U is affinoid perfectoid, U Y γ i, U is invariant under γ i B(Z p )γ i, U lim U m for affinoid open subsets U m m and m large enough. Proof. We first show the assertion for all x Y. So let x Y be arbitrary. Then there exists n N such that π HT (x) X n. Fix such a n. The affinoid open X n D 1 is a rational subset and therefore πht 1 (X n) πht 1 (D1 ) is also a rational subset. Note that πht 1 (X n) XΓ(p ss ), so π 1 HT (X n) is a disjoint union of the affinoids V j defined as the intersection of πht 1 (X n) and a copy of. Let j be such that x V j. Still V j is rational in πht 1 (D1 ) and so U := V j Y is a rational subset of the affinoid perfectoid space Y. In particular U is affinoid perfectoid. As we have seen above, the affinoid opens X n P 1 are invariant under B(Z p ) and the Hodge-Tate period map π HT is GL 2 (Q p )-equivariant, therefore πht 1 (X n) is stable under B(Z p ). Each copy of XΓ(p ss ) is stable under GL 2(Z p ) and Y is also B(Z p ) invariant, which gives that U is invariant under B(Z p ). As U Y is rational, it follows from Lemma 2.4 and Equation (3) that for m large enough there exists U m XΓ(p m ) (ɛ) a affinoid such that U lim the commutativity of diagram (4) implies that U m m., U m. As U 2 It coincides with the subset {[x1 : x 2] P 1 (C p) : x 1 x 2 } denoted by Fl {2} in [10].

17 A QUOTIENT OF THE LUBIN TATE TOWER 17 Now let x be arbitrary. Then there exists an integer i 0 such that x Y γ i and an integer n 0 such that π HT (x) X n γ i. The property of being affinoid perfectoid is stable under the action of GL 2 (Q p ), therefore π 1 HT (D1 γ i ), π 1 HT (X nγ i ) and Y γ i are all affinoid perfectoid. Define U := πht 1 (X nγ i ) Y γ i. The same argument as above shows that this is affinoid perfectoid. It is stable under γ i B(Z p )γ i as γ i leaves each copy XΓ(p ss ) invariant. For the final assertion note that the property for affinoids in XΓ(p ) is stable under GL 2 (Q p ) (see [10] p.59) therefore U lim U m for affinoid opens U m XΓ(p m ) and m large enough, and we argue as above to deduce that U m Definition 3.8. For i 0 define U i to be the collection of affinoid perfectoid open subsets U such that (1) U Y γ i is a rational subset and γ i B(Z p )γ i -stable, (2) U lim U m m for affinoid open subsets U m m, m large enough. By the above proposition every point in has a neighbourhood U U i for some i. We now construct the candidate quotients U B for any U U i. Proposition 3.9. Let U U 0. There exists a perfectoid space U B whose underlying topological space is homeomorphic to U /B(Z p ) and such that U B lim U m /B(Z/p m Z). Proof. By Proposition 3.5, the natural map φ : Y = X Γ(p ) (ɛ) a X Γ 0 (p ) (ɛ) a of affinoid perfectoid spaces is open. Define U B := φ(u). This is a perfectoid space. Let (V k ) k I be a cover of U B by rational subsets V k X Γ 0 (p ) (ɛ) a. Then V k lim V m k for affinoid subsets Vm k XΓ 0 (p m ) (ɛ) a. Note U k := φ 1 (V k ) is still rational in X Γ(p ) (ɛ) a. Recall that at finite level we have a finite étale morphism φ m : X Γ(p m ) (ɛ) a X Γ 0 (p m ) (ɛ) a, which is Galois with Galois group B(Z/p m Z). Furthermore we have a commutative diagram (5) X Γ(p ) (ɛ) a φ XΓ 0 (p ) (ɛ) a. m. q m X Γ(p m ) (ɛ) a φ m q 0 m X Γ 0 (p m ) (ɛ) a

18 18 JUDITH LUDWIG Then U k = qm 1 (φ 1 m (Vm)) k and we get a map Um k := φ 1 m (Vm) k Vm k of affinoid spaces with Um/B(Z/p k m Z) = Vm. k In particular V k lim U k m/b(z/p m Z). This also implies that on topological spaces we get a homeomorphism V k = U k /B(Z p ) = lim U k m /B(Z/p m Z). From the proof of the previous proposition we also see that after passing to rational subsets we may assume that U as well as U B are both affinoid perfectoid. Proposition Let i 1 and U U i. There exists a perfectoid space U B with underlying topological space U /B(Z p ) and such that U B lim m U m /B(Z/p m Z). Proof. As U is invariant under γ i B(Z p )γ i, the affinoid perfectoid space W := Uγ i Y is B(Z p )-invariant. Furthermore we may assume W lim W m for m large enough. Arguing as above we get a perfectoid space φ(uγ i ) lim W m /B(Z/p m Z) and after possibly shrinking U we may assume that φ(uγ i ) is affinoid perfectoid and that φ(uγ i ) lim W m /B(Z/p m Z). In particular, for W m /B(Z/p m Z) = Spa(S m, S m), + the limit (lim S + m m) [p 1 ] is affinoid perfectoid. Define {( ) } H i := γ i B(Z p )γ i a b = B(Z d p ) b p 2i Z p B(Z p ) and let H i,m B(Z/p m Z) be the image of H i under the natural reduction map B(Z p ) B(Z/p m Z). Consider the tower (W m /H i,m ) m 2i of affinoid adic spaces The natural maps W m /H i,m = Spa(R m, R + m). W m /H i,m W m /B(Z/p m Z) are finite étale maps and the degree does not change for m large enough as B(Z p )/H i B(Z/p m Z)/H i,m is a bijection. In fact the diagram W m+1 /H i,m+1 W m+1 /B(Z/p m+1 Z) W m /H i,m W m /B(Z/p m Z) is cartesian. The pullback W/H i := W 2i /H i,2i W2i /B(Z/p 2i Z) φ(w ) φ(w ) is finite étale, therefore affinoid perfectoid say W/H i = Spa(R, R + ).

19 Furthermore ([8, Remark 2.4.3]) and in particular Define We verify A QUOTIENT OF THE LUBIN TATE TOWER 19 W/H i lim W m /H i,m R = (lim R + m) [p 1 ], R + = (lim R + m). U B := Spa(R, R + ). U B lim m U m /B(Z/p m Z) by checking that the towers (U m /B(Z/p m Z)) m 4i and (W m /H i,m ) m 4i are equivalent, i.e., that the systems agree in ( 0 ) proét. For that note that we have the following commutative diagrams γ i γ i K(m) γ i γ i K(m)γ i K(m) γ i γ i K(m)γ i Let m 2i, then we have inclusions γ i K(m + 4i)γ i K(m + 2i) γ i K(m)γ i K(m 2i) and γ i K(m + 4i)γ i K(m + 2i) γ i K(m)γ i K(m 2i). The open subspace W m 2i K(m 2i) is image of U m under the composite γ i K(m) γ i K(m)γ i K(m 2i), where the last map is the natural one from the inclusion γ i K(m)γ i K(m 2i). We get an induced map Similarly we get a map The composite f U W (m, i) : U m /B(Z/p m Z) W m 2i /H i,m 2i. f W U (m 2i, i) : W m 2i /H i,m 2i U m 4i /B(Z/p m 4i Z). f W U (m 2i, i) f U W (m, i) is the natural projection U m /B(Z/p m Z) U m 4i /B(Z/p m 4i Z). Moreover f U W (m 2i, i) f W U (m, i) agrees with the natural projection map W m /H i,m W m 4i /H i,m 4i. Therefore the pro-étale systems (U m /B(Z/p m Z)) m 4i and (W m /H i,m ) m 4i are isomorphic, which is what we wanted to show. To summarize, we have proved the following theorem.

20 20 JUDITH LUDWIG Theorem (1) There exists a unique (up to unique isomorphism) perfectoid space M B lim m /B(Z/p m Z) and a natural surjective map M B such that the diagram ϕ M B m m /B(Z/p m Z) commutes. (2) There is a cover U LT = (U j ) j J of of affinoid perfectoid spaces U j lim U m j i 0 U i such that the image of ϕ(u j ) under ϕ : M B is affinoid perfectoid and satisfies ϕ(u j ) lim U j m/b(z/p m Z) The quotient /B(Z p ). Finally we argue that the structure sheaf of M B is the expected one, using the proétale site ( 0 ) proét. Define B m := B(Z/p m Z). First note that as M B ( 0 ) proét m /B(Z/p m Z) m 1 /B(Z/pm 1 Z) is finite étale surjective and m /B(Z/p m Z) is étale over 0. Proposition Let U = Spa(R, R + ) U LT with U lim U m = Spa(R m, R m). + Then the natural map of affinoid algebras ( S := S + [ 1 p is an isomorphism. ], S + := ( ) ) lim R+ B m m (R B(Zp), R + B(Z p) ) Proof. We have seen that (S = O MB (ϕ(u)), S + = O + M B (ϕ(u))) is an affinoid perfectoid algebra. Consider the completed structure sheaf Ô 0 and the integral as defined in [11, Def. 4.1]. These are sheaves on completed structure sheaf Ô+ 0 ( 0 ) proét. Now U, ϕ(u) ( 0 ) proét are both affinoid perfectoid therefore by Lemma 4.10 (iii) of [11] S = Ô 0 Furthermore U ϕ(u) is a covering so ( S = Ô(ϕ(U)) = eq 0 and analogously for S +. Now 4.10 (iii) of [11]. Furthermore (ϕ(u)), S + = Ô+ (ϕ(u)). 0 Ô M (0) 0 (U) Ô(U) = R and 0 ) Ô(U ϕ(u) U) 0 Ô(U) = R +, again by Lemma 0 U ϕ(u) U lim m U m Um/B(Z/p m Z) U m = lim U m B m U B(Z p ),

21 A QUOTIENT OF THE LUBIN TATE TOWER 21 where the isomorphism in the middle is implied by Lemma 2.9. Corollary 6.6 of [11] then implies that Ô (0) M (U ϕ(u) U) = Map cont (B(Z p ), R) LT,0 Ô + (U ϕ(u) U) = Map cont (B(Z p ), R + ) LT,0 and the two maps in each of the equalizers above are given as r (g r) and r (g gr). Therefore S = R B(Zp) and S + = R + B(Z p). In particular we get the following corollary, which finishes the proof of Theorem 3.4. Corollary We have an isomorphism in the category V M B = M (0) /B(Z p ). In particular, /B(Z p ) is a perfectoid space The quotient M /B(Q p ). We now use the above results and the fact that the Gross Hopkins period morphism π GH,0 at level zero has local sections to prove the following theorem. Theorem The object of V is a perfectoid space. Proof. Define the subgroup M /B(Q p ) B := {b B(Q p ) det(b) Z p} B(Q p ). It is the kernel of the map val det : B(Q p ) Z and B(Q p ) = B Z. We have M = M (0) Z = i Z and B = Stab B(Qp)( ). This implies that in V we have an isomorphism M /B(Q p ) = /B, so it suffices to show that the latter object is a perfectoid space. The Gross Hopkins period map π GH,0 : 0 P 1 C p has local sections (cf. Lemma in [13]). Let U 0 be open affinoid such that π GH,0 U is an isomorphism onto its image. As before let p 0 : 0 be the natural map and consider p 1 0 (U) M(0). It has an action of B(Z p ). Consider the object p 1 0 (U) B(Zp) B V as defined in Section 2.3. By Lemma 2.16 we get a natural map ι U : p 1 0 (U) B(Zp) B.

22 22 JUDITH LUDWIG in V. As the geometric fibres of π GH,0 are isomorphic to G / GL 2 (Z p ) = B /B(Z p ) and π GH,0 U is an isomorphism onto its image, ι U is an injection, therefore ι U is an open embedding (again by Lemma 2.16). But then (p 1 0 (U) B(Zp) B )/B /B is also an open embedding. Using Proposition 2.14 we see that (p 1 0 (U) B(Zp) B )/B = p 1 0 (U)/B(Z p) is a perfectoid space. This finishes the proof as the (p 1 0 (U) B(Zp) B )/B cover /B. The Gross Hopkins period morphism π GH : M P 1 C p factors through the quotient M /B(Q p ). We denote the induced map by π GH : M /B(Q p ) P 1 C p. Proposition The map π GH : M /B(Q p ) P 1 C p is quasicompact. Proof. Again we use that the Gross Hopkins period map at level zero has local sections. So we can cover P 1 C p by affinoid opens U that admit a section to some affinoid V 0. Now for any such U, π 1 GH (U) = (p 1 0 (U) B(Zp) B )/B as B /B(Z p ) = G / GL 2 (Z p ). So it suffices to show that each (p 1 0 (U) B(Zp) B )/B is quasi-compact. By [11, Prop. 3.12], p 1 0 (U) is quasicompact. But now (p 1 0 (U) B(Zp) B )/B = p 1 0 (U)/B(Z p) and the latter is quasi-compact as p 1 0 (U) is. Remark From Proposition 2.5 we see that the space M is partially proper, so M /B(Q p ) is partially proper and by the last proposition it is quasi-compact, therefore M /B(Q p ) may be called proper. We finish this section by proving that the fibres of π GH are affinoid perfectoid. For that we start with the following result on rank one points. Proposition Let x P 1 C p be a rank one point. Then there exists an affinoid open neighbourhood x U such that the preimage π 1 GH (U) is affinoid perfectoid. Proof. Let U P 1 be an affinoid neighbourhood of x and V 0 an affinoid open such that the map π GH,0 V : V U is an isomorphism. Furthermore there exists a cover of 0 by increasing open affinoid spaces Spa(B i, B i + ) such that p 1 0 (Spa(B i, B i + )) is affinoid perfectoid (cf. [14, Section 2.10]). As V is quasi-compact we can find i I such that V Spa(B i, B i + ) and we can cover V by open affinoids that are rational in Spa(B i, B i + ) so we may assume V Spa(B i, B i + ) is rational, in particular that p 1 0 (V ) is affinoid perfectoid. Using the notation of Section 2.4, p 1 0 (V ) Y γi for some i, therefore we can find a rational subset W Y γ i that is contained in p 1 0 (V ) with x (π GH,0 p 0 )(W ) and such that W U LT. In particular there exists m 0 and an affinoid open

23 A QUOTIENT OF THE LUBIN TATE TOWER 23 W m m with W = W m (0) M. Let x m W m be a lift of x and let m H B(Z/p m Z) be the stabilizer of x m. By Lemma 3.18 below there exists a rational subset V m W m with x m V m, V m H = V m and V m b V m = for all 1 b H\B(Z/p m Z). The affinoid U m := V m b b H\B(Z/p m Z) now comes from level zero: U m = p 1 m,0 (U 0) for U 0 := p m,0 (U m ) and U := π GH,0 (U 0 ) has the property that π GH,0 : U 0 U is an isomorphism, x U, p 1 0 (U 0) U LT and so π 1 GH (U) = p 1 0 (U 0)/B(Z p ) is affinoid perfectoid. Lemma X separated analytic adic space, G finite group acting on X, and let x X be a point of rank 1. Let H := Stab G (x). Then there exists an open affinoid neighbourhood U of x such that H U = U Ug U = for all 1 g H\G. Proof. As x is of rank 1, we have x = x V where the intersection runs over all affinoid opens V containing x. Likewise xg = V g for any g. If x g x then V (V g V ) = but the V g V are all quasicompact as they are affinoid as X is separated. So there exists V s.th. V g V =. Let U := h H V h. This is affinoid as X is separated and has the properties we want. Definition (1) Let (S, S + ) (R, R + ) be a morphism of Huber pairs. We say (R, R + ) is of lightly finite type over (S, S + ) if R + R can be generated by finitely many elements as an open and integrally closed S + -subalgebra, i.e., if there exists a finite subset E R + such that R + is the integral closure of S + [E R ] in R. 3 (2) We say that a morphism f : X Y of analytic adic space is locally of lightly finite type if for any x X there exist open affinoid subspaces U, V of X, Y such that x U, f(u) V, and the induced morphism is of lightly finite type. (O Y (V ), O + Y (V )) (O X(U), O + X (U)) 3 This is similar to Huber s definition of + weakly finite type, but he furthermore demands that the morphism is of topologically finite type.

24 24 JUDITH LUDWIG Remark It then follows that for any pair U, V as in part (2) of the previous definition, the induced morphism is of lightly finite type, as the property passes to rational subsets. Note furthermore, that any affinoid rigid space X = Spa(R, R ) over C p is of lightly finite type. Lemma Assume that X is perfectoid and X Spa(K, K + ) is partially proper and locally of lightly finite type. Then for any perfectoid (K, K + )-algebra (S, S + ) X(S, S + ) = X(S, S ). Proof. Assume we have a morphism f : Y := Spa(S, S ) X over Spa(K, K + ). We extend it to a morphism f : Y := Spa(S, S + ) X over Spa(K, K + ) such that the diagram Spa(S, S + ) j f Spa(S, S f ) X commutes. For that we define f as a morphism in V as follows. We first define a map f of underlying topological spaces. For that let y Spa(S, S + ) be a point. Then there is a morphism p y : Spa(k(y), k(y) + ) Y such that y is the image of the unique closed point in Spa(k(y), k(y) + ). Note that so we get a morphism p y (Spa(k(y), k(y) )) Y, f p y Spa(k(y),k(y) ) : Spa(k(y), k(y) ) X. As X is partially proper over Spa(K, K + ), this extends to a morphism f y : Spa(k(y), k(y) + ) X. Define f (y) as the image of f y of the unique closed point in Spa(k(y), k(y) + ). To show that this defines a continuous map let U := Spa(R, R + ) X be open affinoid. The preimage f (U) in Y is open, so by passing to rational subsets we may assume WLOG that we have a diagram over Spa(K, K + ) Y j f Y f U X where Y = Spa(S, S ) and all arrows except f are maps of adic spaces. In particular, we have a morphism f, : (R, R + ) (S, S ). Define (S + ) as the integral closure of S + f, (R + ) in S. As Spa(R, R + ) Spa(K, K + ) is of lightly finite type, there exists a finite set E R + such that R + is the integral closure of K + [E R ] in R. But now (S + ) agrees with the integral closure of S + f, (E) in S, in particular, Spa(S, (S + ) ) Spa(S, S + ) = Y is open. By construction if y Y such that f (y) U, then x Spa(S, (S + ) ), i.e., f 1 (U) = Spa(S, (S + ) ) is open in Y.

25 A QUOTIENT OF THE LUBIN TATE TOWER 25 Constructing the map of structure sheaves is now easy. Note that j O Y = O Y and f gives a morphism O X f O Y. But so we get a natural map f O Y = f j O Y = f O Y, f : O X f O Y. Finally one checks that the morphism ( f, f ) is compatible with the valuations on the stalks, therefore indeed defines a morphism in V. Lemma Consider a cartesian diagram X X Spa(K, K ) Spa(K, K + ) where (K, K + ) is a non-archimedean field, X is a perfectoid space and f is partially proper and locally of lightly finite type. Assume that X = Spa(R, R + ) is affinoid perfectoid. Then X is affinoid perfectoid. Proof. We claim that X = Spa(R, (R + ) ), where (R + ) is the integral closure of K + + R in R. To see this let (S, S + ) be a perfectoid (K, K + )-algebra. Then by the previous lemma X(S, S + ) = X(S, S ) = X (S, S ) = Hom cont (R, S). On the other hand ( Spa(R, (R + ) ) ) (S, S + ) = Hom cont (R, S, (R + ) S + ) is given by continuous homomorphisms R S that send (R + ) into S +. But any continuous homomorphism R S has this property as the image of the topologically nilpotent elements R lands in S S +, K + is mapped to S + and S + is integrally closed. The map M /B(Q p ) C p is locally of lightly finite type as locally M /B(Q p ) is pro-finite étale over a rigid space. Therefore π GH is locally of lightly finite type as well. Let x = Spa(K, K + ) be a point of P 1 C p, then we define the fibre π 1 GH (x) as the space π 1 GH (x) := f Uq.c. open: U π GH 1 (x) Note that we have an equality of uniform adic spaces x = V V q.c. open: V x U.

26 26 JUDITH LUDWIG and therefore the diagram π 1 GH (x) h x = Spa(K, K + ) f M /B(Q p ) P 1 C p is cartesian in the category of uniform adic spaces. If U = Spa(S, S + ) = π 1 GH (V ) M /B(Q p ) for an affinoid open V = Spa(R, R + ) P 1 C p, then f 1 (U) = Spa(T, T + ) where T is the uniform completion of R S K and T + is generated by R + K +. Therefore h is locally of lightly finite type. We can now show that the fibres of π GH are all affinoid perfectoid. Proposition Let x = Spa(K, K + ) be a point of P 1 C p, then the fibre π 1 GH (x) = U is affinoid perfectoid. Uq.c. open: U π GH 1 (x) Proof. For points of rank one this follows from Proposition 3.17 and the fact that given a direct limit lim R i of perfectoid algebras (R i, Ri ) the uniform completion R := (lim R i ) is again perfectoid. For points x = Spa(K, K + ) of higher rank the claim follows from the previous lemma and the fact that π 1 GH (x 0) π 1 GH (x) is cartesian. x 0 = Spa(K, K ) x h 4. Cohomological Consequences Let F/Q p be a finite extension, n 1 an integer and let D be the division algebra with center F and invariant 1/n. In [12], Scholze constructs a functor smooth admissible { } F p -representations D -equivariant sheaves on (P n 1 )ét of GL n (F ) π F π and shows that the cohomology groups H í et(p n 1 C p, F π ) come equipped with an action of Gal(F /F ), are admissible D -representations and vanish for all i > 2(n 1). In particular he constructs a candidate for the mod p local Langlands correspondence and the mod p Jacquet Langlands correspondence. Here, we specialize to the case of F = Q p and n = 2 and we abbreviate S i (π) := H í et(p 1 C p, F π ).

27 A QUOTIENT OF THE LUBIN TATE TOWER 27 For a pair χ = (χ 1, χ 2 ) of characters Q p F q we write Ind(χ) or Ind(χ 1, χ 2 ) for the smooth parabolic induction Ind GL 2(Q p) B(Q p) (χ), so Ind GL 2(Q p) B(Q p) (χ) := {f : GL 2 (Q p ) F q f cont. and f(bx) = χ(b)f(x) b B(Q p )}. This smooth admissible representation is irreducible unless χ 1 = χ 2. When χ 1 = χ 2 we have a short exact sequence 0 χ 1 Ind(χ 1, χ 1 ) St χ 1 0 where St denotes the Steinberg representation, which is irreducible. In this section we show that for π = Ind(χ 1, χ 2 ) with χ 1 χ 2 and for π = St µ any twist of the Steinberg representation the cohomology S i (π) is concentrated in degree one. The vanishing of S 0 (π) is easy. Proposition 4.1. (1) Let χ i : Q p F q, i = 1, 2 be two distinct smooth characters. Then S 0 (Ind(χ 1, χ 2 )) = 0. (2) Let π := St µ be a twist of the Steinberg representation by a character µ : Q p F q. Then S 0 (St µ) = 0. Proof. Note that Ind(χ 1, χ 2 ) SL 2(Q p) = 0 as well as (St µ) SL 2(Q p) = 0. The claim now follows from [12, Prop. 4.7], where it is proved in particular that for any admissible F p [GL 2 (Q p )]-module V the natural map is an isomorphism. H 0 ét(p 1 C p, F V SL 2 (Qp)) H 0 ét(p 1 C p, F V ) To show vanishing of cohomology in degree two we use the quotient constructed above. Again let χ i : Q p F q, i = 1, 2, be two smooth characters. Let F χ be the sheaf on (M /B(Q p ))ét defined as F χ (U) = Map cont,b(qp)( U M /B(Q p) M, χ) for any étale U M /B(Q p ). Here one turns the right action of B(Q p ) on M into a left action and then the subscript B(Q p ) denotes the set of all maps which are equivariant for this action. Proposition 4.2. The sheaf F χ on (M /B(Q p ))ét is a F q -local system of rank one. Proof. The characters χ i are smooth, so there exists an open subgroup K B(Q p ) such that χ 1 K = χ 2 K = 1 is trivial. Fix such K B(Z p ). Then f : M /K M /B(Q p ) is an étale covering and f F χ is the constant sheaf F q. Lemma 4.3. Let X be a topological space with a left action of G such that G X X is continuous. Then Map cont,g (X, Ind G B(χ)) = Map cont,b (X, χ).