l-adic Cohomology of the Dual Lubin-Tate Tower via the Exterior Power Citation Yeo, Foo Yee (2016) l-adic Cohomology of the Dual Lubin-Tate Tower via the Exterior Power. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9F769JT. https://resolver.caltech.edu/CaltechTHESIS:05272016-150420774 Abstract We study the l-adic cohomology of the dual Lubin-Tate tower by using the exterior power of a pi-divisible O-module to relate it to the cohomology of the Lubin-Tate tower. By using a result of Harris-Taylor on the cohomology of the Lubin-Tate tower, we show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower realizes the local Langlands and the Jacquet-Langlands correspondences, up to certain twists. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: dual Lubin-Tate tower; Rapoport-Zink spaces; l-adic cohomology; local Langlands; Jacquet-Langlands Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Mantovan, Elena Thesis Committee: Mantovan, Elena (chair) Ramakrishnan, Dinakar Flach, Matthias Zhu, Xinwen Defense Date: 18 May 2016 Record Number: CaltechTHESIS:05272016-150420774 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05272016-150420774 DOI: 10.7907/Z9F769JT Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9795 Collection: CaltechTHESIS Deposited By: Foo Yee Yeo Deposited On: 31 May 2016 16:05 Last Modified: 07 Nov 2022 23:13 Thesis Files Preview PDF See Usage Policy. 366kB Repository Staff Only: item control page

l-adic Cohomology of the Dual Lubin-Tate Tower via the Exterior Power Thesis by Foo Yee Yeo In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2016 Defended May 18, 2016 ii © 2016 Foo Yee Yeo All rights reserved iii ACKNOWLEDGEMENTS I would like to thank my advisor, Elena Mantovan, for suggesting this problem, and for her invaluable help, guidance and support throughout my work on this problem, without which this thesis would not have been possible. I would also like to thank her for carefully reading through various drafts of this thesis, and suggesting numerous improvements and corrections. I would also like to thank Hadi Hedayatzadeh for his work on the exterior power of π L -divisible OL -modules and for helpful discussions I had with him. iv ABSTRACT In this thesis, we study the l-adic cohomology of the dual Lubin-Tate tower by using the exterior power of a π L -divisible OL -module to relate it to the cohomology of the Lubin-Tate tower. By using a result of Harris-Taylor on the cohomology of the Lubin-Tate tower, we show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower realizes the local Langlands and the Jacquet-Langlands correspondences, up to certain twists. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: Background on π L -divisible OL -modules . . . . . . . . . . . . . 2.1 Ramified Witt vectors, Dieudonné OL -modules and OL -displays . . . 2.2 Isoclinic π L -divisible OL -modules and their endomorphism algebras 2.3 Serre dual and exterior powers of π L -divisible OL -modules . . . . . Chapter III: Moduli spaces of π L -divisible OL -modules . . . . . . . . . . . . 3.1 Basic Rapoport-Zink spaces . . . . . . . . . . . . . . . . . . . . . . 3.2 Results on the Lubin-Tate tower . . . . . . . . . . . . . . . . . . . . Chapter IV: The endomorphism algebras, GL h and representation theory . . . 4.1 The endomorphism algebras and GL h . . . . . . . . . . . . . . . . . 4.2 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . Chapter V: The duality map and the exterior power map on moduli spaces . . 5.1 Definitions of the duality map and the exterior power map . . . . . . 5.2 Properties of the exterior power map and group actions . . . . . . . . Chapter VI: The geometry and cohomology of the dual Lubin-Tate tower . . . 6.1 Geometry of the dual Lubin-Tate tower . . . . . . . . . . . . . . . . 6.2 Cohomology of the dual Lubin-Tate tower . . . . . . . . . . . . . . . 6.3 Geometric realization of local correspondences and vanishing results Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v 1 9 9 10 12 15 15 17 19 19 27 31 31 32 35 35 38 43 46 1 Chapter 1 INTRODUCTION Generalizing earlier work of Lubin-Tate ([LT66]) and Drinfeld ([Dri74]), RapoportZink ([RZ96]) constructed moduli spaces of deformations of p-divisible groups for data of type EL and PEL. These spaces, which are now known as Rapoport-Zink spaces, are local models of Shimura varieties. Kottwitz (cf. [Rap95]) conjectured that the l-adic cohomology of basic Rapoport-Zink spaces should partly realize the local Langlands and the Jacquet-Langlands correspondences. A particularly well-known example of a Rapoport-Zink space is the Lubin-Tate tower. Let p be an odd prime, L be a finite extension of Q p with uniformizer π L and residue field k L = Fq , and let G1,h−1 be the isoclinic π L -divisible OL -module over F p of dimension 1 and height h ≥ 2. Rapoport-Zink ([RZ96]) showed that, in this special case, the functor of deformations of G1,h−1 with a quasi-isogeny is representable by a formal scheme LTh , which is non-canonically isomorphic to ` Lnr i∈Z Spf O L̆ [[t 1, . . . , t h−1 ]], where O L̆ is the ring of integers of L̆ = L · Q p . We get a tower of étale covers over the rigid analytic generic fiber of LTh by trivializing the Tate module of the universal π L -divisible OL -module. Let LThn be the étale cover that trivializes the Tate module of the universal π L -divisible OL -module mod Γn , where Γn is the subgroup of GL h (OL ) consisting of matrices congruent to Ih mod π Ln . The projective system LTh∞ = {LThn }n is known as the Lubin-Tate tower. Let C be the completion of an algebraic closure of L̆. After base change to C, the Lubin-Tate tower has actions by the three groups GL h (L), D × (where D = End(G1,h−1 ) ⊗Z p Q p is the central division algebra over L of invariant 1/h) and W L (the Weil group of L). By using global methods involving Shimura varieties, Harris-Taylor ([HT01]) showed that the supercuspidal part of the cohomology of the Lubin-Tate tower realizes the local Langlands correspondence for GL h and the Jacquet-Langlands correspondence for D× , up to some twists, thus verifying Kottwitz’s conjecture in this particular case. If, in the above moduli problem, we replace G1,h−1 by G h−1,1 , the isoclinic π L divisible OL -module G h−1,1 of dimension h − 1 and height h, we obtain another projective system of rigid analytic spaces {Mhn }n , and we call this the dual LubinTate tower. Again, after base change to C, the dual Lubin-Tate tower Mh∞ has actions 2 by the groups GL h (L), E × (where E = End(G h−1,1 ) ⊗Z p Q p ) and W L . The aim of this paper is to understand the cohomology of the dual Lubin-Tate tower Mh∞ by relating it to the cohomology of the Lubin-Tate tower LTh∞ . We will show that the cohomology of the dual Lubin-Tate tower also realizes the Jacquet-Langlands and the local Langlands correspondences, up to certain twists. For the rest of the introduction, we will slightly abuse notation, and not distinguish between the Lubin-Tate tower LTh∞ , which is defined over L̆, and its base change to C. We will also do the same for the dual Lubin-Tate tower Mh∞ . Although some of the statements still hold even without base change, for simplicity, the reader may assume (for the rest of the introduction) that we are always working with the base change to C. As usual, let l be an odd prime different from p. We consider the functor Hci (Mh∞ ) : Rep E × → Rep(GL h (L) × W L ) ρ 7→ HomE × (Hci (Mh∞, Ql ), ρ). The following are the main results: Theorem A. For π ∈ Cusp(GL h (L)), h−1 Hch−1 (Mh∞ )cusp (JL−1 (π)) = π ⊗ rec(π ∨ ⊗ ( χ π ◦ det) ⊗ (| · | ◦ det) 2 )(h − 1), where χ π is the central character of π, Hch−1 (Mh∞ )cusp is the supercuspidal part of Hch−1 (Mh∞ ), rec is the local Langlands correspondence and JL is the JacquetLanglands correspondence. Theorem A is a consequence of the following Theorem which expresses the cohomology of the dual Lubin-Tate tower Mh∞ in terms of the cohomology of the Lubin-Tate tower LTh∞ . For convenience, we will fix certain embeddings of the division algebras D× and E × into GL h (L h ) (see Section 4.1), where L h is the unramified extension of degree h over L. Theorem B. Define θ : D× → E × φ 7→ (det φ)(φ−1 )T , ψ : GL h (L) → GL h (L) g 7→ (det g)(g −1 )T . 3 Let θ ∗ Hci (LTh∞, Ql ) be the pushforward of the representation Hci (LTh∞, Ql ) under the map θ : D×  θ(D × ). Then for all i ≥ 0,   Hci (LTh∞, Ql ) E× i ∞ h (L) * + Q )  Ind θ ψ ∗ ResGL H (M , l θ(D × ) ∗ h ψ(GL h (L)) c K , as E × × GL h (L) × W L representations, where K = ker(θ). To prove Theorem B, we first define a map from LTh∞ to Mh∞ . Since the dual of G1,h−1 is G h−1,1 , the Serre dual of a π L -divisible OL -module can be used to define a map ∨ : LTh∞ → Mh∞ . However, this map is not W L -equivariant. For example, the Tate module of the Serre dual of (L/OL ) h is OLh (1), and not OLh . As such, while the duality map ∨ could be used to study the cohomology of Mh∞ as a E × × GL h (L)-representation, the action of W L appears to be difficult to understand. V Instead, we will use the exterior power r , which was introduced by Pink and studied by Hedayatzadeh in [Hed10], [Hed13] and [Hed14] for certain π L -divisible OL -modules. The idea of using the exterior power has previously been explored by Chen in [Che13] where she used the top exterior power to define a determinant map from Rapoport-Zink spaces to a tower of dimension 0 rigid analytic spaces. In this thesis, however, we will consider the h − 1 exterior power, and not the top exterior V power. As h−1 G1,h−1 ≈ G h−1,1 , using the h − 1 exterior power, we can define a map h−1 ^ : LTh∞ → Mh∞ . V Unlike the duality map ∨, the map h−1 is W L -equivariant, and the other group V actions can be understood as well. Under the map h−1 , the action of φ ∈ D× on LTh∞ corresponds to the action of θ(φ) on Mh∞ , and the action of g ∈ GL h (L) on LTh∞ corresponds to the action of ψ(g) on Mh∞ . Before proving Theorem B, it is useful to first look at the level 0 case. Using Grothendieck-Messing theory and the fact that at level 0, the duality map ∨ induces an isomorphism on the connected components, we can show that the same is true for V the map h−1 . Let (LThn )m be the subspace of LThn corresponding to quasi-isogenies 4 of height m, and define (Mhn )m similarly. By the above, we can identify (LTh0 )0 with V (Mh0 )0 using the exterior power map h−1 . For simplicity, we will first specialize to the case where h − 1 is coprime to p(q − 1). Under this condition, we can check that K = ker(θ) is trivial, the map θ gives a × to O × , and θ maps a uniformizer of D × to an element of E × bijection from OD E × to O × with normalized valuation h − 1. The fact that θ gives a bijection from OD E Vh−1 n shows that induces a bijection from the connected components of (LTh )0 to the connected components of (Mhn )0 . Using this, and the fact that (LThn )0 and (Mhn )0 V are both étale covers of (LTh0 )0 of the same degree, we deduce that h−1 induces an isomorphism from (LThn )0 to (Mhn )0 . By considering the group actions, it is V then clear that h−1 induces an isomorphism from (LThn )m to (Mhn )(h−1)m . With V the above knowledge of the geometry of h−1 : LTh∞ → Mh∞ , we can then prove Theorem B in this special case. Now, let us look at the general case. Unlike in the previous case where h − 1 is V coprime to p(q − 1), the map h−1 no longer induces a bijection from the connected components of (LThn )0 to the connected components of (Mhn )0 . However, it is still true that each connected component of LThn is mapped isomorphically onto some connected component of Mhn . Let d = (h − 1, q − 1) and e = v p (h − 1). We will separately consider the cases where (i) d , 1 and e = 0, (ii) d = 1 and e , 0. It should then be clear how to handle the general case. First, we recall that (LThn )0 has q n−1 (q − 1) connected components and a result of × on the connected components Strauch ([Str06]) tells us that the action of φ ∈ OD depends on det(φ) mod (1 + π Ln OL ). We would like to understand the map on V the connected components induced by h−1 . To do so, we study the map θ 0,n : × → O × under the det mod (1 + (OL /π Ln OL ) × → (OL /π Ln OL ) × induced by θ : OD E π Ln OL ) map. In fact, θ 0,n is simply the map x 7→ x h−1 . × → O × is cyclic of order d and this maps In case (i), the kernel K of the map θ : OD E isomorphically to ker(θ 0,n ) under the det mod (1 + π Ln OL ) map. Hence, if we write V Yn for the image of the map h−1 : LThn → Mhn , then we have ψ Hci (Yn, Ql )  θ ∗ * ∗ , Hci (LThn, Ql ) K + - 5 as θ(D× ) × GL h (L) × W L representations. Let E θ be the subgroup of E × generated by Im(θ) and OL×h , and let X n be the ` subspace m∈Z (Mhn )(h−1)m of Mhn . It is easy to check that, in fact, E θ = {ϕ ∈ E × : (h − 1)| val(ϕ)}. Using that E θ / Im(θ) is a finite abelian group, and that X n is equal to the disjoint union a Xn = ϕ(Yn ), ϕ∈E θ / Im(θ) θ E i × θ we can show that Hci (X n, Ql )  Indθ(D × ) (Hc (Yn, Ql )). Similarly, E /E is finite abelian (in fact cyclic of order d), and a Mhn = ϕ(X n ), ϕ∈E × /E θ × so we have Hci (Mhn, Ql )  IndEE θ (Hci (X n, Ql )). Putting these together gives us Theorem B for case (i). We now look at case (ii). In this case, the kernel of the map θ : D× → E × is K = µ pe (L), the group of pe roots of unity in L. Unlike in the previous case, K does not surject onto ker(θ 0,n ) under the det mod (1 + π Ln OL ) map. But if we let Kn = 1 + π Ln−ee L OL ≤ D× , then the det mod (1 + π Ln OL ) map gives a surjection of K × Kn onto ker(θ 0,n ), and we can show that ψ Hci (Yn, Ql )  θ ∗ * ∗ Hci (LThn, Ql ) K × Kn , +. - Now, by following the argument in case (i), we have ψ ∗  i n h (L) ResGL ψ(GL h (L)) Hc (Mh , Ql )  Eθ *  Indθ(D × ) θ∗ Hci (LThn, Ql ) K × Kn , The map Hci (LThn−ee L , Ql ) → Hci (LThn, Ql ) factors through 1 + π Ln−ee L OL acts trivially on LThn−ee L , so we have lim −−→ n Hci (LThn, Ql ) Kn +. - Hci (LThn,Ql ) since Kn = Kn = Hci (LTh∞, Ql ). Piecing together the above, we obtain Theorem B for case (ii). It is not hard to see that the methods used to deal with cases (i) and (ii) can be combined to give a proof of Theorem B in the general case. 6 Using Theorem B, and applying Frobenius reciprocity, we see that the following diagram commutes: Rep E ×   × ρ7→θ ∗ ResθE(D × ) ρ / Rep D × Hci (Mh∞ ) Hci (LTh∞ )  Rep(GL h (L) × W L )  GL (L)  h π ⊗r π⊗r7→ψ ∗ Resψ(GL (L))  / Rep(GL h (L) × W L ) h Although the above diagram only gives us Hci (Mh∞ )[ρ] as a ψ(GL h (L))-representation, we can use the duality map to understand the GL h (L)-action. Alternatively, in the special case where p(q −1) is coprime to h−1, the group GL h (L) is generated by the subgroup ψ(GL h (L)) and the element π L ∈ GL h (L), so we just need to understand the action of π L , but the action of π L Ih ∈ GL h (L) on Mh∞ corresponds to the action of π L−1 ∈ E × .     E× ∗ ResGL h (L) π can be written as Finally, we observe that θ ∗ Resθ(D × ) ρ and ψ ψ(GL h (L)) twists of ρ and π respectively by their central characters. This is useful since both the local Langlands and the Jacquet-Langlands correspondences are compatible with twists. Using the theorem of Harris-Taylor on the cohomology of the Lubin-Tate tower, we can then prove Theorem A. We start by introducing some background material on π L -divisible OL -modules in Chapter 2. In particular, we recall the Serre dual of a π L -divisible OL -module and introduce the exterior power for π L -divisible OL -modules of dimension 1, whose existence has been proven in [Hed10]. In Chapter 3, we introduce moduli spaces of certain π L -divisible OL -modules, which includes, as special cases, the Lubin-Tate and the dual Lubin-Tate towers. We also state the well known result of Harris-Taylor on the cohomology of the Lubin-Tate tower. We collect some useful results on the endomorphism algebras and on representation theory in Chapter 4. We will describe in this chapter embeddings of D× and E × into GL h (L h ), and study properties of the maps θ and ψ. In Chapter 5, we use the exterior power introduced in Chapter 2 to define a map from the Lubin-Tate tower LTh∞ to the dual Lubin-Tate tower Mh∞ . We also study how the group actions behave under this map. 7 And finally, in Chapter 6, we study the geometry of the dual Lubin-Tate tower, and then derive results on its cohomology, including Theorems A and B. 8 Notation p: odd prime l: odd prime different from p L: finite extension of Q p π L : uniformizer of L q: order of k L , the residue field of L n L : degree of L over Q p e L : ramification degree of L over Q p k: perfect field of characteristic p containing k L = Fq W L : Weil group of L WOL (k): ramified Witt vectors of k D(G): Dieudonné OL -module of the π L -divisible OL -module G G m,n : isoclinic π L -divisible OL -module over F p of dim m and height m + n Dm,n : endomorphism algebra of G m,n G∨ : Serre dual of the π L -divisible OL -module G Vr G: r-th exterior power of the π L -divisible OL -module G D: endomorphism algebra of G1,h−1 E: endomorphism algebra of G h−1,1 LTh : Lubin-Tate space, moduli space of deformations of G1,h−1 with a quasi-isogeny LThn : étale cover over the generic fiber of LTh parameterizing level n structure LTh∞ : Lubin-Tate tower, the projective system {LThn }n∈N Mh : dual Lubin-Tate space, moduli space of deformations of G h−1,1 with a quasiisogeny Mhn : étale cover over the generic fiber of Mh parameterizing level n structure Mh∞ : dual Lubin-Tate tower, the projective system {Mhn }n∈N Nrd: reduced norm of a division algebra 9 Chapter 2 BACKGROUND ON π L -DIVISIBLE OL -MODULES 2.1 Ramified Witt vectors, Dieudonné OL -modules and OL -displays Let p be an odd prime, L be a finite extension of Q p with uniformizer π L and residue field k L = Fq . For an OL algebra R, the ring of ramified Witt vectors WOL (R) (cf. [Haz80]) is WOL (R) = RN with the unique OL -algebra structure such that 1. the Witt polynomials wn : WOL (R) → R qn q n−1 (x 0, x 1, x 2, . . .) 7→ x 0 + π L x 1 + · · · + π Ln x n are OL algebra homomorphisms for all n ≥ 0, 2. OL -algebra homomorphisms R → S induce OL -algebra homomorphisms WOL (R) → WOL (S). There is an endomorphism of WOL (R), defined by wn (σ(x)) = wn+1 (x), called the Frobenius endomorphism. In fact, σ is a homomorphism of OL -algebras. Let k ⊇ k L be a perfect field of characteristic p. Then WOL (k)  OL D ⊗W (k L ) W (k), where W (k) = WZ p (k) is the usual ring of Witt vectors of k. It is a complete discrete valuation ring with residue field k and uniformizer π L . In particular, WOL (F p ) is Lnr isomorphic to the ring of integers OL̆ of L̆, where L̆ = L · Q p is the completion of the maximal unramified extension of L. We now introduce Dieudonné OL -modules and OL -displays. A Dieudonné OL module over k is a free WOL (k)-module of finite rank, together with two maps F and V such that F is σ-linear, and V is σ −1 -linear, and FV = π L = V F. There is an equivalence of categories between the category of π L -divisible formal OL modules over k and the category of Dieudonné OL -modules over k such that V is topologically nilpotent in the π L -adic topology (cf. [Ahs11], Proposition 2.2.3 and Theorem 5.3.8). For a π L -divisible OL -module G, we will denote its covariant Dieudonné OL -module by D(G). Under this equivalence of categories, D(G) ht(G) = rankW OL (k) D(G), dim(G) = dim k . V D(G) 10 Generalizing earlier work of Zink ([Zin02]) and Lau ([Lau08]), Ahsendorf ([Ahs11]) introduced the notion of an OL -display. An OL -display over an OL -algebra R is a quadruple P = (P, Q, F, V −1 ) with P a finitely generated projective WOL (R)module, Q ⊂ P a submodule and F : P → P and V −1 : Q → P Frobenius linear maps, which satisfy some additional conditions. If π L is nilpotent in R, then there is an equivalence of categories between the category of nilpotent OL -displays over R and the category of formal π L -divisible OL -modules over R (cf. [Ahs11], Theorem 5.3.8). 2.2 Isoclinic π L -divisible OL -modules and their endomorphism algebras We now introduce certain isoclinic π L -divisible OL -modules which will be useful later. Suppose m, n are coprime non-negative integers. We shall write G m,n for the π L -divisible OL -module over F p with covariant Dieudonné OL -module D(G m,n ) having basis {b0, b1, . . . , bm+n−1 } such that Fbi = bi+n, V bi = bi+m and b(m+n)+i = π L bi . G m,n is an isoclinic π L -divisible OL -module of dimension m and height m + n. In particular, D(G1,h−1 ) has a basis {e0, . . . , e h−1 } such that V ei = ei+1 (0 ≤ i ≤ h − 2), V e h−1 = π L e0, and D(G h−1,1 ) has a basis { f 0, . . . , f h−1 } such that F f i = f i−1 (1 ≤ i ≤ h − 1), F f 0 = π L f h−1 . Let ODm,n = End(G m,n ) be the endomorphism ring of G m,n and let Dm,n = ODm,n ⊗Z p Q p = End(G m,n ) ⊗Z p Q p be the endomorphism algebra of G m,n . The following description of the endomorphism ring ODm,n is a straightforward generalization of [JO99, Lemma 5.4]. We include a proof here for lack of a suitable reference. Lemma 2.2.1. Let a, b ∈ Z be such that am + bn = 1, and let OL m+n = WOL (Fm+n ) be the ring of integers of L m+n , the unramified extension of degree m + n over L. Then ODm,n = End(G m,n )  OL m+n [πm,n ], where λ · πm,n = πm,n · σ b−a (λ) for λ ∈ OL m+n , and σ is the Frobenius map. λ ∈ OL m+n acts on b0 via multiplication by λ, and πm,n is a uniformizer of the discrete valuation ring End(G m,n ). 11 Proof. Using the above basis {b0, b1, . . . , bm+n−1 } of D(G m,n ), we easily check that πm,n (bi ) = bi+1 (0 ≤ i ≤ h − 2), πm,n (bh−1 ) = π L b0, is an element of End(G m,n ). Suppose φ ∈ End(G m,n ) is such that φ(b0 ) = λb0 for some λ ∈ OL̆ . Note that (F bV a )i (b0 ) = bi , so φ(bi ) = φ((F bV a )i b0 ) = (F bV a )i φ(b0 ) = (F bV a )i (λb0 ) = σ (b−a)i (λ)bi . In particular, λb0 = φ(b0 ) = φ(π L−1 bm+n ) = σ (b−a)(m+n) (λ)π L−1 bm+n = σ (b−a)(m+n) (λ)b0, which shows that σ (b−a)(m+n) (λ) = λ. If we let a0 = a + n and b0 = b − m, then 0 0 a0 m + b0 n = 1, so similarly, we have σ (b −a )(m+n) (λ) = λ. But (b − a, b0 − a0 ) = 1 since (n + a)(b − a) − a(b0 − a0 ) = am + bn = 1, so σ m+n (λ) = λ and λ ∈ OL m+n . The fact that such an element does indeed lie in End(G m,n ) is again easy to check. We shall denote this element of End(G m+n ) by λ. Pm+n−1 Now, let φ be an arbitrary element of End(G m,n ). Then φ(b0 ) = i=0 αi bi for some αi ∈ OL̆ . A similar argument to that above shows, in fact, that αi ∈ Pm+n−1 i (b ) = 0, from which it follows that φ = OL m+n . Clearly, φ(b0 ) − i=0 αi πm,n 0 Pm+n−1 i αi πm,n . This proves that End(G m,n )  OL m+n [πm,n ]. i=0 To show that OL m+n [πm,n ] is a discrete valuation ring with uniformizer πm,n , we first note that OL m+n [πm,n ] is a subring of the division ring L m+n (πm,n ). Now, define a map v : L m+n (πm,n ) → Z ∪ {∞} by  m+n−1 α0 + α1 πm,n + · · · + α m+n−1 πm,n 7→ min (m + n)v L m+n (αi ) + i : 0 ≤ i < m + n . It is easy to check that v is a valuation on L m+n (πm,n ). Since OL m+n [πm,n ] = {φ ∈ L m+n (πm,n ) : v(φ) ≥ 0}, and v(πm,n ) = 1, we have shown that End(G m,n ) = OL m+n [πm,n ] is a discrete valuation ring with uniformizer πm,n . Corollary 2.2.2. Dm,n is the central division algebra over L with invariant n/(m+n). 12 Let us introduce some notation for the cases (m, n) = (1, h−1) and (m, n) = (h−1, 1). We write OD = OD1,h−1 = End(G1,h−1 ), D = D1,h−1 = End(G1,h−1 ) ⊗Z p Q p, OE = ODh−1,1 = End(G h−1,1 ), E = Dh−1,1 = End(G h−1,1 ) ⊗Z p Q p . Also let $ (respectively $0) be the uniformizer of OD (respectively OE ) as in Lemma 2.2.1. 2.3 Serre dual and exterior powers of π L -divisible OL -modules Let G be a π L -divisible OL -module over a scheme S. The dual π L -divisible OL module G∨ is 0 → (G[π L ]) ∨ → (G[π L2 ]) ∨ → (G[π L3 ]) ∨ → · · · where (G[πiL ]) ∨ = HomS (G[πiL ], Gm ) is the Cartier dual of G[πiL ]. We have ht(G) = dim(G) + dim(G∨ ) = ht(G∨ ). If S = Spec k for some perfect field k of characteristic p, then G∨ has Dieudonné OL -module D(G∨ )  D(G) ∨ = HomW OL (k) (D(G), WOL (k)) with (Fl)(v) = σ(l (V v)), (V l)(v) = σ −1 (l (Fv)), for v ∈ D(G), l ∈ D(G) ∨ . We will now look at the exterior power of π L -divisible OL -modules, which has been shown to exist by Hedayatzadeh in [Hed10] in certain situations. For G and H r L π L -divisible OL -modules over a base scheme S, we will write AltO S (G , H) for the group of alternating OL -multilinear morphisms from Gr to H. Definition 2.3.1. [Hed10, Definition 5.3.3] Let G, G0 be π L -divisible OL -modules over a scheme S, with an alternating OL -multilinear morphism λ : Gr → G0 such that for all π L -divisible OL -modules H over S, the induced morphism r L λ ∗ : HomS (G0, H) → AltO S (G , H) ψ 7→ ψ ◦ λ is an isomorphism. Then G0 (or, more precisely, λ : Gr → G0) is called the r-th V exterior power of G, and we denote it by r G. 13 Theorem 2.3.2. [Hed10, Theorem 9.2.36] Let S be a locally Noetherian OL -scheme and G be a π L -divisible OL -module of height h and dimension at most 1. Then there   V exists a π L -divisible OL -module r G over S of height hr , and an alternating V morphism λ : Gr → r G such that for every morphism f : S0 → S and every π L -divisible OL -module H over S0, the homomorphism ∗ L HomO S0 ( f r ^ ∗ r L G, H) → AltO S 0 (( f G) , H) V induced by f ∗ λ is an isomorphism. In other words, r G is the r-th exterior power of G over S, and taking the exterior power commutes with arbitrary base change. V Moreover, the dimension of r G is a locally constant function ( !) h−1 dim : S → 0, r −1   if G s is étale, 0 s 7→      h−1 otherwise.  r−1 In the cases that S is the spectrum of a perfect field of char p > 2, or the spectrum of a local Artin OL -algebra with residue char p > 2, we can write down the exterior power in terms of its Dieudonné OL -module or its OL -display. Theorem 2.3.3. [Hed10, Construction 6.3.1, Lemma 6.3.2, Proposition 6.3.3, Section 9.1] Let P = (P, Q, F, V −1 ) be an OL -display of height h with tangent module of rank one. Fix a normal decomposition P = L ⊕ T, Then Q = L ⊕ I RT . r ^ r r r−1 r ^ ^ ^ ^ P = * P, Q, V −1 ∧ F, V −1 + ,  h  h−1 is an OL -display of height r and rank r−1 . The map λ:P → r r ^ P (x 1, . . . , x r ) 7→ x 1 ∧ · · · ∧ x r is an alternating morphism of OL -displays and satisfies the universal property that the map r ^ Hom( P, P 0 ) → Alt(P r , P 0 ) induced by λ is an isomorphism. 14 Theorem 2.3.4. [Hed10, Proposition 9.2.24] Let R be a local Artin OL -algebra, and G a π L -divisible OL -module over R with special fiber that is a connected π L -divisible OL -module of dim 1. Let P be the OL -display of the π L -divisible OL -module G. V V Then the OL -display of r G is r P. As an immediate corollary, for G a π L -divisible OL -module of dimension 1 over a V V perfect field of characteristic p, the Dieudonné OL -module of r G is just r D(G), where F and V are as in Theorem 2.3.3. So using the above description of D(G1,h−1 ), we see that the Dieudonné OL -module V of h−1 G1,h−1 has a basis ẽ0, . . . , ẽ h−1 , where ẽi = (−1)i e0 ∧ e1 ∧ · · · ∧ ei−1 ∧ ei+1 ∧ · · · ∧ e h−1, and F ẽi = (−1) h−1 ẽi−1 V ẽi = (−1) h−1 π L ẽi+1 (1 ≤ i ≤ h − 1), (0 ≤ i ≤ h − 2), F ẽ0 = (−1) h−1 π L ẽ h−1, V ẽ h−1 = (−1) h−1 ẽ0 . Comparing this with the above description of D(G h−1,1 ), we see that isomorphic to G h−1,1 . Vh−1 G1,h−1 is 15 Chapter 3 MODULI SPACES OF π L -DIVISIBLE OL -MODULES 3.1 Basic Rapoport-Zink spaces Let C be the category of schemes S over Spec OL̆ such that p is locally nilpotent on S. Let S̄ be the closed subscheme of S defined by the ideal sheaf pOS . We consider a special case of the moduli spaces constructed by Rapoport-Zink in [RZ96]. Theorem 3.1.1. [RZ96, Theorem 3.25 and Proposition 3.79] Let h, d ∈ Z+ coprime, and G be an isoclinic π L -divisible OL -module over F p of height h and dim d. Let F be the functor C −→ Set ,    X is a connected π L -divisible OL -module/S    S 7−→  (X, β) :  ∼. and β : G S̄ → X S̄ a quasi-isogeny   Then F is representable by a formal scheme M d , which is formally locally of finite h type over Spf OL̆ . For d = 1, the formal scheme M 1 is (non-canonically) isomorphic h ` to i∈Z Spf OL̆ [[t 1, . . . , t h−1 ]]. Let M 0d be the rigid analytic generic fiber of M d . We consider a special case of h h the tower of rigid-analytic coverings of M 0 introduced in [RZ96, 5.34]. We write d h G for the universal π L -divisible OL -module on M d , and TπL G for its Tate module. h Let Γn be the subgroup of GL h (OL ) consisting of matrices congruent to Ih mod π Ln . Then we have a finite étale cover M nd of M 0d that parameterizes trivializations h h  α n : OLh − → TπL G mod Γn . n Define M ∞ d to be the projective system (M d )n∈N with maps h h f mn : M nd → M md h (m ≤ n) h (X, β, α n ) 7→ (X, β, α n mod Γm ). 16 Recall from Section 2.2 that G d,h−d is an isoclinic π L -divisible OL -module of height h and dim d. Any two π L -divisible OL -modules with the same Newton polygon are isogenous, so in the definition of Mnd above, we might as well take G = G d,h−d . h The groups Dd,h−d = End(G d,h−d ) ⊗Z p Q p and GL h (L) act on M ∞ d . Let C be the h completion of an algebraic closure of L̆. We also have an action of the Weil group W L on M ∞ d after base change to C. We describe the actions here. h 1. Action of Dd,h−d : γ ∈ Dd,h−d acts on M nd by h (X, β, α n ) 7→ (X, β ◦ γ −1, α n ). 2. Action of GL h (L): Any element g0 ∈ GL h (L) can be written as g0 = π Lk g with k ∈ Z and g ∈ GL h (L) ∩ Math (OL ). Hence it is sufficient to describe the action of such elements. So let g ∈ GL h (L) ∩ Math (OL ) and S → M d be a formal scheme. We have h an isomorphism α rig : (L/O ) h → (L/O  )h − → X rig . Let ker(g) be the kernel of the map L X rig h rig be the π -divisible O -module L ) , and Y L L rig α (ker(g)) g : (L/OL over S rig . In other words, ker(g) _  (L/OL ) h  g (L/OL α rig  α rig  / α rig (ker(g)) _  / X rig qrig )h  α rig / rig X rig Y = αrig (ker(g))  Then g acts on M ∞ d by h (X, β, α) 7→ (Y, q ◦ β, α) where • Y is a π L -divisible OL -module over some admissible blow-up of S with rigid analytic generic fiber Y rig , • q is the special fiber of the quotient map X → Y which has generic fiber X rig X rig → αrig (ker(g)) = Y rig , 17   • α : OLh − → TπL Y has generic fiber α rig : OLh − → TπL Y rig which corre sponds to α rig : (L/OL ) h − → Y rig . 3. Action of W L : Suppose w ∈ W L 7→ σ n ∈ Gal(k L /k L ) = Gal(F p /Fq ) ⊆ Gal(F p /F p ) under the reduction map, where σ = {x 7→ x p } ∈ Gal(F p /F p ) is the arithmetic Frobenius. Then the action of w on M ∞ d × L̆ C is h w : (X, β, α) 7→ (X w, β w ◦ F n G/F p , αw ) where • X w = X × S S w, (pn ) • β w : G S̄ n G (p ) = X ×F p,σ n F p , → X S̄w is induced by β : G S̄ → X S̄ , • α w is the map αw OLh = (OLh ) w −−−−−−−→ TπL X w .  Let us introduce some notation for the cases d = 1 and d = h − 1. For d = 1, we will write LThn for M n1 and LTh∞ for the Lubin-Tate tower M ∞ 1 . h h And for d = h − 1, we write Mhn for M nh−1 , Mh∞ for the dual Lubin-Tate tower M ∞ h−1 . h h 3.2 Results on the Lubin-Tate tower A well known result of Harris-Taylor tells us that the l-adic cohomology (cf. [Ber93]) of LTh∞ × L̆ C realizes the local Langlands and the Jacquet-Langlands correspondence. To be precise, let us consider the functor Hci (LTh∞ ) : Rep D× → Rep(GL h (L) × W L ) ρ 7→ Hom D× (Hci (LTh∞ × L̆ C, Ql ), ρ), and let [Hc (LTh∞ )( ρ)] denote the virtual representation (−1) h−1 i i ∞ i=0 (−1) [Hc (LTh )( ρ)]. Ph−1 Theorem 3.2.1. [HT01, Theorem VII.1.3] Let π be an irreducible supercuspidal representation of GL h (L). Then h−1 [Hc (LTh∞ )(JL−1 (π))] = [π ⊗ rec(π ∨ ⊗ (| · | ◦ det) 2 )(h − 1)] in the Grothendieck group, where rec is the local Langlands correspondence and J L is the Jacquet-Langlands correspondence. 18 Theorem 3.2.2. [Fal02, Section 6] For i , h − 1, Hci (LTh∞ )cusp ( ρ) = 0 for any ρ ∈ Rep D × , where Hci (LTh∞ )cusp is the supercuspidal part of Hci (LTh∞ ). An immediate corollary of Theorems 3.2.1 and 3.2.2 is: Corollary 3.2.3. Let π be an irreducible supercuspidal representation of GL h (L). Then h−1 Hch−1 (LTh∞ )cusp (JL−1 (π)) = π ⊗ rec(π ∨ ⊗ (| · | ◦ det) 2 )(h − 1). Let (LThn )m be the subspace of LThn corresponding to quasi-isogenies of height × × GL (O ) on the m. The following result of Strauch describes the action of OD h L n connected components of (LTh )0 × L̆ C. Theorem 3.2.4. [Str06, Theorem 4.4(i)] There exists a bijection   π0 (LThn )0 × L̆ C → (OL /π Ln OL ) × × × GL (O )-equivariant if we let (φ, g) ∈ O × × GL (O ) act on which is OD h L h L D n × −1 n (OL /π L OL ) by Nrd(φ) det(g) mod (1 + π L OL ), where Nrd is the reduced norm. 19 Chapter 4 THE ENDOMORPHISM ALGEBRAS, GL H AND REPRESENTATION THEORY 4.1 The endomorphism algebras and GL h Recall from Lemma 2.2.1 that ODm,n = End(G m,n )  OL m+n [πm,n ], where πm,n is a uniformizer of the discrete valuation ring ODm,n . Using the basis {e0, e1, . . . , e h−1 } of D(G1,h−1 ) introduced in Section 2.2, the proof of Lemma 2.2.1 shows that we get the following embedding of D ×  L h ($)\{0} into GL h (L h ): ι D : D× ,→ GL h (L h ) *. λ . σ −1 (λ) λ→ 7 ... .. . .. , *.0 ..1 . $→ 7 ..0 .. . .. .. ,0 +/ // // // σ −(h−1) (λ) - 0 πL + / 0 0 // / 0 0 // . .. .. // . . / / 0 ... 1 0 0 0 1 .. . ... ... ... .. . Similarly, using the basis { f 0, f 1, . . . , f h−1 } of D(G h−1,1 ) in Section 2.2, we get the 20 embedding ιE : E × ,→ GL h (L h ) *. λ . σ −1 (λ) λ→ 7 ... .. . .. , *. 0 .. 0 . . $0 → 7 .. .. .. .. 0 ,π L 1 0 .. . +/ // // // σ −(h−1) (λ) - 0 ... 1 ... .. . . . . 0+ / 0// .. // ./ / 0 0 . . . 1// / 0 0 . . . 0- of E ×  L h ($0 )\{0} into GL h (L h ). Lemma 4.1.1. (a) For any φ ∈ D× , Nrd(φ) = det(ι D (φ)) where Nrd is the reduced norm of the central division algebra D over L. In particular, we have det(ι D (φ)) ∈ L × . Furthermore, val(φ) = v L (det(ι D (φ))) where val is the normalized valuation on the division algebra D, and v L is the usual normalized π L -adic valuation on L. (b) The result in (a) holds with D replaced by E, and ι D replaced by ιE . Proof. We shall prove (a). The proof of (b) is almost identical. The map D × L h → Mh (L h ) (φ, µ) 7→ µ · ι D (φ) is L-bilinear, so we have an L-linear map f : D ⊗ L L h → Mh (L h ) φ ⊗ µ 7→ µ · ι D (φ). It is clear that this map respects multiplication, so it is in fact a ring homomorphism. 21 Let us show that f is injective. Suppose f (φ ⊗ µ) = Ih . Then µ · ι D (φ) = Ih , so ι D (φ) = µ−1 Ih . But the only scalar matrices in Im(ι D ) are of the form µIh where µ ∈ L. Hence, φ ⊗ µ = µ−1 ⊗ µ = 1, proving injectivity. Since the dimensions of D ⊗ L h and Mh (L h ) over L h are both equal to h2 , f is an isomorphism. Hence Nrd(φ) = det( f (φ ⊗ 1)) = det(ι D (φ)), proving the first assertion. The second part follows immediately since val(φ) = v L (Nrd(φ)). We shall view D × and E × as subgroups of GL h (L h ) using the embeddings ι D and ιE . Define the maps θ : D× → E × ψ : GL h (L) → GL h (L) φ 7→ (det φ)(φ−1 )T , g 7→ (det g)(g −1 )T . Lemma 4.1.2. Let d = (h − 1, q − 1) and e = v p (h − 1). The map × θ 0 : OD → OE× φ 7→ (det φ)(φ−1 )T (a) has kernel c ker(θ 0 ) = µ h−1 (OL ) = µ pc d (OL ) = {ζ ∈ OL : ζ p d = 1} of order pc d, where c is the maximum integer ≤ e such that OL contains the pc roots of unity. If L is unramified over Q p , then c = 0 and ker(θ 0 ) = µ d (OL ) has order d. (b) image Im(θ 0 ) = {ϕ ∈ OE× : det(ϕ) ∈ (OL× ) h−1 } which is a normal subgroup of OE× . Proof. (a) Let φ ∈ ker(θ 0 ). Since 1 = det(θ 0 (φ)) = (det φ) h−1, 22 det φ must be a (h − 1) root of unity in OL . By Hensel’s lemma, any root of h−1 x pe − 1 = 0 in Fq lifts uniquely to a root in OL . Since F×q is cyclic of order h−1 q − 1, and (h − 1, q − 1) = d, the map x 7→ x pe d is an automorphism of F×q , so µ h−1e (F p ) = µ d (F p ), which shows that µ h−1 (OL ) = µ pe d (OL ) = µ pc d (OL ). So p φ ∈ ker(θ 0 ) ⇒ det φ ∈ µ pc d (OL ) ⇒ (φ−1 )T ∈ µ pc d (OL ) ⇒ φ ∈ µ pc d (OL ). If L is unramified over Q p , then L does not contain any non-trivial p-th roots of unity, so c = 0 and µ h−1 (OL ) = µ d (OL ). (b) Since det(θ 0 (φ)) = (det φ) h−1 , and det φ ∈ OL× , the inclusion Im(θ 0 ) ⊆ {ϕ ∈ OE× : det(ϕ) ∈ (OL× ) h−1 } is clear. To show the reverse inclusion, we need to show that any ϕ satisfying det(ϕ) ∈ (OL× ) h−1 lies in Im(θ 0 ). Let γ ∈ OL× be a (h − 1) root of det(ϕ). Then θ(γ(ϕT ) −1 ) = det(γ(ϕT ) −1 )γ −1 ϕ = γ h−1 (det ϕ) −1 ϕ = ϕ. This proves the reverse inclusion. Proposition 4.1.3. Let c, d, e be as in Lemma 4.1.2. The map θ : D× → E × φ 7→ (det φ)(φ−1 )T has (a) kernel c ker(θ) = µ pc d (OL ) = {ζ ∈ OL : ζ p d = 1} of order pc d, and (b) image Im(θ) = {ϕ ∈ E × : (h − 1)| val(ϕ), det(ϕ) ∈ (L × ) h−1 }. Let E θ be the subgroup of E × generated by the subgroups OL×h and Im(θ), i.e. E θ = hOL×h , Im(θ)i. Then (c) E θ = {ϕ ∈ E × : (h − 1)| val(ϕ)}, 23 (d) the cokernel coker(θ) = with Eθ E× E×  × θ Im(θ) Im(θ) E OE× Eθ  Im(θ) Im(θ 0 ) and E× Z  . θ (h − 1)Z E Proof. Parts (a) and (b) follow almost immediately from Lemma 4.1.2. (c) It is clear that E θ ⊆ {ϕ ∈ E × : (h − 1)| val(ϕ)} since det(θ(φ)) = (det φ) h−1 and OL×h ⊆ OE× . Let us prove the reverse inclusion. It suffices to show that OE× ⊆ E θ . An arbitrary element of OE× is of the form ϕ = a0 + a1 $0 + a2 ($0 ) 2 + · · · + a h−1 ($0 ) h−1, with a0 ∈ OL×h and ai ∈ OL h for all i. We observe that the determinant map on D × , when restricted to the subgroup OL×h ≤ D× , is equal to the norm map NL h /L : OL×h → OL× . Since L h is an unramified extension of L, the map NL h /L : OL×h → OL× is surjective. Now, given any ϕ ∈ OE× , since det ϕ ∈ L × by Lemma 4.1.1, we must have det ϕ ∈ OL× . By surjectivity of the map NL h /L : OL×h → OL× , we can find u ∈ OL×h ≤ D × such that det(u) = det(ϕ). Then θ(u(ϕT ) −1 ) = det(u(ϕT ) −1 )u−1 ϕ = u−1 ϕ, so u−1 ϕ ∈ E θ and hence ϕ ∈ E θ . This proves the reverse inclusion. (d) Define Eθ E× E× → × θ Im(θ) Im(θ) E ϕ Im(θ) 7→ (ϕ$ −b Im(θ), $ b E θ ) where b is any integer congruent to val(ϕ) mod (h − 1). Since $ h−1 ∈ Im(θ) ⊆ E θ , it is clear that this is well defined and is an homomorphism. By (iii), E θ = {ϕ ∈ E × : (h − 1)| val(ϕ)}, so the valuation map gives an isomorphism E× Z  , (h − 1)Z Eθ 24 from which surjectivity of the above map follows easily. For injectivity, we note that $ b ∈ E θ = {ϕ ∈ E × : (h − 1)| val(ϕ)} ⇒ (h − 1)|b ⇒ $ b ∈ Im(θ). Hence ϕ$ −b ∈ Im(θ) and $ b ∈ E θ ⇒ ϕ$ −b ∈ Im(θ) and $ b ∈ Im(θ) ⇒ ϕ ∈ Im(θ). Now we consider the map Eθ  OE× Im(θ 0 ) ϕ 7→ ϕ$ − val(ϕ) Im(θ 0 ) Then ϕ ∈ E θ lies in the kernel ⇔ ϕ$ − val(ϕ) ∈ Im(θ 0 ) ⇔ det(ϕ$ − val(ϕ) ) ∈ (OL× ) h−1 (since (h − 1)| val(ϕ) ⇒ det($ − val(ϕ) ) ∈ (L × ) h−1 ) ⇔ det(ϕ) ∈ (L × ) h−1 ⇔ ϕ ∈ Im(θ), so OE× Eθ  . Im(θ) Im(θ 0 ) Proposition 4.1.4. Let c, d, e be as in Lemma 4.1.2, and let e L be the ramification × → O × induces a map θ n × degree of L over Q p . θ 0 : OD 0,n : (O L /π L O L ) → E (OL /π Ln OL ) × making the following diagram  × OL × det / O × // OD n π OL L L θ0  OE× det  / O× L  //  θ 0,n OL π Ln OL × commute. θ 0,n is the map θ 0,n : (OL /π Ln OL ) × → (OL /π Ln OL ) × x 7→ x h−1 . For n > 2ee L , 25 (a) Im(θ 0 ) is the preimage of Im(θ 0,n ) under the det mod (1 + π Ln OL ) map, (b) the det mod (1 + π Ln OL ) map induces an isomorphism coker(θ 0 ) = OE×  (OL /π Ln OL ) × = coker(θ 0,n ), − → Im(θ 0 ) Im(θ 0,n ) 0 × K 0 , where (c) ker(θ 0,n ) can be written as a direct product ker(θ 0,n ) = Kn,1 n,2 0 = (1 + π n−ee L O )/(1 + π n O ), and the det mod (1 + π n O ) map restricts Kn,2 L L L L L L to an isomorphism det mod (1+π Ln OL ) 0 . ker(θ 0 ) −−−−−−−−−−−−−−→ Kn,1  In particular, if e = 0, then the det mod (1 + π Ln OL ) map restricts to an isomorphism from ker(θ 0 ) to ker(θ 0,n ). × → O × induces a map Proof. To show that θ 0 : OD E  ×  × θ 0,n : OL /π Ln OL → OL /π Ln OL × satisfies making the above diagram commute, we need to check that if φ ∈ OD det φ ≡ 1 mod (1 + π Ln OL ), then det θ 0 (φ) ≡ 1 mod (1 + π Ln OL ). But this is clear since det(θ 0 (φ)) = (det φ) h−1 . The fact that θ 0,n is the (h − 1) power map is also immediate from this. For n1 ≥ n2 , consider the map  ×  × OL /π Ln2 OL OL /π Ln1 OL coker(θ 0,n1 ) =   h−1    h−1 = coker(θ 0,n2 ). (OL /π Ln1 OL ) × (OL /π Ln2 OL ) × Suppose a ∈ OL× is a (h − 1) power mod π Ln2 . By Hensel’s lemma, for n2 > 2ee L , a is also a (h − 1) power mod π Ln1 , so the above map is a bijection for all n1 ≥ n2 > 2ee L . So, for all n1, n2 > 2ee L , we have coker(θ 0,n1 ) = coker(θ 0,n2 ) , and hence ker(θ 0,n1 ) = ker(θ 0,n2 ) . Now, let us assume that n > 2ee L . Since n > 2ee L ⇔ 2(n − ee L ) > n, it is clear that 0 ker(θ 0,n ) ⊇ (1 + π Ln−ee L OL )/(1 + π Ln OL ) = Kn,2 . 26 We note that ker(θ 0,n 0 ) = ker(θ 0,n ) for all n0 ≥ n ⇒ ker(θ 0,n 0 ) n 0 −ee 1 + πL L = OL ker(θ 0,n ) for all n0 ≥ n. n−ee L 1 + πL OL Hence for any a0 + · · · + an−ee L −1 π Ln−ee L −1 ∈ ker(θ 0,n ), there exists a unique an−ee L ∈ OL /π L such that a0 +· · ·+an−ee L −1 π Ln−ee L −1 +an−ee L π Ln−ee L ∈ ker(θ 0,n+1 ). Therefore, the set of elements a0 + · · · + an−ee L −1 π Ln−ee L −1 ∈ ker(θ 0,n ) are given precisely by 0 = ker(θ )/(1 + π n O ) ≤ ker(θ ). We truncations of elements in ker(θ 0 ). Let Kn,1 0,n 0 L L 0 0 0 0 have shown that ker(θ 0,n ) = Kn,1 Kn,2 . As Kn,1 ∩ Kn,2 = {1}, this is a direct product, proving (c). To prove (b), we first show that det OL×  (OL× ) h−1 coker(θ 0 ) −−→ . Consider the commutative diagram 1 /K / Im(θ 0 ) _ 1 /K  / O× E det / (OL× ) h−1 _ det  / O× L /1 /1 with exact rows, where K = {ϕ ∈ OE× : det ϕ = 1}. By the snake lemma, coker(θ 0 ) = OE× det OL× −−→ . Im θ 0  (OL× ) h−1 We have an isomorphism OL×  µ q−1 × µ pa × ZnpL where n L = [L : Q p ] and a is such that the group of p-power roots of unity in OL× is µ pa . So µ q−1 µ pa ! nL Zp  × × coker(θ 0 )  (h − 1)Z p (OL× ) h−1 (µ q−1 ) h−1 (µ pa ) h−1 !n µ q−1 µ pa Z L  × . e × pe Z (µ q−1 ) d (µ pa ) p OL× 27 Clearly, the kernel of the map  mod (1+π Ln OL ) OL× −−−−−−−−−−−→  OL /π Ln OL ×  h−1 (OL /π Ln OL ) × = coker(θ 0,n ) contains (OL× ) h−1 , hence is equal to (OL× ) h−1 for n > 2ee L , since OL× (OL× ) h−1 = penL +c d = ker(θ 0,n ) = coker(θ 0,n ) . Therefore det coker(θ 0 ) −−→  (OL× ) mod (1+π Ln OL ) −−−−−−−−−−−→ coker(θ 0,n ).  (OL× ) h−1 (a) then follows immediately from this. Results analogous to those in Lemma 4.1.2 and Propositions 4.1.3, 4.1.4 also hold for the maps ψ0 : GL h (OL ) → GL h (OL ) ψ : GL h (L) → GL h (L) g 7→ (det g)(g −1 )T , 4.2 g 7→ (det g)(g −1 )T . Representation theory We start this section with a useful result that expresses the cohomology of certain rigid analytic spaces in terms of the cohomology of a subspace. Proposition 4.2.1. Let X be a rigid analytic space with an action of the group G, and let Y be a subspace of X on which the subgroup H of G acts. Suppose (i) H is normal in G, G (ii) H is a finite abelian group, (iii) X is the disjoint union X= a g(Y ). g∈ G H Then Hci (X, Ql )  IndGH Hci (Y, Ql ) as G-representations. 28 G Proof. We first prove the result in the case that H is cyclic of order m, generated by g0 ∈ G. In this case, (iii) becomes X = Y t g0 (Y ) t · · · t g0m−1 (Y ). So we have Hci (X, Ql )  m−1 M Hci (Y, Ql ) j=0 Lm−1 i as Ql -vector spaces. Let x k be the element of j=0 Hc (Y, Ql ) with k-th coordinate equal to x and which is 0 everywhere else. We pick the above isomorphism so that g0 · x k = x k+1, 0 ≤ k < m − 1. Define f : IndGH Hci (Y, Ql ) → Hci (X, Ql )  m−1 M Hci (Y, Ql ) j=0 gok x 7→ x k . We will show that this is an isomorphism of representations. Let g ∈ G. Then we can write g = hg0l with h ∈ H, 0 ≤ l ≤ m − 1. So it suffices to show that f is compatible with both the actions of h ∈ H and of g0 . For 0 ≤ k < m − 1, f g0 · (g0k x) = g0k+1 x 7−→ x k+1 = g0 · x k = g0 · f (g0k x) and since g0m ∈ H, f g0 · (g0m−1 x) = g00 (g0m x) 7−→ (g0m · x)0 = g0m · x 0 = g0 · x m−1 = g0 · f (g0m−1 x). And for h ∈ H, we have hg0k = g0k h0 for some h0 ∈ H since H is normal in G, so f h·(g0k x) = g0k (h0·x) 7−→ (h0·x) k = g0k ·(h0·x)0 = (g0k h0 )·x 0 = (hg0k )·x 0 = h·x k = h· f (g0k x). So f is an isomorphism of G-representations. G G Now, for the general case where H is finite abelian, we can write H as a direct Gn G1 G2 G product H = H × H × · · · × H , where each G j is a subgroup of G containing H and 29 G G each Hj is finite cyclic. Let H j = G1 · · · G j . Since H is abelian, g j1 G j2 = G j2 g j1 for all g j1 ∈ G j1 , 1 ≤ j1, j2 ≤ n, so H j is a subgroup of G and H j−1 is normal in H j . Hj Gj G ···G j Gj Furthermore, we have Hn = G and H j−1 = G1 1···G j−1  (G1 ···G j−1 )∩G j = H . Define X j recursively by X0 = Y and a Xj = g j (X j−1 ). H gj ∈ H j j−1 (iii) implies that the above is indeed a disjoint union for all 1 ≤ j ≤ n, and that X = X n . Applying the previous case repeatedly shows that Hn−1 H1 i G i n Hci (X, Ql )  Ind H Hn−1 Ind Hn−2 · · · Ind H0 Hc (Y, Ql )  Ind H Hc (Y, Ql ). Next, we include a result that will be needed later in Section 6.3 to show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower realizes the local Langlands and the Jacquet-Langlands correspondences. This result is useful as local Langlands and Jacquet-Langlands both behave well with respect to twists. Lemma 4.2.2. (a) Let ( ρ, V ) be an irreducible representation of E × , and let ι : D× → E × be the isomorphism D× _ ι  / E × _  GL h (L h ) φ7→ (φT ) −1  / GL h (L h ).   E× ∗ ∨ Then θ ∗ Resθ(D × ) ρ  ι ρ ⊗ ( χ ι∗ ρ ◦ Nrd) , where χ ι∗ ρ is the central character of ι∗ ρ, and Nrd : D → L is the reduced norm map of the division algebra D. (b) Let π be an irreducible representation of GL h (L), and let j be the isomorphism  j : GL h (L) − → GL h (L) g 7→ (gT ) −1 .   h (L) Then ψ ∗ ResGL π  j ∗ π ⊗ ( χ j ∗ π ◦ det) ∨ , where χ j ∗ π is the central ψ(GL h (L)) character of j ∗ π. Proof. We will prove (a). The proof of (b) is almost identical. 30 First, note that, by Lemma 4.1.1, for our embedding of D× into GL h (L h ), the determinant map on GL h (L h ), when restricted to D × , has image in L, and is in fact the reduced norm map Nrd : D× → L. In other words, the diagram / L Nrd D× _  GL h (L h ) det _  / Lh commutes. For φ ∈ D× , v ∈ V , ((θ ∗ Res ρ)(φ))(v) = ( ρ((det φ)(φT ) −1 ))(v) = (ι∗ ρ((det φ) −1 φ))(v). Let ι∗ ρ ⊗ ( χ ι∗ ρ ◦ Nrd) ∨ act on the vector space V ⊗Q HomQ (Ql, Ql ). Consider the l l isomorphism V → V ⊗Q HomQ (Ql, Ql ) l αv 7→ v ⊗ f α l (where f α : c 7→ αc). For φ ∈ D× , ((( χ ι∗ ρ ◦ Nrd) ∨ (φ))( f α ))(c) = f α (( χ ι∗ ρ ◦ Nrd)(φ−1 )(c)) = f α (ι∗ ρ(det φ−1 )(c)) = α(ι∗ ρ(det φ−1 ))(c), so (( χ ι∗ ρ ◦ Nrd) ∨ (φ))( f α ) = f α(ι∗ ρ(det φ−1 )) , and (ι∗ ρ(φ))(v) ⊗ (( χ ι∗ ρ ◦ Nrd) ∨ (φ))( f 1 ) = (ι∗ ρ(φ))(v) ⊗ f ι∗ ρ(det φ−1 ) 7→ (ι∗ ρ((det φ) −1 φ))(v), as required. 31 Chapter 5 THE DUALITY MAP AND THE EXTERIOR POWER MAP ON MODULI SPACES 5.1 Definitions of the duality map and the exterior power map In this section, we will use the Serre dual and the exterior power of a π L -divisible OL -module to define maps from LTh∞ to Mh∞ . We start with the Serre dual. Note that G∨m,n  G m,n , so G∨1,h−1  G h−1,1 has dimension h − 1 and height h. Let us fix such an isomorphism. The dual of a π L divisible OL -module X has Tate module (TπL X ) ∨ (1). In particular, the Tate module of the dual of (OL /π Ln OL ) h is equal to OLh (1). Over L̆(ζ p∞ ), we have OLh (1)  OLh since L̆(ζ p∞ ) contains the p∞ roots of unity. Definition 5.1.1. Define ∨ : LTh∞ × L̆ L̆(ζ p∞ ) → Mh∞ × L̆ L̆(ζ p∞ ) by (X, β, α) 7→ (X ∨, ( β ∨ ) −1, (α ∨ζ p n ) −1 ) where α ∨ζ p n is given by the composition    TπL X ∨ −−−−−−→ (TπL X ) ∨ (1) −−−−−−→ OLh (1) −−−−−−→ OLh . Let us now look at the exterior power. Recall that fix such an isomorphism. V Definition 5.1.2. Define h−1 : LTh∞ → Mh∞ by (X, β, α) 7→ * , h−1 ^ X, In the above, by a slight abuse of notation, given by h−1 ^ Vh−1 β, Vh−1 h−1 ^ G1,h−1  G h−1,1 . We will α+ . - α is the level structure on h−1 h−1 h−1 Vh−1 ^ ^ φ α ^ h h * X+, OL −−−−−−→ OL −−−−−−→ Tp X −−−−−−→ Tp    , where φ is the isomorphism vi 7→ (−1)i v1 ∧ v2 ∧ · · · ∧ vi−1 ∧ vi+1 ∧ · · · ∧ vh, v1, . . . , vh being the standard basis vectors for OLh . - Vh−1 X 32 5.2 Properties of the exterior power map and group actions In order to study the cohomology of the dual Lubin-Tate tower using the exterior V power map h−1 , it is necessary to first understand how the group actions behave V with respect to h−1 . Recall from Section 4.1 that we have embeddings of D × and E × in GL h (L h ). We shall view D × and E × as subgroups of GL h (L h ) using these embeddings. We also recall the maps θ : D× → E × ψ : GL h (L) → GL h (L) φ 7→ (det φ)(φ−1 )T , V Proposition 5.2.1. The map h−1 : LTh∞ → Mh∞ is g 7→ (det g)(g −1 )T . (a) D× -equivariant if we let D × act on Mh∞ via θ, (b) GL h (L)-equivariant if we let GL h (L) act on Mh∞ via ψ, (c) W L -equivariant. Proof. (a) Suppose φ ∈ D × is given by φei = h−1 X α ji e j . j=0 Then h−1 ^ * φ+ ẽi = (−1)i φe0 ∧ φe1 ∧ · · · ∧ φei−1 ∧ φei+1 ∧ · · · ∧ φe h−1 , h−1 h−1 h−1 h−1 X X X X = (−1)i α j0 e j ∧ · · · ∧ α j,i−1 e j ∧ α j,i+1 e j ∧ · · · ∧ α j,h−1 e j j=0 = h−1 X j=0 j=0 j=0 β k (e0 ∧ e1 ∧ · · · ∧ e k−1 ∧ e k+1 ∧ · · · ∧ e h−1 ) k=0 for some β k ∈ L h . To write down an expression for β k , it is helpful to first introduce the order-preserving bijections bl : {0, 1, . . . , l − 1, l + 1, . . . , h − 1} → {1, 2, . . . , h − 1}    j + 1 if j < l, j→ 7   j if j > l.  33 For τ ∈ Sh−1 , let τik = b−1 k ◦ τ ◦ bi . Then X β k = (−1)i sgn(τ)α τik (0),0 · · · α τik (i−1),i−1 α τik (i+1),i+1 · · · α τik (h−1),h−1 . τ∈Sh−1 So the coefficient of ẽ k in V h−1  φ ẽi is given by (−1)i+k det(φ with row and column containing α k,i deleted) = Ck,i, the cofactor of the element a k,i of φ. So h−1 ^ ··· ··· .. . C0,1 *. C0,0 . C1,0 C1,1 φ = ... . .. . .. .. ,Ch−1,0 Ch−1,1 C0,h−1 + / C1,h−1 // // = (det φ)(φ−1 )T . .. . // · · · Ch−1,h−1 - (b) It suffices to prove the proposition for g ∈ GL h (L) ∩ Math (OL ). For such g, it acts on LTh∞ by (X, β, α) 7→ (Y, q ◦ β, α) where Y, q, α are as defined in Section 3.1. Recall that Y rig , qrig and α rig (which  corresponds to α rig : OLh − → TπL Y rig ) satisfy the commutative diagram α rig  (L/OL ) h / X rig g qrig  Applying the functor α rig   / Y rig = X rig . α rig (ker(g)) (L/OL )h Vh−1 to the above diagram gives Vh−1 Vh−1 (L/OL Vh−1 )h α rig /  Vh−1 X rig Vh−1 g Vh−1  (L/OL ) h Vh−1 α rig  / Vh−1 qrig Y rig . By an argument similar to that in (a), the following diagram commutes: OLh /  Vh−1 ψ(g)=(det g)(g −1 )T OLh Vh−1  OLh  / Vh−1 OLh . g 34 V V Since h−1 α rig corresponds to ( h−1 α) rig , using the above 2 diagrams, we get the commutative diagram OLh Vh−1 α  / Tp V ψ(g)  OLh Vh−1 α  / Tp X  Vh−1 q h−1 V  h−1  Y . Hence ψ(g) acts on Mh∞ by h−1 ^ * , X, h−1 ^ β, h−1 ^ α + 7→ * , h−1 ^ Y, h−1 ^ q◦ h−1 ^ β, h−1 ^ α+ , - as required. (c) Let w ∈ W L . By the universal property of the exterior power, we have F(Vh−1 G1,h−1 )/F p = h−1 ^ FG1,h−1 /F p , and h−1 h−1 ^ ^ * + Tp X = Tp X . , V V The first equality shows that ( h−1 β) w = h−1 β w , and the second gives V V V ( h−1 α) w = h−1 α w , so the map h−1 : LTh∞ → Mh∞ is W L -equivariant. V Proposition 5.2.1 tells us that the exterior power map h−1 is W L -equivariant. Unlike Vh−1 however, the duality map ∨ is not W L -equivariant, one of the reasons being that the Tate module of the dual of (L/OL ) h is OLh (1). As such, it is difficult to understand the W L -action on Mh∞ by considering the duality map. However, the duality map ∨ can still be used to study the cohomology of Mh∞ as a E × × GL h (L)-representation, and hence can be used to show that the supercuspidal part of the cohomology of Mh∞ in the middle degree realizes the Jacquet-Langlands correspondence up to certain twists. 35 Chapter 6 THE GEOMETRY AND COHOMOLOGY OF THE DUAL LUBIN-TATE TOWER 6.1 Geometry of the dual Lubin-Tate tower Before looking at the tower, it is helpful to first understand the level 0 situation. Let us write LTh for the formal scheme M 1 , and Mh for M h−1 . h Proposition 6.1.1. The map Vh−1 h−1 ^ h : LTh → Mh induces an isomorphism : (LTh )0 → (Mh )0 . Proof. By Theorem 3.1.1, we have a non-canonical isomorphism (LTh )0 ≈ Spf OL̆ [[t 1, . . . , t h−1 ]], which shows, by considering the duality map ∨, that (Mh )0 ≈ Spf OL̆ [[T1, . . . , Th−1 ]]. Let P = (P, Q, F, V −1 ) be the OL -display corresponding to the π L -divisible OL module G1,h−1 , so that P = D(G1,h−1 ) and Q = V D(G1,h−1 ) with F and V −1 as given in the definition of D(G m,n ). Let S ∈ C, and let Spec A ⊆ S be an open affine subset of S. By Grothendieck-Messing theory (cf. [Mes72]), π L -divisible OL -modules over A lifting G1,h−1 correspond to s.e.s. 0 −→ M −→ P ⊗OL̆ A −→ N −→ 0 of A-modules lifting the Hodge filtration 0 −→ P P Q −→ −→ −→ 0. πL P πL P Q Since V D(G1,h−1 ) Q hV e0, . . . , V e h−1 i hπ L e0, e1, . . . , e h−1 i = = = , π L P π L D(G1,h−1 ) hπ L e0, . . . , π L e h−1 i hπ L e0, . . . , π L e h−1 i the above condition is equivalent to Q he1, . . . , e h−1 i M = = . m A M π L P hπ L e1, . . . , π L e h−1 i 36 Applying Nakayama’s lemma, we see that M must be of the form he1 + t 1 π L e0, . . . , e h−1 + t h−1 π L e0 i for some t 1, . . . , t h−1 ∈ A. V Applying h−1 to the s.e.s. 0 −→ he1 + t 1 π L e0, . . . , e h−1 + t h−1 π L e0 i −→ he0, . . . , e h−1 i −→ he0 i −→ 0, (6.1) we get the s.e.s. 0 −→ h(e1 +t 1 π L e0 )∧· · ·∧(e h−1 +t h−1 π L e0 )i −→ hẽ0, . . . , ẽ h−1 i −→ hẽ1, . . . , ẽ h−1 i −→ 0 where (e1 + t 1 π L e0 ) ∧ · · · ∧ (e h−1 + t h−1 π L e0 ) = ẽ0 − t 1 π L ẽ1 − t 2 π L ẽ2 − · · · − t h−1 π L ẽ h−1 . Let us rewrite (6.1) as follows: ρ 0 −→ h f 1, . . . , f h−1 i − → f i 7→ τ he0, . . . , e h−1 i → − hg0 i −→ 0 ei + t i π L e0 e0 7→ g0 ei 7→ −t i π L gi, i , 0. Applying ∨, we get the s.e.s. τ∨ ρ∨ ∨ 0 −→ hg0∨ i −−→ he0∨, . . . , e∨h−1 i −−→ h f 1∨, . . . , f h−1 i→0 where   g ∨ (g0 ) = 1,  0 ∨ ∨ ∨ (τ g0 )(ei ) = g0 (τei ) =    g ∨ (−t i π L g0 ) = −t i π L,  0    1, ( ρ∨ ei∨ )( f j ) = ei∨ ( ρ f j ) = ei∨ (e j + t j π L e0 ) =    0,  if i = 0, if i , 0, if i = j, if i , j. So the above s.e.s. is the same as 0 −→ he0∨ − t 1 π L e1∨ − · · · − t h−1 π L e∨h−1 i −→ he0∨, . . . , e∨h−1 i −→ he1∨, . . . , e∨h−1 i −→ 0. 37 Therefore we have a commuting diagram (Mh )0 ≈ Spf OL̆ [[T1, . . . , Th−1 ]] 2 Vh−1 O Ti  7→ (LTh )0 ≈ Spf OL̆ [[t 1, . . . , t h−1 ]] Si ∨  , (Mh )0 ≈ Spf OL̆ [[S1, . . . , Sh−1 ]] V which shows that h−1 induces an isomorphism from (LTh )0 to (Mh )0 since the duality map ∨ : LTh → Mh clearly does. Corollary 6.1.2. The map Vh−1 h−1 ^ : LTh → Mh induces an isomorphism : (LTh )m → (Mh )(h−1)m for any m ∈ Z. Proof. We recall the definition of the map θ : D × → E × . Using the above embeddings of D× and E × into GL h (L h ), we have θ(φ) = (det φ)(φ−1 )T . Let m ∈ Z. Fix some φ ∈ D × with val(φ) = m, then    val(θ(φ)) = v L (det(θ(φ))) = v L det (det φ)(φ−1 )T = v L ((det φ) h−1 ) = (h − 1)v L (det φ) = (h − 1)m. So the action of φ induces an isomorphism from (LTh0 )0 to (LTh0 )m , and the action of θ(φ) induces an isomorphism from (Mh0 )0 to (Mh0 )(h−1)m . By Proposition 5.2.1, we have the following commutative diagram: (LTh0 )0 Vh−1  φ  h  θ(φ)  (LTh0 )m / (M 0 )0 Vh−1  / (M 0 ) (h−1)m h where the top map is an isomorphism by Proposition 6.1.1. Hence isomorphism from (LTh0 )m to (Mh0 )(h−1)m , as desired. Vh−1 induces an Proposition 6.1.3. For any n ≥ 0, m, n ∈ Z, each connected component of (LThn )m is mapped isomorphically onto some connected component of (Mhn )(h−1)m under the V map h−1 : LTh∞ → Mh∞ . 38 Vh−1 Proof. By Proposition 6.1.1, induces an isomorphism on the formal schemes Vh−1 : (LTh )0 → (Mh )0 , hence an isomorphism on their generic fibers. × on the connected components of (LT n ) , we see By considering the action of OD h 0 n 0 that each connected component of (LTh )0 is finite étale over (LTh ) of the same degree. Using duality, we see that each connected component of (Mhn )0 is also finite étale over (Mh0 )0 of the same degree. Since (LThn )0 finite étale  (LTh0 )0 Vh−1 / (M n )0 h  Vh−1  finite étale / (M 0 )0 h V commutes, this shows that h−1 induces an isomorphism from each connected component of (LThn )0 to some connected component of (Mhn )0 . Now, by considering the action of the groups D× and E × , we see that, for any m ∈ Z, V the map h−1 induces an isomorphism from each connected component of (LThn )m to some connected component of (Mhn )(h−1)m . Corollary 6.1.4. Suppose (p(q − 1), h − 1) = 1. Then an isomorphism (LThn )m → (Mhn )(h−1)m Vh−1 : LTh∞ → Mh∞ induces for any n ≥ 0, m, n ∈ Z. Proof. We first consider the case m = 0. Since (p(q − 1), h − 1) = 1, by Proposition 4.1.4, the map θ 0,n is bijective. By Theorem 3.2.4, this means that the map induced V by h−1 on the connected components is a bijection. Together with Proposition V 6.1.3, this shows that h−1 induces an isomorphism from (LThn )0 to (Mhn )0 . The result then follows for all m ∈ Z by considering the action of the groups D × and E×. 6.2 Cohomology of the dual Lubin-Tate tower In this section, we will reinterpret the results of the previous section in terms of the cohomology of the dual Lubin-Tate tower. In order to avoid unnecessarily cumbersome notation, from here on, all cohomology is understood to mean l-adic cohomology with Ql coefficients, where l , p is an odd prime. We will also slightly abuse notation, and whenever the Lubin-Tate tower or the dual Lubin-Tate tower appears, we actually mean its change base from L̆ to C. 39 Lemma 6.2.1. The kernel ker(θ) acts trivially on LTh0 , so the action of D× on LTh0 induces an action of θ(D × ) = Im(θ) ≤ E × on LTh0 . This action can be extended to an action of E θ = hOL×h , Im(θ)i by letting OL×h act trivially on LTh0 . Proof. Suppose k ∈ ker(θ). The action of k on LTh0 is given by (X, β) 7→ (X, β ◦ k −1 ). But k ∈ ker(θ) = µ h−1 (OL ) ⊂ OL× and X has multiplication by OL×h , so (X, β) = (X, β ◦ k −1 ). So k acts trivially on LTh0 , and we get an action of Im(θ) on LTh0 induced by the action of D× . To see that we can extend the above action of Im(θ) to an action of E θ = hOL×h , Im(θ)i where OL×h acts trivially, we just need to check that OL×h ∩Im(θ) ⊆ Im(θ) acts trivially on LTh0 , but this is clear. Proposition 6.2.2. Let E θ ≤ E × act on LTh0 as in Lemma 6.2.1. Then for all i ≥ 0, × Hci (Mh0 )  IndEE θ Hci (LTh0 ) as E × × W L representations. Proof. The proposition is clear for i > 0 since Hci (LTh0 ) = 0 = Hci (Mh0 ) for i > 0. Consider i = 0. Let Y0 be the image of h−1 ^ : LTh0 → Mh0 . By Proposition 5.2.1, and the fact that OL×h acts trivially on Mh0 , we have Hc0 (Y0 )  Hc0 (LTh0 ) as E θ ×W L -representations, where the action of E θ on LTh0 is as described in Lemma 6.2.1. By Proposition 4.1.3, E θ = {ϕ ∈ E × : (h−1)| val(ϕ)} and a full set of representatives for E × /E θ is given by ($0 ) k for k ∈ {0, 1, . . . , h − 2}. It is clear that Mh0 = Y0 t $0 (Y0 ) t · · · t $0 h−2 (Y0 ). So by Proposition 4.2.1, × Hci (Mh0 )  IndEE θ Hci (LTh0 ) as E × × W L representations, as desired. 40 We now prove an analogous result for Hci (Mh∞ ). Theorem 6.2.3. Let d = (q − 1, h − 1), e = v p (h − 1) and c be the maximum integer ≤ e such that OL contains the pc roots of unity. Let θ ∗ Hci (LTh∞ ) be the pushforward of the representation Hci (LTh∞ ) under the map θ : D×  θ(D × ). Then for all i ≥ 0, ψ ∗  i ∞ h (L) ResGL ψ(GL h (L)) Hc (Mh )  E× i ∞ E× *  Indθ(D × ) θ ∗ Hc (LTh )  Indθ(D × ) θ ∗ E× i ∞ K  Indθ(D × ) θ ∗ Hc (LTh ) , Hci (LTh∞ ) K + - as E × × GL h (L) × W L representations, where K = ker(θ 0 ) = µ pc d (OL ) and Hci (LTh∞ ) K is the subrepresentation of Hci (LTh∞ ) fixed by K. Proof. Let Yn be the image of h−1 ^ : LThn → Mhn . By Lemma 4.1.2 (a) and Proposition 4.1.4 (c), for n > 2ee L , the kernel of θ 0,n can be written as a direct product 0 0 ker(θ 0,n ) = Kn,1 × Kn,2 , and the det mod (1 + π Ln OL ) map gives an isomorphism det mod (1+π Ln OL ) 0 K = ker(θ 0 ) = µ pc d (OL ) −−−−−−−−−−−−−−→ Kn,1 .  Let Kn = 1 + π Ln−ee L OL ≤ D× . Proposition 4.1.4 (c) tells us that we have an exact sequence det mod (1+π Ln OL ) 0 1 −→ 1 + π Ln OL −→ Kn −−−−−−−−−−−−−−→ Kn,2 −→ 1. V Recall from Proposition 6.1.3 that h−1 maps each connected component of (LThn )m isomorphically onto some connected component of (Mhn )(h−1)m . Since det mod (1+π Ln OL ) 0 0 K × Kn −−−−−−−−−−−−−−−− Kn,1 × Kn,2 = ker(θ 0,n ), by Theorem 3.2.4, two connected components of LThn will each be mapped isomorV phically to the same connected component of Mhn under the h−1 map if and only 41 if they are in the same orbit of K × Kn . Furthermore, 1 + π Ln OL ≤ D × acts trivially on LThn , so Hci (LThn ) + ψ ∗ Hci (Yn )  θ ∗ * K × K n , × as θ(D ) × GL h (L) × W L -representations. Claim: θ × × E i E i Hci (Mhn )  IndEE θ Indθ(D × ) Hc (Yn )  Indθ(D × ) Hc (Yn ). Proof of claim: Let X n be given by Xn = a (Mhn )(h−1)m . m∈Z Note that X n contains Yn as a subspace. By Propositions 4.1.3 (d) and 4.1.4 (b), OE× det mod (1+πLn OL ) (OL /π Ln OL ) × Eθ  coker θ 0 = −−−−−−−−−−−−−−→ = coker θ 0,n  Im θ Im θ 0 Im θ 0,n is a finite abelian group. D× acts transitively on the connected components of LThn , so for any ϕ < Im(θ), ϕ(Yn ) will be disjoint from Yn . Furthermore, the orbit of Yn under OE× is X n since OE× acts transitively on the connected components of (Mhn )(h−1)m for any m ∈ Z. This shows that a Xn = ϕ(Yn ). θ E ϕ∈ Im(θ) By Proposition 4.2.1, we have θ E i Hci (X n )  Indθ(D × ) Hc (Yn ). Recall from Proposition 4.1.3 (d) that the valuation map gives an isomorphism E× Z  . θ (h − 1)Z E × So a full set of representatives for EE θ is {1, $0, $02, . . . , $0 h−2 }, and it is clear that Mhn = X n t $0 (X n ) t · · · t $0 h−2 (X n ). So, again, by Proposition 4.2.1, × Hci (Mhn )  IndEE θ Hci (X n ). 42 This proves the claim. The map Hci (LThn−ee L ) → Hci (LThn ) Hci (LThn ) since Kn = 1 + π Ln−ee L OL acts trivially on LThn−ee L . Kn H i (LT n ) lim c Kn h = Hci (LTh∞ ), and −−→n factors through So Hci (LThn ) Hci (LTh∞ ) = . lim −−→ K × Kn K n H i (LT ∞ ) It remains to show that c K h  Hci (LTh∞ ) K . For m sufficiently large, the determinant of the elements of K are distinct mod (1 + π Lm OL ), so the orbits of the induced action of K on the connected components of LThm each have size d. We define a subspace Zm of LThm by taking one connected component from each orbit. Let k be a generator of K. Then LThm is the disjoint union LThm = Zm t k (Zm ) t · · · t k d−1 (Zm ), so we have an identification Hci (LThm )  d−1 M Hci (Zm ) j=0 Ld−1 i of Ql -vector spaces such that the action of K on j=0 Hc (Z m ) is given by k · (z0, z1, . . . , z d−1 ) = (z d−1, z0, . . . , z d−2 ). Define an isomorphism of vector spaces Hci (LThm )  −−→ Hci (LThm ) K K [(z0, . . . , z d−1 )] 7−→ (z, . . . , z), P × where z = d1 d−1 j=0 z j . Using the fact that K is in the center of D , an easy computation shows that the above is also an isomorphism of representations. Putting these together, we have   Hci (LThn ) i n E× ∗ i E× h (L) * +, ψ ∗ ResGL H (M )  Ind ψ H (Y )  Ind θ × × c n θ(D ) θ(D ) ∗ h ψ(GL h (L)) c K × K n , and  GL (L)  Hci (LThn ) Hci (LTh∞ ) ∗ i ∞ E× E× h * + * + ψ Resψ(GL h (L)) Hc (Mh )  Indθ(D× ) θ ∗ lim  Indθ(D× ) θ ∗ −−→ K × Kn K , n , E× i ∞ K  Indθ(D× ) θ ∗ Hc (LTh ) , as desired. 43 Corollary 6.2.4. Suppose (q − 1, h − 1) = 1. There is an action of θ(D× ) ⊂ E × on Hci (LTh∞ ) induced from the action of D× . Then for all i ≥ 0,   i ∞ E× i ∞ h (L) ψ ∗ ResGL H (M )  Indθ(D × ) Hc (LTh ) h ψ(GL h (L)) c as E × × GL h (L) × W L representations. Just like for LTh∞ , we will consider the functor Hci (Mh∞ ) : Rep E × → Rep(GL h (L) × W L ) ρ 7→ HomE × (Hci (Mh∞ ), ρ). The following is an easy corollary of Theorem 6.2.3. Corollary 6.2.5. For all i ≥ 0,   f  g i ∞ i ∞ ∗ E× h (L) ψ ∗ ResGL H (M )[ρ] = H (LT ) θ Res ρ . × c θ(D ) h h ψ(GL h (L)) c Proof. By Theorem 6.2.3 and Frobenius reciprocity,    GL (L)    i ∞ ∗ i ∞ h (L) h × ψ Res (M ) , ρ H (M )[ρ] = Hom H ψ ∗ ResGL E c c h h ψ(GL h (L)) ψ(GL h (L)) H i (LTh∞ ) E× * c + , ρ+ = HomE × *Indθ(D × ) θ∗ K , , - Hci (LTh∞ ) + , ResE × × ρ+ = Homθ(D× ) *θ ∗ * θ(D ) K , , i ∞   Hc (LTh ) ∗ E× +. , θ Resθ(D = Hom D× * ×) ρ K ,   × E Since K acts trivially on θ ∗ Resθ(D × ) ρ , the above is equal to f  g    E× E× . Hom D× Hci (LTh∞ ), θ ∗ Resθ(D = Hci (LTh∞ ) θ ∗ Resθ(D ×) ρ ×) ρ 6.3 Geometric realization of local correspondences and vanishing results In this section, we will use Corollary 6.2.5 and results on the cohomology of the Lubin-Tate tower to deduce results for the cohomology of the dual Lubin-Tate tower. In particular, we will show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower in the middle degree realizes the local Langlands and the Jacquet-Langlands correspondences (up to appropriate twists). 44 Theorem 6.3.1. For π ∈ Cusp(GL h (L)), h−1 Hch−1 (Mh∞ )cusp (JL−1 (π)) = π ⊗ rec(π ∨ ⊗ ( χ π ◦ det) ⊗ (| · | ◦ det) 2 )(h − 1), where χ π is the central character of π and Hch−1 (Mh∞ )cusp is the supercuspidal part of Hch−1 (Mh∞ ). Proof. By Corollary 6.2.5, we have the commutative diagram   × ρ7→θ ∗ ResθE(D × ) ρ Irr0 E × / Irr0 D × Hci (LTh∞ )cusp Hci (Mh∞ )cusp  Cusp(GL h (L)) × Irr(W L )  GL (L)  h π⊗r7→ψ ∗ Resψ(GL π ⊗r (L))  / Cusp(GL h (L)) × Irr(W L ) h where Irr0 D× ⊂ Irr D× is the subset consisting of representations of the form JL−1 (π) for some π ∈ Cusp(GL h (L)) (similarly for Irr0 E × ). Note that   E× 0 × • θ ∗ Resθ(D × ) ρ ∈ Irr D since ρ = JL−1 (π) with π ∈ Cusp(GL h (L)) ⇒ ρ ⊗ ( χ π ◦ Nrd) ∨ = JL−1 (π ⊗ ( χ π ◦ det) ∨ ), • π ∈ Cusp(GL h (L)) since   h (L) π ψ ∗ ResGL = π ⊗ ( χ π ◦ det) ∨ ∈ Cusp(GL h (L)). ψ(GL h (L)) The commutative diagram only gives us Hci (Mh∞ )cusp [ρ] as a ψ(GL h (L)) × W L representation. But by considering the duality map ∨ : LTh∞ → Mh∞ , we know that Hci (Mh∞ )cusp realizes the Jacquet-Langlands correspondence, so we already understand Hci (Mh∞ )cusp [ρ] as a GL h (L) representation. Let us define F : Irr0 E × → Cusp(GL h (L)) × Irr(W L ) h−1 JL−1 (π) 7→ π ⊗ rec(π ∨ ⊗ ( χ π ◦ det) ⊗ (| · | ◦ det) 2 )(h − 1). 45 By the above, it will suffice to check that F makes the diagram commute. Applying Theorem 3.2.1, and using the fact that JL and rec are compatible with twists, we see f g   h (L) that Hch−1 (LTh∞ )cusp ρ ⊗ ( χ ρ ◦ Nrd) ∨ and ψ ∗ ResGL F ( ρ) are both equal ψ(GL h (L)) to h−1 (JL( ρ) ⊗ ( χJL( ρ) ◦ det) ∨ ) ⊗ rec(JL( ρ) ∨ ⊗ ( χJL( ρ) ◦ det) ⊗ (| · | ◦ det) 2 )(h − 1), as desired. We can also use Theorem 3.2.2 to deduce that Theorem 6.3.2. For i , h − 1, ρ ∈ Rep E × , Hci (Mh∞ )cusp ( ρ) = 0. Proof. This is immediate from Theorem 3.2.2 and the commutative diagram Rep E ×   × ρ7→θ ∗ ResθE(D × ) ρ / Rep D × Hci (Mh∞ )cusp Hci (LTh∞ )cusp  Rep(GL h (L) × W L )  GL (L)  h π ⊗r π⊗r7→ψ ∗ Resψ(GL (L)) h  / Rep(GL h (L) × W L ). 46 BIBLIOGRAPHY [Ahs11] T. Ahsendorf, O-displays and π-divisible formal O-modules. PhD thesis, Universität Bielefeld, 2011. [Ber93] V. G. 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