On Hodge-Newton Reducible Local Shimura Data of Hodge Type Citation Hong, Serin (2018) On Hodge-Newton Reducible Local Shimura Data of Hodge Type. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4F4W-Y024. https://resolver.caltech.edu/CaltechTHESIS:05262018-184212374 Abstract Rapoport-Zink spaces are formal moduli spaces of p -divisible groups which give rise to local analogues of certain Shimura varieties. In particular, one can construct them from purely group theoretic data called local Shimura data . The primary purpose of this dissertation is to study Rapoport-Zink spaces whose underlying local Shimura datum is of Hodge type and Hodge-Newton reducible. Our study consists of two main parts: the study of the l -adic cohomology of Rapoport-Zink spaces in relation to the local Langlands correspondence and the study of deformation spaces of p -divisible groups via the local geometry of Rapoport-Zink spaces. The main result of the first part is a proof of the Harris-Viehmann conjecture in our setting; in particular, we prove that the l -adic cohomology of Rapoport-Zink spaces contains no supercuspidal representations under our assumptions. In the second part, we obtain a generalization of Serre-Tate deformation theory for Shimura varieties of Hodge type. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Shimura varieties, Local Langlands correspondence, Rapoport-Zink spaces Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2018. Apostol Award for Excellence in Teaching in Mathematics, 2016. Scott Russell Johnson Prize for Excellence in Graduate Studies, 2014. Thesis Availability: Public (worldwide access) Research Advisor(s): Mantovan, Elena Thesis Committee: Mantovan, Elena (chair) Ramakrishnan, Dinakar Zhu, Xinwen Amir Khosravi, Zavosh Defense Date: 9 April 2018 Record Number: CaltechTHESIS:05262018-184212374 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05262018-184212374 DOI: 10.7907/4F4W-Y024 ORCID: Author ORCID Hong, Serin 0000-0002-0410-9041 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10943 Collection: CaltechTHESIS Deposited By: Serin Hong Deposited On: 30 May 2018 18:49 Last Modified: 04 Oct 2019 00:21 Thesis Files Preview PDF - Final Version See Usage Policy. 439kB Repository Staff Only: item control page

On Hodge-Newton reducible local Shimura data of Hodge type Thesis by Serin Hong In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2018 Defended April 9, 2018 ii © 2018 Serin Hong ORCID: 0000-0002-0410-9041 All rights reserved except where otherwise noted iii ACKNOWLEDGEMENTS First and foremost, I would like to express my deepest gratitude to my advisor, Elena Mantovan, for always being supportive and inspirational. Her invaluble advice on both research and life has greatly helped me grow as a mathematician and a person. This thesis would have never seen the light of day without her guidance. My research has been inspired and aided by many faculty members and fellow students in the Caltech Mathematics Department. I especially thank Dinakar Ramakrishnan for all the help he has given me both personally and academically. I sincerely thank Xinwen Zhu and Zavosh Amir Khosravi for serving as committe members and teaching me wonderful mathematics. I would also like to thank my parents for their unconditional support and care. They have encouraged and helped me at every stage of my life, and I am extremely grateful to share this achievement with them. Finally, my dearest thanks go to Yoonhyung and our upcoming baby, for always reminding me of the true meaning of life. You have given me the strength to overcome numerous challenges I faced during the course of my research. I am extremely grateful for all the memories that we share, and also looking forward to the future that we will shape together. iv ABSTRACT Rapoport-Zink spaces are formal moduli spaces of p-divisible groups which give rise to local analogues of certain Shimura varieties. In particular, one can construct them from purely group theoretic data called local Shimura data. The primary purpose of this dissertation is to study Rapoport-Zink spaces whose underlying local Shimura datum is of Hodge type and Hodge-Newton reducible. Our study consists of two main parts: the study of the l-adic cohomology of RapoportZink spaces in relation to the local Langlands correspondence and the study of deformation spaces of p-divisible groups via the local geometry of Rapoport-Zink spaces. The main result of the first part is a proof of the Harris-Viehmann conjecture in our setting; in particular, we prove that the l-adic cohomology of Rapoport-Zink spaces contains no supercuspidal representations under our assumptions. In the second part, we obtain a generalization of Serre-Tate deformation theory for Shimura varieties of Hodge type. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation: the local Langlands correspondence and local Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 The strategy: EL realization of Hodge-Newton reducibility . . . . . . 5 1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter II: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Group theoretic preliminaries . . . . . . . . . . . . . . . . . . . . . 10 2.3 F-isocrystals with G-structure . . . . . . . . . . . . . . . . . . . . . 13 2.4 Unramified local Shimura data of Hodge type . . . . . . . . . . . . . 16 2.5 Deformation Spaces of p-divisible groups with Tate tensors . . . . . 22 2.6 Rapoport-Zink spaces of Hodge type . . . . . . . . . . . . . . . . . 26 Chapter III: The Hodge-Newton filtration for Hodge-Newton reducible local Shimura data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1 EL realization of Hodge-Newton reducibility . . . . . . . . . . . . . 32 3.2 The Hodge-Newton decomposition and the Hodge-Newton filtration . 35 Chapter IV: Harris-Viehmann conjecture for Hodge-Newton reducible RapoportZink spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1 Harris-Viehmann conjecture: statement . . . . . . . . . . . . . . . . 39 4.2 Rigid analytic tower associated to the parabolic subgroup . . . . . . 41 4.3 Harris-Viehmann conjecture: proof . . . . . . . . . . . . . . . . . . 45 Chapter V: Serre-Tate deformation theory for local Shimura data of Hodge type 50 5.1 The slope filtration of µ-ordinary p-divisible groups . . . . . . . . . 50 5.2 The canonical deformation of µ-ordinary p-divisible groups . . . . . 51 5.3 Structure of deformation spaces . . . . . . . . . . . . . . . . . . . . 53 1 Chapter 1 INTRODUCTION 1.1 Motivation: the local Langlands correspondence and local Shimura varieties Let G be a connected reductive group over a nonarchimedean local field F. The local Langlands correspondence asserts that there should be a natural surjection with finite fibers from the set Π(G) of irreducible admissible representations of G(F) to the set Φ(G) of Langlands parameters, which are certain analogues of Galois representations. For G = GLn and F a p-adic field, the correspondence is constructed by Harris and Taylor in [HT01] and Henniart in [Hen00]. The study of the local Langlands correspondence motivated the theory of local Shimura varieties, originally formulated by Rapoport and Viehmann in [RV14]. The expectation is that the l-adic cohomology of local Shimura varieties should realize many cases of the local Langlands correspondence. For example, in the work of Harris and Taylor [HT01] the correspondence for GLn over a p-adic field is constructed via the l-adic cohomology of the Lubin-Tate space, which can be regarded as a local Shimura variety for GLn . In fact, there are two main conjectures, namely the Kottwitz conjecture and the Harris-Viehmann conjecture, which relates the l-adic cohomology of local Shimura varieties to the local Langland correspondence. The main motivation of this work is to study certain local Shimura varieties with emphasis on their relation to the local Langlands correspondence. 1.2 Overview of the results We fix a prime p > 2, and write F p and Q p respectively for a fixed algebraic closure of F p and Q p . For a p-adic local field F, we denote by F̆ the p-adic completion of the maximal unramified extension F un , and by σ ∈ Gal(F̆/F) the relative Frobenius automorphism. We also write C p for the p-adic completion of Q p and, and Z̆ p for the ring of integers of Q̆ p . Given a connected reductive group G over Q p , we say that two elements b and b0 of G(Q̆ p ) are σ-conjugate if b0 = gbσ(g)−1 for some g ∈ G(Q̆ p ). To an element 2 b ∈ G(Q̆ p ) we associate an algebraic group Jb over Q p with functor of points Jb (R) = {g ∈ G(R ⊗Q p Q̆ p ) : gbσ(g)−1 = b} for any Q p -algebra R. The isomorphism class of Jb depends only on the σ-conjugacy class of b. A local Shimura datum is a triple (G, [b], {µ}) consisting of a connected reductive group G over Q p , a σ-conjugacy class [b] of G(Q̌ p ), and a geometric conjugacy class {µ} of cocharacters of GQ p satisfying certain axioms (see 2.4.1 and [RV14], §5 for details). Let E denote the field of definition of {µ}, referred to as the local reflex field, which is an unramified finite extension of Q p . The local Shimura variety associated to the datum (G, [b], {µ}) is a tower of rigid analytic spaces over Ĕ M(G, [b], {µ}) = {MKp }, where Kp ranges over all open compact subgroups of G(Q p ), with the following properties: (i) each space MKp is equipped with an action of Jb (Q p ), (ii) the group G(Q p ) acts on the tower as a group of Hecke correspondences, (iii) the tower is equipped with a Weil descent datum down to E. The l-adic cohomology groups H i (MKp ) := Hci MKp ⊗Ĕ C p, Ql (dim MKp )
for i > 0 fit into a tower {H i (MKp )} with a natural action of G(Q p ) × WE × Jb (Q p ), where WE is the Weil group of E. For an l-adic admissible representation ρ of Jb (Q p ), we define a virtual representation of G(Q p ) × WE Õ

j H • M(G, [b], {µ}) ρ := (−1)i+ j lim Ext J (Q ) H i (MKp ), ρ . p b −−→ i,j≥0 Kp This is the object of our interest for the relation between the l-adic cohomology of local Shimura varieties and the local Langlands correspondence. In this thesis, we study local Shimura varieties and their l-adic cohomology under the following assumptions on the underlying local Shimura datum (G, [b], {µ}): (A1) the datum (G, [b], {µ}) is unramified and of Hodge type. 3 (A2) the datum (G, [b], {µ}) is Hodge-Newton reducible with respect to a parabolic subgroup P of G with Levi factor L. The first assumption means that the datum (G, [b], {µ}) satisfy the following two conditions: (i) the group G admits a reductive model over Z p , (ii) there is an embedding of local Shimura data (G, [b], {µ}) ,→ (GLn, [b]GLn , {µ}GLn ). Under these conditions, the associated local Shimura variety M(G, [b], {µ}) arises from a certain moduli space of p-divisible groups, known as a Rapoport-Zink space, constructed by Rapoport and Zink [RZ96] (for local Shimura data of EL/PEL type) and Kim [Kim13] (for unramified local Shimura data of Hodge type). We will write write RZG,[b],{µ} for the Rapoport-Zink space associated to the datum Kp ∞ (G, [b], {µ}), and often write RZG,[b],{µ} = {RZG,[b],{µ} } in lieu of the corresponding local Shimura variety M(G, [b], {µ}). Any F p -valued point x ∈ RZG,[b],{µ} (F p ) represents an isomorphism class of p-divisible groups over F p with some additional structures induced by the underlying local Shimura datum (G, [b], {µ}). Let us denote this isomorphism class by X x and its underlying p-divisible group by Xx . When G = GLn , the class X x simply represents the isomorphism class of Xx since local Shimura data for GLn induce no additional structures. The second assumption roughly means that the datum (G, [b], {µ}) naturally reduces to a local Shimura datum (L, [b] L , {µ} L ) for the specified Levi subgroup L of G. A precise definition of Hodge-Newton reducibility is given in terms of two invariants, namely the Newton point and the σ-invariant Hodge point, attached to the datum (G, [b], {µ}). When G = GLn , the Newton point and the σ-invariant Hodge point of (GLn, [b]GLn , {µ}GLn ) can be identified with the Newton polygon and the Hodge polygon of the p-divisible group Xx for any x ∈ RZGLn,[b]GLn ,{µ}GLn (F p ). We refer the readers to 4.1.4 for a precise definition of Hodge-Newton reducibility. Our first main result verifies the Harris-Viehmann conjecture under the assumptions (A1) and (A2). The Harris-Viehmann conjecture gives a parabolic inductive formula
for H • M(G, [b], {µ}) ρ when the underlying local Shimura datum (G, [b], {µ}) is not basic. An important implication of the conjecture is that the virtual representa
tion H • M(G, [b], {µ}) ρ contains no supercuspidal representations of G(Q p ) if the 4 datum (G, [b], {µ}) is not basic. When the datum (G, [b], {µ}) is basic, the Kottwitz
conjecture predicts how the virtual representation H • M(G, [b], {µ}) ρ should realize supercuspidal representations. The readers can find a precise statement of these conjectures in [RV14], Conjecture 7.3 and Conjecture 8.4. In our setting, we verify the Harris-Viehmann conjecture as follows: Theorem 1. Let (G, [b], {µ}) be an unramified local Shimura datum of Hodge type which is Hodge-Newton reducible with respect to a parabolic subgroup P of G with Levi factor L. For any admissible Ql -representation ρ of Jb (Q p ), we have the following equality of virtual representations of G(Q p ) × WE :

G(Q p ) • ∞ ∞ H • RZG,[b],{µ} ρ = IndP(Q ) H RZ L,[b] L ,{µ} L ρ . p
∞ In particular, the virtual representation H • RZG,[b],{µ} ρ contains no supercuspidal representations of G(Q p ). The earliest result of this form is Boyer’s work in [Boy99] for Drinfeld’s modular varieties. For Rapoport-Zink spaces of PEL type, Mantovan in [Man08] and Shen in [Sh13] verified the conjecture assuming Hodge-Newton reducibility. Hansen in [Han16] gave another proof of the conjecture for G = GLn , also under the HodgeNewton reducibility assumption, using the general construction of local Shimura varieties by Scholze in his Berkely lectures [Sch14]. Our second main result establishes a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. The classical Serre-Tate deformation theory states that the formal deformation space of an ordinary p-divisible group X over F p has a canonical structure of a formal torus over Z̆ p , whose identity section corresponds to a unique deformation X with the property that all endomorphisms of X lift to endomorphisms of X . The deformation X is referred to as the canonical deformation of X. The theory is based on the fact that an ordinary p-divisible group over F p admits a canonical filtration, called the slope filtration, which can be uniquely lifted to Z̆ p . These results first appeared in the Woods Hole reports of Lubin, Serre and Tate [LST64]. For our generalization of Serre-Tate deformation theory, we consider the case when the local Shimura datum (G, [b], {µ}) is µ-ordinary. This case corresponds to a special case of Hodge-Newton reducibility, and is characterized by the property that the Newton point and the σ-invariant Hodge point coincide. When G = GLn , this condition precisely means that the Newton polygon and the Hodge polygon of the 5 p-divisible group Xx are equal for all x ∈ RZGLn,[b]GLn ,{µ}GLn (F p ); in other words, the p-divisible group Xx is ordinary for all x ∈ RZGLn,[b]GLn ,{µ}GLn (F p ). For any x ∈ RZG,[b],{µ} (F p ), we denote by Def X x the formal deformation space of X x , which classifies deformations of Xx that lift the additional structures on Xx . We can naturally identify the space Def X x with the the formal completion of RZG,[b],{µ} at x. When G = GLn , this space is the classical deformation space of Xx . In this setting, our generalization of Serre-Tate deformation theory states that the formal deformation space Def X x for any x ∈ RZG,[b],{µ} (F p ) has an explicit natural “group-like” structure with an identity element corresponding to the canonical deformation X x of X x . Moreover, as in the classical setting, the p-divisible group X x with additional structures for any x ∈ RZG,[b],{µ} (F p ) admits a “slope filtration” that can be uniquely lifted to Z̆ p . When the slope filtration has two steps, the formal deformation space indeed has an explicit natural structure of a formal group as stated in the following theorem: Theorem 2. Let X be a p-divisible group over F p with additional structures that arises from a µ-ordinary local Shimura datum of Hodge type. If X has two steps in the slope filtration, the formal deformation space Def X of X has an explicit natural structure of a p-divisible group over Z̆ p . More precisely, there exist two positive integers h and d (which can be explicitly computed) such that Def X  Yhd as p-divisible groups over Z̆ p , where Yh is the Lubin-Tate formal group of height h. Moreover, the identity element corresponds to the canonical deformation X of X. When G = GLn , this theorem recovers the classical Serre-Tate deformation theory. For local Shimura data of EL/PEL type, this theorem agrees with the result of Moonen in [Mo04]. 1.3 The strategy: EL realization of Hodge-Newton reducibility We retain the assumptions (A1) and (A2) on the datum (G, [b], {µ}). Our main strategy is to study the datum (G, [b], {µ}) by embedding it into another Hodge-Newton reducible local Shimura datum of EL type, for which most of our results are previously known. We embody this strategy in the following lemma: 6 Lemma 3. Let (G, [b], {µ}) be an unramified local Shimura datum of Hodge type which is Hodge-Newton reducible with respect to a parabolic subgroup P of G with e of EL type with the following properties: Levi factor L. Then there exists a group G e (i) the embedding G ,→ GLn factors through G, e [b] e, {µ} e) is Hodge-Newton reducible with (ii) the local Shimura datum (G, G G e e and its Levi factor e respect to a parabolic subgroup P ( G L such that e e P = P ∩ G and L = L ∩ G. e [b] e, {µ} e) is referred to as an EL realization of the datum (G, [b], {µ}). The datum (G, G G An important consequence of Lemma 3 is that, by functoriality of Rapoport-Zink spaces, we have a closed embedding RZG,[b],{µ} ,−→ RZG,[b] e e ,{µ} e G G where both spaces come from Hodge-Newton reducible local Shimura data. We remark that, if you consider the embedding (G, [b], {µ}) ,→ (GLn, [b]GLn , {µ}GLn ) that comes from the assumption (A1), the datum (GLn, [b]GLn , {µ}GLn ) is not HodgeNewton reducible unless G is split. For a local Shimura datum of EL type, the Hodge-Newton reducibility condition is relatively easy to study as the condition has an alternative simple characterization. The key fact is that the Newton point and the σ-invariant Hodge point for such a datum can be identified with convex polygons, called the Newton polygon and the σ-invariant Hodge polygon. These polygons have the same endpoints and satisfy a relation that the Newton polygon lies above the σ-invariant Hodge polygon. The points at which the Newton polygon changes slope are called its break points. e [b] e, {µ} e), the Hodge-Newton reducibility condition For the EL realization (G, G G means that the σ-invariant Hodge polygon passes through some break points of e These the Newton polygon which are specified by the Levi subgroup e L of G. contact points divide the Newton polygon and the σ-invariant Hodge polygon into subpolygons, which we denote by ν1, ν2, · · · , νr and µ1, µ2, · · · , µr . Here we choose our numbering so that the slopes of νi are less than the slopes of νi+1 . 7 e be a p-divisible group over F p with additional structures that arise from the Let X e [b] e, {µ} e); in other words, X e = X for some x ∈ RZ e datum (G, x G G G,[b]Ge ,{µ}Ge (F p ). A e is a decomposition of the form Hodge-Newton decomposition of X e= X e1 × X e2 × · · · × X er , X ei is a p-divisible group over F p with additional structures that arise where each X ei, [bi ], {µi }) of EL type with the Newton polygon from some local Shimura datum (G νi and the σ-invariant Hodge polygon µi . Note that we require the decomposition to e and the factors X ei in an appropriate be compatible with additional structures on X sense. Given such a decomposition, we set X (i) := Xi × · · · × Xr and get a filtration of the underlying p-divisible group 0 ⊂ X (r) ⊂ X (r−1) ⊂ · · · ⊂ X (1) = X such that each quotient X (i) /X (i+1) ≃ Xi is equipped with additional structures that ei, [bi ], {µi }). We refer to this filtration as a Hodge-Newton arise from the datum (G e By the work of Katz [Ka79], Mantovan and Viehmann [MV10] and filtration of X. Shen [Sh13], we have the following facts: e (1) There exists a Hodge-Newton decomposition of X. e admits a unique filtration that lifts the Hodge-Newton (2) Every deformation of X e filtration of X. Lemma 3 allows us to extend these facts to the datum (G, [b], {µ}). For simplicity, e = ResK |Q p GLn for some finite unramified extension K of Q p . we may assume that G e is of the form The Levi subgroup e L(G e L = ResK |Q p GLnn1 × · · · × ResK |Q p GLnnr . For i = 1, 2, · · · , r, we denote by e Li the i-th factor in the above decomposition, and by Li the image of L = e L ∩ G under the projection e Le Li . The datum (G, [b], {µ}) induces local Shimura data (Li, [bi ], {µi }) via the projections L  Li . Then for any p-divisible group X over F p with additional structures that arise from the datum (G, [b], {µ}), we prove the following facts (Theorem 3.2.2 and Theorem 3.2.4): (1) There exists a Hodge-Newton decomposition X = X1 × · · · × Xr 8 where X i is a p-divisible group over F p with additional structures that arises from the datum (Li, [bi ], {µi }). (2) Given a deformation X of X over a ring of the form R = Z̆ p [[u1, · · · , u N ]] or R = Z̆ p [[u1, · · · , u N ]]/(pm ), there exists a unique filtration 0 ⊂ X (r) ⊂ X (r−1) ⊂ · · · ⊂ X (1) = X with the following properties: (i) each X (i) is a deformation of X (i) := Xi × · · · × Xr over R, (ii) each X (i) /X (i+1) is a deformation of Xi over R, which carries additional structures that lift the additional structures on Xi . Let us now briefly explain how these technical results are used in the proof of our main results. rig For the proof of Theorem 1, we show that the rigid analytic generic fiber RZG,[b],{µ} of the space RZG,[b],{µ} is “parabolically induced” from the rigid analytic generic rig fiber RZ L,[b] ,{µ} of RZ L,[b]L ,{µ}L , as stated in the following lemma: L L Lemma 4. There exists an analogue of Rapoport-Zink space RZP,[b] P ,{µ} P , associated to the parabolic subgroup P, with a diagram of the rigid analytic generic fibers rig RZP,[b] ,{µ} s P π1 rig RZ L,[b] ,{µ} L L with the following properties: (i) s is a closed immersion, (ii) π1 is a fibration in balls, (iii) π2 is an isomorphism. P π2 rig RZG,[b],{µ} 9 e Over the space RZG,[b] e e ,{µ} e , we can construct the space RZP,[b] e e,{µ} e associated to P P P G G following Mantovan in [Man08]. Then we construct the desired space RZP,[b] P ,{µ} P by taking the pull-back of RZG,[b] e e ,{µ} e via the closed embedding G G RZG,[b],{µ} ,−→ RZG,[b] e e ,{µ} e G G e [b] e, {µ} e). To construct the diagram induced by the embedding (G, [b], {µ}) ,→ (G, G G in the lemma, we use existence of a Hodge-Newton decomposition and a unique lifting of the Hodge-Newton filtration for the datum (G, [b], {µ}). By construction, the space RZP,[b] P ,{µ} P comes with a tower of étale coverings K p0 RZ∞ := {RZ } which is an analogue of the local Shimura varieties P,[b] P ,{µ} P P,[b] P ,{µ} P ∞ ∞ RZG,[b],{µ} and RZ L,[b]L ,{µ}L corresponding to the spaces RZG,[b],{µ} and RZ L,[b]L ,{µ}L . Our proof of Theorem 1 is based on comparing the cohomology of the three towers ∞ ∞ RZG,[b],{µ} , RZ∞ P,[b] P ,{µ} P and RZ L,[b] L ,{µ} L using the diagram in Lemma 4. ex for the p-divisible For our second result, we take x ∈ RZG,[b],{µ} (F p ) and write X group over F p with additional structures corresponding to x considered as an F p valued point of RZG,[b] e e ,{µ} e via the embedding G G RZG,[b],{µ} ,−→ RZG,[b] e e ,{µ} e . G G Taking the formal completions at x yields a closed embedding of deformation spaces Def X x ,−→ Def Xex . e [b] e, {µ} e) is µ-ordinary. Hence Our key observation is that the EL realization (G, G G we know the structure of the latter space as studied by Moonen in [Mo04]. For our generalization of Serre-Tate deformation theory, it sufficecs to prove that the space Def X x is closed under (some of) the structures of the space Def Xex . For this, we use existence of the slope decomposition and a unique lifting of the slope filtration for X x , which follows from a special case of our general results on the Hodge-Newton decomposition and the Hodge-Newton filtration. 1.4 Structure of the thesis In section 2, we introduce general notations and recall some preliminaries, such as the definition of local Shimura data and the construction of Rapoport-Zink spaces. Section 3 serves as the framework of the thesis where we develop the notion of EL realiation and prove related technical results. In section 4 and 5 we prove our main results. 10 Chapter 2 PRELIMINARIES 2.1 General notations 2.1.1. Throughout this paper, k is a perfect field of positive characteristic p. We write W(k) for the ring of Witt vectors over k, and K0 (k) for its quotient field. We will often write W = W(k) and K0 = K0 (k). We generally denote by σ the Frobenius automorphism over k, and also its lift to W(k) and K0 (k). We also fix the following standard notations: • F p is a fixed algebraic closure of F p ; • Q p is a fixed algebraic closure of Q p ; • C p is the p-adic completion of Q p ; • Q̆ p is the p-adic completion of the maximal unramified extension of Q p in Q p ; • Z̆ p is the ring of integers of Q̆ p . We remark that Z̆ p = W(F p ) and Q̆ p = K0 (F p ). 2.1.2. Given a Noetherian ring R and a free R-module Λ, we denote by Λ⊗ the direct sum of all the R-modules which can be formed from Λ using the operations of taking duals, tensor products, symmetric powers and exterior powers. An element of Λ⊗ is called a tensor over Λ. Note that there is a natural identification Λ⊗ ≃ (Λ∗ )⊗ , ∼ where Λ∗ is the dual R-module of Λ. Any isomorphism Λ −→ Λ0 of free R-modules ∼ of finite rank naturally induces an isomorphism Λ⊗ −→ (Λ0)⊗ . For a p-divisible group X over a Z p -scheme S, we write D(X) for its (contravariant) Dieudonné module and Fil1 (D(X)) ⊂ D(X)S for its Hodge filtration. We generally denote by F the Frobenius map on D(X). 2.2 Group theoretic preliminaries 11 2.2.1. Let Λ be a finitely generated free module over Z p . Then σ acts on ΛW = Λ ⊗Z p W and on GL(ΛW ) = GL(Λ) ⊗Z p W via 1 ⊗ σ. Alternatively, we may write this action as σ(g) = (1 ⊗ σ) ◦ g ◦ (1 ⊗ σ −1 ) for g ∈ GL(ΛW ). We also have an induced action of σ on the group of cocharacters HomW (Gm, GL(ΛW )) defined by σ(µ)(a) = σ(µ(a)). For two Z p -algebras R ⊆ R0, we will denote by Res R0 |R GLn the Weil restriction of GLn ⊗ R R0. If O is a finite unramified extension of Z p , a choice of σ-invariant basis of O over Z p determines an embedding of affine Z p -groups ResO |Z p GLn ,→ GLmn, where m = |O : Z p |. If Λ is a free module over O of rank n, then there is a natural identification ResO |Z p GL(Λ) ⊗Z p W  GLO ⊗Z p W (ΛW ) where the latter is identified with a product of m copies of GLn ⊗Z p W after choosing a σ-invariant basis of O over Z p . 2.2.2. Let G be a connected reductive group over Q p with a Borel subgroup B ⊆ G and a maximal torus T ⊆ B. We will write (X ∗ (T), Φ, X∗ (T), Φ∨ ) for the associated root datum, and Ω for the associated Weyl group. The choice of B determines a set of positive roots Φ+ ⊆ Φ and a set of positive coroots Φ∨+ ⊆ Φ∨ . The group Ω naturally acts on X∗ (T) (resp. X ∗ (T)), and the dominant cocharacters (resp. dominant characters) form a full set of representatives for the orbits in X∗ (T)/Ω (resp. X ∗ (T)/Ω). Except for 2.3, we will always assume that G is unramified. This means that G satisfies the following equivalent conditions: (i) G is quasi-split and split over a finite unramified extension of Q p . (ii) G admits a reductive model over Z p . When G is unramified, we fix a reductive model GZ p over Z p , and will often write G = GZ p if there is no risk of confusion. We also fix a Borel subgroup B ⊆ G and a maximal torus T ⊆ B which are both defined over Z p . We use the standard notation RepZ p (G) (resp. RepQ p (G)) to denote the category of finite rank G-representations of over Z p (resp. Q p ). 12 2.2.3. For any local, strictly Henselian Z p -algebra R and a cocharacter λ : Gm,R → G R , we denote the G(R)-conjugacy class of λ by {λ}G , or usually by {λ} if there is no risk of confusion. We have identifications X∗ (T)  Hom R (Gm,TR ) and Ω  NG (T)(R)/T(R), which induce a bijection between X∗ (T)/Ω and the set of G(R)conjugacy classes of cocharacters for G R . We will be mostly interested in the case R = W(k) for some algebraically closed k, where we also have a bijection ∼ HomW (Gm, GW )/G(W)  HomK0 (Gm, G K0 )/G(K0 ) → G(W)\G(K0 )/G(W) induced by {λ} 7→ G(W)λ(p)G(W); indeed, the first bijection follows from the fact that G is split over W, while the second bijection is the Cartan decomposition. Let Λ ∈ RepZ p (G) be a faithful G-representation over Z p . By [Ki10], Proposition 1.3.2, we can choose a finite family of tensors (si )i∈I on Λ such that G is the pointwise stabilizer of the si ; i.e., for any Z p -algebra R we have G(R) = {g ∈ GL(Λ ⊗Z p R) : g(si ⊗ 1) = si ⊗ 1 for all i ∈ I}. We say that a grading gr• (Λ R ) is induced by λ if the following conditions are satisfied: (i) the Gm -action on Λ R via λ leaves each grading stable, (ii) the resulting Gm -action on gri (Λ R ) is given by z7→ z −i z7→ z·id Gm −−−−−→ Gm −−−−−→ GL(gri (Λ R )). Let S be an R-scheme, and E a vector bundle on S. For a finite family of global sections (ti ) of E ⊗ , we define the following scheme over S   PS := IsomOS [E , (ti )], [Λ ⊗ R OS, (si ⊗ 1)] . In other words, PS classifies isomorphisms of vector bundles E  Λ ⊗ R OS which match (ti ) and (si ⊗ 1). Let Fil• (E ) be a filtration of E . When PS is a trivial G-torsor, we say that Fil• (E ) is a {λ}-filtration with respect to (ti ) if there exists an isomorphism E  Λ ⊗ R OS , matching (ti ) and (1 ⊗ si ), which takes Fil• (E ) to a filtration of Λ ⊗ R OS induced by gλg −1 for some g ∈ G(R). More generally, when PS a G-torsor, we say that Fil• (E ) is a {λ}-filtration with respect to (ti ) if it is étale-locally a {λ}-filtration. 13 2.3 F-isocrystals with G-structure We review the theory of F-isocrystals with G-structure due to R. Kottwitz in [Ko85] and [Ko97]. We do not assume that G is unramified for this section. 2.3.1. Let k be a perfect field of positive characteristic p. An F-isocrystal over k ∼ is a vector space V over K0 (k) with an isomorphism F : σ ∗V → V. The dimension of V is called the height of the isocrystal. Let F-Isoc(k) denote the category of F-isocrystals over k. For a connected reductive group G over Q p , we define an F-isocrystal over k with G-structure as an exact faithful tensor functor RepQ p (G) → F-Isoc(k). Example 2.3.2. (i) An F-isocrystal with GLn -structure is an F-isocrystal of height n. (ii) If G = ResE |Q p GLn where E |Q p is a finite extension of degree m, an Fisocrystal with G-structure is an F-isocrystal V of height mn together with a Q p homomorphism ι : E → End k (V). (iii) If G = GSp2n , an F-isocrystal with G-structure is an F-isocrystal V of height 2n together with a non-degenerate alternating pairing V ⊗ V → 1, where 1 is the unit object of the tensor category F-Isoc(k). 2.3.3. Let us now assume that k is algebraically closed. We say that b, b0 ∈ G(K0 ) are σ-conjugate if there exists g ∈ G(K0 ) such that b0 = gbσ(g)−1 . We denote by B(G) the set of all σ-conjugacy classes in G(K0 ). The definition of B(G) is independent of k in the sense that any inclusion k ,→ k 0 into another algebraically closed field of characteristic p induces a bijection between the σ-conjugacy classes of G(K0 (k)) and those of G(K0 (k 0)). We will write [b]G , or simply [b] when there is no risk of confusion, for the σ-conjugacy class of b ∈ G(K0 ). The set B(G) classifies the F-isocrystals over k with G-structure up to isomorphism. We describe this classification as explained in [RR96], 3.4. Given b ∈ G(K0 ) and a G-representation (V, ρ) over Q p , set Nb (ρ) to be V ⊗Q p K0 with a σ-linear automorphism F = ρ(b) ◦ (1 ⊗ σ). Then Nb : RepQ p (G) → F-Isoc(k) is an exact faithful tensor functor. It is evident that two elements b1, b2 ∈ G(K0 ) give an isomorphic functor if and only if they are σ-conjugate. One can also prove that any 14 F-isocrystal on k with G-structure is isomorphic to a functor Nb for some b ∈ G(K0 ). Thus the association b 7→ Nb induces the desired classification. 2.3.4. Let D be the pro-algebraic torus over Q p with character group Q. We introduce the set N (G) := (Int G(K0 )\HomK0 (D, G K0 ))hσi . If we fix a Borel subgroup B ⊆ G and a maximal torus T ⊆ B, we can also write N (G) = (X∗ (T)Q /Ω)hσi . We can define a partial order  on N (G) as follows. Let C̄ be the closed Weyl chamber. First we define a partial order 1 on X∗ (T)R by declaring that α 1 α0 if and only if α0 − α is a nonnegative linear combination of positive coroots. Each orbit in X∗ (T)R /Ω is represented by a unique element in C̄, so the restriction of 1 to C̄ induces a partial order 2 on X∗ (T)R /Ω. Then we take  to be the restriction of 2 to (X∗ (T)Q /Ω)hσi . Remark. A closed embedding G1 ,→ G2 of connected reductive algebraic groups over Q p induces an order-preserving map N (G1 ) → N (G2 ), which is not necessarily injective. 2.3.5. Kottwitz studied the set B(G) by introducing two maps νG : B(G) → N (G), κG : B(G) → π1 (G)hσi called the Newton map and the Kottwitz map of G. We refer the readers to [Ko85], §4 or [RR96], §1 for definition of the Newton map, and [Ko97], §4 and §7 for definition of the Kottwitz map. Both maps are functorial in G; more precisely, they induce natural transformations of set-valued functors on the category of connected reductive groups ν : B(·) → N (·), κ : B(·) → π1 (·)hσi . Given [b] ∈ B(G) (and its corresponding F-isocrystal with G-structure), we will often refer to two invariants νG ([b]) and κG ([b]) respectively as the Newton point and the Kottwitz point of [b]. Kottwitz proved that a σ-conjugacy class is determined by its Newton point and Kottwitz point; in other words, the map νG × κG : B(G) → N (G) × π1 (G)hσi 15 is injective ([Ko97], 4.13). Example 2.3.6. We describe the Newton map for G = GLn . Let T be the diagonal torus contained in the Borel subgroup of lower triangular matrices. Then using the identification X∗ (T)  Zn we can write N (GLn ) = {(r1, r2, · · · , rn ) ∈ Qn : r1 ≤ r2 ≤ · · · ≤ rn }, which can be identified with the set of convex polygons with rational slopes. We l Õ have (ri )  (si ) if and only if (ri − si ) ≥ 0 for all l ∈ {1, 2, · · · , n}, so the ordering i=1  coincides with the usual “lying above" order for convex polygons. If V is an F-isocrystal V of height n associated to [b] ∈ B(GLn ), its Newton point νGLn ([b]) is the same as its classical Newton polygon. In this case, the Kottwitz point κGLn ([b]) is determined by the Newton point νGLn ([b]). Hence V and [b] are determined by the Newton point νGLn ([b]), and we recover Manin’s classification of F-isocrystals by their Newton polygons in [Ma63]. 2.3.7. Let µ ∈ X∗ (T) be a dominant cocharacter. Then µ represents a unique conjugacy class of cocharacters of G(K0 ) which we denote by {µ}. We identify µ with its image in X∗ (T)/Ω, and define m−1 1 Õ i σ (µ) ∈ N (G), µ̄ = m i=0 where m is some integer such that σ m (µ) = µ. Note that our definition of µ̄ does not depend on the choice of m. We also let µ\ ∈ π1 (G)hσi be the image of µ under the natural projection X∗ (T) → π1 (G)hσi = (X∗ (T)/hα∨ : α∨ ∈ Φ∨ i)hσi . The characterization of the Newton map in [Ko85], 4.3 shows that µ̄ is the image of [µ(p)] under νG . It also follows directly from the definition of κG that µ\ is the image of [µ(p)] under κG . Let us now define the set B(G, {µ}) := {[b] ∈ B(G) : κG ([b]) = µ\, νG ([b])  µ̄}. This set is known to be finite (see [RR96], 2.4.). It is also non-empty since we have [µ(p)] ∈ B(G, {µ}) by the discussion in the previous paragraph. Since the Newton map is injective on B(G, {µ}) (see 2.3.5), the partial order  on N (G) induces a partial order on B(G, {µ}). We will also use the symbol  to 16 denote this induced partial order. Note that [µ(p)] is a unique maximal element in B(G, {µ}) as the inequality [b]  [µ(p)] clearly holds for all [b] ∈ B(G, {µ}). We refer to the σ-conjugacy class [µ(p)] as the µ-ordinary element of B(G, {µ}). We say that an F-isocrystal over k with G-structure is µ-ordinary if it corresponds to [µ(p)] in the sense of 2.3.3. Note that a σ-conjugacy class [b] ∈ B(G, {µ}) is µ-ordinary if and only if νG ([b]) = µ̄. 2.4 Unramified local Shimura data of Hodge type In this subsection, we review the notion of unramified local Shimura data of Hodge type and describe F-crystals with additional structures that arise from such data. 2.4.1. Assume that k is algebraically closed. By an unramified (integral) local Shimura datum of Hodge type, we mean a tuple (G, [b], {µ}) where • G is an unramified connected reductive group over Q p ; • [b] is a σ-conjugacy class of G(K0 ); • {µ} is a G(W)-conjugacy class of cocharacters of G, which satisfy the following two conditions: (i) [b] ∈ B(G, {µ}), (ii) there exists a faithful G-representation Λ ∈ RepZ p (G) (with its dual Λ∗ ) such that, for all b ∈ [b] and µ ∈ {µ} satisfying b ∈ G(W)µ(p)G(W), the W-lattice M := Λ∗ ⊗Z p W ⊂ Nb (Λ∗ ⊗Z p Q p ) satisfies the property pM ⊂ F M ⊂ M (where F is defined from b as explained in 2.3.3). Here Nb : RepQ p (G) → F-Isoc(k) is the functor defined in 2.3.3 which is uniquely determined by [b]. The set G(W)µ(p)G(W) is independent of the choice µ ∈ {µ} as explained in 2.2.2. The property pM ⊂ F M ⊂ M means that M is an F-crystal over k (with a σ-linear endomorphism F). The requirement b ∈ G(W)µ(p)G(W) ensures that the Hodge filtration of M is induced by σ −1 (µ). 17 In practice when one tries to check that a given tuple (G, [b], {µ}) is an unramified local Shimura datum, it is often more convenient to work with the following equivalent conditions of (i) and (ii): (i’) [b] ∩ G(W)µ(p)G(W) is not empty for some (and hence for all) µ ∈ {µ}, (ii’) there exists a faithful G-representation Λ ∈ RepZ p (G) (with its dual Λ∗ ) such that, for some b ∈ [b] and µ ∈ {µ} satisfying b ∈ G(W)µ(p)G(W), the W-lattice M := Λ∗ ⊗Z p W ⊂ Nb (Λ∗ ⊗Z p Q p ) satisfies the property pM ⊂ F M ⊂ M. The equivalence of (i) and (i’) is due to work of several authors, including KottwitzRapoport [KR03], Lucarelli [Lu04] and Gashi [Ga10]. Note that (i’) ensures that the condition (ii) is never vacuously satisfied. The equivalence of (ii) and (ii’), one observes that both conditions are equivalent to the condition that the linearization of F has an integer matrix representation after taking some σ-conjugate, which depends only on [b]. Remark. When {µ} is minuscule, an unramified local Shimura datum of Hodge type as defined above is a local Shimura datum as defined by Rapoport and Viehmann in [RV14], Definition 5.1. In fact, since G is split over W, we may view geometric conjugacy classes of cocharacters as G(W)-conjugacy classes of cocharacters. Using the conditions (i’) and (ii’) one easily verifies the following functorial properties of unramified local Shimura data of Hodge type: Lemma 2.4.2. Let (G, [b], {µ}) be an unramified local Shimura datum of Hodge type. (1) If (G0, [b0], {µ0 }) is another unramified local Shimura datum of Hodge type, the tuple (G × G0, [b, b0], {µ, µ0 }) is also an unramified local Shimura datum of Hodge type. (2) For any homomorphism f : G −→ G0 of unramified connected reductive group defined over Z p , the tuple (G0, [ f (b)], { f ◦ µ}) is an unramified local Shimura datum of Hodge type. 18 2.4.3. For the rest of this section, we fix our unramified local Shimura datum of Hodge type (G, [b], {µ}) and also a faithful G-representation Λ ∈ RepZ p (G) in the condition (ii) of 2.4.1. By Lemma 2.4.2, we obtain a morphism of unramified local Shimura data of Hodge type (G, [b], {µ}) −→ (GL(Λ), [b]GL(Λ), {µ}GL(Λ) ). Let us now choose an element b ∈ [b] ∩ G(W)µ(p)G(W) and take M := Λ∗ ⊗Z p W as in the condition (ii) of 2.4.1. We also choose a finite family of tensors (si )i∈I on Λ as in 2.2.3. Then M = Λ∗ ⊗Z p W is equipped with tensors (ti ) := (si ⊗ 1), which are F-invariant since the linearization of F on M[1/p] = Nb (Λ∗ ⊗Z p Q p ) is given by an element b ∈ G(K0 ) in the conjugacy class [b]. We may regard the tensors (ti ) as additional structures on M induced by the group G. Following the terminology of 2.3, we will often refer to these additional structures as G-structure. We will also write M := (M, (ti )), which will often be referred to as an F-crystal with G-structure (induced by b). When {µ} is minuscule, we have a unique p-divisible group X over k with D(X) = M. The Hodge filtration Fil1 (D(X)) ⊂ D(X) is a {σ −1 (µ−1 )}-filtration with respect to (ti ), as explained in [Kim13], Lemma 2.5.7 and Remark 2.5.8. In this case, we will often write X := (X, (ti )) and refer to it as a p-divisible group with G-structure (induced by b). We will sometimes use the phrase “tensors on X” to indicate the tensors (ti ), although strictly speaking they are tensors on the Dieudonne module D(X) = M. 2.4.4. For the datum (G, [b], {µ}), we can define its Newton point and Kottwitz point by νG ([b]) and κG ([b]). Taking a unique dominant representative µ of {µ}, we can also define µ̄ as in 2.3.7, which we call the σ-invariant Hodge point of (G, [b], {µ}). We say that (G, [b], {µ}) is µ-ordinary if [b] is µ-ordinary. For the F-crystal with G-structure M, we define its Newton point, Kottwitz point and σ-invariant Hodge point to be the corresponding invariants for (G, [b], {µ}). We say that M is ordinary if (G, [b], {µ}) is ordinary. When {µ} is minuscule, these definitions obviously extend to the corresponding p-divisible group with G-structure X. Remark. We can further extend most of the notions defined in this section to the case when k is not algebraically closed. For example, we may define an F-crystal 19 over k with G-structure as an F-crystal M over k equipped with tensors (ti ) such that the pair (M ⊗W(k) W( k̄), (ti ⊗ 1)) is an F-crystal over k̄ with G-structure as defined in 2.4.3. Then we have natural notions of the Newton point, Kottwitz point, σ-invariant Hodge point, and µ-ordinariness induced by the corresponding notions for (M ⊗W(k) W( k̄), (ti ⊗ 1)). This explains why we may safely focus our study on the case when k is algebraically closed. Example 2.4.5. As a concrete example, let us consider the case G = ResO |Z p GLn , where O is the ring of integers of some finite unramified extension E of Q p . Choosing a family of tensors (si ) on Λ whose pointwise stabilizer is G amounts to choosing a Z p -basis of O. Hence M = (M, (ti )) can be identified with an F-crystal M with an action of O (cf. Example 2.3.2.(ii)). Following Moonen in [Mo04], we will often say O-module structure in lieu of G-structure. We now take I := Hom(O, W(k)) and m := |E : Q p |. Note that I has m elements. For convenience, we will write i + s := σ s ◦ i for any i ∈ I and s ∈ Z. Then M, Î being a module over O ⊗Z p W(k) = i∈I W(k), decomposes into character spaces Ê M= Mi where Mi = {x ∈ M : a · x = i(a)x}. (2.4.5.1) i∈I For each i ∈ I , the Frobenius map F restricts to a σ-linear map Fi : Mi → Mi+1 . Then the map F m restricts to a σ m -linear endomorphism φi of Mi , thereby yielding a σ m -F-crystal (Mi, φi ) over k. By construction, Fi induces an isogeny from σ ∗ (Mi, φi ) to (Mi+1, φi+1 ). This implies that the rank and the Newton polygon of (Mi, φi ) is independent of i ∈ I . We will write d for the rank of (Mi, φi ). The decomposition (2.4.5.1) yields a decomposition Ê M/F M = Mi /Fi−1 Mi . i∈I Define a function f : I → Z by setting f(i) to be the rank of Mi /Fi−1 Mi . We refer to the datum (d, f) as the type of M. Let us describe the Newton point in this setting. Using the identifications GW  Î md we can write i∈I GL(Mi ) and X∗ (T)  Z X∗ (T)Q /Ω = {(x1, · · · , xmd ) ∈ Qmd : xds+1 ≤ · · · ≤ xd(s+1) for s = 0, 1, · · · , m − 1}. For µ = (x1, · · · , xmd ) ∈ X∗ (T)Q /Ω the action of σ is given by σ(µ) = (y1, · · · , ymd ) where yt = xt+d . Therefore we obtain an identification N (G) = {(r1, r2, · · · , rd ) ∈ Qd : r1 ≤ r2 ≤ · · · ≤ rd }. (2.4.5.2) 20 Under this identification, the Newton point νG ([b]) of M coincides with the Newton polygon of (Mi, φi ) which was already seen to be independent of i ∈ I . We will refer to this polygon as the Newton polygon of M. The polygon νG ([b]) is closely related with the Newton polygon of M (without O-module structure) as follows: a rational number λ appears with multiplicity α in νG ([b]) (viewed as a d-tuple) if and only if it appears with multiplicity mα in the Newton polygon of M (viewed as an md-tuple). We can also regard the σ-invariant Hodge point µ̄ as a polygon under the identification (2.4.5.2). We will refer to this polygon as the σ-invariant Hodge polygon of M. The inequality νG ([b])  µ̄ serves as a generalized Mazur’s inequality, which says that the Newton polygon νG ([b]) lies above the σ-invariant Hodge polygon µ̄. Note that M is µ-ordinary if and only if the two polygons coincide. When {µ} is minuscule, we also identify X = (X, (ti )) with a p-divisible group X with an action of O. All of the discussions above evidently apply to X. Namely, we can define the type, the Newton polygon and the σ-invariant Hodge polygon of X. In addition, when {µ} is minuscule we have the following facts: (1) The σ-invariant Hodge polygon µ̄ of X is determined by the type (d, f) as follows: if we write µ̄ = (a1, a2, · · · , ad ), the slopes a j are given by a j = #{i ∈ I : f(i) > d − j} (see [Mo04], 1.2.5.). (2) There exists a unique isomorphism class of µ-ordinary p-divisible groups with O-module structure of a fixed type (d, f) (see [Mo04], Theorem 1.3.7.). Remark. As seen in 2.2.1, we have an embedding GW = ResO |Z p GLn ⊗Z p W ,→ GL(M) where the image is identified with a product of m copies of GLn ⊗Z p W. The decomposition (2.4.5.1) shows that these copies are given by GL(Mi ). In particular, we have n = d. 2.4.6. The isomorphism class of M = (M, (ti )) depends on the choice b ∈ [b], even though M[1/p] ≃ Nb (Λ∗ ⊗Z p Q p ) is independent of this choice. To see this, let M 0 = (M 0, (ti0)) be the F-crystal over k with G-structure that arises from another choice b0 = gbσ(g)−1 ∈ [b] ∩ G(W)µ(p)G(W) for some g ∈ G(K0 ). Then g gives 21 an isomorphism ∼ M[1/p] ≃ Nb (Λ∗ ⊗Z p Q p ) −→ Nb0 (Λ∗ ⊗Z p Q p ) ≃ M 0[1/p], which also matches (ti ) with (ti0) since g ∈ G(K0 ). However, this isomorphism does not induce an isomorphism between M and M 0 unless g ∈ G(W). The above discussion motivates us to consider the set G X{µ} ([b]) := {g ∈ G(K0 )/G(W)|gbσ(g)−1 ∈ G(W)µ(p)G(W)}. This set is clearly independent of our choice of b ∈ [b] up to bijection. It is also independent of the choice of µ ∈ {µ} as we already noted that the set G(W)µ(p)G(W) G ([b]) is called the affine only depends on the conjugacy class of µ. The set X{µ} Deligne-Lusztig set associated to (G, [b], {µ}). Proposition 2.4.7. Fix an element b ∈ [b], and let M = (M, (ti )) denote the F-crystal with G-structure induced by b (as defined in 2.4.3). Then the affine Deligne-Lusztig G ([b]) classifies isomorphism classes of tuples (M 0, (t 0), ι) where set X{µ} i • (M 0, (ti0)) is an F-crystal over k with G-structure; ∼ • ι : M 0[1/p] −→ M[1/p] is an isomorphism which matches (ti0) with (ti ). When {µ} is minuscule, take X to be the p-divisible group with D(X) = M. Then G ([b]) also classifies isomorphism classes of tuples (X 0, (t 0), ι) where the set X{µ} i • (X 0, (ti0)) is a p-divisible group over k with G-structure; ∼ • ι : X → X 0 is a quasi-isogeny such that the induced isomorphism D(X 0)[1/p] −→ D(X)[1/p] matches (ti0) with (ti ). Proof. The second part follows immediately from the first part using Dieudonné theory, so we need only prove the first part. G ([b]). Then as discussed in 2.4.6, the Let g be a representative of gG(W) ∈ X{µ} element b0 := g −1 bσ(g) gives rise to an F-crystal over k with G-structure (M 0, (ti0)) ∼ and an isomorphism ι : M 0[1/p] −→ M[1/p] which matches (ti0) with (ti ). It is clear that the isomorphism class of (M 0, (ti0), ι) does not depend on the choice of the representative g. 22 Conversely, let (M 0, (ti0), ι) be a tuple as in the statement. Let b0 ∈ G(K0 ) be the linearization of the Frobenius map on M 0[1/p]. Then the isomorphism ι : ∼ M 0[1/p] −→ M[1/p] determines an element g ∈ G(K0 ) such that b0 = gbσ(g)−1 . Moreover, we have b0 ∈ G(W)µ(p)G(W) since (M 0, (ti0)) is an F-crystal over k with G-structure. Changing (M 0, (ti0), ι) to an isomorphic tuple will change g to gh for G ([b]). some h ∈ G(W), so we get a well-defined element gG(W) ∈ X{µ} These associations are clearly inverse to each other, so we complete the proof.  We now describe some functorial properties of affine Deligne-Lusztig sets which are compatible with the functorial properties of unramified local Shimura data of Hodge type described in Lemma 2.4.2. Lemma 2.4.8. Let G0 be an unramified connected reductive group over Q p . (1) If (G0, [b0], {µ0 }) is an unramified local Shimura datum of Hodge type, we have an isomorphism ∼ 0 0 0 0 G G G×G X{µ,µ 0 } ([b, b ]) −→ X{µ} ([b]) × X{µ 0 } ([b ]) induced by the natural projections. (2) For any homomorphism f : G −→ G0 defined over Z p , we have a natural map 0 G X{µ} ([b]) −→ X{Gf ◦µ} ([ f (b)]) induced by gG(W) 7→ f (g)G0(W), which is injective if f is a closed immersion. Proof. The only possibly non-trivial assertion is the injectivity of the natural map G ([b]) −→ X G 0 X{µ} ([ f (b)]) in (2) when f is a closed immersion. To see this, one { f ◦µ} 0 may assume that G = GLn by embedding G0 into some GLn . Then the assertion follows from the fact that the map G(K0 )/G(W) −→ GLn (K0 )/GLn (W) is injective (see [HP17], 2.4.4.). 2.5  Deformation Spaces of p-divisible groups with Tate tensors In this subsection, we review Faltings’s construction of a “universal" deformation of p-divisible groups with Tate tensors, given in [Fal99], §7. We refer readers to [Mo98], §4 for a more detailed discussion of these results. 23 2.5.1. Let R be a ring of the form R = W[[u1, · · · , u N ]] or R = W[[u1, · · · , u N ]]/(pm ). We can define a lift of the Frobenius map on R, which we also denote by σ, by p setting σ(ui ) = ui . We define a filtered crystalline Dieudonné module over R to be a 4-tuple (M , Fil1 (M ), ∇, F) where • M is a free R-module of finite rank; • Fil1 (M ) ⊂ M is a direct summand; b R/W is an integrable, topologically quasi-nilpotent connec• ∇:M →M ⊗Ω tion; • FM : M → M is a σ-linear horizontal endomorphism, which satisfy the following conditions:
∼ (i) FM induces an isomorphism M + p−1 Fil1 (M ) ⊗ R,σ R −→ M , and
(ii) Fil1 (M ) ⊗ R (R/p) = Ker F ⊗ σR/p : M ⊗ R (R/p) → M ⊗ R (R/p) . Combining the work of de Jong in [dJ95] and Grothendieck-Messing theory, we obtain an equivalence between the category of filtered crystalline Dieudonné modules over R and the (opposite) category of p-divisible groups over R (see also [Mo98], 4.1.). 2.5.2. Let X be a p-divisible group over k. We write CW for the category of artinian local W-algebras with residue field k. By a deformation or lifting of X over R ∈ CW , we mean a p-divisible group X over R with an isomorphism α : X ⊗ R k  X. We define a functor Def X : CW → Sets by setting Def X (R) to be the set of isomorphism classes of deformations of X over R. We take M := D(X), the contravariant Dieudonné module of X, and write F for the Frobenius map and Fil1 (M) ⊂ M for its Hodge filtration. We choose a cocharacter µ : Gm → GLW (M) such that σ −1 (µ) induces this filtration; for instance, we take µ to be the dominant cocharacter that represents the Hodge polygon of X under the identification of the Newton set N (GLn ) in Example 2.3.6. The stabilizer of the complement of Fil1 (M) is a parabolic subgroup of GLW (M). We let U µ be its 24 µ bµ = SpfR of U µ at the identity unipotent radical, and take the formal completion U GL µ section. Then RGL is a formal power series ring over W, so we can define a lift of µ Frobenius map on RGL . µ bµ (R ) be the tautological point. Define Proposition 2.5.3 ([Fal99], §7). Let ut ∈ U GL µ M := M ⊗W RGL, µ Fil1 (M ) := Fil1 (M) ⊗W RGL, FM := ut ◦ (F ⊗W σ). (1) There exists a unique topologically quasi-nilpotent connection ∇ : M → b Rµ /W that commutes with FM , and this connection is integrable. M ⊗Ω GL (2) If p > 2, the filtered crystalline Dieudonné module (M , Fil1 (M ), ∇, FM ) corresponds to the universal deformation of X via the equivalence described in 2.5.1. µ In particular, (2) implies that we have an identification Def X  SpfRGL . We will µ write XGL for the universal deformation of X. 2.5.4. We now consider deformations of p-divisible groups with G-structure. We fix an unramified local Shimura datum of Hodge type (G, [b], {µ}) with minuscule {µ}. We also fix a faithful G-representation Λ ∈ RepZ p (G) in the condition (ii) of 2.4.1, and choose b ∈ [b] and µ ∈ {µ} such that b ∈ G(W)µ(p)G(W). Then we obtain an F-crystal with G-structure M = (M, (ti )) as explained in 2.4.3, which gives rise to a p-divisible group with G-structure X = (X, (ti )) since {µ} is minuscule. The condition b ∈ G(W)µ(p)G(W) ensures that the Hodge filtration Fil1 (M) ⊂ M is induced by σ −1 (µ), so all the constructions from 2.5.2 and Proposition 2.5.3 are valid for X. µ µ µ b = SpfR Let UG := U µ ∩GW , which is a smooth unipotent subgroup of GW . Take U G G µ to be its formal completion at the identity section. Then RG is a formal power series µ ring over W, so we get a lift of Frobenius map to RG . Alternatively, we get this µ µ µ lift from the lift on RGL via the surjection RGL  RG induced by the embedding bµ ,−→ U bµ . U G bµ (R µ ) be the tautological point. Define Let ut,G ∈ U G G µ MG := M ⊗W RG, µ Fil1 (MG ) := Fil1 (M) ⊗W RG, FMG := ut,G ◦ (F ⊗W σ). Then we have an integrable, topologically quasi-nilpotent connection ∇G : MG → b Rµ /W induced by ∇ : M → M ⊗ Ω b Rµ /W from Proposition 2.5.3. In MG ⊗ Ω G GL 25 addition, ∇G clearly commutes with FMG by construction. Hence we have a filtered crystalline Dieudonné module (MG, Fil1 (MG ), ∇G, FMG ). Note that MG is equipped with tensors (tiuniv ) := (ti ⊗ 1), which are evidently FMG invariant by construction. If p > 2, one can prove that these tensors lie in the 0th filtration (see [Kim13], Lemma 2.2.7 and Proposition 2.5.9.). µ µ Let XG be the p-divisible group over RG corresponding to (MG, Fil1 (MG ), ∇G, FMG ) µ via the equivalence described in 2.5.1. Alternatively, one can get XG by simply µ µ µ pulling back XGL over RG . Then XG is the “universal deformation" of (X, (ti )) in the following sense: Proposition 2.5.5 ([Fal99], §7). Assume that p > 2. Let R be a ring of the form R = W[[u1, · · · , u N ]] or R = W[[u1, · · · , u N ]]/(pm ). Choose a deformation X of X µ µ over R, and let f : RGL → R be the morphism induced by X via SpfRGL  Def X . µ Then f factors through RG if and only if the tensors (ti ) can be lifted to tensors (ti ) ∈ D(X )⊗ which are Frobenius-invariant and lie in the 0th filtration with respect to the Hodge filtration. If this holds, then we necessarily have ( f ∗ tiuniv ) = (ti ). µ µ We define Def X,G to be the image of the closed immersion SpfRG ,−→ SpfRGL  Def X . Then Def X,G classifies deformations of (X, (ti )) over formal power series rings over W or W/(pm ) in the sense of Proposition 2.5.5. Note that our definition of Def X,G is independent of the choice of (ti ) and µ ∈ {µ}; indeed, the independence of the choice of (ti ) is clear by construction, and the independence of the choice of µ follows from the universal property. We close this section with some functorial properties of deformation spaces, which are compatible with the functorial properties of unramified local Shimura data of Hodge type described in Lemma 2.4.2. The proof is straightforward and thus omitted. Lemma 2.5.6. Let (G0, [b0], {µ0 }) be another unramified local Shimura datum of Hodge type. Choose b0 ∈ [b0] and µ0 ∈ {µ0 } such that b0 ∈ G0(W)µ0(p)G0(W), and let (X 0, (ti0)) be a p-divisible group with G0-structure that arises from this choice. (1) The natural morphism Def X × Def X 0 −→ Def X×X 0 , defined by taking the product of deformations, induces an isomorphism ∼ Def X,G × Def X 0,G 0 −→ Def X×X 0,G×G 0 . 26 (2) For any homomorphism f : G → G0 defined over Z p such that f (b) = b0, we have a natural morphism Def X,G → Def X 0,G 0 bµ → U b f ◦µ induced by the map U G G0 . Remark. With some additional work, one can show that the natural morphism Def X,G → Def X 0,G 0 in (2) is independent of the choice of µ ∈ {µ}. See [Kim13], Proposition 3.7.2 for details. 2.6 Rapoport-Zink spaces of Hodge type In this section, we discuss the construction and key properties of Rapoport-Zink spaces of Hodge type, following [Kim13]. 2.6.1. Let us fix some notations for this section. We set k = F p so that W = Z̆ p and K0 = Q̆ p . We fix an unraified local Shimura datum (G, [b], {µ}) such that {µ} is minuscule. We also choose b ∈ [b] ∩ G(Z̆ p )µ(p)G(Z̆ p ) and take X := (X, (ti )) as in 2.4.3. Let NilpZ̆ p denote the category of Z̆ p -algebra where p is nilpotent. For any R ∈ NilpZ̆ p we set RZb (R) to be the set of isomorphism classes of pairs (X, ι) where • X is a p-divisible group over R; • ι : XR/p −→ XR/p is a quasi-isogeny, i.e., an invertible global section of Hom(XR/p, XR/p ) ⊗Z Q. Then RZb defines a covariant set-valued functor on NilpZ̆ p , which does not depend on the choice of b ∈ [b] ∩ G(Z̆ p )µ(p)G(Z̆ p ) up to isomorphism. Rapoport and Zink in [RZ96] proved that the functor RZb is represented by a formal scheme which is locally formally of finite type and formally smooth over Z̆ p . We write RZb also for the representing formal scheme, and XGL,b for the universal p-divisible group over RZb . 2.6.2. Given a pair (X, ι) ∈ RZb (R) with R ∈ NilpZ̆ p , we have an isomorphism ∼ D(ι) : D(XR/p )[1/p] −→ D(XR/p )[1/p] 27 induced by ι. We write (tX,i ) for the inverse image of the tensors (ti ) R under this isomorphism. Let Nilpsm denote the full subcategory of NilpZ̆ p consisting of formally smooth and Z̆ p formally finitely generated algebra over Z̆ p /pm for some positive integer m. For any (si ) (R) ⊂ HomZ̆ p (Spf(R), RZb ) as follows: for a R ∈ Nilpsm , we define the set RZG,b Z̆ p morphism f : Spf(R) → RZb and a p-divisible group X over Spec(R) which pulls (si ) (R) if and only if there exists a back to f ∗ XGL,b over Spf(R), we have f ∈ RZG,b (unique) family of tensors (ti ) on D(X) with the following properties: (i) for some ideal of definition J of R containing p, the pull-back of (ti ) over R/J agrees with the pull-back of (tX,i ) over R/J, (ii) for a p-adic lift R of R which is formally smooth over Z̆ p , the R-scheme   ∗ PR := IsomR [D(X)R , (ti )R ], [Λ ⊗Z p R, (si ⊗ 1)] defined as in 2.2.3 is a G-torsor, (iii) the Hodge filtration of X is a {σ −1 (µ−1 )}-filtration with respect to (ti ). (si ) Then RZG,b defines a set-valued functor on Nilpsm . Z̆ p Proposition 2.6.3 ([Kim13], Theorem 4.9.1.). Assume that p > 2. Then there exists a closed formal subscheme RZG,b ⊂ RZb , which is formally smooth over (si ) Z̆ p and represents the functor RZG,b for any choice of the tensors (si ) in 2.4.3. Moreover, the isomorphism class of the formal scheme RZG,b depends only on the datum (G, [b], {µ}). We let XG,b denote the “universal p-divisible group" over RZG,b , obtained by taking the pull-back of XGL,b . Then we obtain a family of “universal tensors" (tiuniv ) on D(XG,b ) by applying the universal property to an open affine covering of RZG,b . For the rest of this section, we assume that p > 2 and take RZG,b as in Proposition 2.6.3. Example 2.6.4. Consider the case G = ResO |Z p GLn where O is the ring of integers of some finite unramified extension of Q p . As explained in Example 2.4.5, we can regard X = (X, (ti )) as a p-divisible group X with an action of O. In this setting, the construction of RZG,b agrees with the construction of Rapoport-Zink spaces of EL type in [RZ96] (see [Kim13], Proposition 4.7.1.). In other words, for any R ∈ NilpZ̆ p the set RZG,b (R) classifies the isomorphism classes of pairs (X, ι) where 28 • X is a p-divisible group over R, endowed with an action of O such that det R (a, Lie(X)) = det(a, Fil0 (D(X))Q̆ p ) for all a ∈ O, • ι : XR/p → XR/p is a quasi-isogeny which commutes with the action of O. 2.6.5. Let us give a concrete description of the set RZG,b (F p ). Consider a pair (X, ι) ∈ RZb (F p ) with a family of tensors (ti ) on D(X). Then (ti ) has the property (i) of 2.6.2 if and only if it is matched with the family (ti ) under the isomorphism ∼ D(ι) : D(X)[1/p] −→ D(X)[1/p] induced by ι. In addition, it satisfies the properties (ii) and (iii) of 2.6.2 if and only if (X, (ti )) is a p-divisible group with G-structure that arises from the datum (G, [b], {µ}). Hence the set RZG,b (F p ) classifies the isomorphism classes of tuples (X, (ti ), ι) where • (X, (ti )) is a p-divisible group over F p with G-structure; ∼ • ι : X −→ X is a quasi-isogeny such that the induced isomorphism D(X)[1/p] −→ D(X)[1/p] matches (ti ) with (ti ). By Proposition 2.4.7, we have a natural bijection ∼ G X{µ} ([b]) −→ RZG,b (F p ). Now we consider an F p -valued point x ∈ RZG,b (F p ). Let us write (Xx , (t x,i ), ι x ) for œ the corresponding tuple under the above classification, and (RZ G,b ) x for the formal completion of RZG,b at x. Then we have a natural isomorphism œ Def Xx ,G ≃ (RZ G,b ) x as explained in [Kim13], 4.8. Proposition 2.6.6 ([Kim13], Theorem 4.9.1.). Let (G0, [b0], {µ0 }) be another unramified local Shimura datum of Hodge type, and choose b0 ∈ [b0] ∩ G(Z̆ p )µ0(p)G(Z̆ p ) that gives rise to a p-divisible group over F p with G0-structure as in 2.4.3. (1) The natural morphism RZb ×Spf(Z̆ p ) RZb0 −→ RZ(b,b0) , defined by the product of p-divisible groups with quasi-isogeny, induces an isomorphism ∼ RZG,b ×Spf(Z̆ p ) RZG 0,b0 −→ RZG×G 0,(b,b0) 29 (2) For any homomorphism f : G −→ G0 with f (b) = b0, there exists an induced morphism RZG,b −→ RZG 0,b0, which is a closed embedding if f is a closed embedding. Moreover, via the natural bijections given in 2.6.5, the functorial properties (1) and (2) induce the functorial properties of affine Deligne-Lusztig sets and the formal deformation spaces described in Lemma 2.4.8 and Lemma 2.5.6. 2.6.7. We define a group valued functor Jb on the category of Q p -algebras by setting for any Q p -algebra R Jb (R) := {g ∈ G(R ⊗Q p Q̆ p ) : gbσ(g)−1 = b}. This functor is represented by an algebraic group over Q p which is an inner form of some Levi subgroup of GQ p (see [RZ96], Corollary 1.14.). The isomorphism class of Jb does not depend on the choice b ∈ [b] since any g ∈ G(Q̆ p ) induces an isomorphism Jb  Jgbσ(g)−1 via conjugation. Note that Jb (Q p ) can be identified with the group of quasi-isogenies γ : X −→ X that preserve the tensors (ti ). One can show that RZG,b carries a natural left Jb (Q p )-action defined by γ(X, ι) = (X, ι ◦ γ −1 ) for any R ∈ NilpZ̆ p , (X, ι) ∈ RZG,b (R) and γ ∈ Jb (Q p ) (see [Kim13], 7.2.). 2.6.8. Let E be the field of definition of the G(Q̆ p )-conjugacy class of µ, and let OE denote its ring of integers. Note that E is a finite unramified extension of Q p since GQ p is split over a finite unramified extension of Q p . Let d be the degree of the extension, and write τ for the Frobenius automorphism of Q̆ p relative to E. For any formal scheme S over Spf(Z̆ p ), we write S τ := S ×Spf(Z̆ p ),τ Spf(Z̆ p ). By ∼ a Weil descent datum on S over OE , we mean an isomorphism S −→ S τ . If S  S0 ×Spf(OE ) Spf(Z̆ p ) for some formal scheme S0 over Spf(OE ), then there exists a natural Weil descent datum on S over OE , called an effective Weil descent datum. For any R ∈ NilpZ̆ p , we define Rτ to be R viewed as a Z̆ p -algebra via τ. Note that we have a natural identification RZτb (R) = RZb (Rτ ). Following Rapoport and Zink in [RZ96], 3.48, we define a Weil descent datum Φ on RZb over OE by sending (X, ι) ∈ RZb (R) with R ∈ NilpZ̆ p to (X Φ, ιΦ ) ∈ RZb (Rτ ), where 30 • X Φ is X viewed as a p-divisible group over Rτ ; • ιΦ is the quasi-isogeny ι Frob−d Φ ιΦ : XRτ /p = (τ ∗ X) R/p −−−−−→ XR/p, −→ XR/p = XR/p where Frobd : X → τ ∗ X is the relative q-Frobenius with q = pd . One can check that Φ restricts to a Weil descent datum ΦG on RZG,b over OE by looking at F p -points and the formal completions thereof. The Weil descent datum ΦG clearly commutes with the Jb (Q p )-action defined in 2.6.7. 2.6.9. Since RZG,b is locally formally of finite type over Spf(Z̆ p ), it admits a rigid rig analytic generic fiber which we denote by RZG,b (see [Ber96].). The Jb (Q p )-action rig and the Weil descent datum ΦG on RZG,b induce an action of Jb (Q p ) on RZG,b and rig ∼ rig an Weil descent datum ΦG : RZG,b −→ (RZG,b )τ over E. Recall that we have a universal p-divisible group XG,b over RZG,b and a family of universal tensors (tiuniv ) on D(XG,b ). In addition, the family (tiuniv ) has a “étale univ ) on the Tate module T (X ) (see [Kim13], Theorem 7.1.6.). realization” (ti,ét p G,b For any open compact subgroup Kp of G(Z p ), we define the following rigid analytic rig étale cover of RZG,b :  . Kp univ )] Kp . RZG,b := IsomRZrig [Λ, (si )], [Tp (XG,b ), (ti,ét G,b rig K p The Jb (Q p )-action and the Weil descent datum over E on RZG,b pull back to RZG,b . K p As the level Kp varies, these covers form a tower {RZG,b } with Galois group G(Z p ). ∞ We denote this tower by RZG,b . ∞ By [Kim13], Proposition 7.4.8, there exists a right G(Q p )-action on the tower RZG,b extending the Galois action of G(Z p ), which commutes with the natural Jb (Q p )action and the Weil descent datum over E. In addition, there is a well-defined rig ∞ period map on RZG,b as explained in [Kim13], 7.5. Hence the tower RZG,b is a local Shimura variety in the sense of Rapoport and Viehmann in [RV14], 5.1. 2.6.10. We fix a prime l , p, and let WE denote the Weil group of E. For any level Kp ⊂ G(Z p ), we consider the cohomology groups K K K p p p ) = Hci (RZG,b ⊗Q̆ p C p, Ql (dim RZG,b )). H i (RZG,b 31 K p As the level Kp varies, these cohomology groups form a tower {H i (RZG,b )} for each i, endowed with a natural action of G(Q p ) × WE × Jb (Q p ). Let ρ be an admissible l-adic representation of Jb (Q p ). The groups j K p ∞ H i,j (RZG,b ) ρ := lim Ext J (Q ) (H i (RZG,b ), ρ) p b −−→ Kp satisfy the following properties (see [RV14], Proposition 6.1 and [Man08], Theorem 8): ∞ (1) The groups H i,j (RZG,b ) ρ vanish for almost all i, j. ∞ (2) There is a natural action of G(Q p ) × WE on each H i,j (RZG,b )ρ. ∞ (3) The representations H i,j (RZG,b ) ρ are admissible. Hence we can define a virtual representation of G(Q p ) × WE Õ ∞ ∞ H • (RZG,b ) ρ := (−1)i+ j H i,j (RZG,b )ρ. i,j≥0 32 Chapter 3 THE HODGE-NEWTON FILTRATION FOR HODGE-NEWTON REDUCIBLE LOCAL SHIMURA DATA In this chapter, we state and prove our results on the Hodge-Newton decomposition and the Hodge-Newton filtration in the setting of unramified local Shimura data of Hodge type. Throughout this chapter, k is assumed to be algebraically closed. 3.1 EL realization of Hodge-Newton reducibility 3.1.1. Let (G, [b], {µ}) be an unramified local Shimura datum of Hodge type. Choose a maximal torus T ⊆ G and a Borel subgroup B ⊆ G containing T, both defined over Z p . Let P be a proper standard parabolic subgroup of G with Levi factor L and unipotent radical U. We say that (G, [b], {µ}) is Hodge-Newton reducible (with respect to P and L) if there exist µ ∈ {µ} which factors through L and an element b ∈ [b] ∩ L(K0 ) which satisfy the following conditions: (i) [b] L ∈ B(L, {µ} L ), (ii) in the action of µ and νG ([b]) on Lie(U) ⊗Q p K0 , only non-negative characters occur. Since G is unramified, one can give an alternative definition in terms of some specific choice of b ∈ [b] ∩ L(K0 ) and µ ∈ {µ} (see [RV14], Remark 4.25.). Example 3.1.2. Consider the case G = ResO |Z p GLn , where O is the ring of integers of some finite unramified extension of Q p . Then L is of the form L = ResO |Z p GL j1 × ResO |Z p GL j2 × · · · × ResO |Z p GL jr . Recall from Example 2.4.5 that we have an identification N (G) = {(r1, r2, · · · , rn ) ∈ Qd : r1 ≤ r2 ≤ · · · ≤ rn }. 33 Using this, we may write νG ([b]) = (ν1, ν2, · · · , νn ) and µ̄ = (µ1, µ2, · · · , µr ). In this setting, the conditions (i) and (ii) in 3.1.1 are equivalent to the following conditions: (i’) ν1 + ν2 + · · · + ν jk = µ1 + µ2 + · · · + µ jk (ii’) ν jk < ν jk +1 for each k = 1, 2, · · · , r, for each k = 1, 2, · · · , r. In other words, (G, [b], {µ}) is Hodge-Newton reducible (with respect to P and L) if and only if the Newton polygon νG ([b]) and the σ-invariant Hodge polygon µ̄ have contact points which are break points of νG ([b]) specified by L. We refer the readers to [MV10], §3 for more details. 3.1.3. For the rest of this chapter, we fix an unramified local Shimura datum of Hodge type (G, [b], {µ}) which is Hodge-Newton reducible with respect to P and L. Let us also fix a faithful G-representation Λ ∈ RepZ p (G) in the condition (ii) of 2.4.1 and choose a finite family of tensors (si ) on Λ as in 2.4.3. Our strategy is to e of EL type such that the study (G, [b], {µ}) by embedding G into another group G e [b], {µ}) is also Hodge-Newton reducible. datum (G, Note that if G is not split, the datum (GL(Λ), [b], {µ}) may not be Hodge-Newton reducible in general. In fact, the map on the Newton sets N (G) −→ N (GL(Λ)) induced by the embedding G ,−→ GL(Λ) does not map µ̄G to the Hodge polygon µGL(Λ) since it does not respect the action of σ. e of EL type with the following properties: Lemma 3.1.4. There exists a group G e (i) the embedding G ,→ GL(Λ) factors through G. e [b], {µ}) is Hodge-Newton reducible with respect to a parabolic (ii) the datum (G, e( G e and its Levi factor e e ∩ G and L = e subgroup P L such that P = P L ∩ G. Proof. Write V := Λ ⊗Z p Qun where Qun is the maximal unramified extension of Q p in a fixed algebraic closure. We know that G is split over Qun for being unramified over Q p . Hence V admits a decomposition into character spaces Ê V= Vχ (3.1.4.1) χ∈X ∗ (T) with the property that σ(Vχ ) = Vσ χ . 34 For each χ ∈ X ∗ (T), let h χi denote the Ω-conjugacy class of χ and write Vh χi := ⊕ω∈ΩVω· χ . Since V is a G-representation, we can rewrite the decomposition (3.1.4.1) as Ê V= Vh χi , h χi∈X ∗ (T)/Ω where Vh χi ’s are sub G-representations (see [Se68], Theorem 4.) with the property that Vhσ χi = σ(Vh χi ). If a Ω-conjugacy class h χi ∈ X ∗ (T)/Ω is in an orbit of size m under the action of σ, the G-representation m−1 Ê Vhσi χi i=0 is also a ResE |Q p GLn -representation where E is the field of definition of h χi, which is a degree m unramified extension of Q p (cf. (2.4.5.1) in Example 2.4.5). Hence the Î embedding GQ p ,→ GL(ΛQ p ) factors through a group of the form ResE j |Q p GLn j where each E j is the field of definition of an orbit in X ∗ (T)/Ω. Then by [Se68], Theorem 5, we can take the pull-back of this embedding over Z p to obtain Ö G ,−→ ResO j |Z p GLn j ,−→ GL(Λ) where O j is the ring of integers of E j . We take e := G Ö ResO j |Z p GLn j . e T) e of G e such that B ⊆ B e and T ⊆ T. e Then we get a proper Choose a Borel pair ( B, e( G e with Levi factor e e ∩ G and standard parabolic subgroup P L such that P = P L=e L ∩ G (e.g. by using [SGA3], Exp. XXVI, Cor. 6.10.). e respects the action It is evident from the construction that the embedding G ,−→ G e of σ on cocharacters. Hence the induced map on the Newton sets N (G) −→ N (G) maps µ̄G to the σ-invariant Hodge polygon µ̄Ge. Combining this fact with the functoriality of the Kottwitz map and the Newton map, we verify that the datum e [b], {µ}) is Hodge-Newton reducible with respect to P e and e (G, L.  e [b], {µ}) in Lemma 3.1.4 as an EL realization of the We will refer to the datum (G, Hodge-Newton reducible datum (G, [b], {µ}). e = GL(Λ). Remark. If G is split, the construction in the proof above yields G 35 3.2 The Hodge-Newton decomposition and the Hodge-Newton filtration e [b], {µ}) of our datum (G, [b], {µ}), and take P e and 3.2.1. Fix an EL realization (G, e L as in Lemma 3.1.4. In a view of the functorial properties in Lemma 2.4.2, Lemma 2.4.8, Lemma 2.5.6 and Proposition 2.6.6, we will always assume for simplicity that e is of the form G e := ResO |Z p GLn, G where O is the ring of integers of some finite unramified extension E of Q p . Then e L is of the form e L = ResO |Z p GL j1 × ResO |Z p GL j2 × · · · × ResO |Z p GL jr . (3.2.1.1) Let us now choose b ∈ [b] ∩ L(K0 ) and µ ∈ {µ} as in (i) of 3.1.1. After taking σ-conjugate in L(K0 ) if necessary, we may assume that b ∈ L(W)µ(p)L(W). Let M = (M, (ti )) be the corresponding F-crystal over k with G-structure (in the sense of 2.4.3). If {µ} is minuscule, we let X = (X, (ti )) denote the corresponding p-divisible group over k with G-structure. Note that the tuple (L, [b] L , {µ} L ) is an unramified local Shimura datum of Hodge type; indeed, with our choice of b ∈ [b] L and µ ∈ {µ} L one immediately verifies the conditions (i’) and (ii’) of 2.4.1. Theorem 3.2.2. Notations as above. In addition, we set the following notations: • e L j denotes the j-th factor in (3.2.1.1), • L j is the image of L under the projection e Le Lj, • b j is the image of b under the projection L  L j , • µ j is the cocharacter of L j obtained by composing µ with the projection L  Lj. Then M can be naturally regarded as an F-crystal with L1 × L2 × · · · × Lr -structure, and admits a decomposition M = M 1 × M 2 × · · · × Mr, (3.2.2.1) 36 where M j is an F-crystal with L j -structure that arises from an unramified local Shimura datum of Hodge type (L j , [b j ], {µ j }). When {µ} is minuscule, we also have a decomposition X = X 1 × X 2 × · · · × Xr, (3.2.2.2) where X j is a p-divisible group with L j -structure corresponding to M j . Proof. We need only prove the first part, as the second part follows immediately from the first part via Dieudonné theory. We first note that M has a natural L1 × L2 × · · · × Lr -structure as follows: our choice of b ∈ [b] L and µ ∈ {µ} L gives rise to an L-structure on M, which can be regarded as an L1 × L2 × · · · × Lr -structure via the embedding L ,→ L1 × L2 × · · · × Lr . e Now considering b as an element of [b]Ge, we get an F-crystal over k with Ge from an unramified local Shimura datum of Hodge type (G, e [b], {µ}). structure M e As explained in Example 2.4.5, we can regard the G-structure as an action of O e [b], {µ}) is Hodge-Newton which we refer to as O-module structure. Since (G, reducible, [MV10], Corollary 7 yields a decomposition e=M e1 × M e2 × · · · × M er , M (3.2.2.3) ej is an F-crystal over k with O-module structure which arises from an unwhere M ej corresponds ramified local Shimura datum of Hodge type (e L j , [b j ], {µ j }). In fact, M to the choice b j ∈ [b j ] (and µ j ∈ {µ j }). A priori, it is not clear that the tuple (e L j , [b j ], {µ j }) is an unramified local Shimura datum of Hodge type. This is indeed implied in the statement and the proof of [MV10], Corollary 7. We check that the tuple (L j , [b j ], {µ j }) is an unramified local Shimura datum of Hodge type by verifying the conditions (i’) and (ii’) of 2.4.1. For (i’), we simply observe that b j ∈ L j (W)µ(p)L j (W), which follows from our assumption that b ∈ L(W)µ(p)L(W) using the decomposition e L=e L1 × e L2 × · · · × e Lr . Then the condition (ii’) immediately follows since we already know that M j gives the desired W-lattice for b j and µ j . Since (L j , [b j ], {µ j }) is an unramified local Shimura datum of Hodge type, we can equip each M j with an L j -structure corresponding to the choice b j ∈ [b j ] (and µ j ∈ {µ j }). We thus get the desired decomposition (3.2.2.1) from the decomposition (3.2.2.3).  37 Remark. We give an alternative proof of Theorem 3.2.2 using affine DeligneLusztig sets. After proving that the tuples (L j , [b j ], {µ j }) are unramified local Shimura data of Hodge type, we find the following maps of affine Deligne-Lusztig sets: ∼ L1 L2 Lr G L X{µ} ([b]) −→ X{µ} ([b]) ,→ X{µ ([b1 ]) × X{µ ([b2 ]) × · · · × X{µ ([br ]). } } } 1 2 r Here the first isomorphism is given by [MV10], Theorem 6, whereas the second map is induced by the embedding L ,→ L1 × L2 × · · · × Lr as in Lemma 2.4.8. Now the desired decomposition follows from the composition of these two maps via the moduli interpretation of affine Deligne-Lusztig sets given in Proposition 2.4.7. 3.2.3. We will refer to the decomposition (3.2.2.1) in Theorem 3.2.2 as the HodgeNewton decomposition of M (associated to P and L). For 1 ≤ a ≤ b ≤ r, we define b Ö Ms . Ma,b := s=a Then we obtain a filtration 0 ⊂ M1,1 ⊂ M1,2 ⊂ · · · ⊂ M1,r = M (3.2.3.1) such that each quotient M1,s /M1,s−1 ≃ Ms carries an Ls -structure. We call this filtration the Hodge-Newton filtration of M (associated to P and L). When {µ} is minuscule, we will refer to the decomposition (3.2.2.2) in Theorem 3.2.2 as the Hodge-Newton decomposition of X (associated to P and L). For 1 ≤ a ≤ b ≤ r, we define b Ö Xa,b := Xs . s=a Then via (contravariant) Dieudonné theory, the filtration (3.2.3.1) yields a filtration 0 ⊂ Xr,r ⊂ Xr−1,r ⊂ · · · ⊂ X1,r = X, (3.2.3.2) where each quotient Xs,r /Xs+1,r ≃ Xs carries an Ls -structure. We call this filtration the Hodge-Newton filtration of X (associated to P and L). Theorem 3.2.4. Assume that p > 2 and {µ} is minuscule. Let R to be a ring of the form R = W[[u1, · · · , u N ]] or R = W[[u1, · · · , u N ]]/(pm ). Let X be a deformation of X over R with an isomorphism α : X ⊗ R k  X. Then there exists a unique filtration of X 0 ⊂ Xr,r ⊂ Xr−1,r ⊂ · · · ⊂ X1,r = X 38 which lifts the Hodge-Newton filtration (3.2.3.2) in the sense that α induces isomorphisms Xs,r ⊗ R k  Xs,r and Xs,r /Xs+1,r ⊗ R k  X s for s = 1, 2, · · · , r. Note that we require each quotient Xs,r /Xs+1,r to carry tensors that lift those on X s . Proof. We will only consider the case r = 2 as the argument easily extends to the general case. Take unramified local Shimura data of Hodge type (L j , [b j ], {µ} j ) and (e L j , [b j ], {µ j }) e be the p-divisible group over k with Oas in Theorem 3.2.2. In addition, let X e [b], {µ}) with the choice b ∈ [b], module structure that arises from the datum (G, ej be the p-divisible group over k with O-module structure that arises from and let X the datum (e L j , [b j ], {µ j }) with the choice b j ∈ [b j ]. Then the filtration e2 ⊆ X e 0⊆X e is the Hodge-Newton filtration of X. By the functorial properties of deformation spaces in Lemma 2.5.6, the closed e induces a closed embedding embedding G ,−→ G Def X,G ,−→ Def X,Ge. f of X f e over R. Then by [Sh13], Theorem 5.4, X Thus X yields a deformation X admits a (unique) filtration 0 ⊆ X2 ⊆ X Ÿ f2 ⊗ R k  X e1 and α2 : X e2 . such that α induces isomorphisms α1 : X /X2 ⊗ R k  X It remains to show that X /X2 and X2 are equipped with tensors which lift the tensors of X 1 and X 2 respectively in the sense of Proposition 2.5.5. Note that we have isomorphisms of Dieudonné modules β : D(X ⊗ R k)  D(X), β1 : D((X /X2 )⊗ R k)  D(X1 ), β2 : D(X2 ⊗ R k)  D(X2 ) corresponding to the isomorphisms α, α1 and α2 . We may regard β as an element of G(W) by identifying both modules with Λ∗ ⊗Z p W. Similarly, we may regard each β j as an element of e L j (W). Then β j should be in the image of e L(W) ∩ G(W) = L(W) under the projection e Le L2 since it is induced by β. Hence we have β j ∈ L j (W) for each j = 1, 2. This implies that X /X2 and X2 respectively lift the tensors of X 1 and X 2 via α1 and α2 , completing the proof.  39 Chapter 4 HARRIS-VIEHMANN CONJECTURE FOR HODGE-NEWTON REDUCIBLE RAPOPORT-ZINK SPACES 4.1 Harris-Viehmann conjecture: statement 4.1.1. Throughout this chapter, we fix a prime p > 2 and set k = F p so that W = Z̆ p and K0 = Q̆ p . We also fix an unramified local Shimura datum of Hodge type (G, [b], {µ}) such that {µ} is minuscule. We choose a faithful G-representation Λ ∈ RepZ p (G) in the condition (ii) of 2.4.1 and a finite family of tensors (si ) on Λ as in 2.4.3. In addition, we fix a maximal torus T ⊆ G and a Borel subgroup B ⊆ G containing T, both defined over Z p . Let P be a proper standard parabolic subgroup of G with Levi factor L and unipotent radical U. For any element b ∈ [b] ∩ L(Q̆ p ), we define Ib,{µ},L to be the set of L(Z̆ p )conjugacy classes of cocharacters of L with a representative µ0 such that (i) µ0 ∈ {µ}G , (ii) [b] L ∩ L(Z̆ p )µ0(p)L(Z̆ p ) is not empty. Then Ib,{µ},L is finite and nonempty (see [RV14], Lemma 8.1.). Lemma 4.1.2. For any {µ0 } L ∈ Ib,{µ},L , the tuple (L, [b] L , {µ0 } L ) is an unramified local Shimura datum of Hodge type. Proof. By construction, the tuple (L, [b] L , {µ0 } L ) satisfies the conditions (i’) of 2.4.1. After taking σ-conjugate in L(Q̆ p ) if necessary, we may assume that b ∈ L(Z̆ p )µ0(p)L(Z̆ p ). Then we have b ∈ G(Z̆ p )µ(p)G(Z̆ p ) since µ0 ∈ {µ}G . Now we verify the condition (ii’) with b since (G, [b], {µ}) is an unramified local Shimura datum of Hodge type.  We can now state the Harris-Viehmann conjecture in the setting of Rapoport-Zink spaces of Hodge type. Conjecture 4.1.3 ([RV14], Conjecture 8.4.). Let P be a parabolic subgroup of G with Levi factor L. Assume that b ∈ [b] ∩ G(Z̆ p )µ(p)G(Z̆ p ) satisfies the following properties: 40 (i) [b] ∩ L(Q̆ p ) is not empty, (ii) Jb is an inner form of a Levi subgroup of G contained in L. Choose representatives µ1, µ2, · · · , µs of the L(Z̆ p )-conjugacy classes of cocharacters in Ib,{µ},L , and also choose b k ∈ [b] L ∩ L(Z̆ p )µ k (p)L(Z̆ p ) for each k = 1, 2, · · · , s. Then for any admissible Ql -representation ρ of Jb (Q p ), we have an equality of virtual representations of G(Q p ) × WE ∞ H (RZG,b )ρ = • s Ê G(Q ) IndP(Q p) H • (RZ∞ L,bk ) ρ . p k=1 ∞ In particular, the virtual representation H • (RZG,b ) ρ contains no supercuspidal representations of G(Q p ). Here we consider the groups H • (RZ∞ L,bk ) ρ as a virtual representation of P(Q p ) × WE by letting the unipotent radical of P(Q p ) act trivially. Note that the choice of b k ’s (or µ k ’s) is unimportant since the isomorphism class of the spaces RZ∞ L,bk only depend on the tuples (L, [b] L , {µ k } L ). 4.1.4. For the rest of this chapter, we assume that the datum (G, [b], {µ}) is HodgeNewton reducible with respect to P and L. We also choose µ ∈ {µ} and b ∈ L(Z̆ p )µ(p)L(Z̆ p ) as in 3.2.1, and denote by X = (X, (ti )) the corresponding pdivisible group over F p with G-structure. Let us interpret the statement of Conjecture 4.1.3 under our assumption. Note that b and L clearly satisfy the condition (i) of Conjecture 4.1.3. One can also check that b and L satisfy the condition (ii) of Conjecture 4.1.3 (see [RV14], Remark 8.9.). Moreover, under our assumption the set Ib,{µ},L consists of a single element, namely {µ} L (see [RV14], Theorem 8.8.). We now state the main result for this chapter, which proves Conjecture 4.1.3 under our assumption. Theorem 4.1.5. Assume that (G, [b], {µ}) is Hodge-Newton reducible with respect to a standard parabolic subgroup P with Levi factor L. Choose µ ∈ {µ} and b ∈ L(Z̆ p )µ(p)L(Z̆ p ) as in 3.2.1. Then for any admissible Ql -representation ρ of Jb (Q p ), we have an equality of virtual representations of G(Q p ) × WE G(Q ) ∞ H • (RZG,b ) ρ = IndP(Q p) H • (RZ∞ L,b ) ρ . p 41 ∞ In particular, the virtual representation H • (RZG,b ) ρ contains no supercuspidal representations of G(Q p ). 4.2 Rigid analytic tower associated to the parabolic subgroup For our proof of Theorem 4.1.5, we construct an intermediate tower of rigid analytic spaces associated to the parabolic subgroup P. e [b], {µ}) of our datum 4.2.1. For the rest of this chapter, we fix an EL realization (G, e and e e is (G, [b], {µ}) and take P L as in Lemma 3.1.4. We continue to assume that G of the form e = ResO |Z p GLn, G where O is the ring of integers of some finite unramified extension of Q p . We also take e L j , L j , b j , µ j for j = 1, 2, · · · , r as in Theorem 3.2.2. Then Theorem 3.2.2 gives a Hodge-Newton decomposition of X X = X1 × X2 × · · · × Xr (4.2.1.1) and the induced Hodge-Newton filtration of X 0 ⊂ X (r) ⊂ X (r−1) ⊂ · · · ⊂ X (1) = X, (4.2.1.2) where each quotient X ( j) /X ( j+1) ≃ X j carries L j -structure that arises from the datum (L j , [b j ], {µ j }) with the choice b j ∈ [b j ]. 4.2.2. Following Mantovan in [Man08], Definition 9, we define a set-valued functor RZP,b e on NilpZ̆ p as follows: for any R ∈ NilpZ̆ p , we set RZP,b e (R) to be the set of isomorphism classes of triples (X, X •, ι) where • X is a p-divisible group over R with an action of O (see Example 2.4.5); • X • is a filtration of p-divisible groups over R 0 ⊂ X (r) ⊂ X (r−1) ⊂ · · · ⊂ X (1) = X, which is preserved by the action of O such that the quotients X ( j) /X ( j+1) are p-divisible groups (with the induced action of O); 42 • ι : XR/p → XR/p is a quasi-isogeny which is compatible with the action of O ( j) ( j) and induces quasi-isogenies ι( j) : XR/p −→ XR/p for j = 1, 2, · · · , r, such that for all a ∈ O and j = 1, 2, · · · , r, det R (a, Lie(X ( j) )) = det(a, Fil0 (X ( j) )Q̆ p ). Mantovan in [Man08], Proposition 11 proved that the functor RZP,b e is represented by a formal scheme which is formally smooth and locally formally of finite type rig over Z̆ p . We write RZP,b e also for this representing formal scheme, and RZ e for P,b • respectively its rigid analytic generic fiber. In addition, we write XP,b e and XP,b e for the universal filtered p-divisible group over RZP,b e and the associated “universal filtration”. Remark. As in0 [Man08], Definition 10, we can also define a tower of étale covers fp K rig e RZ∞ e = {RZ e} over RZ e with a natural action of P(Q p ) × Jb (Q p ) and a Weil P,b X,P P,b fp0 runs over open and compact subgroups of P(Z e p ). descent datum over E, where K 4.2.3. By the functoriality of Rapoport-Zink spaces described in Proposition 2.6.6, e induces a closed embedding the embedding G ,−→ G RZG,b ,−→ RZG,b e . In addition, we have a natural map e π2 : RZP,b e e −→ RZG,b defined by (X, X •, ι) 7→ (X, ι) on the points. We define RZP,b := RZP,b RZG,b . e ×RZG,b e Then we have the following Cartesian diagram: RZP,b RZP,b e π2 e π2 RZG,b RZG,b e Moreover, π2 is a local isomorphism which gives an isomorphism on the rigid analytic generic fiber since e π2 has the same property (see [Man08], Theorem 36 and [Sh13], Proposition 6.3.). 43 We want to describe the universal property of the closed embedding RZP,b ,−→ RZP,b e in an analogous way to the universal property of RZG,b ⊂ RZb described in 2.6.2. For this, we choose a decomposition of Λ Λ = Λ1 ⊕ Λ2 ⊕ · · · ⊕ Λr corresponding to the decomposition of e L in (3.2.1.1). We set Λ( j) = Λ1 ⊕ · · · ⊕ Λ j for j = 1, 2, · · · , r, and denote by Λ• the filtration 0 ⊂ Λ(1) ⊂ · · · ⊂ Λ(r) = Λ. Then for any Z p -algebra R we have P(R) = {g ∈ G(R) : g(Λ•R ) = Λ•R }. • Now consider a morphism f : Spf(R) → RZP,b e for some R ∈ NilpZ̆ p . Let (X, X ) be ∗ • ) a p-divisible group over Spec(R) with a filtration which pulls back to ( f ∗ XP,b e , f XP,b e • over Spf(R). We denote by D(X ) the filtration of Dieudonné modules 0 = D(X/X (1) ) ⊂ D(X/X (2) ) ⊂ · · · ⊂ D(X/X (r) ) ⊂ D(X) induced by X • via (contravariant) Dieudonne theory. We choose tensors (tˆi ) on D(X)[1/p] as in 2.6.2. Then f factors through RZP,b if and only if e π2 ◦ f factors through RZG,b ,−→ RZG,b e , which is equivalent to existence of a (unique) family of tensors (ti ) on D(X) such that (i) for some ideal of definition J of R containing p, the pull-back of (ti ) over R/J agrees with the pull-back of (tˆi ) over R/J, (ii) for a p-adic lift R of R which is formally smooth over Z̆ p , the R-scheme   PR := IsomR [D(X)R , (ti )R ], [Λ∗ ⊗Z p R, (si ⊗ 1)] defined in 2.6.2 is a G-torsor, and consequently the R-scheme   PR0 := IsomR [D(X • )R , (ti )R ], [(Λ• )∗ ⊗Z p R, (si ⊗ 1)] is a P-torsor, (iii) the Hodge filtration of X is a {µ}-filtration with respect to (ti ). 44 Here the scheme PR0 in (ii) classifies the isomorphisms D(X)R  Λ∗R which map the tensors (ti ) to (si ⊗ 1) and the filtration D(X • )R to (Λ• )∗ ⊗Z p R. We obtain the “universal p-divisible group" XP,b over RZP,b with the associated • by taking the pull-back of X • “universal filtration” XP,b e and X e over RZP,b . P,b P,b univ,P We also obtain a family of “universal tensors" (ti ) on D(XP,b ) by applying the universal property to an open affine covering of RZP,b . Moreover, this family has univ,P a “étale realization” (ti,ét ) on the Tate module Tp (XP,b ) (see [Kim13], Theorem 7.1.6.). 4.2.4. The formal scheme RZP,b is formally smooth and locally formally of finite type over Z̆ p by construction. Hence it admits a rigid analytic generic fiber, which rig we denote by RZP,b . Moreover, since π2 gives an isomorphism on the rigid analytic rig generic fiber, we have a Jb (Q p )-action and a Weil descent datum over E on RZP,b rig induced by the corresponding structures on RZG,b . For any open compact subgroup Kp0 of P(Z p ), we define the following rigid analytic rig étale cover of RZP,b :  . K 0 univ,P • RZP,bp := IsomRZrig [Λ•, (si )], [Tp (XP,b ), (ti,ét )] Kp0 . P,b rig K 0 The Jb (Q p )-action and the Weil descent datum over E on RZP,b pull back to RZP,bp . K 0 p We denote by RZ∞ P,b := {RZP,b } the tower of these covers with Galois group P(Z p ). The Galois action on this tower gives rise to a natural P(Q p )-action which commutes with the Jb (Q p )-action and the Weil descent datum over E (cf. [Kim13], Proposition 7.4.8.). Hence the cohomology groups K 0 K 0 K H i (RZP,bp ) = Hci (RZP,bp ⊗Q̆ p C p, Ql (dim RZP,bp )) K 0 form a tower {H i (RZP,bp )} for each i, which are endowed with a natural action of P(Q p ) × WE × Jb (Q p ). Moreover, for any admissible l-adic representation ρ of Jb (Q p ), the groups j K 0 H i,j (RZ∞ Ext J (Q ) (H i (RZP,bp ), ρ) P,b ) ρ := lim p b −−→0 Kp satisfy the following properties (cf. 2.6.10): (1) The groups H i,j (RZ∞ P,b ) ρ vanish for almost all i, j. 45 (2) There is a natural action of P(Q p ) × WE on each H i,j (RZ∞ P,b ) ρ . (3) The representations H i,j (RZ∞ P,b ) ρ are admissible. We can thus define a virtual representation of P(Q p ) × WE Õ • ∞ H (RZP,b ) ρ := (−1)i+ j H i,j (RZ∞ P,b ) ρ . i,j≥0 Remark. Alternatively, we can obtain the tower RZ∞ P,b as the pull-back of the tower rig ∞ RZ e over RZP,b . P,b 4.3 Harris-Viehmann conjecture: proof Let us now present our proof of Theorem 4.1.5. We retain all the notations from 4.2. Lemma 4.3.1. There exists a diagram rig RZP,b s π1 rig RZ L,b π2 rig RZG,b such that (i) s is a closed immersion, (ii) π1 is a fibration in balls, (iii) π2 is an isomorphism. Proof. For notational simplicity, we assume that r = 2, i.e., the decomposition of e L in (3.2.1.1) has two factors. Our argument will naturally extend to the general case. Note that we have already constructed π2 and proved (iii) in 4.2.3. 46 Let us now construct s and prove (i). From the decomposition e L=e L1 × e L2 we obtain a natural isomorphism RZeL,b ≃ RZeL1,b1 × RZeL2,b2 by Proposition 2.6.6. Consider the map e s : RZeL,b ≃ RZeL1,b1 × RZeL2,b2 −→ RZP,b e , where the second arrow is defined by (X1, ι1, X2, ι2 ) 7→ (X1 × X2, 0 ⊂ X2 ⊂ X1 × X2, ι1 × ι2 ) on the points. Then e s gives a closed immersion on the rigid analytic generic fibers by [Man08], Proposition 14. We define s to be the restriction of e s on RZ L,b . Since s also gives a closed immersion on the rigid analytic generic fibers by construction, it suffices to show that s factors through the embedding RZP,b ,−→ RZP,b π2 ◦ s factors through RZG,b . In e , which amounts to proving that e fact, e π2 ◦ s is the natural closed embedding RZ L,b ,−→ RZG,b e which is functorially e in the sense of Proposition 2.6.6. Hence e induced by the embedding L ,−→ G π2 ◦ s e factors through G. factors through RZG,b as the embedding L ,−→ G It remains to construct π1 and prove (ii). Note that we have a natural embedding RZ L,b ,−→ RZeL,b which is functorially induced by the embedding L ,−→ e L in the sense of Proposition 2.6.6. Consider the map ∼ e π1 : RZP,b e −→ RZe L1,b1 × RZe L2,b2 −→ RZe L,b defined by (X, X •, ι) 7→ (X/X (2), ι/ι(2), X (2), ι(2) ) 7→ ((X/X (2) ) × X (2), (ι/ι(2) ) × ι(2) ) on the points, where ι/ι(2) : (X1 ) R/p = (X/X (2) ) R/p −→ (X/X (2) ) R/p denotes the quasi-isogeny induced by ι and ι(2) . We define π1 be the restriction of e π1 on RZP,b . We claim that π1 factor through the embedding RZ L,b ,−→ RZeL,b . It suffices to show that (locally) the map π2−1 ◦ π1 factors through RZ L,b ,−→ RZeL,b . Moreover, we only need to check this on the set of F p -valued points and the completions thereof. Recall that we have natural identifications ∼ G X{µ} ([b]) −→ RZG,b (F p ), ∼ L X{µ} ([b]) −→ RZ L,b (F p ) as explained in 2.6.5. Under these identifications, π2−1 ◦ π1 on the F p -valued points coincides with the map ∼ L1 L2 G L X{µ} ([b]) −→ X{µ} ([b]) ,→ X{µ ([b1 ]) × X{µ ([b2 ]) } } 1 2 induced by the Hodge-Newton decompositon of X as explained in the remark following Theorem 3.2.2. Hence we see that π2−1 ◦ π1 factors through RZ L,b on the set of F p -valued points. 47 We now take an F p -valued point x ∈ RZG,b (F p ) and consider the map induced œ by π2−1 ◦ π1 on the formal completion (RZ G,b ) x at x. We denote by (Xx , (t x,i ), ι x ) the tuple corresponding to x under the description of RZG,b (F p ) in 2.6.5. Then
X x := Xx , (t x,i ) admits a Hodge-Newton decomposition X x = X x,1 × X x,2 and the induced Hodge-Newton filtration 0 ⊂ Xx(2) ⊂ Xx(1) = Xx . Take R to be a ring of the form R = Z̆ p [[u1, · · · , u N ]] or R = Z̆ p [[u1, · · · , u N ]]/(pm ). Then for any X ∈ Def Xx ,G (R) with an isomorphism α : X ⊗ R k  X x , Theorem 3.2.4 gives a filtration 0 ⊂ X (2) ⊂ X (1) = X with isomorphisms α1 : X x /X x(2) ⊗ R F p  X x,1 and α(2) : X x(2) ⊗ R F p  Xx(2) . Under the identifications œ Def Xx ,G ≃ (RZ G,b ) x , œ Def Xx ,L ≃ (RZ L,b ) x described in 2.6.5, the map π2−1 ◦ π1 gives rise to a map Def Xx ,G −→ Def Xx ,eL1 × Def Xx ,eL2 ≃ Def Xx ×Xx ,eL1 ×eL2 = Def Xx ,eL 1 2 1 2 induced by the association X x 7→ (X x /X x(2) ) × X x(2) . Note that (X x /X x(2) ) × X x(2) is a deformation of Xx via the isomorphism α1 × α(2) . Since this isomorphism is induced by α, we see that (X x /X x(2) ) × X x(2) lifts the tensors that define G-structure on Xx . Hence the image of this map must lie in Def Xx ,eL ∩ Def Xx ,G = Def Xx ,L . Finally, we easily see that π1 is a fibration in balls. In fact, for any point x ∈ RZ L,b (F p ) the completion of RZP,b at s(x) is isomorphic to Def Xx ,G , which is isomorphic to a formal spectrum of a power series ring over Z̆ p by Proposition 2.5.5.  Proposition 4.3.2. For any admissible l-adic representation ρ of Jb (Q p ), we have • ∞ H • (RZ∞ L,b ) ρ = H (RZP,b ) ρ as virtual representations of P(QP ) × WE . 48 Proof. For any open compact subgroups Kp0 ⊆ P(Z p ), we get morphisms of rigid analytic spaces K 0 ∩L(Q p ) sKp 0 : RZ L,bp K 0 −→ RZP,bp and K 0 K 0 ∩L(Q p ) π1,Kp 0 : RZP,bp −→ RZ L,bp which are P(Q p ) × Jb (Q p )-equivariant and compatible with the Weil descent datum. Moreover, sKp 0 ’s are closed immersions and satisfy π1,Kp 0 ◦ sKp 0 = id K p 0∩L(Q p ) . RZ L,b Recall that we have a universal p-divisible group XP,b e over RZP,b e with the associated • −→ filtration X e . By [Man08], Proposition 30, we have a formal scheme RZ(m) e P,b P,b RZP,b e for each integer m > 0 with the following properties: (m) (i) a morphism f : Spf(R) −→ RZP,b e for some R ∈ NilpZ̆ p factors through RZ e P,b if and only if the filtration f ∗ X e• [pm ] is split, P,b (ii) the formal schemes RZ(m) and RZP,b e become isomorphic when considered as e P,b π1 : RZP,b formal schemes over RZeL,b via the map e e −→ RZe L,b . over RZP,b , we obtain a formal scheme RZ(m) −→ Taking the pull back of RZ(m) P,b e P,b (m),rig RZP,b for each integer m > 0 with analogous properties. We write RZP,b for the rigid analytic generic fiber of RZ(m) . P,b
For each integer m > 0, we set Kp0(m) := ker P(Z p )  P(Z p /pm Z p ) and define two distinct covers Pm −→ RZ(m) and Pm0 −→ RZ(m) by the following Cartesian P,b P,b diagrams: Pm (m),rig RZP,b (m),rig Pm0 RZP,b π1 K 0(m) RZP,bp rig RZP,b K 0(m) rig RZ L,bp RZ L,b Since π1 is a fibration in balls, we obtain quasi-isomorphisms K 0(m) RΓc (Pm0 ⊗Q̆ p C p, Ql )  RΓc (RZ L,bp ⊗Q̆ p C p, Ql (−D))[−2D] for all m > 0, where D = dim RZP,b − dim RZ L,b . Moreover, we can argue as in [Man08], Lemma 31 and Proposition 32 to deduce quasi-isomorphisms K 0(m) RΓc (RZP,bp ⊗Q̆ p C p, Ql )  RΓc (Pm0 ⊗Q̆ p C p, Ql ) for all m > 0. 49 Thus we have quasi-isomorphisms K 0(m) RΓc (RZP,bp K 0(m) ⊗Q̆ p C p, Ql )  RΓc (RZ L,bp ⊗Q̆ p C p, Ql (−D))[−2D] which yield the desired equality. for all m > 0,  Proposition 4.3.3. For any admissible l-adic representation ρ of Jb (Q p ), we have G(Q ) ∞ H • (RZG,b ) ρ = IndP(Q p) H • (RZ∞ P,b ) ρ p as virtual representations of P(QP ) × WE . Proof. For any open compact subgroup Kp ⊆ G(Z p ), we have natural morphisms of rigid analytic spaces K ∩P(Q p ) π2,Kp : RZP,bp K p −→ RZG,b which are P(Q p ) × Jb (Q p )-equivariant and compatible with the Weil descent datum. Moreover, these maps are evidently closed immersions. Hence we have isomorphisms Þ Kp Kp K ∩P(Q p ) rig RZG,b  RZG,b ×RZrig RZP,b  RZP,bp for all Kp ⊆ G(Z p ), G,b K p \G(Q p )/P(Q p ) thereby obtaining the desired identity. Proposition 4.3.2 and 4.3.3 together imply Theorem 4.1.5.  50 Chapter 5 SERRE-TATE DEFORMATION THEORY FOR LOCAL SHIMURA DATA OF HODGE TYPE Our goal for this chapter is to establish a generalization of Serre-Tate deformation theory for p-divisible groups that arise from µ-ordinary local Shimura data of Hodge type. There are two main ingredients for our theory, namely (a) existence of a “slope filtration” which admits a unique lifting over deformation rings; (b) existence of a “canonical deformation”. We prove (a) by applying Theorem 3.2.2 and Theorem 3.2.4 to µ-ordinary local Shimura data of Hodge type. To prove (b), we first embed our deformation space into a deformation space that arises from an EL realization of our local Shimura datum (cf. the proof of Theorem 3.2.4), then use the existence of a canonical deformation in the latter space proved by Moonen in [Mo04]. Throughout this section, we fix a prime p > 2 and assume that k is algebraically closed. 5.1 The slope filtration of µ-ordinary p-divisible groups 5.1.1. Let us first fix some notations for this chapter. We fix a µ-ordinary unramified local Shimura datum of Hodge type (G, [b], {µ}). We assume that {µ} is minuscule, and take a unique dominant representative µ ∈ {µ}. Then we have [b] = [µ(p)] by definition of µ-ordinariness, so we may take b = µ(p) and write X for the p-divisible group over k with G-structure that arises from this choice b ∈ [b] ∩ G(W)µ(p)G(W). Let m be a positive integer such that σ m (µ) = µ, and take L to be the centralizer of m · µ̄ in G which is a Levi subgroup (see [SGA3], Exp. XXVI, Cor. 6.10.). We set P to be a proper standard parabolic subgroup of G with Levi factor L. 51 5.1.2. One can check that (G, [b], {µ}) is Hodge-Newton reducible with respect to P and L (see [Wo13], Proposition 7.4.). Hence Theorem 3.2.2 gives us the Hodge-Newton decomposition associated to P and L X = X1 × X2 × · · · × Xr (5.1.2.1) which we call the slope decomposition of X. If we set Xa,b := b Ö Xs s=a for 1 ≤ a ≤ b ≤ r, we obtain the induced Hodge-Newton filtration 0 ⊂ Xr,r ⊂ Xr−1,r ⊂ · · · ⊂ X1,r = X, (5.1.2.2) which we refer to as the slope filtration of X. Now Theorem 3.2.4 readily gives us the first main ingredient of the theory, namely the unique lifting of the slope filtration. Proposition 5.1.3. Let R be a W-algebra of the form R = W[[u1, · · · , u N ]] or R = W[[u1, · · · , u N ]]/(pm ). Let X be a deformation of X over R with an isomorphism α : X ⊗ R k  X. Then there exists a unique filtration of X 0 ⊂ Xr,r ⊂ Xr−1,r ⊂ · · · ⊂ X1,r = X which lifts the slope filtration (5.1.2.2) in the sense that α induces isomorphisms Xs,r ⊗ R k  Xs,r and Xs,r /Xs+1,r ⊗ R k  X s for s = 1, 2, · · · , r. Proof. This is an immediate consequence of Theorem 3.2.4. 5.2  The canonical deformation of µ-ordinary p-divisible groups 5.2.1. We now aim to find the canonical deformation X can of X over W, which has the property that all endomorphisms of X lifts to endomorphisms of X can . When G is of EL type, we already know existence of such a deformation thanks to the work of Moonen in [Mo04]. Our strategy is to deduce the existence of X can from Moonen’s result by means of an EL realization of the datum (G, [b], {µ}). The following lemma is crucial for our strategy. e [b], {µ}) be an EL realization of the datum (G, [b], {µ}). Then Lemma 5.2.2. Let (G, e [b], {µ}) is µ-ordinary. (G, 52 Proof. Consider the map on the Newton sets e N (G) −→ N (G) e It maps µ̄G to µ̄ e by the proof of Lemma induced by the embedding G ,−→ G. G 3.1.4, and νG ([b]) to νGe([b]) by the functoriality of the Newton map. On the other hand, we have νG ([b]) = µ̄G since (G, [b], {µ}) is µ-ordinary. Hence we deduce that νGe([b]) = µ̄Ge, which implies the assertion.  e [b], {µ}) of the datum (G, [b], {µ}). Then 5.2.3. Let us now fix an EL realization (G, e [b], {µ}) is Hodge-Newton reducible with respect to some parabolic subgroup P e (G, e with Levi factor e e ∩ G and L = e of G L such that P = P L ∩ G. In fact, since L is the e such that e e centralizer of m · µ̄ in G, we may take P L is the centralizer of m · µ̄ in G. e is of the form As in 3.2.1, we assume for simplicity that G e := ResO |Z p GLn, G where O is the ring of integers of some finite unramified extension of Q p . Then e L takes the form e L = ResO |Z p GL j1 × ResO |Z p GL j2 × · · · × ResO |Z p GL jr . (5.2.3.1) We define e L j , L j , b j , µ j as in Theorem 3.2.2. Then by the proof of Theorem 3.2.2 we have the following facts: (1) The tuples (L j , [b j ], {µ j }) and (e L j , [b j ], {µ j }) are unramified Shimura data of Hodge type, (2) Each factor X j in the slope decomposition (5.1.2.1) arises from the datum (L j , [b j ], {µ j }) with the choice b j ∈ [b j ]. e be the p-divisible group over k with O-module structure that arises from Let X e [b], {µ}) with the choice b ∈ [b]. It admits the Hodge-Newton the datum (G, decomposition e= X e1 × X e2 × · · · × X er , X (5.2.3.2) which gives rise to the slope decomposition (5.1.2.1) of X. By Lemma 5.2.2, the e coincide. Newton polygon νGe([b]) and the σ-invariant Hodge polygon µ̄Ge of X e each factor in the decompositions (5.2.3.1) Since e L is the centralizer of m · µ̄ in G, and (5.2.3.2) corresponds to a unique slope in the polygon µ̄Ge = νGe([b]). Hence the e decomposition (5.2.3.2) is in fact the slope decomposition of X. 53 Proposition 5.2.4. Each factor X j in the slope decomposition (5.1.2.1) is rigid, i.e., Def X j ,L j is pro-represented by W. ej arises from the datum (e Proof. Note that X L j , [b j ], {µ j }) with the choice b j ∈ [b j ] (see the proof of Theorem 3.2.2). It corresponds to a unique slope in the polygon µ̄Ge = νGe([b]), so it is µ-ordinary with single slope. By [Mo04], Corollary 2.1.5, its deformation space Def X j ,eL j is pro-represented by W. Now the assertion follows from the closed embedding of deformation spaces Def X j ,L j ,−→ Def X j ,eL j induced by the embedding L j ,−→ e L j (Lemma 2.5.6).  be the universal deformation of X j in the sense of Proposition 2.5.5. Let X can j is defined over W. Hence for any ring R of the Proposition 5.2.4 says that X can j form R = W[[u1, · · · , u N ]] or R = W[[u1, · · · , u N ]]/(pm ), there exists a unique deformation of X j over R, namely X can j ⊗W R. We define the canonical deformation of X to be a deformation of X over W given by can can X can := X can 1 ×X 2 ×···×X r . It is clear from this construction that all endomorphisms of X lifts to endomorphisms of X can ⊗W R for any ring R of the form R = W[[u1, · · · , u N ]] or R = W[[u1, · · · , u N ]]/(pm ). 5.3 Structure of deformation spaces 5.3.1. When r = 1, we have Def X,G ≃ Spf(W) by Proposition 5.2.4. Let us now consider the case r = 2. Then we have the slope decompositions X = X1 × X2 and e= X e1 × X e2 . X es for s ∈ {1, 2} (see Example 2.4.5 for definition). Define Let (ds, fs ) be the type of X a function f0 : I → {0, 1} by    0 if f1 (i) = f2 (i) = 0;     0 f (i) = 0 if f1 (i) = d1 and f2 (i) = d2 ;      1 if f1 (i) = 0 and f2 (i) = d2 .  54 As noted in Example 2.4.5 for definition, there exists a unique isomorphism class of µ-ordinary p-divisible group over k with O-module structure of type (1, f0). We fcan (1, f0) denote its canonical lifting. let X Theorem 5.3.2. Notations above. The deformation space Def X,G has a natural structure of a p-divisible group over W. More precisely, we have an isomorphism fcan (1, f0)d Def X,G  X 0 as p-divisible groups over W with O-structure for some integer d 0 ≤ d1 d2 . Proof. Consider the category CW of artinian local W-algebra with residue field k. fcan denote the canonical deformation of X ej for j = 1, 2. We define the functor Let X j fcan, X fcan ) : CW → Sets Ext(X 1 2 fcan, X fcan )(R) to be the set of isomorphism classes of extensions by setting Ext(X 1 2 fcan ⊗W R by X fcan ⊗W R as fppf sheaves of O-module. of X j 2 By [Mo04], Theorem 2.3.3, we have the following isomorphisms: fcan, X fcan ) as smooth formal groups over W, (a) Def X,Ge  Ext(X 1 2 fcan (1, f0)d1 d2 as p-divisible groups over W with O-module struc(b) Def X,Ge  X ture. On the other hand, by Lemma 2.5.6 we have a closed embedding of deformation spaces Def X,G ,−→ Def X,Ge. (5.3.2.1) Our first task is to show that Def X,G is a subgroup of Def X,Ge with O-module structure. Let R be a smooth formal W-algebra of the form R = W[[u1, · · · , u N ]] or R = W[[u1, · · · , u N ]]/(pm ), and take two arbitrary deformations X and X 0 of X over R. By Proposition 5.1.3, we have exact sequences can 0 −→ X can 1 ⊗W R −→ X −→ X 2 ⊗W R −→ 0, 0 can 0 −→ X can 1 ⊗W R −→ X −→ X 2 ⊗W R −→ 0. We denote by X X 0 the underlying p-divisible group of their Baer sum taken in fcan, X fcan )(R). Ext(X 1 2 55 We wish to show that X X 0 ∈ Def X,G (R). By the isomorphism (a), we already know that X X 0 ∈ Def X,Ge(R). Hence it remains to show that we have tensors on (the Dieudonné module of) X X 0 which lift the tensors (ti ) on X in the sense of Proposition 2.5.3. Unfortunately, it is not easy to explicitly find these tensors in terms of the tensors on X and X 0. Instead, we start with the family of all tensors (s j ) on Λ which are fixed by G. Then we have a family (t j ) := (s j ⊗ 1) on Λ∗ ⊗Z p W = M, where M denotes the Dieudonne module of X as before. Since the formal deformation space Def X,G is independent of the choice of tensors (ti ), we get tensors (t̄ j ) on X and (t̄0j ) on X 0 which lift (t j ) (in the sense of Proposition 2.5.3). Moreover, the families (t̄ j ) and (t̄0j ) map to the same family of tensors on can 0 can X can 2 under the surjections X  X 2 and X  X 2 . Hence the families (t̄ j ) and (t̄0j ) define the same family of tensors on X X 0 which lift (t j ). In particular, there exists a family of tensors on X X 0 which lift (ti ). Since Def X,Ge has a finite p-torsion for being a p-divisible group, we observe from the embedding (5.3.2.1) that Def X,G also has finite p-torsion. Using the same argument as in the proof of [Mo04], Theorem 2.3.3, we deduce that Def X,G is a p-divisible group. fcan (1, f0)d1 d2 with O-module Hence Def X is a p-divisible subgroup of Def X,Ge  X structure. Now the dimension of Def X,G determines an integer d 0 such that fcan (1, f0)d 0 Def X,G  X as p-divisible groups over W with O-module structure.  Remark. From the proof, one sees that the canonical deformation X can corresponds to the identity element in the p-divisible group structure of Def X,G . 5.3.3. We finally consider the case r ≥ 3. For convenience, we write Def Xea,b for ea,b . These spaces fit into a diagram the deformation space of X Def Xe1,r = Def X,Ge Def Xe1,r−1 Def Xe1,r−2 ··· Def Xe2,r Def Xe2,r−1 ··· Def Xe3,r ··· ··· 56 where each map comes from the restriction of the filtration in Proposition 5.1.3 (see [Mo04], 2.3.6.). This diagram carries some additional structures called the cascade structure, as described by Moonen in loc. cit. We denote by Def X a,b the pull back of Def Xea,b over Def X,G . Then Def X a,b classifies deformations of Xa,b with a filtration that comes from the filtration of X in Proposition 5.1.3. If we pull back the above diagram over over Def X,G , we get another diagram Def X 1,r = Def X,G Def X 1,r−1 Def X 1,r−2 ··· Def X 2,r Def X 2,r−1 ··· Def X 3,r ··· ··· where each map comes from the restriction of the filtration in Proposition 5.1.3. With similar arguments as in the proof of Theorem 5.3.2, one can give a group structure on Def X a,b over Def X a,b−1 and Def X a+1,b (cf. [Mo04], 2.3.6.). However, this diagram does not carry the full cascade structure in general. 57 BIBLIOGRAPHY [Ber96] P. Berthelot, Cohomologie rigide et cohomologie rigide á support propre. premiére partie, Prépublication IRMAR 96-03(1996) [Boy99] P. Boyer, Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math. 138(1999), 573-629. [dJ95] A. J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Inst. Hautes Études Sci. Publ. Math. 82(1995) 5-96. [Fal99] G. Faltings, Integral crystalline cohomology over very ramified valuation rings, J. 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