Γ(p)-Level Structure on p-Divisible Groups Citation Frimu, Andrei (2019) Γ(p)-Level Structure on p-Divisible Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8JYH-KT84. https://resolver.caltech.edu/CaltechTHESIS:05302019-185557287 Abstract The main result of the thesis is the introduction of a notion of Γ( p )-level structure for p -divisible groups. This generalizes the Drinfeld-Katz-Mazur notion of full level structure for 1-dimensional p -divisible groups. The associated moduli problem has a natural forgetful map to the Γ 0 ( p )-level moduli problem. Exploiting this map and known results about Γ 0 ( p )-level, we show that our notion yields a flat moduli problem. We show that in the case of 1-dimensional p -divisible groups, it coincides with the existing Drinfeld-Katz-Mazur notion. In the second half of the thesis, we introduce a notion of epipelagic level structure. As part of the task of writing down a local model for the associated moduli problem, one needs to understand commutative finite flat group schemes G of order p 2 killed by p , equipped with an extension structures 0→ H 1 → G→ H 2 → 0, where H 1 ,H 2 are finite flat of order p . We investigate a particular class of extensions, namely extensions of Z/pZ by μ p over Z p -algebras. These can be classified using Kummer theory. We present a different approach, which leads to a more explicit classification. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: arithmetic geometry; algebraic geometry; p-divisible groups; level structures; finite flat group schemes; shimura varieties Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Apostol Award for Excellence in Teaching in Mathematics, 2015. Thesis Availability: Public (worldwide access) Research Advisor(s): Zhu, Xinwen Thesis Committee: Mantovan, Elena (chair) Zhu, Xinwen Ramakrishnan, Dinakar Flach, Matthias Defense Date: 22 May 2019 Record Number: CaltechTHESIS:05302019-185557287 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05302019-185557287 DOI: 10.7907/8JYH-KT84 ORCID: Author ORCID Frimu, Andrei 0000-0002-7782-7204 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 11575 Collection: CaltechTHESIS Deposited By: Andrei Frimu Deposited On: 10 Jun 2019 22:37 Last Modified: 04 Oct 2019 00:26 Thesis Files Preview PDF - Final Version See Usage Policy. 441kB Repository Staff Only: item control page

Γ(p)-level structure on p-divisible groups Thesis by Andrei Frimu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended May 22, 2019 ii © 2019 Andrei Frimu ORCID: 0000-0002-7782-7204 All rights reserved iii ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor, Prof. Xinwen Zhu, for suggesting the problem and for the continuous support and inspiration he provided. I have learned a lot of beautiful mathematics through his recommendations and conversations. I would also like to thank Prof. Elena Mantovan, Prof. Dinakar Ramakrishnan and Prof. Matthias Flach for agreeing to be part of my defense committee. I am also grateful to various other faculty and staff of the Caltech Math Department for all the help they provided in the last several years. Greatest thanks goes to my parents and my sister for their love, support and encouragement throughout all my life. I am very grateful to Prof. Valeriu Baltag and Prof. Marcel Teleuca for nurturing my passion for mathematics from a young age and guiding me through mathematical contests. Finally, I thank my friends (currently or formerly) at Caltech, in Pasadena and many other cities around the world, for making my PhD years so enjoyable. Special thanks goes to fellow student Nathan Lawless, whom I met on day one at Caltech and with whom I shared most of the math PhD journey. iv ABSTRACT The main result of the thesis is the introduction of a notion of Γ(p)-level structure for p-divisible groups. This generalizes the Drinfeld-Katz-Mazur notion of full level structure for 1-dimensional p-divisible groups. The associated moduli problem has a natural forgetful map to the Γ0 (p)-level moduli problem. Exploiting this map and known results about Γ0 (p)-level, we show that our notion yields a flat moduli problem. We show that in the case of 1-dimensional p-divisible groups, it coincides with the existing Drinfeld-Katz-Mazur notion. In the second half of the thesis, we introduce a notion of epipelagic level structure. As part of the task of writing down a local model for the associated moduli problem, one needs to understand commutative finite flat group schemes G of order p2 killed by p, equipped with an extension structures 0 → H1 → G → H2 → 0, where H1, H2 are finite flat of order p. We investigate a particular class of extensions, namely extensions of Z/pZ by µ p over Z p -algebras. These can be classified using Kummer theory. We present a different approach, which leads to a more explicit classification. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Finite flat group schemes . . . . . . . . . . . . . . . . . . . . . . . 2.2 Oort-Tate theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 p-divisible groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Dieudonné theory . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter III: Level Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Oort-Tate generators . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter IV: Various Moduli Problems . . . . . . . . . . . . . . . . . . . . . 4.1 Moduli problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter V: Γ(p)-level structure . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A (non-)example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 1-dimensional p-divisible groups . . . . . . . . . . . . . . . . . . . 5.3 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VI: Epipelagic level and certain group schemes of order p2 . . . . . 6.1 Epipelagic level structure . . . . . . . . . . . . . . . . . . . . . . . 6.2 Certain group schemes of order p2 . . . . . . . . . . . . . . . . . . . 6.3 Extensions of Z/pZ by µ p in characteristic p . . . . . . . . . . . . . 6.4 Extensions of Z/pZ by µ p over any Z p -algebra . . . . . . . . . . . . Chapter VII: Level structures on Dieudonné modules . . . . . . . . . . . . . 7.1 A commutative diagram . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Level structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The case S = Spec(k), where k is a perfect field . . . . . . . . . . . 7.4 Integral domain base . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 From integral domain to general case . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v 1 1 3 5 6 6 9 11 13 16 20 21 21 23 24 26 28 32 32 33 34 40 43 43 45 46 48 51 53 1 Chapter 1 INTRODUCTION 1.1 Motivation The main focus on the thesis is introducing a notion of Γ(p)-level structure on p-divisible groups. Level structures on abelian varieties or their associated pdivisible groups have long been studied, part of the appeal being that PEL type Shimura varieties have been a rich source of representations of interest to the Langlands program. For example, Harris and Taylor [HT01] establish Langlands correspondence for GL(n) over p-adic fields by studying the (cohomology of the) bad reduction of certain PEL type Shimura varieties which arise as moduli problems of abelian varieties with extra data - a polarization and a Drinfeld-Katz-Mazur level structure on an associated p-divisible group, which is 1-dimensional. In the above setting, the p-divisible groups under consideration are 1-dimensional (i.e. their Lie algebra is locally free of rank 1). In this case, the Drinfeld-KatzMazur notion of Γ(p)-level (and Γ(pm )-level) structure is well-behaved. If G is a p-divisible group of height h, a level pm -structure is a map of group schemes ϕ : (Z/pm Z)h → G[pm ] satisfying certain extra properties. These properties have been refined, historically, several times. For example, if G is the p-divisible group associated to an elliptic curve E (so G has height h = 2) over some base S where p is invertible, one simply requires ϕ to be an isomorphism (and one can replace pm by any integer n invertible on S). When p is not necessarily invertible on S, it can happen Õthat no such isomorphism exists and a remedy is to require that G[pm ] = [ϕ(x)], where this equality is an equality of Cartier divisors on E. x∈(Z/pm Z)h One can check that this implies that ϕ is an isomorphism if p is invertible. Katz and Mazur introduce in [KM85] a notion of level structure (which they call ‘‘full set of sections,’’ and which we call in this thesis a Drinfeld-Katz-Mazur level structure) that generalizes the above scenarios and does not depend on an ambient curve where an equality of Cartier divisors is to take place. That is the definition used successfully by Harris and Taylor in [HT01]. The moduli space of 1-dimensional p-divisible groups with such level is well-behaved. The notion of Drinfeld-Katz-Mazur level structure, applies, in principle, to higher- 2 dimensional p-divisible groups (for instance, arising from abelian varieties of dimension > 1), but it is no longer well-behaved. One of the main problems is the fact that points are no longer Cartier divisors - if A is an abelian variety of dimension d, A[n] is a relative Cartier divisor of A only if d = 1. Loosely speaking, 1-dimensionality allows us to locally embed our (truncated) p-divisible group G in a curve (see lemma 5.2.1), thus the sections of G(S) in the image of a group homomorphism (Z/pm Z)h → G correspond to Cartier divisors on this curve, allowing us to make sense of equalities as Cartier divisors, as in the elliptic curve example above. For higher dimensions, among other issues, the arising moduli spaces are no longer flat, as shown for example by Chai and Norman in [CN90]. It is thus an interesting question to define such a notion of ‘‘full’’ Γ(p) (and Γ(pm ))-level structure for higher-dimensional p-divisible groups. We propose such a notion for Γ(p)-level, show that it agrees with the existing definition for 1-dimensional p-divisible groups, and prove that the arising moduli space is flat over Z p . In the second part of the thesis, motivated by the study of epipelagic representations, we introduce a notion of epipelagic level structure. We show that the moduli space of p-divisible groups equipped with this level structure is flat. When trying to write down the local model for the moduli space, a useful gadget would be a classification of commutative finite flat group schemes G of order p2 killed by p, equipped with an extension structure 0 → H1 → G → H2 → 0; or at least of such extensions when H2 = Z/pZ. We give some partial results in this direction, by explicitly classifying extensions G of Z/pZ by µ p over Z p -algebras. Locally on the base, it is known that 1 (S, µ ) and one can compute this group using such extensions are classified by Hfppf p Kummer theory. We arrive at the same result without using Kummer theory, but by studying in depth the structure of the affine algebra of G, making use of Oort-Tate theory. We believe that a detailed such study is necessary if one is to write down an explicit moduli problem classifying general extensions 0 → H1 → G → H2 → 0 as above. In the third part of the thesis, we present an interpretation of the notion of level structure on p-divisible groups over perfect bases using Dieudonné theory. It is known that over perfect rings there is an anti-equivalence of categories between p-divisible groups (truncated or not) and Dieudonné modules. Thus, the notion of Drinfeld-KatzMazur level structure for p-divisible groups should have a corresponding notion in terms of Dieudonné modules. We give such an interpretation. 3 1.2 Overview of the results In the first part of the thesis, we introduce the following notion of Γ(p)-level structure on p-divisible group: Definition 1.2.1. A Γ(p)-level structure on a p-divisible group G of height h is the following data: 1. A filtration 0 = H0 ⊂ H1 ⊂ . . . ⊂ Hh = G[p] of G[p] by finite flat subgroup schemes. 2. Oort-Tate generators (see section 3.1) γi : (Z/pZ) → Hi /Hi−1 for i = 1, . . . , h. 3. Group homomorphism ϕi : (Z/pZ)i → Hi such that the following diagram commutes: 0 0 = H0 / Z/pZ  ϕ1 / H1 / (Z/pZ)2  / ... ϕ2 / H2 / ... / (Z/pZ) h−1  ϕh−1 / Hh−1 / (Z/pZ) h  ϕh / Hh = G[p] so that the induced maps Z/pZ → Hi /Hi−1 are the Oort-Tate generators γi . We show that if G is 1-dimensional, the above notion agrees with the (usual) DrinfeldKatz-Mazur level structure. Our main result is that the moduli space of p-divisible groups with the above notion of Γ(p)-level structure is flat (for any dimension), which we prove in section 5.3. We establish this by examining the forgetful map to moduli space M Γ0 (p) of p-divisible groups with Γ0 (p)-level structure, which is known to be flat over Z p by [Goe01]. We recall some of the facts about M Γ0 (p) we use in chapter 4. The forgetful map factors through several moduli problems defined by ‘‘intermediate’’ data. We show that each of these intermediate moduli problems is flat over the next one. In the second part of the thesis, motivated by the study of epipelagic representations, we give a notion of epipelagic level structure and show that the associated moduli space is flat. A useful ingredient in writing down local models for the associated moduli space would be a classification of commutative finite flat group schemes of order p2 killed by p, equipped with an extension structure of the form 0 → H1 → G → H2 → 0 where H1, H2 are finite flat group schemes of order p. The problem seems hard even when H2 = Z/pZ. 4 A general family of such extensions over any ring R is the following: let  ∈ R be a unit and consider the R-algebra B := p−1 Ê p R[Xi ]/(Xi −  i ). i=0 This is the Hopf algebra of a group scheme G := Spec(B ). If S = Spec A is a connected R-scheme, then G (S) consists of pairs (t, i) with t ∈ A such that t p =  i . The group law is given by (t, i) · (s, j) =    (st, i + j), if i + j < p    (st/, i + j − p), if i + j ≥ p  We shall prove that locally over a Z p -algebra R, every extension of Z/pZ by µ p is of 1 (S, µ ) the form G , for some unit  ∈ R. One can show that Ext1S (Z/pZ, µ p ) = Hfppf p ∗ ∗ p and using Kummer theory, the latter group is (locally) isomorphic R /(R ) . We take a different route and look explicitly at the action of Hom(H2, H1 ) = Hom(Z/pZ, µ p ) = µ p on such groups. Using results of [Tzi17] and [DG70], the action of µ p on any scheme that’s nice enough (not necessarily group scheme) can be written quite explicitly. In characteristic p, such actions are in bijection with global derivations D satisfying the relation D p = D. Over a general Z p -algebra we also obtain a linear map D. It no longer is a derivation, but still has special properties that we can exploit. In both cases, in the presence of (p − 1)th roots of unity, D is diagonalizable and we can analyze the structure of the (group) schemes acted upon explicitly. Finally, in the last part, we give an auxiliary result - an interpretation of the notion of Drinfeld-Katz-Mazur level structure using Dieudonné theory. Over a perfect field k of characteristic p, it is classical that p-divisible groups are classified by Dieudonné modules - free W(k)-modules with semilinear actions of Frobenius and Verschiebung. In fact, this follows from a more general equivalence: finite flat group schemes of p-power order over k and W(k)-modules of finite length with actions of Frobenius and Verschiebung. This equivalence has more recently been established more generally over any perfect ring R (unpublished results of Gabber, see also [Lau10]). Interpreting a level structure on a p-divisible group G of height h as a group homomorphism between the constant group scheme (Z/pm Z)h → G[pm ] with certain extra properties (see chapter 7 for a precise statement), it corresponds then to a homomorphism of (truncated) Dieudonné modules with some corresponding properties. The main result is theorem 7.2.1. 5 1.3 Structure of the thesis In Chapter II, we recall background material on finite flat group schemes, p-divisible groups and Dieudonné theory. In Chapter III, we introduce various existing notions of level structures and their properties. Chapter IV contains background material regarding moduli spaces of p-divisible groups with Γ0 (p) and Γ1 (p) level structures. Chapters V, VI and VII contain our main results. In Chapter V, we introduce our notion of Γ(p)-level structure and establish its properties. In Chapter VI, we introduce the notion of epipelagic level structure and classify extensions of Z/pZ by µ p . Chapter VII contains our auxiliary result about interpreting the notion of level structure in the Dieudonné setting. 6 Chapter 2 PRELIMINARIES In this chapter, we recall various definitions, examples and results that we will use throughout the thesis. Most of the results are well-known and can be found in a variety of sources, for example [Dem72] or [DG70]. We fix a prime number p. S will typically denote a base scheme. Following the conventions of Messing [Mes72], a group G over S is a commutative fppf sheaf of groups on the site Sch/S. 2.1 Finite flat group schemes Let R be a ring. By a finite flat group scheme of order m over R we mean a commutative group scheme G which is locally free of rank m over R. In this paper m will usually be a power of a fixed prime p. Locally on the base, G is the spectrum of a Hopf algebra A and is endowed with maps describing the group structure: comultiplication ∆ : A → A ⊗ A, neutral element map  : A → R and an automorphism i : A → A corresponding to the inverse automorphism. These maps are required to satisfy various conditions which make G(T) into a group for any test scheme T. The kernel of the map  : A → R is a locally free ideal called the augmentation ideal and has rank m − 1 over the base. Examples 1. Constant group scheme: for any abstract group Γ, the functor associating each connected scheme S the group Γ is representable by the constant group scheme associated to Γ. Its Hopf algebra is R(Γ) = R × . . . × R, where the factors are indexed by the elements of Γ. This can also be written as R[eg ]g∈Γ , where eg are orthogonal idempotents, i.e. eg2 = eg and eg eg 0 = 1. The Hopf algebra structure is given by comultiplication Õ eg 7→ ∆(eg ) = eh ⊗ eg−h, h∈Γ The neutral element map e0 7→ 1 and eg 7→ 0 where 0 is the zero element of Γ. The inverse automorphism is eg 7→ e−g . 7 In the section on Oort-Tate theory below, we shall see a different description of Z/pZ under certain assumptions on the base scheme. 2. µm : this represents the functor sending an R-algebra A to the group {x ∈ A| x m − 1 = 0} (under multiplication). The Hopf algebra can be written as R[Z]/(Z m − 1). Comultiplication sends Z 7→ Z ⊗ Z, neutral element sends Z 7→ 1 and inverse Z 7→ Z −1 . In the section on Oort-Tate theory below, we shall see a different description of µ p under certain assumptions on the base scheme. k 3. αpk : consider the functor sending an R-algebra A to the set {x ∈ A| x p = 0}. This set is empty if A is reduced. If R is an F p -algebra, then this set is actually a group and the functor over F p is representable. The Hopf algebra is given k by R[X]/(X p ) with comultiplication sending X 7→ X ⊗ 1 + 1 ⊗ X, neutral element X 7→ 0 and inverse element X 7→ −X. In the section on Oort-Tate theory below, we shall see another perspective on αp under certain assumptions on the base scheme. Cartier duality If G = Spec B is a finite flat group scheme over a ring R, then B∨ = Hom R (B, R) has a natural structure of a Hopf algebra. Indeed, using the identification (B ⊗ R B)∨ = B∨ ⊗ R B∨ , dualizing the multiplication m : B ⊗ R B → B, comultiplication c : B ⊗ R B, inverse map i : B → B, identity section  : R → B and structure map B → R one obtains five ring maps m∨ : B ∨ → B ∨ ⊗ R B ∨ ∆∨ : B∨ ⊗ R B∨ → B∨ i ∨ : B∨ → B∨  ∨ : R → B∨ B∨ → R One can check that these maps define, in order, comultiplication, multiplication, inverse, structure map and identity section map on B∨ , i.e. endow B∨ with the structure of a Hopf algebra (which is also finite and flat over R). Spec(B∨ ) is denoted G∨ and is called the Cartier dual of G. One can also define G∨ as G∨ (S) := Homgroup (G S, GmS ). 8 Cartier duality induces an anti-equivalence from the category of finite flat group schemes to itself. Examples: The dual of Z/pZ is µ p and the dual of µ p is Z/pZ. The group scheme αp introduced above is self-dual. Frobenius and Verschiebung Assume in this subsection that p = 0. Any F p -ring R has a ring endomorphism FR : R → R given by FR (x) = x p , for x ∈ R. If A is an R-algebra, then we can form the FR -base change A(p) := A ⊗ R,FR R. This naturally extends to F p -schemes. In algebraic-geometric language, starting with a group scheme G over an F p -scheme S, we can then form the S-scheme G(p) := G ×S,FS S. G(p) is also called the Frobenius twist of G. It is also a group scheme. We then have an induced map FG/S : G → G(p) , called the relative Frobenius, arising from the following diagram: G FG FG/S !  /G G(p) #  S FS  /S The relative Frobenius is functorial and commutes with products, so in particular is a group homomorphism. It also commutes with taking Cartier duals. Let G be a group over a base scheme S over F p . Then, we can form the relative Frobenius G∨ → (G∨ )(p) = (G(p) )∨ . Taking the duals again, one obtains a group homomorphism G(p) → G, called Verschiebung and commonly denoted by V. One can then prove Proposition 2.1.1. V ◦ F = p · idG and F ◦ V = p · idG(p) . Étale-connected sequence Let R be a Henselian local ring (for example R = Z p ) and G = Spec B a finite flat group scheme over R. Then, one has a connected-étale sequence 0 → G0 → G → Gét → 0 9 Ö This is obtained as follows: B can be written as a product B = Bi of local rings. 0 One of these local rings, typically denoted B , has the property that the identity section e : S → G lands in Spec(B0 ). Then, G0 := Spec(B0 ) is called the connected component of the identity in G. One shows that G0 is actually a (sub)group scheme. Using the hypothesis that R is a Henselian ring, one can show that the quotient G/G0 exists (as a group scheme) and is étale. It is typically denoted by Gét . One thus has an exact sequence 0 → G0 → G → Gét → 0. If R is a perfect field, the homomorphism G → Gét has a section and thus G can be written as a product G = G0 × Gét . This is a very special phenomenon, relying crucially on R being perfect. See, for example, section 6.2 below for a family of finite flat group schemes G of order p2 . If the base is a field of characteristic p, then we have an exact sequence 0 → µ p → G → Z/pZ → 0 which only splits if  is a pth power. 2.2 Oort-Tate theory In this subsection, we review Oort-Tate theory. Finite flat group schemes of prime order p are (relatively) well-understood. Namely, we have the following modern reformulation (see for example, [HR12]) of the results of Oort and Tate in [TO70]: Theorem 2.2.1. Let OT be the Z p -stack of finite flat group schemes of order p. Then (i) OT is an Artin stack isomorphic to  
Spec Z p [X,Y ]/(XY − w p ) /Gm , where Gm acts via λ.(X,Y ) = (λ p−1 X, λ1−pY ). w p is an explicit element of pZ×p . (ii) The universal group scheme G over OT is equal to G=  
SpecOT O[Z]/(Z p − X Z) /Gm where Gm acts via Z 7→ λZ, with zero section Z = 0. In particular, the above theorem classifies finite flat group schemes of order p over Z p and F p . In fact, Oort and Tate classify group schemes of order p over the ring 10   1 ∩ Z p , where ζ is a primitive (p − 1)th root of unity in Z p , but for p(p − 1) our purposes, this classification over Z p -algebras suffices. If S is a Z p -scheme, then finite flat group schemes G of order p over S correspond to morphisms ϕ : S → OT, such that G = ϕ∗ (G). Λ = Z ζ, Let G = Spec(B) be a group scheme of order p over a base S over Spec(Z p ). An important idea in Oort and Tate’s paper is that the action of (Z/pZ)× = F×p on the augmentation ideal decomposes this ideal into a direct sum of invertible modules (this crucially relies on the presence of p − 1 roots of unity). Indeed, the action allows us to write the augmentation ideal I of B as a direct sum I = I1 ⊕ . . . ⊕ I p−1 of modules I j . These can be obtained as I j = e j I, where e j are certain idempotents of the group ring Λ[F∗p ]. By inspecting what happens at geometric points one then j concludes that each I j has rank 1. Moreover, I1 = I j . Let χ be the Teichmuller character χ : F p → Z p . Thus χ(0) = 0 and for i , 0, χ(i) is the unique (p − 1)th root of unity in Z p which is congruent to i mod p. We have the following useful description of the modules I j : I j consists of sections f of I such that [m] f = χ j (m) f , for every m ∈ (Z/pZ)× . An important part in the proof of the classification result is the following description   1 of the multiplicative group scheme µ p over the ring Λ = Z ζ, ∩ Zp p(p − 1) mentioned above. Let B = Λ[z]/(z p − 1) be the Hopf algebra of µ p over Λ. Define then y j := (p − 1)e j (1 − z). and write y := y1 . Extend the indexing to all integers by defining y j+(p−1)k := y j for any integer k. Using the above notation, the following properties then hold: Proposition 2.2.1 ([TO70]). for 1 ≤ j ≤ p − 1. 1. I1 is generated by y and I j is generated by y j , 2. For 1 ≤ j ≤ p − 1, there are constants w j such that y j = w j y j . We also have y p = w p y, where w p = p · unit. 3. p−1 1 Õ ∆(yi ) = yi ⊗ 1 + 1 ⊗ yi + y j ⊗ yi− j 1 − p j=1 11 and in particular p−1 1 Õ yi y p−i ∆(y) = y ⊗ 1 + 1 ⊗ y + ⊗ 1 − p j=1 wi w p−i 4. w j ≡ j! (mod p) for 1 ≤ j ≤ p − 1.   y2 y p−1 1 y+ +···+ . 5. z = 1 + 1−p w2 w p−1 Examples, revisited In this subsection, we present how the three examples of finite flat group schemes shown above fit in Oort-Tate theory. 1. Constant group scheme Z/pZ: If g = χ(1)e1 + . . . + χ(p − 1)e p−1 , where χ is the Teichmuller character described above, then one checks that the affine ring B = R[eg ]g∈Z/pZ /heg2 = eg, eg eg 0 = 0i of the constant group scheme Z/pZ can also be written as B = R[g]/(g p − g). One recovers the ei ’s, by factoring g p − g into linear factors and omitting the ith one. The augmentation ideal is I = ⊕gi R, and I j = g j R. 2. µ p : This is the content of proposition 2.2.1 above. Note that y is neither z nor z − 1 (which appear often in description of µ p ). 3. αp : As mentioned above, this group scheme only exists over F p -algebras R and has Hopf algebra R[t]/(t p ). The augmentation ideal is I = hti. The invertible modules I j in the decomposition I = I1 ⊕ . . . ⊕ I p−1 are simply I j = t j R. 2.3 p-divisible groups Definition and basic examples Definition 2.3.1. Let S be a base scheme. An S-group G (recall that G is, a priori, an fppf sheaf) is called a p-divisible group on S if it satisfied the following three conditions: (i) The morphism p : G → G is an epimorphism. (ii) G is p-torsion, i.e. G = lim G[pn ], where G[pn ] := ker(pn : G → G). −−→n 12 (iii) The S-groups G[pn ] are representable by finite locally free group schemes over S. If S is connected, then the group schemes G[pn ] have constant rank over the base equal to phn , for some positive integer h not depending on n (only on G). We call h the height of G. Assume now p is locally nilpotent on S. We define the Lie algebra of G as Lie G[pm ] for m large enough. This is a locally free sheaf on S and we call the rank of Lie G the dimension of G. Here are some basic examples: 1. The constant p-divisible group (Q p /Z p )S . The finite flat group scheme G[pn ] is just the constant group scheme (p−n Z p /Z p ) w (Z/pn Z) over S. G has height 1 and dimension 0. 2. G = µ p∞ . This can be defined as Gm [p∞ ] and we have G[pn ](S) = µ pn (S). One can check that G has height 1 and dimension 1. 3. Let A/S be an abelian scheme of dimension g. Then A[p∞ ] is a p-divisible group of height 2g. Étale-connected sequence Let R be a Henselian ring and let G be a p-divisible group over S = Spec R of height h. Since each G[pn ] is a finite flat group scheme over S, we have étale connected sequences (see section 2.1) for every n: 0 → G0 [pn ] → G[pn ] → Gét [pn ] → 0. One can show that these sequences maps are compatible between different levels n and m. Moreover, (G0 [pn ]0 ) and (Gét [pn ]) are themselves p-divisible groups, typically called the connected and the étale part of G and denoted G0 and Gét , respectively. If R is a perfect field, then the exact sequence of p-divisible groups 0 → G0 → G → Gét → 0 splits and one has G w G0 × Gét . 13 Cartier duality Let G be a p-divisible group over a base scheme S. As each G[pn ] is a finite flat group scheme, one can form the Cartier duals G[pn ]∨ . One can check that these duals also satisfy the conditions in the definition of a p-divisible group. The p-divisible group thus obtained is denote G∨ . It has the same height as G. The dimensions of G and G∨ are related by the following result: Proposition 2.3.1. Let G be a p-divisible group of height h and dimension d. Then G∨ has dimension h − d. Common examples are the following: the dual of the constant p-divisible group (Q p /Z p ) is µ p∞ . If G is the p-divisible group arising from an abelian variety A, then G∨ is the p-divisible group associated to the dual abelian variety A∨ . Frobenius and Verschiebung Assume that the base scheme S has characteristic p (i.e. pOS = 0). From the functoriality of the Frobenius F and Verschiebung V, these maps on each group scheme G[pn ] are compatible (in n). The Frobenius twists then G[pn ](p) form a p-divisible group which we denote G(p) . We thus have isogenies of p-divisible groups F : G → G(p) and V : G(p) → G. Similar to proposition 2.1.1, we have Proposition 2.3.2. F ◦ V = p · idG(p) and V ◦ F = p · idG . 2.4 Dieudonné theory In this section, we summarize some results on Dieudonné theory over perfect rings. One can consult [Dem72] or [Fon77] for Dieudonné theory over perfect fields and [Lau10] for Dieudonné theory over perfect rings. In its classical form, Dieudonné theory gives a contravariant equivalence between the category of p-divisible groups over a perfect field k of characteristic p and the category of Dieudonné modules over the Dieudonné ring D = W(k)[F,V], which we detail below. Recall first that to any perfect ring R of characteristic p, one can associate, functorially, the ring of Witt vectors W(R). This is a ring of characteristic 0. As a set it is in bijection with RN , but with more complicated addition and multiplication, given by certain Witt polynomials. If R is a perfect field k, then W(k) is a complete discrete valuation ring with maximal ideal (p) and residue field k. 14 The most common example is k = F p . Then W(k) = Z p , the ring of p-adic integers. If k = F pr , then W(k) is isomorphic to the unique unramified extension of Z p of degree r. The Frobenius automorphism of R lifts to a Frobenius automorphism W(R), usually denoted by σ. On sequences (a0, a1, . . .) it acts by raising each ai to its pthe power. One also has the rings of truncated Witt vectors Wm (R). If R is perfect, these are isomorphic to W(R)/pmW(R). Its elements can be viewed as finite sequences of length m (but again, not with componentwise addition and multiplication) and are endowed with a Frobenius as well. Note that W1 (R) w R. The Dieudonné ring ER is defined as the (mildly) non-commutative ring W(R)[F,V] where F and V satisfy the relations: 1. FV = V F = p. 2. F is σ-linear, i.e. Fw = σ(w)F. 3. V is σ −1 -linear, i.e. wV = V σ(w). Recall the following definition from [Lau10]: Definition 2.4.1. Let R be a perfect ring. A projective Dieudonné module over R is a ER -module which is a projective W(R)-module of finite type. A truncated Dieudonné module of level n is a ER -module M which is projective over Wn (R) and of finite type; if n = 1 we further require that ker F = im V, im F = ker V and M/V M is a projective R-module. We can now state the anti-equivalence. As we are stating it over perfect rings, we follow [Lau10]. Theorem 2.4.1 (Dieudonné equivalence). There is an anti-equivalence of categories between that of p-divisible groups over R of height h and projective Dieudonnémodules of rank h. There is an anti-equivalence of categories between the category of truncated p-divisible groups of height h and truncated Dieudonné modules of rank h. One can also formulate such an equivalence more generally for finite flat group schemes over R. For a p-divisible group, truncated p-divisible group or finite flat group scheme G over R, we denote by M(G) its associated Dieudonné module. 15 This equivalence is functorial. For example, the Frobenius map G → G(p) corresponds to a linear map M(G(p) ) w M(G)(p) → M(G) and analogously for the Verschiebung. If R is a perfect field k, every finite flat group scheme G fits into the étale-connected sequence: 0 → G0 → G → G et → 0 which splits. In that case, M(G) is the direct sum of its local and etale part: M(G) = M(G)loc ⊕ M(G)ét . Here M(G)loc is the Dieudonné module of G0 and M(G)loc the Dieudonné module of G et . One also has the following equalities: M(G)loc = ∩n≥0 ker(F n ) and M(G)ét = ∪n≥0 F n M(G). On connected p-divisible groups (truncated or not), the Frobenius is nilpotent and the Verschiebung is an isomorphism. On étale such groups the behavior is reversed: F is an isomorphism, and V is nilpotent. Examples Let k be a perfect field, and consider the three examples of finite flat group schemes of order p over k: 1. G = Z/pZ: Its Dieudonné module is M(G) = k, with F = σ and V = 0. More generally, if H is the constant p-divisible group (Q p /Z p ), then M(H) is just W(k) (as a free rank 1 module over itself) with F = σ and V = 0. 2. G = µ p : Its Dieudonné module is M(G) = k, with F = 0 and V = σ −1 . 3. G = αp : M(αp ) = k, with F = V = 0. A more interesting application is the classification of group schemes of order p2 , killed by p, over a perfect field k of characteristic p. This is done in [Buz12], by classifying instead the corresponding Dieudonné modules, which are just 2dimensional vector spaces over k with various actions of F and V. There are 9 such group schemes: direct products of any two of Z/pZ, αp and µ p , and also three non-split extensions of αp by αp : αp2 , its Cartier dual and a third one, which is self-dual and on which F and V are both nilpotent but nonzero. 16 Chapter 3 LEVEL STRUCTURES The notion of level structure was originally introduced by Drinfeld in [Dri74]. It was later generalized by Katz and Mazur and, in that formulation, among other applications, was used by Harris and Taylor to establish local Langlands correspondence for GLn . Historically, the notion has been refined several times. In one of its more basic instances, for an elliptic curve E over a base scheme S over Z[1/n], a level-n structure, ∼ or Γ(n)-structure, is simply an isomorphism (Z/nZ)2S − → E[n]. Two further notions of level can be defined: Γ1 (n)-level which is a homomorphism of group schemes Z/nZ → E[n](S), and Γ0 (n)-level structure which is a finite flat subgroup scheme of E[n] locally isomorphic to Z/nZ. Here locally means for the étale topology. These three notions of level structure give rise to three moduli problems (for schemes S over Z[1/n]). By Katz-Mazur [KM85], Theorem 3.7.1 each of these moduli problems are relatively representable and finite étale. For an elliptic curve over an arbitrary scheme S, where n need not be invertible, ∼ there may not exist an isomorphism (Z/nZ)2S − → E[n]. To fix this problem, one introduces the notion of ‘‘full level-n structure’’ as the data of two points P, Q of order n (equivalently, a group homomorphism (Z/nZ)2S → E[n]) so that we have the following equality of Cartier divisors inside E: Õ E[n] = [iP + jQ]. i,j∈(Z/nZ)2 Over Z[1/n] this notion coincides with the previous level-n structure notion. When n is not invertible, this ‘‘correctly’’ counts the zero section with correct multiplicity. A mild generalization, introduced in [KM85] is the following: Definition 3.0.1. Let C/S be a smooth commutative group scheme over S of relative dimension one. Let A be a finite group (in the usual sense). A homomorphism of abstract groups ϕ : A → C(S) is said to be an A-structure on C/S if the effective Cartier divisor of degree #A defined by Õ [ϕ(i)] D= i∈A 17 is a subgroup G of C/S. A disadvantage of all above notions is that, given a finite flat group scheme G, it is not clear how to define a level structure on G intrinsically, i.e. without embedding G in a curve as a Cartier divisor. All the above notions depend on such an ambient curve where we can make sense of Cartier divisors. Katz and Mazur in [KM85] generalize the above definitions to an intrinsic notion of level structure that, i.e. not restricted to groups embeddable in a curve. In what follows, we call this level structure a Drinfeld-Katz-Mazur level structure, or sometimes ‘‘full set of sections.’’ Before we introduce the definition, let’s recall a few basic facts from commutative algebra. Let S be a scheme and Z/S a finite flat S-scheme of finite presentation and rank N ≥ 1 (typically N will be a power of a prime p). Reducing to the noetherian case, one can equivalently assume that Z/S is finite locally free over S of rank N ≥ 1. For every affine S-scheme Spec(R) → S, the base change Z R /Spec(R) is then the spectrum of an algebra B, locally free over R as an R-module. Thus, for every global section f ∈ B, one can view f as an R-linear endomorphism of B, and we can speak of the characteristic polynomial of f : det(T − f ) = T N − trace( f )T n−1 + . . . + (−1)N norm( f ). We are now ready to introduce Katz and Mazur’s notion of full set of sections: Definition 3.0.2 (Full set of sections). Let P1, . . . , PN ∈ Z(S) be N points (not necessarily distinct). One says that they form a full set of sections if for every affine S-scheme Spec(R) → S and for every f ∈ H 0 (Z R, OZR ), we have the equality of polynomials in R[T]: N Ö det(T − f ) = (T − f (Pi )). i=1 Here, we can interpret each Pi as a map of schemes Spec(R) → Z R , or, as a ring homomorphism H 0 (Z R, OZR ) → R, and f (Pi ) denotes the image of f under the aforementioned ring homomorphism. An equivalent statement of the determinant condition above is to require for any affine S-scheme Spec(R) and f ∈ OZR the following equality in R: norm( f ) = N Ö i=1 f (Pi ). 18 In the case that Z is embeddable in a curve C as a closed subscheme, which is finite flat of finite presentation over S, and we are given N not necessarily distinct points P1, . . . , PN of C(S), then Katz and Mazur show in theorem 1.10.1 [KM85] that the N Õ condition Z = [Pi ] is equivalent to the Pi ’s forming a full set of sections of Z/S. i=1 Example: Consider the zero map Z/pZ → µ p over an F p -scheme S. Write µ p as µ p = Spec B with B = OS [t]/(t p − 1). Since we are in characteristic p, one can also write B = OS [z]/z p , where z = t − 1 and z p = 0. The zero section sends z 7→ 0. Let R be an OS -algebra and f ∈ B. Write f (z) = a0 + a1 z + . . . + a p−1 z p−1 . Since z p = 0, multiplication by f is upper triangular with respect to the basis {1, z, . . . , z p−1 } with diagonal entries all equal to a0 and thus det(T − f ) = (T − a0 ) p = (T − f (0)) p , showing that the zero map is a full set of sections. We shall see below another way to analyze full sets of sections on µ p using Oort-Tate theory. Note that the zero map is not a full set of sections for µ p over R if pR , 0. One can show this computationally or otherwise embed µ p in Gm and then test whether the zero map is a level structure using Cartier divisors. If it were, we would have the equality (X − 1) p = X p − 1 in R[X], which is false if p , 0 on R. We shall need the following result from [KM85] (proposition 1.11.3 in loc. cit.): Lemma 3.0.1. Let S be an arbitrary scheme and consider a short exact sequence 0 → G1 → G → G2 → 0 of finite flat commutative S-group-schemes of finite presentation, and ranks N1, N and N2 respectively. Suppose give a short exact sequence 0 → A1 → A → A2 → 0 of (abstract) finite abelian groups of orders N1, N and N2 respectively. Consider a commutative diagram 0 / A1  0 φ1 / G 1 (S) /A  φ / G(S) / A2  /0 φ2 / G 2 (S) If for i = 1, 2 the homomorphisms φi are Drinfeld-Katz-Mazur level structures for Gi /S, then φ is a Drinfeld-Katz-Mazur level structure for G. 19 Proof of lemma: Indeed, let B2, B, B1 be the affine rings of G2, G and G1 respectively. For a ∈ A1, A and A2 respectively, denote by φ1,a , φa and φ2,a the ring maps B1 → R, B → R and B2 → R corresponding to the section that is the image of a in G1 (S), G(S) and G2 (S) respectively. We want to show that if f ∈ B, then NB/R ( f ) = Ö φa ( f ). a∈A By transitivity of the norm, the left hand side equals Ö NB2 /R (NB/B2 ( f )) = φ2,b (NB/B2 ( f )), b∈A2 since φ2 is a level structure. Denote f2 := NB/B2 ( f ). It suffices then to show that Ö φ2,b ( f2 ) = φa ( f ). A3a7→b∈A2 Note that the base change via φ2,b : S → G2 of G → G2 is just G1 → S. Thus, the norm from G to G2 can be computed as the norm from G1 to S. In other words, NB/B2 ( f ) = NB1 /R ( f1 ), where f1 is the image of f in B1 under the ring homomorphism B → B1 , corresponding to the above fiber product. Note that B → B1 , and thus f1 , depends on φ2,b . Thus, φ2,b ( f2 ) = φ2,b (NB/B2 ( f )) = NB1 /B ( f1 ). Since A1 → G1 (S) is a level structure, the right hand side equals Ö φ1,a ( f1 ). But a∈A1 φ1,a ( f1 ) = φa 0 ( f ), for a unique a0 ∈ A which maps to b. Taking the product over all a ∈ A1 and corresponding a0s, the conclusion follows. One can refer to the diagram below, where the parallelogram is cartesian and the triangles are commutative. f2 D ∈ B2 NB/B2  f ∈B φ2,b φa (/ A3a7→b ( R\  NB1 /R f1 ∈ B1  20 3.1 Oort-Tate generators In this section, we introduce the notion of Oort-Tate generators and show that they coincide with Drinfeld-Katz-Mazur level structures for finite flat group schemes of order p. Our main reference is [HR12]. Recall from Section 2.2 that these are classified by the Z p -stack OT = Spec Z p [X,Y ]/(XY − w p )/Gm with the universal group scheme G = SpecOT O[Z]/(Z p − X Z)/Gm of order p. Inside G we define the scheme of generators G × : it is the closed subscheme of G cut out by Z p−1 − X. The morphism G × → OT is relatively representable and finite flat of degree p − 1. Thus if G is a finite flat group scheme over a Z p -scheme S corresponding to a map ϕ : S → OT then G = ϕ∗ (G) and the subscheme of generators of G is G× = ϕ∗ (G × ). Let’s show that this notion is the same as Drinfeld-Katz-Mazur level structure. We check that if α ∈ G(S), then α ∈ G× (S) if and only if α has exact order p in the sense of Katz-Mazur, or, in other words, if and only if {0, α, . . . , (p − 1)α} is a full set of sections of G/S: Lemma 3.1.1. Let α ∈ G× (S) be an Oort-Tate generator. Then {0, α, . . . , (p − 1)α} is a full set of sections of G/S. Proof of lemma: As mentioned above, G corresponds to ϕ : S → OT via G = ϕ∗ (G). By localizing, we reduce to the affine case, so we can take S = Spec R. Then G = Spec R[Z]/(Z p − aZ), where a has the property that there exists b ∈ R so that ab = w p . (Note that in [HR12], a mistake seems to have been made, and a is assumed to be in pR). Since α is a section in G(S), it arises from a ring map R[Z]/(Z p − aZ) → R, i.e. is given by an element c ∈ R such that c p = ac. Note that Z has the property that [m]Z = χ(m)Z, for every 1 ≤ m ≤ p − 1. Thus (see also [HR12]) that the group of sections {0, α, . . . , (p − 1)α} corresponds to {0, c, ζ c, . . . , ζ p−2 c}, where ζ is a primitive p − 1th root of unity in R. Then α defines a Drinfeld-Katz-Mazur level structure (i.e. full set of sections) if Z p − aZ = Z(Z − ζ c)(Z − ζ 2 c) · . . . · (Z − ζ p−2 c) which is easily seen to be equivalent to a = c p−1 , hence α ∈ G× (S).  21 Chapter 4 VARIOUS MODULI PROBLEMS 4.1 Moduli problems In this chapter, we recall the definitions and basic properties of several moduli problems of p-divisible groups with different level structures. We do not use standard notation for these moduli problems. We largely follow the exposition of [HR12]. Denote by M Γ1 the stack of p-divisible groups defined as follows: as a fibered category, if S is a scheme over Z p , then M Γ1 (S) consists of (isomorphism classes) of p-divisible groups over S. The moduli problem M Γ0 (p) classifies p-divisible groups G with the following extra data: a filtration 0 = H0 ⊂ H1 ⊂ · · · ⊂ Hh = G[p], where Hi is a finite flat group scheme of rank pi and h is the height of G (compare with the Γ0 (n)-level in the introduction to the previous chapter). An equivalent datum is a chain of isogenies of p-divisible groups G → G1 → G2 . . . → G h−1 → G, each of degree p and whose composition is p · idG . Let S be a scheme over Z p and consider an S-point of M Γ0 (p) , corresponding to a p-divisible group G over S with a filtration Hi as above. Denote Xi := Hi /Hi−1 . The Xi ’s are finite flat group schemes of order p. Recall from 2.2 that OT denotes the Z p -stack of finite flat group schemes of order p and that over OT we have the universal finite flat group scheme of order p, denoted by G. Inside G we have a closed subscheme G × - the scheme of generators of G. From the definition of M Γ0 (p) there is thus a map M Γ0 (p) → OT ×Z p . . . ×Z p OT sending (G, (Hi )) 7→ (X1, . . . , Xh ). The moduli problem M Γ1 (p) is then defined as the 2-fiber product M Γ1 (p) / G× × . . . × G×   / OT × . . . × OT . M Γ0 (p) In other words, if S is a Z p -scheme, an S-point of M Γ1 (p) is given by a p-divisible group G over S, with a filtration 0 = H0 ⊂ H1 ⊂ · · · ⊂ Hh = G[p] as above and 22 with Oort-Tate generators for each quotient Hi /Hi−1 , which are finite flat of order p (compare with the Γ1 (n)-level in the introduction to the previous chapter). Note that we have forgetful morphisms M Γ1 (p) → M Γ0 (p) → M Γ1 . It follows from [Goe01] that M Γ0 (p) is flat over Spec(Z p ). In loc. cit. Gortz also writes down local model equations. From the 2-fiber product defining M Γ1 (p) and the fact that G × is flat over OT (or from [Sha15]) it follows that M Γ1 (p) is flat over M Γ0 (p) and is thus flat over Spec(Z p ). Let’s also introduce the moduli space M Γ(p)DKM . For any Z p -scheme S, M Γ(p)DKM (S) will consist of (isomorphism classes of) p-divisible groups G equipped with a Drinfeld-Katz-Mazur level structure on G[p]. The moduli space M Γ(p)DKM is not well-behaved for higher-dimensional p-divisible groups. For example, it is not even flat, as Chai and Norman show in [CN90]. This contradicts for example the flatness conjectures of Rapoport and Zink [RZ96]. The non-flatness suggests that the Drinfeld-Katz-Mazur notion of level structure is inadequate for higher-dimensional p-divisible groups. Another example of this phenomenon is presented in [CN90], where the authors show, that the scheme S of all (Z/pZ)2 -level structures on µ p × µ p over Z(p) is not flat by computing the dimensions of its special and of its generic fibers and showing they are distinct. Note that µ p × µ p is a truncated two-dimensional p-divisible group. We shall see the case of µ p × µ p again in the next chapter, as a non-example for the notion of level structure we are going to introduce. 23 Chapter 5 Γ(P)-LEVEL STRUCTURE Let S be a scheme over Z p and G a p-divisible group of height h and dimension d over S. We introduce the following notion of Γ(p)-level structure on G: Definition 5.0.1. A Γ(p)-level structure on G is the following data: 1. A filtration 0 = H0 ⊂ H1 ⊂ . . . ⊂ Hh = G[p] of G[p] by finite flat subgroup schemes, so that the rank of Hi is pi . 2. Oort-Tate generators γi : (Z/pZ) → Hi /Hi−1 . 3. Group homomorphism ϕi : (Z/pZ)i → Hi such that the following diagram commutes: 0 0 = H0 / Z/pZ  ϕ1 / H1 / (Z/pZ)2  / ... ϕ2 / H2 / ... / (Z/pZ) h−1  ϕh−1 / Hh−1 / (Z/pZ) h  ϕh / Hh = G[p] so that the induced maps Z/pZ → Hi /Hi−1 are the Oort-Tate generator γi . We denote it succinctly by ϕ := (ϕi ) = (ϕ1, . . . , ϕh ). Note that data 1 defines a Γ0 (p) level structure and G. Data 1 and 2 define a Γ1 (p) level on G[p]. Consider the functor which to an Z p -scheme S associated the set of p-divisible groups of height h and dimension d together with a Γ(p)-level structure, up to isomorphism. This is relatively representable by M Γ(p) which has a natural map to M Γ1 (p) obtained by ‘‘forgetting’’ the (vertical) group homomorphism in the above diagram. Sometimes, to emphasize that we are using our new definition, we write M Γ(p)new instead of M Γ(p) . Let’s establish some basic properties of this new notion of level structure. We first see that (Z/pZ)h → G[p] forms a full set of sections i.e. is a Drinfeld-Katz-Mazur level structure. In fact, we show: 24 Lemma 5.0.1. Let ϕ = (ϕ1, . . . , ϕh ) be a Γ(p)-level structure on a p-divisible group G. Then each ϕi is a Drinfeld-Katz-Mazur level structure on Hi . In particular, ϕh is a Drinfeld-Katz-Mazur level structure on G[p]. Proof of lemma: We use induction on i. For i = 1, it is part of the definition of a Γ(p)-level structure that ϕ1 : Z/pZ → H1 is an Oort-Tate generator, which by lemma 3.1.1 means that ϕ1 is a Drinfeld-Katz-Mazur level structure. For the induction step, apply Lemma 3.0.1 to the diagram 0 / (Z/pZ)i−1  0 / (Z/pZ)i ϕi−1  / Hi−1 (S) / Z/pZ ϕi /0  / (Hi /Hi−1 )(S). / Hi (S) The left vertical arrow is a Drinfeld-Katz-Mazur level structure by induction and the right one by definition. Lemma 3.0.1 then implies that ϕi also is a Drinfeld-KatzMazur level structure.  The content of Lemma 5.0.1 and the forgetful maps mentioned above and in the previous chapter can be summarized in the following diagram: M Γ(p)new  M Γ1 (p)
M Γ(p)DKM  M Γ0 (p) & 5.1  M Γ1 A (non-)example In this section, we will see that our notion of Γ(p)-level structure is different from than of Drinfeld-Katz-Mazur level structure. We give an example of a p-divisible group and a Drinfeld-Katz-Mazur level structure which does not arise as a Γ(p)-level structure. In a later section, we prove that for 1-dimensional p-divisible groups our notion of Γ(p)-level structure does agree with the Drinfeld-Katz-Mazur one, so the smallest (non-)example should arise from a p-divisible group of dimension at least 2. 25 Let G be the p-divisible group µ p∞ × µ p∞ over S = Spec(Z/p2 Z). This is a p-divisible group of height 2 and dimension 2. Let H := G[p] = µ p × µ p . Consider the map α : (Z/pZ)2 → G(S) given by sending every element to the zero section. We check that this is a Drinfeld-Katz-Mazur level structure. The argument is presented in Katz and Mazur [KM85], as a counter-example for a converse of Lemma 3.0.1. It also serves the purpose of an example of a Drinfeld-Katz-Mazur level structure which is not a Γ(p)-level structure, as introduced by us. For completeness, we present the argument from [KM85] here. The Hopf algebra of H is B = (Z/p2 Z)[X,Y ]/(X p − 1,Y p − 1). The zero section on B corresponds to the ring homomorphism sending X and Y to 1. Thus the p2 sections Pi in the definition of full set of sections, on the level of rings, do the following: for any Z/p2 Z-algebra R and f ∈ R ⊗Z/p2 Z B = R[X,Y ]/(X p − 1,Y p − 1) it sends f 7→ f (1, 1) ∈ R. Thus, to check that α is a Drinfeld-Katz-Mazur level structure, we need to verify the following: if R is a Z/p2 Z-algebra, and f ∈ R ⊗Z/p2 Z B = R[X,Y ]/(X p − 1,Y p − 1), then det(T − f ) = (T − f (1, 1)) p 2 or, equivalently, that norm( f ) = f (1, 1) p 2 In other words, it suffices to check that if R is any algebra, then norm( f ) ≡ f (1, 1) p 2 (mod p2 ). NX NY By transitivity of the norm, it factors as R[X,Y ]/(X p −1,Y p −1) −−→ R[Y ]/(Y p −1) −−→ R. Since the zero map is a full set of sections for µ p in characteristic p (as seen above), it follows that NX ( f ) = f (1,Y ) p + p · g(Y ), for some polynomial g ∈ R[Y ]/(Y p − 1). Note now that NY (h + pg) ≡ NY (h) mod p2 . Indeed, consider more generally the norm map NY (h + T g) : R[T][Y ]/(Y p − 1) → R[T] for an indeterminate T and express this as a polynomial in T: NY (h + T g) = NY (h) + T NY (g) + p p−1 Õ T i Qi (h, g). i=1 Since mod p the zero map is a full set of sections, NY (h + T g) mod p is just (h(1) + T g(1)) p = h(1) p + T p g(1) p . This implies that Qi (h, g) ≡ 0 (mod p) for every 26 i. Thus, NY (h+T g) ≡ NY (h) (mod pT,T p ). Taking T = p yields NY (h+pg) ≡ NY (h) (mod p2 ), as claimed. Thus, N( f ) = NY ( f (1,Y ) p + p · g(Y )) ≡ NY ( f (1,Y ) p ) = (NY ( f (1,Y ))) p = ( f (1, 1) p + p · α) p (mod p2 ) = 2 (mod p2 ) ≡ f (1, 1) p (mod p2 ), as wanted. We then check that this does not arise as a Γ(p)-level structure as in our definition. Indeed, it suffices to verify that the zero map Z/pZ → µ p does not form a full set of sections over the base S = Spec(Z/p2 Z). Indeed, we have already seen this in Chapter 3, the example following the definition of full set of sections - the zero map Z/pZ → µ p is not a full set of sections unless p = 0. 5.2 1-dimensional p-divisible groups In this subsection, we show that if G is 1-dimensional (i.e. d = 1) then our new notion of Γ(p)-level structure agrees with the existing notion of Drinfeld-Katz-Mazur level structure, as used for example in [HT01]. Using Lemma 5.0.1, we saw that given a Γ(p)-level structure ϕ = (ϕi ) as in our definition gives a Drinfeld-Katz-Mazur level structure, namely ϕh . This holds true for any dimension d. This gives us a morphism M Γ(p)new → M Γ(p)DKM . We show that a Drinfeld-Katz-Mazur level structure on a 1-dimensional p-divisible group G (over a base scheme S/Z p ) gives us a Γ(p)-level structure and these two associations are inverse to each other, in other words our notion of Γ(p)-level structure coincides with the existing notion of ‘‘full level structure’’ for 1-dimensional p-divisible groups. We shall need a preliminary lemma, sketched in [HT01]: Lemma 5.2.1. Let G be a 1-dimensional p-divisible group over a locally noetherian base scheme S. Then, for any i, locally on S, G[pi ] is embeddable in a smooth curve C/S. Proof of lemma: As we are working locally on the base, we can assume that S is affine. Let S = Spec A and G[pi ] = Spec R. Our goal is to find a surjection A[T]  R. It suffices to find, for each maximal ideal m of A and each f ∈ A − m a surjection A f [T]  R f . Using Nakayama’s lemma, it suffices then to find a 27 surjection (A/m)[T]  R/m, i.e. we have reduced to the case where the base S is the spectrum of a field. In fact, we can assume that this field is algebraically closed. Indeed, let N = pih be the rank of R over A as a module. The map A[T] → R is surjective if and only if the images 1,T, . . . ,T N−1 are linearly independent (so the image of A[T] → R has full rankÕ and is thus all of R). All these images are determined by the image of T. If T 7→ t j e j , where (e j ) is some basis of R over A, then linear independence is equivalent to a certain polynomial in the t j ’s being nonzero. This condition stays invariant when passing to algebraic closure. We have thus reduced the setup of the lemma to a 1-dimensional p-divisible group G over an algebraically closed field A. Recall that over perfect fields, we have a (split) connected-étale sequence (see section 2.3 in chapter 2), hence G[pi ] is a disjoint union of copies of G0 [pi ] and thus we reduce to the case where G is connected. But hi in this case we know that G[pi ] = Spec R for a ring R of the form A[[T]]/(T p ), and the map A[T] → A[[T]]  R is the desired surjective map.  Proposition 5.2.1. Let S be a base scheme over Z p and let G be a 1-dimensional p-divisible group over S. Consider a Drinfeld-Katz-Mazur level structure ϕ : (Z/pZ)h → G[p] on G. Then there is a unique filtration 0 = H0 ⊂ H1 ⊂ . . . ⊂ Hh = G[p] of G[p], so that each Hi has rank pi , ϕ restricted to (Z/pZ)i factors through Hi and is a Drinfeld level structure for Hi and the induced maps Z/pZ → Hi /Hi−1 for each i are Oort-Tate generators. Proof. We follow and elaborate on details of the proof of Lemma II.4.1 in [HT01]. Note that we do not need to assume that S is locally Noetherian and has a dense set of points with residue field algebraic over F p - we can always reduce to this case using standard results from EGA (see also [Wed00], page 306). By the lemma above, we are justified in applying Corollary I.10.3 in [KM85], which says that there is a unique closed subscheme Hi of G[p], locally free over S, for which ϕ|(Z/pZ)i → Hi is a Drinfeld level structure for Hi . We have to check that Hi is a subgroup of G[p]. The condition of Hi being a subgroup and the conditions imposed on the quotients Hi /Hi−1 , are both closed conditions, defined in fact by finitely many relations and generators. There is thus a closed subscheme S0 ⊂ S so that for any scheme T/S, Hi ×S T ,→ G[p] ×S T has the required properties if and only if T → S factors though S0. 28 Our goal is then to prove that S0 = S. We establish this via a series of reductions. The first one is to the case where S = Spec A for an Artinian local ring A. Indeed, to check equality of two closed subschemes, it suffices to do so at the local rings of a dense set of points. We then reduce to the Artinian local case and thus look at A = OS,s /mis as s runs through a dense set of points and i ∈ Z≥0 , A having residue field k algebraic over F p (recall that we can assume S has these properties, e.g. [Wed00], page 306). Recall that if R is any perfect ring of characteristic p, then W(R) is flat over Z p . Since checking equality of two closed subschemes (defined   by certain ideals) can be done after a flat base change, we can tensor with W k over Z p and it suffices to check S0 = S after this base change. In other words, we can assume S = Spec A where A is a local Artinian ring with algebraically closed residue field F p . But a p-divisible group H/A with a Drinfeld-Katz-Mazur level structure is obtained by pullback from the universal deformation on H × Spec(F p ), together with its level structure. This deformation has been studied explicitly by Drinfeld and Harris and Taylor, and we have thus reduced to proving S = S0 in the case of a p-divisible group H over a complete Noetherian local ring R, together with its level structure. This ring is flat over OQˆp unr , and thus after another flat base change, it suffices to check our assertion over Spec(R) × Spec(Q p ), which is dense in Spec(R). In this case, H is étale, the level structure is an isomorphism and the existence of the filtration as claimed is clear: for example, Õ embed H in a curve, and take the Hi ’s as sum of Cartier divisors: Hi := [ϕ(x)], viewed as a closed subscheme of H[pm ]. x∈(Z/pZ)i  5.3 Flatness The goal of this section is to prove that Theorem 5.3.1. M Γ(p)new is flat over Z p . Recall from the previous chapter that the maps M Γ1 (p) → M Γ0 (p) and M Γ0 (p) → Z p are flat. We will verify that M Γ(p)new is flat over M Γ0 (p) . Since the latter is flat over Z p , our result will follow. Theorem 5.3.2. M Γ(p)new is flat over M Γ0 (p) . 29 Proof. We will construct a tower of moduli spaces Mi for i = 0, 1, . . . , h − 1, h, so that M0 = M Γ0 (p) , M h = M Γ(p)new and such that Mi is flat over Mi−1 for every i = 1, . . . , h. Mi will be moduli space representing the following moduli problem: For a base scheme S/Z p , we consider the moduli problem classifying p-divisible groups of height h over S, together with a filtration 0 = H0 ⊂ H1 ⊂ . . . ⊂ Hh = G[p] together with i compatible group homomorphisms ϕ1, . . . , ϕi such that the induced maps Z/pZ → H j /H j−1 , 1 ≤ j ≤ i are Oort-Tate generators: 0 H0 = 0 / Z/pZ  ϕ1 / H1 / (Z/pZ)2  / ... / (Z/pZ)i−1 ϕ2 / H2  / ... / (Z/pZ)i ϕi−1 / Hi−1  ϕi / Hi Note that M0 is just M Γ0 (p) and M h = M Γ(p)new . There is a natural map Mi → Mi−1 obtained by forgetting ϕi and the Oort-Tate generator Z/pZ → Hi /Hi−1 . We will show that Mi is flat over Mi−1 for each i. It will follow that M Γ(p)new = M h is flat over M0 = M Γ0 (p) , as desired. The case i = 1 is slightly different from the others, and we treat it first: Let i = 1. By definition, M1 represents the functor over M Γ0 (p) whose S-points are given by a p-divisible group G/S together with a filtration (Hi ) as in the definition of Γ0 (p)-level, together with an Oort-Tate generator of H1 . This means that M1 is the 2-fiber product of the following diagram: M1 / G×   / OT M0 = M Γ0 (p) where the bottom arrow extracts H1 . Since G × is flat over OT (see 3.1), it follows that M1 is flat over M Γ0 (p) . Assume now that 2 ≤ i ≤ h and let’s show that Mi is flat over Mi−1 . Note that the forgetful map Mi → Mi−1 factors through Mi−1 ×OT G × . Indeed, this corresponds to forgetting ϕi , but not the induced Oort-Tate generator Z/pZ → Hi /Hi−1 . By a similar argument as in the i = 1 case, Mi−1 ×OT G × → Mi−1 is flat. It thus suffices to establish flatness of Mi → Mi−1 ×OT G × . 30 We show that Mi is in fact a fppf torsor over Mi−1 ×OT G × under Hi−1 and thus flat. Consider the diagram: 0 / (Z/pZ)i−1 / Z/pZ   ϕi−1  / Hi−1 0 / (Z/pZ)i / Hi /0 γi / Hi /Hi−1 /0 Observe that since ϕi−1 is a Drinfeld-Katz-Mazur level structure and Z/pZ → Hi /Hi−1 is an Oort-Tate generator, Lemma 3.1.1 implies that any group homomorphism, if it exists, completing the diagram as above is automatically a Drinfeld-KatzMazur level structure. Group homomorphisms ϕi : (Z/pZ)i → Hi yielding the above diagram commutative, if they exist, are classified by H := Hom(Z/pZ, Hi−1 ) = Hi−1 . We claim that Mi is a Hi−1 -torsor over Mi−1 ×OT G × . For that we need to verify that locally, the above diagram can be completed with a dotted arrow. Any extension filling in the above diagram arises in the following form: 0 / (Z/pZ)i−1 ϕi−1 / (Z/pZ)i / Z/pZ /0 0 / Hi−1  / ?1 / (Z/pZ) /0 0 / Hi−1  / ?2 / Z/pZ /0    / Hi−1 0 / Hi γi / Hi /Hi−1 /0 where the second row is the pushout under ϕi : (Z/pZ)i−1 → Hi−1 and the third row is the pull-back under γi : Z/pZ → Hi /Hi−1 . We thus need to verify that a dotted arrow such as above exists, locally on the base. Note that the top row 0 → (Z/pZ)i−1 → (Z/pZ)i → Z/pZ → 0 is a split extension, and thus corresponds to the trivial element in the corresponding Ext group. Thus the extension / Hi−1 /0 / ?1 / (Z/pZ) 0 is trivial, as it is obtained by pushout from a trivial extension. A dotted arrow as above then exists if and only if the class of 0 / Hi−1 / ?2 / (Z/pZ) /0 31 in the Ext group is trivial, i.e. if the sequence splits locally. Note that ?2 is the fiber product Hi ×Hi /Hi−1 (Z/pZ). Since Hi (and Z/pZ) are killed by p (see [TO70]), ?2 is also killed by p. We then invoke following lemma: Lemma 5.3.1. Let G be a commutative finite flat group scheme killed by p. Every short exact sequence 0 → H → G → (Z/pZ) → 0 splits locally in the fppf topology. Proof of lemma: If S is our base scheme, then any section in (Z/pZ)S (S) has, fppf locally, a preimage in G. Say T → S is such a scheme, i.e. 1 ∈ (Z/pZ)(T) acquires a preimage. But G(T), by assumption is killed by p. Note that G(T), as an abstract group, could be infinite (for example G = αp × αp over an infinite field k of characteristic p and T = k[]/( p )). However, any pre-image of 1 ∈ (Z/pZ)(T) gives a splitting, as G(T) is killed by p. In other words, every exact sequence of (abstract) abelian groups 0 → A → G(T) → Z/pZ → 0, where A and G(T) are killed by p, splits.   32 Chapter 6 EPIPELAGIC LEVEL AND CERTAIN GROUP SCHEMES OF ORDER P2 6.1 Epipelagic level structure In this section, we discuss yet another notion of level structure on p-divisible groups. The motivation comes from the study of epipelagic representations. Just like our notion of Γ(p)-level, it ‘‘lives’’ over M Γ1 (p) , however it is not ‘‘comparable’’ to it. To introduce the moduli problem on p-divisible groups, we first define: Definition 6.1.1. Let S be a scheme over Z p and G be a p-divisible group of height h over S. An epipelagic level structure on G is the data consisting of 1. A filtration 0 = H0 ⊂ H1 ⊂ . . . ⊂ Hh = G[p] of G[p] by finite flat group w schemes Hi of rank pi . Note that the isomorphism G[p2 ]/G[p] − → G[p] induced by multiplication by p allows us to extend this filtration to the right Hh−1 ⊂ Hh = G[p] = Hh+1 ⊂ . . . ⊂ G[p2 ], such that Hh+1 /G[p] w H1 . 2. Oort-Tate generators γi for the quotients Hi /Hi−1 (which are finite flat of rank p), for i = 1, . . . , h + 1 such that γh+1 = γ1 . 3. For 2 ≤ i ≤ h + 1, group homomorphisms Z/pZ → Hi /Hi−2 such that the composition Z/pZ → Hi /Hi−2  Hi /Hi−1 is the Oort-Tate generator γi : Z/pZ → Hi /Hi−1 . Denote by M I++ the set-valued functor on Sch/Z p whose values on S are given by p-divisible groups with an epipelagic level structure. Note that we have a natural map M I++ → M Γ1 (p) by forgetting the group homomorphisms Z/pZ → Hi /Hi−2 . We now show that M I++ is flat over Z p by showing that it flat over M Γ1 (p) , which is flat over Z p (see chapter 4). In fact we prove that Proposition 6.1.1. M I++ is a torsor over M Γ1 (p) under h+1 Ö Hi /Hi−1 . i=2 Proof. The data of a map Z/pZ → Hi /Hi−2 such that the composition with Hi /Hi−2 → Hi /Hi−1 is the Oort-Tate generator γi is the same as a dotted group 33 homomorphism making the following diagram commute: 0 / Z/pZ  0 / (Z/pZ)2 / Z/pZ   γi−1 / Hi−1 /Hi−2 / Hi /Hi−2 /0 γi / Hi /Hi−1 /0 If a group homomorphism completing the above exists, then all such group homomorphisms are classified by Hom(Z/pZ, Hi−1 /Hi−2 ) = Hi−1 /Hi−2 . The argument for i = 2 used in the proof of flatness for Γ(p)-level structures shows that existence is indeed satisfied.  Corollary 6.1.1. M I++ is flat over Z p . We now take a different turn. Recall that in writing down the local models of Shimura varieties for Γ1 (p)-level structure, a crucial gadget was the classification of group schemes of order p due to Oort-Tate and level structures on them (see [Sha15] and [HR12]). Similarly, when trying to write down the local model for the moduli space of p-divisible groups with epipelagic level structure, a useful gadget would be a classification of group schemes G of order p2 , killed by p, equipped with an extension structure 0 → H1 → G → H2 → 0, or at least of such extensions when H2 = Z/pZ. We will give a partial result in this direction, giving an explicit classification of extensions of Z/pZ by µ p over Z p -algebras R. Note that one can compute (abstractly) 1 (Spec(R), µ ). Using Kummer theory, the latter the group Ext1 (Z/pZ, µ p ) as Hfppf p ∗ ∗ p is (locally) isomorphic to R /(R ) . We will arrive at this result from a different direction, by analyzing in detail what happens on the level of Hopf algebras, using Oort-Tate theory along the way. It is our hope that such an approach can be generalized to classify other extensions. 6.2 Certain group schemes of order p2 Let R be any algebra and  ∈ R× a unit. Consider the following group schemes, which we denote by G , introduced in [KM85]. The affine ring of G is B := p−1 Ê p R[Xi ]/(Xi −  i ). i=0 For any R-algebra A with connected spectrum T = Spec A, we have G (T) = {(t, i)| t p =  i, 0 ≤ i ≤ p − 1} 34 Addition is defined as (t, i) · (s, j) =    (st, i + j), if i + j < p    (st/, i + j − p), if i + j ≥ p  Comultiplication is given by Xi = (0, . . . , 0, Xi, 0, . . . , 0) 7→ i Õ X j ⊗ Xi− j + j=0 p−1 Õ X j ⊗ Xp+i− j j=i+1  and Õ ei = (0, . . . , 0, 1, 0, . . . , 0) 7→ j+k=i e j ⊗ ek , (mod p) as is usual for constant group schemes. Here we view ei and Xi as elements in the p ith component R[Xi ]/(Xi −  i ). The augmentation ideal has rank p2 − 1 and equals p p hX0 − 1i ⊕ R[X1 ]/(X1 − ) ⊕ · · · ⊕ R[Xp−1 ]/(Xp−1 −  p−1 ). p Projection onto the first factor B  R[X0 ]/(X0 − 1) and the map sending Xi → 0 for every i (on functor of points, this sends (a, i) 7→ i) induces a short exact sequence of group schemes 0 → µ p → G → Z/pZ → 0 This sequence splits if and only if  is a pth power of a unit in R. 6.3 Extensions of Z/pZ by µ p in characteristic p In this section we prove that in characteristic p, locally on the base, the group schemes G defined above are the only finite flat extensions of Z/pZ by µ p . Let R be a (connected) F p -algebra. Since Z/pZ, as a scheme over S = Spec(R), has p disjoint Ý Ý Ý components, an extension G of Z/pZ by µ p is of the form G0 S1 . . . Sp−1 , where G0 = µ p and Si are schemes, flat over S = Spec(R) of rank p. For each i = 1, . . . , p − 1 we have an action G0 × Si → Si . Let us investigate this action. We need to use certain results about µ p actions from [Tzi17]. That paper studies actions of αp and µ p on schemes and the result we need is only proved for αp , so we prove it for µ p here: Lemma 6.3.1. Let X be a scheme of finite type over a ring R of characteristic p > 0. X admits a nontrivial µ p action if and only if X has a nontrivial global vector field D such that D p = D. Before we begin the proof, let us summarize the results of Proposition 2.2.1 in characteristic p: 35 Proposition 6.3.1 ([TO70]). Let R be a ring of characteristic p and consider the Hopf algebra B = R[z]/(z p − 1) of µ p /Spec R. Then, there is an element y such that: 1. I1 is generated by y and I j is generated by y j , for 1 ≤ j ≤ p − 1 (notation from chapter II). 2. For 1 ≤ j ≤ p − 1, we have y j = j!y j . We denote w j := j!. Moreover y p = w p y = 0, for some w p ∈ Z p equal to p · (unit). 3. ∆(yi ) = yi ⊗ 1 + 1 ⊗ yi + p−1 Õ y j ⊗ yi− j j=1 and in particular ∆(y) = y ⊗ 1 + 1 ⊗ y + p−1 i Õ y j=1 4. z = 1 + y + wi ⊗ y p−i w p−i y p−1 y2 +···+ . 2! (p − 1)! Let us go back now to the proof of the Lemma 6.3.1. Proof. Suppose first that X has a nontrivial vector field D such that D p = D. Let y be the element in the Hopf algebra of µ p described above (so the Hopf algebra of µ p is Spec R[y]/(y p − w p y) = Spec R[y]/(y p ); however y is not z − 1 in the traditional representation of µ p as R[z]/(z p − 1)). Consider the map Φ : O X → O X [y]/(y p ) by setting Φ(a) = p−1 Õ D(m) (a) m=0 m! ym Let’s check that this defines an action of µ p on X. Recall that the comultiplication p−1 i Õ y y p−i p p p ∆ : R[y]/(y ) → R[y]/(y ) ⊗ R[y]/(y ) sends y 7→ y ⊗ 1 + 1 ⊗ y + ⊗ . w w p−i i=1 i 36 We need to verify that the following diagram commutes: / O X ⊗ R R[y]/(y p ) Φ OX Φ  1⊗∆ / O X ⊗ R R[y]/(y p )  Φ⊗1 O X ⊗ R R[y]/(y p ) ⊗ R R[y]/(y p ) p−1 Õ D(m+n) (a) y m ⊗ y n under the above-diagonal m!n! m,n=0 composition of the above diagram. The below-diagonal composition sends Indeed, a section a ∈ O X gets sent to a 7→ p−1 Õ D(m) (a) m=0 m! ∆(yi ) Since p−1 Õ © ª ∆(y ) = ∆(wi yi ) = wi ∆(yi ) =wi ­ yi ⊗ 1 + 1 ⊗ yi + y j ⊗ yi− j ® = j=1 « ¬ p−1 j Õ 1 y yi− j ª ©1 ⊗ = wi ­ yi ⊗ 1 + 1 ⊗ yi + ® wi wi w j wi− j j=1 « ¬ i we can see that the coefficients of y n ⊗ y m is equal to   D(m+n) (a) 1 1 · wm+n · ⊗ , (m + n)! wn w m if m + n < p and   1 1 D(m+n−p+1) (a) · wm+n · ⊗ , (m + n)! wn w m if m + n ≥ p respectively. The result then follows since wi = i! (mod p) for 1 ≤ i ≤ p − 1 and D p = D. Assume now X admits a non-trivial µ p -action given by α : µ p × X → X. This corresponds to a co-action map α∗ : O X → O X ⊗ R R[y]/(y p ). By definition, we then have the commutative diagram: OX  α∗ α∗ O X ⊗ R R[y]/(y p ) 1⊗∆ / / O X ⊗ R R[y]/(y p )  α∗ ⊗1 O X ⊗ R R[y]/(y p ) ⊗ R R[y]/(y p ) 37 Write α∗ (a) = a ⊗ 1 + Φ1 (a) ⊗ y + p−1 Õ Φ j (a) ⊗ y j . j=2 It is clear that the Φi ’s are R-linear. Since y p = 0 and α∗ is a ring homomorphism, the relation α∗ (ab) = α∗ (a)α∗ (b) implies that Φ1 (ab) = aΦ1 (b) + bΦ1 (a), i.e. that Φ(i) Φ1 is a derivation. We now prove by induction that Φi = 1 . i! Indeed, the coefficient of y ⊗ yi under the above-diagonal compositions is Φ1 (Φi (a)). Under the below-diagonal composition, this is wi+1 Φi+1 (a) . wi w1 By induction, Φi (a) = Φ(i) 1 (a) i! , thus Φi+1 (a) = Φ(i+1) (a) 1 (i + 1)! since wi = i! for every i. Finally, equating the coefficients of y ⊗ y p−1 under the two compositions, one obtains D p = D, proving the lemma. We will also need the following result (mild generalization of lemma 4.1 in [Tzi17]): Lemma 6.3.2. Let A be an Z p -algebra and M and A-module. Let Φ : M → M p−1 be an A-module homomorphism such that Φ p = Φ. Then M = ⊕i=0 Mi , where Mi = {m ∈ M |Φ(m) = χ(i)m}, where χ : F p → Z p is the Teichmuller character defined in section 2.2. We remark that this lemma is a purely algebraic result, similar to the one used in [TO70] to decompose the augmentation ideal of a group scheme of order p into a direct sum of p − 1 modules. The only prerequisite is the presence of p − 1 roots of unity. Note that the proof referenced in [Tzi17] is for the case of F p -algebras, and is based on Lemma 1, A.V. 104 in [Bou03]. The Z p case is nearly Õ identical to the F p case: while in loc. cit., over F p one has the identity 1 = Pi (x), where X − Xp Pi (x) = ; over Z p the sum is still a unit in Z p [X]/(X p − X), so the argument X − χ(i)  Õ p of loc. cit. goes through. Indeed, one has the identity 1 + X p−1 Pi (x) = 1. 1−p Returning to our problem, the action of µ p on Si = Spec(Ai ) thus gives R-derivations p Di on Ai such that Di = Di . Using the above lemma, we decompose each Ri as a direct sum of p submodules Mi,j , j = 0, . . . , p − 1. We shall analyze these derivations and decompositions. Because we repeat this process for every Ai , we 38 drop (temporarily) the subscript i and thus write A, D, M j instead of Ai, Di and Mi,j respectively. We check that each M j is locally free of rank 1. Indeed, it suffices to prove that at every geometric point x, the vector space (M j ) x is 1-dimensional. But at geometric points all extensions of Z/pZ by µ p are split and thus the schemes Si are actually copies of µ p = Spec k[X]/(X p − 1), where k is an algebraically closed field. In this case one checks that Mi is spanned by yi , where y is, as usual, the generating Oort-Tate section in the Oort-Tate description of µ p (see Proposition 6.3.1). Thus M1 is free of rank 1 and let f be a generating section of M1 . Then D( f ) = f by definition and an easy induction shows that D( f i ) = i f i . In fact because the image of f i at geometric points is nonzero, it follows that f i , 0. Moreover, because the decomposition of OSi as a sum of eigenspaces of D is direct, it follows that the f i s are linearly independent for 0 ≤ i ≤ p − 1 (compare also Lemma 2 on page 9 of [TO70]). Since however Si is finite flat of rank p over S = Spec(R), the section f satisfies a monic polynomial of degree p (e.g. the characteristic polynomial), and this polynomial is killed by D. On the other hand, D( f i ) = i f i for 0 ≤ i ≤ p − 1. It follows that the only possible polynomial relation must be of the form f p = , where  is some constant in R. Thus each Si of the form Spec R[X]/(X p − i ). Note that we don’t know yet i are units in R. However, we have also not fully used the fact that G is a group scheme of order p2 - we have only exploited the action of µ p on the Si ’s. We will now use p the ‘‘global’’ group structure to conclude that each i is a unit and in fact i = 1i · ui where the ui are some units (i.e. i =  i up to multiplication by pth power of units). Denote by αi the action map µ p × Si → Si . The group structure on G induces further actions αi j : Si × S j → Si+ j if i + j < p and actions αi j : Si × S j → Si+ j−p if i + j ≥ p. All these actions commute with the S0 = G0 = µ p action. In other words, we have, for i + j < p, a commutative diagram G0 × Si × S j αi ×id  Si × S j id ×αi j / G 0 × Si+ j αi j  / Si+ j 39 Let’s analyze the ring maps induced by the above diagram: id ×αi j R[y]/(y p ) ⊗ R[Xi ]/(Xi − i ) ⊗ R[X j ]/(X j −  j ) o p p p R[y]/(y p ) ⊗ R[Xi+ j ]/(Xi+ j − i+ j ) O O αi∗ ×id R[Xi ]/(Xi − i ) ⊗ R[X j ]/(X j −  j ) o p p ∗ αi+j p R[Xi+ j ]/(Xi+ j − i+ j ) αi j Note that we can write the maps αi∗ explicitly. They are determined by derivations p Di on Spec R[Xi ]/(Xi − i ) such that Di (Xik ) = k Xik . An easy induction shows that Dim (Xik ) = k m Xik . Thus the co-actions αi∗ satisfy: αi∗ (Xik ) = p−1 Õ Dm (X k ) i i m! m=0 ⊗y = m p−1 m k Õ k X i m! m=0 Fix some indices i and j. Assume αi j (Xi+ j ) = Õ ⊗ ym . ckl Xik ⊗ X jl . Let’s look at what kl happens to Xi+ j in the above commutative diagram. Its image in the top-left ring is a sum of monomials Xik ⊗ X jl ⊗ y m with various coefficients. For each k, l, m, equating these coefficients from the two compositions yields ckl · k m ckl = . m! m! Since this is true for any k, l, m, we conclude that ckl = 0 if k , 1. By symmetry p ckl = 0 if l , 1. Thus αi j (Xi+ j ) = c11 Xi ⊗ X j . Since Xi = i , this implies p i+ j = c11 i  j . Recall that the indices were fixed and c11 depends on the pair (i, j). A straightforward p induction then shows that i = 1i · ui , for some ui ∈ R. It remains to check that i are units. By the above, we can assume that i =  i where  = 1 . Note that for every 1 ≤ i ≤ p − 1 we also have maps βi : Si × Sp−i → G0 = µ p . These commute with the actions of G0 on the Si ’s, making the following diagram commutative: id ×βi / G0 × G0 G0 × Si × Sp−i αi ×id  Si × Sp−i βi  m / G0 40 The induced maps on rings are id ×βi R[z]/(z p − 1) ⊗ R[Xi ]/(Xi −  i ) ⊗ R[Xp−i ]/(Xp−i −  p−i ) o p p O R[z]/(z p − 1) ⊗O R[z]/(z p − 1) αi∗ ×id ∆ R[Xi ]/(Xi −  i ) ⊗ R[Xp−i ]/(Xp−i −  p−i ) o p p βi R[z]/(z p − 1) Note that we have switched, for computational convenience, to the presentation R[z]/(z p − 1) of µ p , rather than R[y]/(y p − w p y). The relations between y and z are given in proposition 2.2.1. Õ l Assume βi (z) = dkl Xik ⊗ Xp−i . k,l Then the image of z in the top-left ring under the above-diagonal composition is Õ l dkl z ⊗ Xik ⊗ Xp−i . k,l Recall that αi∗ (Xik ) = D(m) (Xik ) Õ k m m y ⊗ = y ⊗ Xik . m! m! m≥0 m≥0 Õ m Thus, the image of z under the below-diagonal composition is  Õ Õ km m ∗ k l l dkl αi (Xi ) ⊗ Xp−i = dkl · y ⊗ Xik ⊗ Xp−i . m! k,l k,l,m l , we obtain Equating the coefficients of Xik ⊗ Xp−i Õ km dkl z = dkl On the other hand, from 2.2.1, z = m! m≥0 ym . Õ 1 y m . Thus, if k , 1, we must have dkl = 0. m! m≥0 By symmetry (i.e. letting G0 act on Sp−i instead of Si ), we obtain that dkl = 0 if p l , 1. Thus βi (z) = d11 Xi ⊗ Xp−i . Since z p = 1, we must have d11  i ·  p−i = 1, and thus  is a unit, as wanted.  6.4 Extensions of Z/pZ by µ p over any Z p -algebra Most of the above results go through, with minor modifications which we explain here. We still have the commutative diagram: OX  α∗ α∗ O X ⊗ R R[y]/(y p ) 1⊗∆ / / O X ⊗ R R[y]/(y p )  α∗ ⊗1 O X ⊗ R R[y]/(y p ) ⊗ R R[y]/(y p ) 41 Write α∗ (a) = a ⊗ 1 + Φ1 (a) ⊗ y + p−1 Õ Φ j (a) ⊗ y j . j=2 We show that Φi (a) = i−1 Φ(i) 1 (a)(1 − p) wi When p = 0, we recover the formula Φi (a) = . Φi1 (a) . The proof is nearly identical to i! the characteristic p case - one looks at the coefficient of y ⊗ yi−1 . One picks up an extra (1 − p) because it is present in the comultiplication rule for y (see 2.2.1). By looking at the coefficient of y ⊗ y p−1 , we deduce that  p D(1 − p) = D(1 − p). Thus in the presence of p − 1 roots of unity, we still have a nice decomposition of O X into eigenspaces of D(1 − p), with eigenvalues 0, χ(1) = 1, χ(2), . . . , χ(p − 1). In our case X = S1 is of rank p and looking at geometric points just as in the characteristic p case, since every extension of Z/pZ by µ p splits, one sees that these are all of rank 1. A difference from the case when p = 0 is that α∗ (ab) = α∗ (a)α∗ (b) no longer implies that D = Φ1 is a derivation. However we see below that it still has strong properties. The relation α∗ (ab) = α∗ (a)α∗ (b) together with y p = w p y then implies D(ab) = aD(b) + bD(a) + w p p−1 Õ Φ j (a)Φ p− j (a) j=1 = aD(b) + bD(a) + w p p−1 Õ (1 − p) p−2 D( j) (a)D(p− j) (a) . w w j p− j j=1 Let f generate the invertible module corresponding to eigenvalue 1, so (1 − p)D( f ) = f . Using the above relation, we show that (1 − p)D( f ) = f implies (1 − p)D( f 2 ) = χ(2) f and more generally (1 − p)D( f i ) = χ(i) f i . Indeed, (1 − p)D( f ) = f implies (1 − p)i D(i) ( f ) = f , thus D( j) ( f )D(p− j) ( f ) = 1 f 2 . Thus, plugging a = b = f in the above relation yields: (1 − p) p p−1 wp Õ 2 1 ª D( f ) = f ­ + ® 2 1 − p (1 − p) j=1 w j w p− j « ¬ 2 2© 42 This implies that (1 − p)D( f 2 ) = c2 f 2 for some constant c2 . Since (1 − p)D is fully diagonalizable with a full set of distinct eigenvalues 0, χ(1), . . . , χ(p − 1), it follows that c2 is one of those. Since w p = p · unit, we have c2 ≡ 2 (mod p), thus c2 must be χ(2). In fact, an easy induction then shows that there are constants ci , congruent to i mod p, such that D( f i ) = ci f i and then a similar argument shows that ci = χ(i). Alternatively, to show directly D( f 2 ) = χ(2) 2 f , it suffices to verify the identity: 1−p p−1 wp Õ 1 χ(2) = 2 + . 1 − p j=1 w j w p− j A series of similar identities can be derived for each χ(i), 2 ≤ i ≤ p − 1. Computing then D( f p ) (for example from α∗ ( f p ) = α∗ ( f )α∗ ( f p−1 )), one finds that it vanishes (even if we are not in characteristic p and D is not a derivation!). Thus, any polynomial relation involving f must be of the form f p =  as before. p Thus, Si = Spec(R[Xi ]/(Xi − i )). The rest of the argument in the characteristic p p case goes through unchanged, and we deduce that i = 1i · ui for some unit ui . Moreover, the global group structure implies, by a nearly identical argument, that 1 is a unit. The only changes are the appearance of (1 − p) factors in the formulas, the replacement of m! by wm throughout the formulas and D(m) (Xik ) = k m Xik is replaced χ(k)m k by X . (1 − p)m i Thus G is isomorphic to a group scheme of the form G , as we wanted to show. 43 Chapter 7 LEVEL STRUCTURES ON DIEUDONNÉ MODULES In this chapter, we present an auxiliary result about interpreting Drinfeld-Katz-Mazur level structures using (contravariant) Dieudonné theory. Gabber (unpublished) and Eike Lau [Lau10] have shown the Dieudonné equivalence over perfect bases. We briefly recalled this equivalence in Section 2.4. Thus, the notion of Drinfeld-KatzMazur level structure should have a (semi-)linear algebraic analogue. We restrict our attention to a perfect base schemes S, locally given by S = Spec R, where R is a perfect ring (over F p ). The goal of this chapter is to give an interpretation of the notion of Drinfeld-Katz-Mazur level structure in the Dieudonnś setting. We will use some results of [HT01] but to simplify the exposition, we will work with (usual) p-divisible groups instead of Barsotti-Tate OK -modules as is done in [HT01] (so K = Q p and OK = Z p ). Let thus S be a base scheme over Z p and H/S a p-divisible group of height h endowed with an embedding Z p ,→ End(H). We do not assume H is 1-dimensional. Recall that a Drinfeld-Katz-Mazur pm -level structure on H/S is then a morphism of Z p -modules: α : (Z/pm Z)⊕h w (p−m Z p /Z p )⊕h → H[pm ](S) so that the set of α(x) for x ∈ (Z/pm Z)h forms a full set of sections of H[pm ]. Note that, from the definition of endomorphisms of p-divisible groups, Z p acts on each ‘‘level’’ H[pm ], i.e. H[pm ](T) is a Z p -module for any scheme T over S. By embedding Z p into End(H), we embed Z p into End(MH ), hence the Dieudonné module MH of H becomes itself a Z p -module. Then an abstract group homomorphism (Z/pm Z)h → H[pm ](S) is a homomorphism of Z p -modules if and only if the induced homomorphism of Dieudonné modules MH = Wm (R)h → Mconst = Wm (R)h is a Z p -module homomorphism. 7.1 A commutative diagram Let H be a p-divisible group as above, m ≥ 1 an integer and G := H[pm ]. Consider a group homomorphism f : (p−m Z/Z)⊕h → G. This must commute with 44 the action of Frobenius (and Verschiebung), so we have the following diagram: /G (p−m Z/Z)⊕h w Frob   (7.1) Frob / G (p) (p−m Z/Z)⊕h (p) Locally on S, the Dieudonné module of H is free of rank h over W(R) (and thus the Dieudonné module of G is free of rank h over Wm (R)). Choose a basis. The corresponding diagram of contravariant Dieudonné modules is ⊕h o Wm (R) O ⊕h Wm (R) O Frob w Wm (R)⊕h o (7.2) B Frob A Wm (R)⊕h Note that the left side of the diagram are the Dieudonné modules of the constant group and the ones on the right those of G. The diagram can be interpreted and used in two (equivalent) ways: we either view the bottom two Dieudonné modules as identical to the ones above, and the vertical maps are semi-linear maps; or the vertical maps are linear and the modules on the bottom row are Frobenius twists of the ones above. We will mainly work in the linear setting. Here A and B are h × h matrix with entries in Wm (R). The condition for the map to ‘‘commute’’ with the Frobenius is Frob A = AB, where Frob A is obtained by raising each entry of A to the pth power. To see this one can take a basis e1, . . . , eh of the Dieudonné module corresponding to G (which is free over Wm (R)) and look at the action of each map. Let’s make this a bit more explicit as follows: write MG for the Dieudonné module of G and Mconst for that of (p−m Z/Z)Sh . Then, in the semi-linear point of view, we have: o A Mconst MO G O Frob w Mconst o B Frob A MG 45 In the linear viewpoint, we have o Mconst O A MO G Frob wid B Frob Mconst ⊗W(R),σ W(R) o A⊗1 MG ⊗W(R),σ W(R) Note that in the linear case, the matrix of A ⊗ 1 is obtained by applying the Frobenius automorphism of W(R) to each entry of A (note that the entries are of the matrices are elements of Wm (R)). So we do indeed get, as matrices Frob A = AB, (p) as maps from MG → Mconst . Note that id corresponds to the identity matrix but not the identity homomorphism, as the modules are different. 7.2 Level structures A Drinfeld-Katz-Mazur level structure is, in particular, a group homomorphism, so the commutativity conditions in Section 7.1 hold (in the category of Dieudonné modules). The goal is to further translate what does it mean to be a ‘‘full set of sections’’ into a condition involving A and B. Let B(i) denote the linear homomorphism corresponding to the Frobenius F : i−1 i G(p ) → G(p ) , i.e. B(i) is a linear map MG ⊗W(R),σi W(R) → MG ⊗W(R),σi−1 W(R). Note that B = B(1) . Denote by Ni (B) the composition B(i) ◦ B(i−1) ◦ . . . ◦ B(1) . This is a linear map MG ⊗W(R),σi W(R) → MG . Let s = mh be a large enough integer. Consider the diagram of linear maps. ⊕h o Wm (R) O A id w ⊕h Wm (R) O Ns (B) Wm (R)⊕h ⊗W(R),σ s W(R) o s Frob A Wm (R)⊕h ⊗W(R),σ s W(R) Since the vertical arrow on the left is an isomorphism, it follows that we always have i : ker Ns (B) ,→ ker(Frobs A) for any group homomorphism (Z/pZ)h → G, not from a level structure. We will prove the following result: Theorem 7.2.1. Let R be a perfect ring and H be a p-divisible group of height h over S = Spec R. Consider a group homomorphism α : (p−m Z/Z)h → H[pm ]. Let A be the corresponding homomorphism of associated Dieudonné modules. Then α is a level structure on H if and only if the inclusion i is an isomorphism, i.e. if and only if ker Ns (B) = ker(Frobs A). 46 7.3 The case S = Spec(k), where k is a perfect field Both to motivate our guess and as an aid for the general proof, let’s prove the main theorem 7.2.1 when R is a perfect field k. We shall need parts 4 and 5 of a lemma from Harris-Taylor [HT01]: Lemma 7.3.1 (II.2.1). Let S be an Z p -scheme and H/S a Barsotti-Tate Z p -module of constant height h. 4. Suppose that S is connected and that over S there is an exact sequence of p-divisible groups 0 → H 0 → H → H ét → 0 with H 0 formal and H ét ind-étale. Then α : (Z/pm Z)h → H[pm ](S) is a Drinfeld pm -level structure if and only if there is a direct summand of M ⊂ (Z/pm Z)h such that: 1. α| M : M → H 0 [pm ](S) is a Drinfeld pm -structure. 2. α induces an isomorphism w α : ((Z/pm Z)h /M)S − → H ét [pm ]. 5. Suppose that S is reduced, connected and that p = 0 on S. Suppose also that there is an exact sequence of p-divisible groups (0) → H 0 → H → H ét → (0), over S with H 0 formal and H ét ind-etale. If H/S admits a Drinfeld pm -level structure (with m ≥ 1) then there is a unique splitting H[pm ] w H 0 [pm ] × H ét [pm ] over S. On the other hand if there is a splitting H[pm ] w H 0 [pm ] × H ét [pm ]/S then to give a Drinfeld pm -structure α : (Z/pm Z)h → H[pm ](S) is the same as giving a direct summand M ⊂ (Z/pm Z)h and an isomorphism   w (Z/pm Z)h /M − → H ét [pm ]. S Our goal is thus to establish: 47 Theorem 7.3.1 (Field case). Let k be a perfect field and H be a p-divisible group of height h over S = Spec k. Consider a group homomorphism α : (Z/pm Z)h → H[pm ]. Let A be the corresponding homomorphism of associated Dieudonné modules. Then α is a level structure on H if and only if i : ker Ns (B) ,→ ker(Frobs A) is an isomorphism, i.e. if and only if ker Ns (B) = ker(Frobs A) (see discussion before this subsection). Proof. Over a perfect field k, H (and H[pm ]) has a connected-étale sequence that splits (Demazure [Dem72], page 34). Note that such an exact sequence exists over any field k, but only splits over a perfect field k. Write G := H[pm ] and consider 0 → G0 → G → Gét → 0. Note that G0 and G et are truncated BT groups. The Dieudonné module MG of G can be written as a direct sum Mét ⊕ Mloc (see Fontaine [Fon77]). In fact, Mét = ∩n≥1 F n M and Mloc = ∪n≥1 ker F n . Since the Frobenius commutes with the Z p -action, it is easy to see that Mét and Mloc are Z p -modules. Recall from [Dem72] that a high enough power of the Frobenius kills Mloc (in the semi-linear setting; if we work in the linear setting, this statement says that Ni (B) kills Mloc ⊗W(R),σi W(R) for large enough i). Moreover, the Frobenius is an isomorphism on Mét . In fact, the high enough exponent
that suffices is i = mh =: s works. Thus ker Ns (B) = MG ⊗W(R),σ s W(R) loc .
Assume first that ker Ns (B) = ker(Frobs A) = MG ⊗W(R),σ s W(R) loc . We want to show that α is a level structure. Note that A is injective on the étale part. Consider the image P of Mét , under A, sitting inside Mconst = Wm (R)h . Note that P is stable under Frobenius, hence P is a Dieudonné submodule of Mconst . Moreover P is a Z p -module (because Mét is). The inclusion P ,→ Mconst corresponds thus a quotient map (Z/pm Z)Sh → ((Z/pm Z)h /N)S for a subgroup N of (Z/pm Z)h . The restriction of A to Mét gives an isomorphism between Mét and P, hence it corresponds to an isomorphism ((Z/pm Z)h /N)S w Gét . This shows that ((Z/pm Z)h /N)S is not killed by pm−1 (because Gét is not). Together with the quotient being constant, we get that N cannot be an arbitrary subgroup of (Z/pm Z)h (i.e. there are restrictions), in fact it has to be a direct summand (and also a Z p -module as noticed above). Thus we can use Lemma 7.3.1 (II.2.1 part (5) of Harris-Taylor [HT01]) to conclude that α is a level structure (keep in mind the clash of notation between the theorem and the lemma). 48 Assume now that α is a level structure. Invoking again lemma 7.3.1 (II.2.1 part 4 in [HT01]) we see that there is a summand N of (p−m /OK )⊕h so that α induces an isomorphism α|N : ((p−m /OK )h /N)S → G = H ét [pm ] and α|N : N → H 0 [pKm ](S) is a level structure. But since S = Spec R is reduced, part 3 of the same lemma implies that the latter condition simply means that α|N is the zero homomorphism N → H(S). Thus, the matrix A is an isomorphism from Mét to a Dieudonné OK -submodule of Mconst = Wm (R)⊕h and vanishes on Mloc which gives us ker(Frobs A) = ker Ns (B) (both equal to Mloc ), as wanted.  7.4 Integral domain base Let now R be an perfect integral domain of characteristic p. Denote K := Frac(R) and η the generic point of S = Spec R. The ring R injects into K. Moreover W(R) is an integral domain which injects into W(K) ([Shi14], Lemma 3.7). Lemma 7.4.1. K is flat over R and W(K) is flat over W(R). Proof of lemma: The first part is basic commutative algebra. The second part is more subtle. From [Shi14], Proposition 5.2 it follows that W(K) is flat over W(R). Indeed, Proposition 5.2 in loc. cit. says that W(K) w W(B)(p)
∧ The proof of this proposition shows that W(B)(p) is Noetherian. Thus, W(K) is obtained via a localization and a completion of a Noetherian ring, and both these operations are flat.  Lemma 7.4.2. If R → S is a flat ring homomorphism of perfect F p -rings, then Wm (R) → Wm (S) is flat for any m ≥ 1. Proof of lemma: We apply Theorem 1 in Bourbaki, Alg. com., III, §5 [Bou85]. We reproduce below the part we need. We have to verify that we are justified in applying the result. Using the notation of the Bourbaki lemma, we have A = Wm (R), M = Wm (S) and J = (p). Since the rings are perfect J is indeed nilpotent. M/JM = Wm (S)/pWm (S) w S and A/J = Wm (R)/pWm (R) w R. By assumption R → S is flat, thus M/JM is flat over A/J. 49 Finally gri (A) = pi Wm (R)/pi+1Wm (R) w R, since R is perfect. Similarly, gri (M) = pi Wm (S)/pi+1Wm (S) w S. Thus, the requirement gr(A) ⊗gr0 (A) gr0 (M) → gr(M) being bijective is indeed met, as it simply says that R ⊗ R S → S is bijective.  Lemma 7.4.3 (Bourbaki ACIII§5, Th.1). Let A be a commutative ring, J an ideal of A, and M an A-module. Consider the following properties: (i) M is a flat A-module. (iv) M/JM is a flat A/J-module, and the canonical homomorphism gr(A) ⊗gr0 (A) gr0 (M) → gr(M) is bijective. Then (i) implies (iv). If moreover J is nilpotent, then (i) and (iv) are equivalent. Taking truncations we have Wm (R) ,→ Wm (K) (since at the level of sets, this is just Rm ,→ K m ). Consider the diagram (the subscript c stands for constant): Mc,R = Wm (R)⊕h o A 0 id 0 MG,R = Wm (R)⊕h O O w Ns (B) (σ) Mc,R = (Wm (R)⊕h )(σ) o s Frob A (σ) MG,R = (Wm (R)⊕h )(σ) o ker(A ⊗ 1) 3 a O ker Ns (B)  Mc,K = Wm (K)⊕h o  A MG,K = Wm (K)⊕h O  O w s Frob AK (σ) Mc,K = (Wm (K)⊕h )(σ) o Ns (BK )  (σ) MG,K = (Wm (K)⊕h )(σ) o O  ker Ns (BK ) The bottom-right part of the diagram is obtained by base-changing to the generic point, i.e. using the morphism Spec K → Spec R. The vertical maps are linear. All parallelograms in the diagram are commutative (by functoriality of the Frobenius, Dieudonné module and base change).   Assume first that α is a level structure. We want to show that ker Frobs (A) = ker Ns (B).  ker(Frobs AK ) 50 Let a ∈ ker Frobs A. Then the image e a of a in Wm (K)⊕h(σ) is in the kernel of Frobs AK . Since αK is a level structure for the truncated p-divisible group G K over Spec K, from the main theorem in the case where the base is a field, proved above, we have ker(Frobs AK ) = ker Ns (BK ). So ã ∈ ker Ns (BK ). But the map Wm (R)⊕h → Wm (K)⊕h is injective because it is simply induced by the inclusion R ,→ K (so on the level of sets is just Rm ,→ K m ). Thus a ∈ ker Ns (B), and thus ker Frobs A = ker Ns (B). Assume now that ker Frobs A = ker Ns (B). We want to prove that α is a level structure. We shall need the following simple result from commutative algebra: Lemma 7.4.4. Given a module homomorphism f : M → N of R-modules and a flat R-algebra S, then (ker f ) ⊗ R S = ker( f ⊗ 1S ). Proof of lemma: Let M 0 be the kernel of f , N 0 the image and P the cokernel. We then have a diagram 0 / M0 /M 0 /N > / N0 /P /0 /0 The top row is exact, and the two arising short exact sequences are exact. Tensoring with S, the short exact sequences stay exact. If a tensor w in M ⊗ R S goes to 0 in N ⊗ R S under f ⊗ 1S , then from the injectivity of N 0 ⊗ R S → N ⊗ R S it follows that it goes to zero in N 0 ⊗ R S. Thus, ker( f ⊗ 1) = ker(M ⊗ R S → N 0 ⊗ R S) = M 0 ⊗ S = (ker f ) ⊗ R S, as wanted.  Since W(K) is flat over W(R) (and Wm (K) flat over Wm (R)), it follows the maps induced by base change: Ns (BK ) and Frobs AK have kernels equal to ker(Ns (B)) ⊗W(R) W(K) and ker Frobs A ⊗W(R) W(K) respectively. By assumption, ker Frobs A = ker Ns (B), hence ker Ns (BK ) = ker Frobs AK . But our main theorem over a perfect field precisely says that this equality is equivalent to the group homomorphism αK : (p−m Z/Z)Kh → G K induced by α begin a level structure. Thus the condition ker Frobs A = ker Ns (B) implies that αK is a level structure on G K . It will suffice then to prove the following statement: Proposition 7.4.1. Let α : (Z/pm Z)Sh → G be a homomorphism of group schemes over S = Spec R, with R a perfect integral domain. Let K be the fraction field of 51 R and αK the homomorphism obtained by base-change to Spec K. If αK is a level structure on G K , then α is a level structure on G. In other words, the above proposition says that being a level structure is a closed condition. In fact, we can replace (Z/pm Z)h by an arbitrary (abstract) finite abelian group H and view α : HS → G as a set of |H | sections. Moreover, G need not be a truncated p-divisible group. We keep, however, the essential assumption that R is a (perfect) integral domain. Proof. Let G = Spec B for a locally free R-algebra B. We can assume that B is free over R, of rank N. Choose a basis b1, . . . , bN . Then b1 ⊗ 1, . . . , bN ⊗ 1 form a basis for B ⊗ R K over K. Let ϕ1, . . . , ϕN : S → G be a set of sections, corresponding to ring homomorphisms f1, . . . , fN : B → R. By assumption, (ϕi )K form a full set of sections for G K over K. This isÕ equivalent to the fact (see [KM85], page 38), that the ‘‘universal’’ element Z0 = Ti (bi ⊗ 1) ∈ (B ⊗ R K) ⊗K K[T1, . . . ,TN ] satisfies Norm(Z0 ) = Ö ( fi )K (Z0 ) (7.3) as an equality in (B ⊗ R K)[T1, . . . ,TN ]. Now ϕ1, . . . , ϕÕ N form a full set of sections for G over R if and only if the ‘‘universal’’ element Z = Ti bi ∈ B ⊗ R R[T1, . . . ,TN ] satisfies Norm(Z) = Ö fi (Z). (7.4) So the proposition comes down to checking the equivalence of (7.3) and (7.4). The proposition then follows because Z0 = Z ⊗ 1, so the linear map that is multiplication by Z0 has the ‘‘same’’ matrix (with respect to basis {bi ⊗ 1}) as multiplication by Z does with respect to basis {bi }, so has the same norm / characteristic polynomial.  7.5 From integral domain to general case We proceed as follows: Let q be a minimal prime of R. A basic but key point (from commutative algebra) is that Frac(R/q) = Rq . This helps because localization is exact, hence the composition map R → R/q → Frac(R/q) is flat (even if R → R/q isn’t!). Thus Wm (R) → Wm (Frac(R/q)) is flat (by Lemma 7.4.2). 52 Assume α is a level structure. We want to show ker(Frobs A) = ker Ns (B). As usual, we have ker(Frobs A) ⊃ ker Ns (B). Let a ∈ ker(Frobs A) and assume a < ker Ns (B). Recall that the (twisted or not) Dieudonné module of G is free over Wm (R) of rank h. Consider the image of a under Ns (B) (by assumption it is nonzero) and write h Õ it as ai ei , with ai ∈ Wm (R). Since R is reduced, there is a minimal prime q i=1 so that not all ai ’s are in the ideal Wm (q) of Wm (R). This means that under the map Wm (R)h → Wm (R/q)h ,→ Wm (F)h , the image of Ns (B)(a) is not zero. But via flatness, we know that kernels are preserved when passing from R to F. This means that ker Frobs AF , ker Ns (BF ), which is a contradiction because αF is a level structure on the generic point. Assume now ker Frobs A = ker Ns (B). Let q be a minimal prime and F = Frac(R/q). Passing to the generic point of the irreducible component corresponding to q, and using the flatness of Wm (R) → Wm (F) we have ker(Frobs AF ) = ker(Ns (BF )). Thus, from the field case, αF is a level structure on G F . By the argument in the integral domain case, α is a level structure on the whole irreducible component corresponding to q. In particular, α induces a level structure at every geometric point of every irreducible component, i.e. at every geometric point of S = Spec R. From corollary 1.9.2 of Katz-Mazur, this implies that α is a level structure. 53 BIBLIOGRAPHY [Bou03] N. Bourbaki. Algebra II, Chapter 4-7. Springer, 2003. [Bou85] N. Bourbaki. Éléments de Mathématique: Algebre Commutative. Springer, 1985. [Buz12] Kevin Buzzard. Group schemes of order p2 . 2012. url: http://wwwf. imperial . ac . uk / ~buzzard / maths / research / notes / group _ schemes_of_order_p_squared.pdf. [CN90] Ching-Li Chai and Peter Norman. ‘‘Bad Reduction of the Siegel Moduli Scheme of Genus Two with Γ(p)-Level Structure.’’ In: American Journal of Mathematics 112.6 (1990), pp. 1003–1071. issn: 00029327, 10806377. url: http://www.jstor.org/stable/2374734. [Dem72] M. Demazure. Lectures on p-divisible groups. 1972. [DG70] M. Demazure and P. Gabriel. Groupes Algébriques Tome I. Masson, 1970. [Dri74] V.G. Drinfeld. ‘‘Elliptic Modules.’’ In: Mat. Sb. (N.S.) (1974). [Fon77] J-M Fontaine. Groupes p-divisibles sur les corps locaux. 1977. [Goe01] Ulrich Goertz. ‘‘On the flatness of models of certain Shimura varieties of PEL-type.’’ In: Mathematische Annalen 321 (Nov. 2001), pp. 689–727. doi: 10.1007/s002080100250. [HR12] Thomas Haines and Michael Rapoport. ‘‘Shimura varieties with Γ1 (p)level via Hecke algebra isomorphisms: the Drinfeld case.’’ In: Annales Scientifiques de l’École Normale Supérieure 45 (Sept. 2012), pp. 719– 785. doi: 10.24033/asens.2177. [HT01] M. Harris and R. Taylor. The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, Princeton University Press, 2001. [KM85] N. Katz and B. Mazur. Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies, Princeton University Press, 1985. [Lau10] E. Lau. Smoothness of the truncated display functor. 2010. url: https: //arxiv.org/pdf/1006.2723.pdf. [Mes72] W. Messing. The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes. Vol. 264. Lecture Notes in Mathematics, 1972. [RZ96] M. Rapoport and Th. Zink. Period Spaces for "p"-divisible Groups (AM-141). Princeton University Press, 1996. isbn: 9780691027814. url: http://www.jstor.org/stable/j.ctt1b7x72g. 54 [Sha15] R. Shadrach. Integral models of certain PEL-Shimura varieties with Γ1 (p)-type level structure. 2015. url: https : / / arxiv . org / abs / 1309.3485. [Shi14] K. Shimomoto. On The Witt Vectors Of Perfect Rings In Positive Characteristic. 2014. url: https://arxiv.org/abs/1404.5147. [TO70] J Tate and Frans Oort. ‘‘Group scheme of Prime Order.’’ In: Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 3 (Jan. 1970). [Tzi17] N. Tziolas. ‘‘Quotients of schemes by αp or µ p actions in characteristic p > 0.’’ In: Manuscripta Mathematica (2017). url: https://arxiv. org/pdf/1412.7941.pdf. [Wed00] Torsten Wedhorn. ‘‘The local Langlands correspondence for GL(n) over p-adic fields.’’ In: (Dec. 2000). url: http://users.ictp.it/~pub_ off/lectures/lns021/Wedhorn/Wedhorn.pdf.