Construction of a Rapoport-Zink space for GU(1,1) in the ramified 2 -adic case
CCONSTRUCTION OF A RAPOPORT-ZINK SPACE FOR
GU(1 , INTHE RAMIFIED -ADIC CASE DANIEL KIRCH
Abstract.
Let F | Q be a finite extension. In this paper, we construct an RZ-space N E for split GU(1 ,
1) over a ramified quadratic extension E | F . For this, we firstintroduce the naive moduli problem N naive E and then define N E ⊆ N naive E as acanonical closed formal subscheme, using the so-called straightening condition. Weestablish an isomorphism between N E and the Drinfeld moduli problem, provingthe 2-adic analogue of a theorem of Kudla and Rapoport. The formulation of thestraightening condition uses the existence of certain polarizations on the points ofthe moduli space N naive E . We show the existence of these polarizations in a moregeneral setting over any quadratic extension E | F , where F | Q p is a finite extensionfor any prime p .[2010 Mathematics Subject Classification: 11G18,14G35] Contents
1. Introduction 22. Preliminaries on quaternion algebras and hermitian forms 63. The moduli problem in the case (R-P) 113.1. The definition of the naive moduli problem N naive E N E ⊆ N naive E Date : July 3, 2018. a r X i v : . [ m a t h . AG ] M a y DANIEL KIRCH Introduction
Rapoport-Zink spaces (short RZ-spaces) are moduli spaces of p -divisible groups en-dowed with additional structure. In [17], Rapoport and Zink study two major classes ofRZ-spaces, called (EL) type and (PEL) type. The abbreviations (EL) and (PEL) indi-cate, in analogy to the case of Shimura varieties, whether the extra structure comes inform of E ndomorphisms and L evel structure or in form of P olarizations, E ndomorphismsand L evel structure. [17] develops a theory of these spaces, including important theo-rems about the existence of local models and non-archimedean uniformization of Shimuravarieties, for the (EL) type and for the (PEL) type whenever p = 2.The blanket assumption p = 2 made by Rapoport and Zink in the (PEL) case is byno means of cosmetical nature, but originates to various serious difficulties that arise for p = 2. However, we recall that one can still use their definition in that case to obtain“naive” moduli spaces that still satisfy basic properties like being representable by aformal scheme.In this paper, we construct the 2-adic Rapoport-Zink space N E corresponding to thegroup of unitary similitudes of size 2 relative to any (wildly) ramified quadratic exten-sion E | F , where F | Q is a finite extension. It is given as the closed formal subscheme ofthe corresponding naive RZ-space N naive E described by the so-called “straightening con-dition”, which is defined below. The main result of this paper is a natural isomorphism η : M Dr ∼ −→ N E , where M Dr is Deligne’s formal model of the Drinfeld upper halfplane( cf. [3]). This result is in analogy with [11], where Kudla and Rapoport construct acorresponding isomorphism for p = 2 and also for p = 2 when E | F is an unramified ex-tension. The formal scheme M Dr solves a certain moduli problem of p -divisible groupsand, in this way, it carries the structure of an RZ-space of (EL) type. In particular, M Dr is defined even for p = 2.As in loc. cit., there are natural group actions by SL ( F ) and the split SU ( F ) on thespaces M Dr and N E , respectively. The isomorphism η is hence a geometric realizationof the exceptional isomorphism of these groups. As a consequence, one cannot expect asimilar result in higher dimensions. Of course, the existence of “good” RZ-spaces is stillexpected, but a general definition will probably need a different approach.The study of residue characteristic 2 is interesting and important for the followingreasons: First of all, from the general philosophy of RZ-spaces and, more generally,of local Shimura varieties [16], it follows that there should be uniform approach for allprimes p . In this sense, the present paper is in the same spirit as the recent constructionsof RZ-spaces of Hodge type of W. Kim [10], Howard and Pappas [8] and Bültel andPappas [4]. Second, Rapoport-Zink spaces have been used to determine the arithmeticintersection numbers of special cycles on Shimura varieties [12]; in this kind of problem,it is necessary to deal with all places, even those of residue characteristic 2. Finally,studying the cases of residue characteristic 2 also throws light on the cases previouslyknown. In the specific case at hand, the methods we develop in the present paper alsogive a simplification of the proof for p = 2 of Kudla and Rapoport [11], see Remark 5.3(2).We will now explain the results of this paper in greater detail. Let F be a finiteextension of Q and E | F a ramified quadratic extension. Following [9], we consider thefollowing dichotomy for this extension (see section 2):(R-P) There is a uniformizer π ∈ F , such that E = F [Π] with Π + π = 0. Then therings of integers O F of F and O E of E satisfy O E = O F [Π].(R-U) E | F is given by an Eisenstein equation of the form Π − t Π + π = 0. Here, π isagain a uniformizer in F and t ∈ O F satisfies π | t |
2. We still have O E = O F [Π]. Notethat in this case E | F is generated by a square root of the unit 1 − π /t in F . ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
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An example for an extension of type (R-P) is Q ( √− | Q , whereas Q ( √− | Q is oftype (R-U). Note that for p >
2, any ramified quadratic extension over Q p is of the form(R-P).Our results in the cases (R-P) and (R-U) are similar, but different. We first describethe results in the case (R-P). Let E | F be of type (R-P).We first define a naive moduli problem N naive E , that merely copies the definition from p = 2 ( cf. [11]). Let ˘ F be the completion of the maximal unramified extension of F and˘ O F its ring of integers. Then N naive E is a set-valued functor on Nilp ˘O F , the category of˘ O F -schemes where π is locally nilpotent. For S ∈ Nilp ˘O F , the set N naive E ( S ) is the setof equivalence classes of tuples ( X, ι, λ, % ). Here,
X/S is a formal O F -module of height4 and dimension 2, equipped with an action ι : O E → End( X ). This action satisfies theKottwitz condition of signature (1 , i.e. , for any α ∈ O E , the characteristic polynomialof ι ( α ) on Lie X is given bychar(Lie X, T | ι ( α )) = ( T − α )( T − α ) . Here, α α denotes the Galois conjugation of E | F . The right hand side of this equationis a polynomial with coefficients in O S via the structure map O F , → ˘ O F → O S . Thethird entry λ is a principal polarization λ : X → X ∨ such that the induced Rosatiinvolution satisfies ι ( α ) ∗ = ι ( α ) for all α ∈ O E . (Here, X ∨ is the dual of X as formal O F -module.) Finally, % is a quasi-isogeny of height 0 (and compatible with all previousdata) to a fixed framing object ( X , ι X , λ X ) over k = ˘ O F /π . This framing object isunique up to isogeny under the condition that { ϕ ∈ End ( X , ι X ) | ϕ ∗ ( λ X ) = λ X } ’ U( C, h ) , for a split E | F -hermitian vector space ( C, h ) of dimension 2, see Lemma 3.2.Recall that this is exactly the definition used in loc. cit. for the ramified case with p >
2. There, N E = N naive E and we have natural isomorphism η : M Dr ∼ −→ N E , where M Dr is the Drinfeld moduli problem mentioned above.However, for p = 2, it turns out that the definition of N naive E is not the “correct” onein the sense that it is not isomorphic to the Drinfeld moduli problem. Hence this naivedefinition of the moduli space is not in line with the results from [11] and the generalphilosophy of (conjectural) local Shimura varieties (see [16]). In order to remedy this,we will describe a new condition on N naive E , which we call the straightening condition ,and show that this cuts out a closed formal subscheme N E ⊆ N naive E that is naturallyisomorphic to M Dr . Interestingly, the straightening condition is not trivial on the rigid-analytic generic fiber of N naive E (as originally assumed by the author), but it cuts out an(admissible) open and closed subspace, see Remark 3.13.We would like to explicate the defect of the naive moduli space. For this, let usrecall the definition of M Dr . It is a functor on Nilp ˘O F , mapping a scheme S to the set M Dr ( S ) of equivalence classes of tuples ( X, ι B , % ). Again, X/S is a formal O F -moduleof height 4 and dimension 2. Let B be the quaternion division algebra over F and O B its ring of integers. Then ι B is an action of O B on X , satisfying the special condition ofDrinfeld (see [3] or section 3.3 below). The last entry % is an O B -linear quasi-isogeny ofheight 0 to a fixed framing object ( X , ι X ,B ) over k . This framing object is unique up toisogeny ( cf. [3, II. Prop. 5.2]).Fix an embedding O E , → O B and consider the involution b b ∗ = Π b Π − on B ,where b b is the standard involution. By Drinfeld (see Proposition 3.14 below), thereexists a principal polarization λ X on the framing object ( X , ι X ,B ) of M Dr , such that theinduced Rosati involution satisfies ι X ,B ( b ) ∗ = ι X ,B ( b ∗ ) for all b ∈ O B . This polarization DANIEL KIRCH is unique up to a scalar in O × F . Furthermore, for any ( X, ι B , % ) ∈ M Dr ( S ), the pullback λ = % ∗ ( λ X ) is a principal polarization on X .We now set η ( X, ι B , % ) = ( X, ι B | O E , λ, % ) . By Lemma 3.15, this defines a closed embedding η : M Dr , → N naive E . But η is far frombeing an isomorphism, as the following proposition shows: Proposition 1.1.
The induced map η ( k ) : M Dr ( k ) → N naive E ( k ) is not surjective. Let us sketch the proof here. Using Dieudonné theory, we can write N naive E ( k ) natu-rally as a union N naive E ( k ) = [ Λ ⊆ C P (Λ / ΠΛ)( k ) , where the union runs over all O E -lattices Λ in the hermitian vector space ( C, h ) that areΠ − -modular, i.e. , the dual Λ ] of Λ with respect to h is given by Λ = Π − Λ ] (see Lemma3.7). By Jacobowitz ([9]), there exist different types ( i.e. , U( C, h )-orbits) of such latticesΛ ⊆ C that are parametrized by their norm ideal Nm(Λ) = h{ h ( x, x ) | x ∈ Λ }i ⊆ F . Inthe case at hand, Nm(Λ) can be any ideal with 2 O F ⊆ Nm(Λ) ⊆ O F . It is easily checked(see Chapter 2) that the norm ideal of Λ is minimal, that is Nm(Λ) = 2 O F , if and onlyif Λ admits a basis consisting of isotropic vectors, and hence we call these lattices hyperbolic . Now, the image under η of M Dr ( k ) is the union of all lines P (Λ / ΠΛ)( k )where Λ ⊆ C is hyperbolic. This is a consequence of Remark 3.12 and Theorem 3.16below.On the framing object ( X , ι X , λ X ) of N naive E , there exists a principal polarization e λ X such that the induced Rosati involution is the identity on O E . This polarization isunique up to a scalar in O × E (see Thm. 5.2 (1)). On C , the polarization e λ X induces an E -linear alternating form b , such that det b and det h differ only by a unit (for a fixedbasis of C ). After possibly rescaling b by a unit in O × E , a Π − -modular lattice Λ ⊆ C is hyperbolic if and only if b ( x, y ) + h ( x, y ) ∈ O F for all x, y ∈ Λ. This enables usto describe the “hyperbolic” points of N naive E ( i.e. , those that lie on a projective linecorresponding to a hyperbolic lattice Λ ⊆ C ) in terms of polarizations.We now formulate the closed condition that characterizes N E as a closed formalsubscheme of N naive E . For a suitable choice of ( X , ι X , λ X ) and e λ X , we may assume that ( λ X + e λ X ) is a polarization on X . The following definition is a reformulation of Definition3.11. Definition 1.2.
Let S ∈ Nilp ˘O F . An object ( X, ι, λ, % ) ∈ N naive E ( S ) satisfies the straight-ening condition, if λ = ( λ + e λ ) is a polarization on X . Here, e λ = % ∗ ( e λ X ).We remark that e λ = % ∗ ( e λ X ) is a polarization on X . This is a consequence of Theorem5.2, which states the existence of certain polarizations on points of a larger moduli space M E containing N naive E , see below.For S ∈ Nilp ˘O F , let N E ( S ) ⊆ N naive E ( S ) be the subset of all tuples ( X, ι, λ, % ) thatsatisfy the straightening condition. By [17, Prop. 2.9], this defines a closed formalsubscheme N E ⊆ N naive E . An application of Drinfeld’s Proposition (Proposition 3.14,see also [3]) shows that the image of M Dr under η lies in N E . The main theorem in the(R-P) case can now be stated as follows, see Theorem 3.16. Theorem 1.3. η : M Dr → N E is an isomorphism of formal schemes. This concludes our discussion of the (R-P) case. From now on, we assume that E | F is of type (R-U). ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
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In the case (R-U), we have to make some adaptions for N naive E . For S ∈ Nilp ˘O F , let N naive E ( S ) be the set of equivalence classes of tuples ( X, ι, λ, % ) with (
X, ι ) as in the (R-P)case. But now, the polarization λ : X → X ∨ is supposed to have kernel ker λ = X [Π] (incontrast to the (R-P) case, where λ is a principal polarization). As before, the Rosatiinvolution of λ induces the conjugation on O E . There exists a framing object ( X , ι X , λ X )over Spec k for N naive E , which is unique up to isogeny under the condition that { ϕ ∈ End ( X , ι X ) | ϕ ∗ ( λ X ) = λ X } ’ U( C, h ) , where ( C, h ) is a split E | F -hermitian vector space of dimension 2 (see Proposition 4.1).Finally, % is a quasi-isogeny of height 0 from X to X , respecting all structure.Fix an embedding E , → B . Using some subtle choices of elements in B (theseare described in Lemma 2.3 (2)) and by Drinfeld’s Proposition, we can construct apolarization λ as above for any ( X, ι B , % ) ∈ M Dr ( S ). This induces a closed embedding η : M Dr −→ N naive E , ( X, ι B , % ) ( X, ι B | O E , λ, % ) . We can write N naive E ( k ) as a union of projective lines, N naive E ( k ) = [ Λ ⊆ C P (Λ / ΠΛ)( k ) , where the union now runs over all selfdual O E -lattices Λ ⊆ ( C, h ) with Nm(Λ) ⊆ π O F .As in the (R-P) case, these lattices Λ ⊆ C are classified up to isomorphism by theirnorm ideal Nm(Λ). Since Λ is selfdual with respect to h , the norm ideal can be any idealsatisfying tO F ⊆ Nm(Λ) ⊆ O F . We call Λ hyperbolic when the norm ideal is minimal, i.e. , Nm(Λ) = tO F . Equivalently, the lattice Λ has a basis consisting of isotropic vectors.Recall that here t is the element showing up in the Eisenstein equation for the (R-U)extension E | F and that π | t |
2. Hence there exists at least one type of selfdual latticesΛ ⊆ C with Nm(Λ) ⊆ π O F . In the case (R-U), it may happen that | t | = | π | , in whichcase all lattices Λ in the description of N naive E ( k ) are hyperbolic.The image of M Dr ( k ) under η in N naive E ( k ) is the union of all projective lines corre-sponding to hyperbolic lattices. Unless | t | = | π | , it follows that η ( k ) is not surjectiveand thus η cannot be an isomorphism. For the case | t | = | π | , we will show that η is an isomorphism on reduced loci ( M Dr ) red ∼ −→ ( N naive E ) red (see Remark 4.11), but η is not an isomorphism of formal schemes. This follows from the non-flatness of thedeformation ring for certain points of N naive E , see section 4.4.On the framing object ( X , ι X , λ X ) of N naive E , there exists a polarization e λ X such thatker e λ X = X [Π] and such that the Rosati involution induces the identity on O E . After asuitable choice of ( X , ι X , λ X ) and e λ X , we may assume that t ( λ X + e λ X ) is a polarizationon X . The straightening condition for the (R-U) case is given as follows (see Definition4.10). Definition 1.4.
Let S ∈ Nilp ˘O F . An object ( X, ι, λ, % ) ∈ N naive E ( S ) satisfies the straight-ening condition, if λ = t ( λ + e λ ) is a polarization on X . Here, e λ = % ∗ ( e λ X ).Note that e λ = % ∗ ( e λ X ) is a polarization on X by Theorem 5.2.The straightening condition defines a closed formal subscheme N E ⊆ N naive E thatcontains the image of M Dr under η . The main theorem in the (R-U) case can now bestated as follows, compare Theorem 4.14. Theorem 1.5. η : M Dr → N E is an isomorphism of formal schemes. When formulating the straightening condition in the (R-U) and the (R-P) case, wementioned that e λ = % ∗ ( e λ X ) is a polarization for any ( X, ι, λ, % ) ∈ N naive E ( S ). This fact is DANIEL KIRCH a corollary of Theorem 5.2, that states the existence of this polarization in the followingmore general setting.Let F | Q p be a finite extension for any prime p and E | F an arbitrary quadraticextension. We consider the following moduli space M E of (EL) type. For S ∈ Nilp ˘O F ,the set M E ( S ) consists of equivalence classes of tuples ( X, ι E , % ), where X is a formal O F -module of height 4 and dimension 2 and ι E is an O E -action on X satisfying theKottwitz condition of signature (1 ,
1) as above. The entry % is an O E -linear quasi-isogeny of height 0 to a supersingular framing object ( X , ι X ,E ).The points of M E are equipped with polarizations in the following natural way, seeTheorem 5.2. Theorem 1.6. (1)
There exists a principal polarization e λ X on ( X , ι X ,E ) such that theRosati involution induces the identity on O E , i.e. , ι ( α ) ∗ = ι ( α ) for all α ∈ O E . Thispolarization is unique up to a scalar in O × E . (2) Fix e λ X as in part (1) . For any S ∈ Nilp ˘O F and ( X, ι E , % ) ∈ M E ( S ) , there exists aunique principal polarization e λ on X such that the Rosati involution induces the identityon O E and such that e λ = % ∗ ( e λ X ) . If p = 2 and E | F is ramified of (R-P) or (R-U) type, then there is a canonical closedembedding N E , → M E that forgets about the polarization λ . In this way, it followsthat e λ is a polarization for any ( X, ι, λ, % ) ∈ N naive E ( S ).The statement of Theorem 1.6 can also be expressed in terms of an isomorphism ofmoduli spaces M E, pol ∼ −→ M E . Here M E, pol is a moduli space of (PEL) type, definedby mapping S ∈ Nilp ˘O F to the set of tuples ( X, ι, e λ, % ) where ( X, ι, % ) ∈ M E ( S ) and e λ is a polarization as in the theorem.We now briefly describe the contents of the subsequent sections of this paper. Insection 2, we recall some facts about the quadratic extensions of F , the quaternionalgebra B | F and hermitian forms. In the next two sections, sections 3 and 4, we definethe moduli spaces N naive E , introduce the straightening condition describing N E ⊆ N naive E and prove our main theorem in both the cases (R-P) and (R-U). Although the techniquesare quite similar in both cases, we decided to treat these cases separately, since the resultsin both cases differ in important details. Finally, in Section 5 we prove Theorem 1.6 onthe existence of the polarizations e λ . Acknowledgements.
First of all, I am very grateful to my advisor M. Rapoport forsuggesting this topic and for his constant support and helpful discussions. I thank themembers of our Arbeitsgruppe in Bonn for numerous discussions and I also would liketo thank the audience of my two AG talks for many helpful questions and comments.Furthermore, I owe thanks to A. Genestier for many useful remarks and for pointingout a mistake in an earlier version of this paper. I would also like to thank the refereefor helpful comments.This work is the author’s PhD thesis at the University of Bonn, which was supportedby the SFB/TR45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ ofthe DFG (German Research Foundation). Parts of this paper were written during thefall semester program ‘New Geometric Methods in Number Theory and AutomorphicForms’ at the MSRI in Berkeley.2.
Preliminaries on quaternion algebras and hermitian forms
Let F | Q be a finite extension. In this section we will recall some facts about thequadratic extensions of F , the quaternion division algebra B | F and certain hermitianforms. For more information on quaternion algebras, see for example the book by ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
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Vigneras [19]. A systematic classification of hermitian forms over local fields has beendone by Jacobowitz in [9].Let E | F be a quadratic field extension and denote by O F resp. O E the rings ofintegers. There are three mutually exclusive possibilities for E | F : • E | F is unramified. Then E = F [ δ ] for δ a square root of a unit in F . We can choose δ such that δ = 1 + 4 u for some u ∈ O × F . In this case, O E = O F [ δ ]. The element γ = δ satisfies the Eisenstein equation γ − γ − u = 0. In the following we will write F (2) instead of E and O (2) F instead of O E when talking about the unramified extensionof F . • E | F is ramified and E is generated by the square root of a uniformizer in F . Thatis, E = F [Π] and Π is given by the Eisenstein equation Π + π = 0 for a uniformizingelement π ∈ O F . We also have O E = O F [Π]. Following Jacobowitz, we will say E | F isof type (R-P) (which stands for “ramified-prime”). • Finally, E | F can be given by an Eisenstein equation of the form Π − t Π + π = 0 fora uniformizer π and t ∈ O F such that π | t |
2. Then E | F is ramified and O E = O F [Π].Here, E is generated by the square root of a unit in F . Indeed, for ϑ = 1 − /t wehave ϑ = 1 − π /t ∈ O × F . Thus E | F is said to be of type (R-U) (for “ramified-unit”).We will use this notation throughout the paper. Remark 2.1.
The isomorphism classes of quadratic extension of F correspond to thenon-trivial equivalence classes of F × / ( F × ) . We have F × / ( F × ) ’ H ( G F , Z / Z ) forthe absolute Galois group G F of F and dim H ( G F , Z / Z ) = 2 + d , where d = [ F : Q ]is the degree of F over Q (see, for example, [14, Cor. 7.3.9]).A representative of an equivalence class in F × /F × can be chosen to be either aprime or a unit, and exactly half of the classes are represented by prime elements, theothers being represented by units. It follows that there are, up to isomorphism, 2 d different extensions E | F of type (R-P) and 2 d − ∈ F × /F × and one unit element corresponding to theunramified extension.) Lemma 2.2.
The inverse different of E | F is given by D − E | F = O E in the case (R-P) and by D − E | F = t O E in the case (R-U) .Proof. The inverse different is defined as D − E | F = { α ∈ E | Tr E | F ( αO E ) ⊆ O F } . It is enough to check the condition on the trace for the elements 1 and Π ∈ O E . If wewrite α = α + Π α with α , α ∈ F , we getTr E | F ( α ·
1) = α + α = 2 α + α (Π + Π) , Tr E | F ( α · Π) = α Π + α Π = α (Π + Π) + α (Π + Π ) . In the case (R-P) we have Π + Π = 0 and Π + Π = 2 π , while in the case (R-U),Π + Π = t and Π + Π = t − π . It is now easy to deduce that the inverse differentis of the claimed form. (cid:3) Over F , there exists up to isomorphism exactly one quaternion division algebra B ,with unique maximal order O B . For every quadratic extension E | F , there exists anembedding E , → B and this induces an embedding O E , → O B . If E | F is ramified,a basis for O E as O F -module is given by (1 , Π). We would like to extend this to an O F -basis of O B . Lemma 2.3. (1) If E | F is of type (R-P) , there exists an embedding F (2) , → B suchthat δ Π = − Π δ . An O F -basis of O B is then given by (1 , γ, Π , γ · Π) , where γ = δ . DANIEL KIRCH (2) If E | F is of type (R-U) , there exists an embedding E , → B , where E | F is of type (R-P) with uniformizer Π such that ϑ Π = − Π ϑ . The tuple (1 , ϑ, Π , ϑ Π ) is an F -basis of B .Furthermore, there is also an embedding e E , → B with e E | F of type (R-U) with elements e Π and e ϑ as above, such that ϑ e ϑ = − e ϑϑ and e ϑ = 1 + ( t /π ) · u for some unit u ∈ F .In terms of this embedding, an O F -basis of O B is given by (1 , Π , e Π , Π · e Π /π ) . Also, Π · e Π π = γ (2.1) for some embedding F (2) , → B of the unramified extension and γ − γ − u = 0 . Hence, O B = O F [Π , γ ] as O F -algebra.Proof. (1) This is [19, II. Cor. 1.7].(2) By [19, I. Cor. 2.4], it suffices to find a uniformizer Π ∈ F × \ Nm E | F ( E × ) inorder to prove the first part. But Nm E | F ( E × ) ⊆ F × is a subgroup of order 2 and F × ⊆ Nm E | F ( E × ). On the other hand, the residue classes of uniformizing elements in F × /F × generate the whole group. Thus they cannot all be contained in Nm E | F ( E × ).For the second part, choose a unit δ ∈ F (2) with δ = 1 + 4 u ∈ F × \ F × forsome u ∈ O × F and set γ = δ . Let e E | F be of type (R-U), generated by e ϑ with e ϑ = 1 + ( t /π ) · u . We have to show that e ϑ is not contained in Nm E | F ( E × ).Assume it is a norm, so e ϑ = Nm E | F ( b ) for a unit b ∈ E × . Then b is of the form b = 1 + x · ( t/ Π) for some x ∈ O E . Indeed, let ‘ be the Π-adic valuation of b − i.e. , b = 1 + x · Π ‘ and x ∈ O × E . We have1 + ( t /π ) · u = Nm E | F ( b ) = 1 + Tr E | F ( x Π ‘ ) + Nm E | F ( x Π ‘ ) (2.2)Let v be the π -adic valuation on F . Then v (Nm E | F ( x Π ‘ )) = ‘ and v (Tr E | F ( x Π ‘ )) ≥ v ( t ) + b ‘ c , by Lemma 2.2. On the left hand side, we have v (( t /π ) · u ) = 2 v ( t ) − ‘ < v ( t ) − ‘ ≥ v ( t ) − b = 1 + x · ( t/ Π) for some x ∈ O E . Again,1 + ( t /π ) · u = Nm E | F ( b ) = 1 + Tr E | F ( xt/ Π) + Nm E | F ( xt/ Π) . An easy calculation shows that the residue x ∈ k = O E / Π = O F /π of x satisfies u = x + x . But this equation has no solution in k , since a solution of γ − γ − u = 0generates the unramified quadratic extension of F . It follows that e ϑ cannot be a norm.Using again [19, I. Cor. 2.4], we find an embedding e E , → B such that ϑ e ϑ = − e ϑϑ .We have Π = t (1 + ϑ ) / e Π = π (1 + e ϑ ) /t , thusΠ · e Π π = (1 + ϑ ) · (1 + e ϑ )2 = 1 + ϑ + e ϑ + ϑ · e ϑ , and ( ϑ + e ϑ + ϑ · e ϑ ) = ϑ + e ϑ − ϑ · e ϑ = (1 − π /t ) + (1 + t u/π ) − (1 − π /t )(1 + t u/π )= 1 + 4 u. Hence γ Π · e Π π induces an embedding F (2) , → B .It remains to prove that the tuple u = (1 , Π , e Π , Π · e Π /π ) is a basis of O B as O F -module. By [19, I. Cor. 4.8], it suffices to check that the discriminantdisc( u ) = det(Trd( u i u j )) · O F ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 9 is equal to disc( O B ). An easy calculation shows det(Trd( u i u j )) · O F = π O F and thenthe assertion follows from [19, V, II. Cor. 1.7]. (cid:3) For the remainder of this section, we will consider lattices Λ in a 2-dimensional E -vector space C with a split E | F -hermitian form h . Recall from [9] that, up toisomorphism, there are 2 different E | F -hermitian vector spaces ( C, h ) of fixed dimension n , parametrized by the discriminant disc( C, h ) ∈ F × / Nm E | F ( E × ). A hermitian space( C, h ) is called split whenever disc(
C, h ) = 1. In our case, where (
C, h ) is split ofdimension 2, we can find a basis ( e , e ) of C with h ( e i , e i ) = 0 and h ( e , e ) = 1.Denote by Λ ] the dual of a lattice Λ ⊆ C with respect to h . The lattice Λ is calledΠ i - modular if Λ = Π i Λ ] (resp. unimodular or selfdual when i = 0). In contrast to the p -adic case with p >
2, there exists more than one type of Π i -modular lattices in ourcase ( cf. [9]): Proposition 2.4.
Define the norm ideal
Nm(Λ) of Λ by Nm(Λ) = h{ h ( x, x ) | x ∈ Λ }i ⊆ F. (2.3) Any Π i -modular lattice Λ ⊆ C is determined up to the action of U( C, h ) by the ideal Nm(Λ) = π ‘ O F ⊆ F . For i = 0 or , the exponent ‘ can be any integer such that | | ≤ | π | ‘ ≤ | | (for E | F (R-P), unimodular Λ) , | π | ≤ | π | ‘ ≤ | π | (for E | F (R-P), Π -modular Λ) , | t | ≤ | π | ‘ ≤ | | (for E | F (R-U), unimodular Λ) , | t | ≤ | π | ‘ ≤ | π | (for E | F (R-U), Π -modular Λ) , where | · | is the (normalized) absolute value on F . Two Π i -modular lattices Λ and Λ are isomorphic if and only if Nm(Λ) = Nm(Λ ) . (cid:3) For any other i , the possible values of ‘ for a given Π i -modular lattice Λ are easilyobtained by shifting. In fact, we can choose an integer j such that Π j Λ is either uni-modular or Π-modular. Then Nm(Λ) = π − j Nm(Π j Λ) and we can apply the propositionabove.Since (
C, h ) is split, any Π i -modular lattice Λ contains an isotropic vector v ( i.e. ,with h ( v, v ) = 0). After rescaling with a suitable power of Π, we can extend v to a basisof Λ. Hence there always exists a basis ( e , e ) of Λ such that h is represented by amatrix of the form H Λ = (cid:18) x Π i Π i (cid:19) , x ∈ F. (2.4)If x = 0 in this representation, then Nm(Λ) = π ‘ O F is as small as possible, or in otherwords, the absolute value of | π | ‘ is minimal. On the other hand, whenever | π | ‘ takesthe minimal absolute value for a given Π i -modular lattice Λ, there exists a basis ( e , e )of Λ such that h is represented by H Λ with x = 0. Indeed, this follows because the idealNm(Λ) already determines Λ up to isomorphism. In this case (when x = 0), we call Λa hyperbolic lattice. By the arguments above, a Π i -modular lattice is thus hyperbolicif and only if its norm is minimal. In all other cases, where Λ is Π i -modular but nothyperbolic, we have Nm(Λ) = xO F .For further reference, we explicitly write down the norm of a hyperbolic lattice forthe cases that we need later. For other values of i , the norm can easily be deduced fromthis by shifting (see also [9, Table 9.1]). Here and in the following, sesquilinear forms will be linear from the left and semi-linear from the right.
Lemma 2.5. A Π i -modular lattice Λ is hyperbolic if and only if Nm(Λ) = 2 O F , for E | F (R-P), i = 0 or − , Nm(Λ) = tO F , for E | F (R-U), i = 0 or . The norm ideal of Λ is minimal among all norm ideals for Π i -modular lattices in C . (cid:3) In the following, we will only consider the cases i = 0 or − E | F (R-P) and thecases i = 0 or 1 for E | F (R-U), since these are the cases we will need later. We want tostudy the following question: Question 2.6.
Assume E | F is (R-P). Fix a Π − -modular lattice Λ − ⊆ C (not nec-essarily hyperbolic). How many unimodular lattices Λ ⊆ Λ − are there and whatnorms Nm(Λ ) can appear? Dually, for a fixed unimodular lattice Λ ⊆ C , how manyΠ − -modular lattices Λ − with Λ ⊆ Λ − do exist and what are their norms?Same question for E | F (R-U) and unimodular resp. Π-modular lattices.Of course, such an inclusion is always of index 1. The inclusions Λ ⊆ Λ − of index 1correspond to lines in Λ − / ΠΛ − . Denote by q the number of elements in the commonresidue field of O F and O E . Then there exist at most q + 1 such Π-modular lattices Λ for a given Λ − . The same bound holds in the dual case, i.e. , there are at most q + 1Π − -modular lattices containing a given unimodular lattice Λ . The Propositions 2.7and 2.8 below provide an exhaustive answer to Question 2.6. Since the proofs consist ofa lengthy but simple case-by-case analysis, we will leave it to the interested reader. Proposition 2.7.
Let E | F of type (R-P) . (1) Let Λ − ⊆ C be a Π − -modular hyperbolic lattice. There are q + 1 hyperbolic uni-modular lattices contained in Λ − . (2) Let Λ − ⊆ C be a Π − -modular non-hyperbolic lattice. Let Nm(Λ − ) = π ‘ O F . Then Λ − contains one unimodular lattice Λ with Nm(Λ ) = π ‘ +10 O F and q unimodularlattices of norm π ‘ O F . (3) Let Λ ⊆ C be a unimodular hyperbolic lattice. There are two hyperbolic Π − -modular lattices Λ − ⊇ Λ and q − non-hyperbolic Π − -modular lattices Λ − ⊇ Λ with Nm(Λ − ) = 2 /π O F . (4) Let Λ ⊆ C be unimodular non-hyperbolic. Let Nm(Λ ) = π ‘ O F . There exists one Π − -modular lattice Λ − ⊇ Λ with Nm(Λ − ) = π ‘ O F and, unless ‘ = 0 , there are q non-hyperbolic Π − -modular lattices Λ − ⊇ Λ with Nm(Λ − ) = π ‘ − O F . Note that the total amount of unimodular resp. Π − -modular lattices found forΛ = Λ − resp. Λ is q + 1 except in the case of Proposition 2.7 (4) when ‘ = 0. Inthat particular case, there is just one Π − -modular lattice contained in Λ . The samephenomenon also appears in the case (R-U), see part (2) of the following proposition. Proposition 2.8.
Let E | F of type (R-U) . (1) Let Λ ⊆ C be a unimodular hyperbolic lattice. There are q + 1 hyperbolic Π -modularlattices Λ ⊆ Λ . (2) Let Λ ⊆ C be unimodular non-hyperbolic with Nm(Λ ) = π ‘ O F . There is one Π -modular lattice Λ ⊆ Λ with norm ideal Nm(Λ ) = π ‘ +10 O F and if ‘ = 0 , there are also q non-hyperbolic Π -modular lattices Λ ⊆ Λ with Nm(Λ ) = π ‘ O F . (3) Let Λ ⊆ C be a Π -modular hyperbolic lattice. There are two unimodular hyperboliclattices containing Λ and q − unimodular lattices Λ with Λ ⊆ Λ and Nm(Λ ) = t/π O F . ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 11 (4)
Let Λ ⊆ C be a Π -modular non-hyperbolic lattice and let Nm(Λ ) = π ‘ O F . Thelattice Λ is contained in q unimodular lattices of norm π ‘ − O F and in one unimodularlattice Λ with Nm(Λ ) = π ‘ O F . If E | F is a quadratic extension of type (R-U) such that | t | = | π | , there exist onlyhyperbolic Π-modular lattices in C and hence case (4) of Proposition 2.8 does not appear.3. The moduli problem in the case (R-P)
Throughout this section, E | F is a quadratic extension of type (R-P), i.e. , there existuniformizing elements π ∈ F and Π ∈ E such that Π + π = 0. Then O E = O F [Π] forthe rings of integers O F and O E of F and E , respectively. Let k be the common residuefield with q elements, k an algebraic closure, and ˘ F the completion of the maximalunramified extension of F , with ring of integers ˘ O F = W O F ( k ). Let σ be the lift of theFrobenius in Gal( k | k ) to Gal( ˘ O F | O F ).3.1. The definition of the naive moduli problem N naive E . We first construct afunctor N naive E on Nilp ˘O F , the category of ˘ O F -schemes S such that π O S is locallynilpotent. We consider tuples ( X, ι, λ ), where • X is a formal O F -module over S of dimension 2 and height 4. • ι : O E → End( X ) is an action of O E satisfying the Kottwitz condition : The charac-teristic polynomial of ι ( α ) on Lie X for any α ∈ O E ischar(Lie X, T | ι ( α )) = ( T − α )( T − α ) . Here α α is the non-trivial Galois automorphism and the right hand side is a poly-nomial with coefficients in O S via the composition O F [ T ] , → ˘ O F [ T ] → O S [ T ]. • λ : X → X ∨ is a principal polarization on X such that the Rosati involution satisfies ι ( α ) ∗ = ι ( α ) for α ∈ O E . Definition 3.1. A quasi-isogeny (resp. an isomorphism ) ϕ : ( X, ι, λ ) → ( X , ι , λ ) oftwo such tuples ( X, ι, λ ) and ( X , ι , λ ) over S is an O E -linear quasi-isogeny of height 0(resp. an O E -linear isomorphism) ϕ : X → X such that λ = ϕ ∗ ( λ ).Denote the group of quasi-isogenies ϕ : ( X, ι, λ ) → ( X, ι, λ ) by QIsog(
X, ι, λ ).For S = Spec k we have the following proposition: Proposition 3.2.
Up to isogeny, there exists precisely one tuple ( X , ι X , λ X ) over Spec k such that the group QIsog( X , ι X , λ X ) contains SU(
C, h ) as a closed subgroup. Here SU(
C, h ) is the special unitary group for a -dimensional E -vector space C with split E | F -hermitian form h . Remark 3.3.
If ( X , ι X , λ X ) is as in the proposition, we always have QIsog( X , ι X , λ X ) ∼ =U( C, h ). This follows directly from the proof and gives a more natural way to describe theframing object. However, we will need the slightly stronger statement of the Propositionlater, in Lemma 3.15.
Proof of Proposition 3.2.
We first show uniqueness. Let (
X, ι, λ ) / Spec k be such a tuple.Its (relative) rational Dieudonné module N X is a 4-dimensional vector space over ˘ F withan action of E and an alternating form h , i such that for all x, y ∈ N X , h x, Π y i = −h Π x, y i . (3.1)The space N X has the structure of a 2-dimensional vector space over ˘ E = E ⊗ F ˘ F and we can define an ˘ E | ˘ F -hermitian form on it via h ( x, y ) = h Π x, y i + Π h x, y i . (3.2) The alternating form can be recovered from h by h x, y i = Tr ˘ E | ˘ F (cid:18) · h ( x.y ) (cid:19) . (3.3)Furthermore we have on N X a σ -linear operator F , the Frobenius, and a σ − -linearoperator V , the Verschiebung, that satisfy VF = FV = π . Recall that σ is the lift ofthe Frobenius on ˘ O F . Since h , i comes from a polarization, we have h F x, y i = h x, V y i σ , and h ( F x, y ) = h ( x, V y ) σ , for all x, y ∈ N X . Let us consider the σ -linear operator τ = Π V − . Its slopes are all zero,since N X is isotypical of slope . (This follows from the condition on QIsog( X , ι X , λ X ).)We set C = N τX . This is a 2-dimensional vector space over E and N X = C ⊗ E ˘ E . Now h induces an E | F -hermitian form on C since h ( τ x, τ y ) = h ( − F Π − x, Π V − y ) = − h (Π − x, Π y ) σ = h ( x, y ) σ . A priori, there are up to isomorphism two possibilities for (
C, h ), either h is split on C or non-split. But automorphisms of ( C, h ) correspond to elements of QIsog( X , ι X , λ X ).The unitary groups of ( C, h ) for h split and h non-split are not isomorphic and theycannot contain each other as a closed subgroup. Hence the condition on QIsog( X , ι X , λ X )implies that h is split.Assume now we have two different objects ( X, ι, λ ) and ( X , ι , λ ) as in the propo-sition. These give us isomorphic vector spaces ( C, h ) and ( C , h ) and an isomorphismbetween these extends to an isomorphism between N X and N X (respecting all rationalstructure) which corresponds to a quasi-isogeny between ( X, ι, λ ) and ( X , ι , λ ).The existence of ( X , ι X , λ X ) now follows from the fact that a 2-dimensional E -vectorspace ( C, h ) with split E | F -hermitian form contains a unimodular lattice Λ. Indeed,this gives us a lattice M = Λ ⊗ O E ˘ O E ⊆ C ⊗ E ˘ E . We extend h to N = C ⊗ E ˘ E anddefine the ˘ F -linear alternating form h , i as in (3.3). Now M is unimodular with respectto h , i , because ˘ O E is the inverse different of ˘ E | ˘ F (see Lemma 2.2). We choose theoperators F and V on M such that FV = VF = π and Λ = M τ for τ = Π V − . Thismakes M a (relative) Dieudonné module and we define ( X , ι X , λ X ) as the correspondingformal O F -module. (cid:3) We fix such a framing object ( X , ι X , λ X ) over Spec k . Definition 3.4.
For arbitrary S ∈ Nilp ˘O F , let S = S × Spf ˘ O F Spec k . Define N naive E ( S )as the set of equivalence classes of tuples ( X, ι, λ, % ) over S , where ( X, ι, λ ) as above and % : X × S S −→ X × Spec k S is a quasi-isogeny between the tuple ( X, ι, λ ) and the framing object ( X , ι X , λ X ) (afterbase change to S ). Two objects ( X, ι, λ, % ) and ( X , ι , λ , % ) are equivalent if and onlyif there exists an isomorphism ϕ : ( X, ι, λ ) → ( X , ι , λ ) such that % = % ◦ ( ϕ × S S ). Remark 3.5. (1) The morphism % is a quasi-isogeny in the sense of Definition 3.1, i.e. ,we have λ = % ∗ ( λ X ). Similarly, we have λ = ϕ ∗ ( λ ) for the isomorphism ϕ . We obtain anequivalent definition of N naive E if we replace strict equality by the condition that, locallyon S , λ and % ∗ ( λ X ) (resp. ϕ ∗ ( λ )) only differ by a scalar in O × F . This variant is used inthe definition of RZ-spaces of (PEL) type for p > ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 13 (2) N naive E is pro-representable by a formal scheme, formally locally of finite type overSpf ˘ O F . This follows from [17, Thm. 3.25].As a next step, we use Dieudonné theory in order to get a better understanding ofthe special fiber of N naive E . Let N = N X be the rational Dieudonné module of the basepoint ( X , ι X , λ X ) of N naive E . This is a 4-dimensional vector space over ˘ F , equipped withan E -action, an alternating form h , i and two operators V and F . As in the proof ofProposition 3.2, the form h , i satisfies condition (3.1): h x, Π y i = −h Π x, y i . (3.4)A point ( X, ι, λ, % ) ∈ N naive E ( k ) corresponds to an ˘ O F -lattice M X ⊆ N . It is stableunder the actions of the operators V and F and of the ring O E . Furthermore M X isunimodular under h , i , i.e. , M X = M ∨ X , where M ∨ X = { x ∈ N | h x, y i ∈ ˘ O F for all y ∈ M X } . We can regard N as a 2-dimensional vector space over ˘ E with the ˘ E | ˘ F -hermitian form h defined by h ( x, y ) = h Π x, y i + Π h x, y i . (3.5)Let ˘ O E = O E ⊗ O F ˘ O F . Then M X ⊆ N is an ˘ O E -lattice and we have M X = M ∨ X = M ]X , where M ]X is the dual lattice of M X with respect to h . The latter equality follows fromthe formula h x, y i = Tr ˘ E | ˘ F (cid:18) · h ( x.y ) (cid:19) (3.6)and the fact that the inverse different of E | F is D − E | F = O E (see Lemma 2.2). Wecan thus write the set N naive E ( k ) as N naive E ( k ) = { ˘ O E -lattices M ⊆ N X | M ] = M, π M ⊆ V M ⊆ M } . (3.7)Let τ = Π V − . This is a σ -linear operator on N with all slopes zero. The elementsinvariant under τ form a 2-dimensional E -vector space C = N τ . The hermitian form h is invariant under τ , hence it induces a split hermitian form on C which we denoteagain by h . With the same proof as in [11, Lemma 3.2], we have: Lemma 3.6.
Let M ∈ N naive E ( k ) . Then: (1) M + τ ( M ) is τ -stable. (2) Either M is τ -stable and Λ = M τ ⊆ C is unimodular (Λ ] = Λ ) or M is not τ -stable and then Λ − = ( M + τ ( M )) τ ⊆ C is Π − -modular (Λ ] − = ΠΛ − ) . Under the identification N = C ⊗ E ˘ E , we get M = Λ ⊗ O E ˘ O E for a τ -stable Dieudonnélattice M . If M is not τ -stable, we have M + τ M = Λ − ⊗ O E ˘ O E and M ⊆ Λ − ⊗ O E ˘ O E is a sublattice of index 1. The next lemma is the analogue of [11, Lemma 3.3]. Lemma 3.7. (1)
Fix a Π − -modular lattice Λ − ⊆ C . There is an injective map i Λ − : P (Λ − / ΠΛ − )( k ) , −→ N naive E ( k ) mapping a line ‘ ⊆ (Λ − / ΠΛ − ) ⊗ k to its preimage in Λ − ⊗ ˘ O E . Identify P (Λ − / ΠΛ − )( k ) with its image in N naive E ( k ) . Then P (Λ − / ΠΛ − )( k ) ⊆ P (Λ − / ΠΛ − )( k ) is the set of τ -invariant Dieudonné lattices M ⊆ Λ − ⊗ ˘ O E . (2) The set N naive E ( k ) is a union N naive E ( k ) = [ Λ − ⊆ C P (Λ − / ΠΛ − )( k ) , (3.8) ranging over all Π − -modular lattices Λ − ⊆ C . The projective lines corresponding to thelattices Λ − and Λ intersect in N naive E ( k ) if and only if Λ = Λ − ∩ Λ is unimodular.In this case, their intersection consists of the point M = Λ ⊗ ˘ O E ∈ N naive E ( k ) .Proof. We only have to prove that the map i Λ − is well-defined. Denote by M thepreimage of ‘ ⊆ (Λ − / ΠΛ − ) ⊗ k in Λ − ⊗ ˘ O E . We need to show that M is an elementin N naive E ( k ) under the identification of (3.7). It is clearly a sublattice of index 1 inΛ − ⊗ ˘ O E , stable under the actions of F , V and O E .Let e ∈ Λ − ⊗ ˘ O E such that e ⊗ k generates ‘ . We can extend this to a basis ( e , e )of Λ − and with respect to this basis, h is represented by a matrix of the form (cid:18) x − Π − Π − y (cid:19) , with x, y ∈ Π − ˘ O E ∩ ˘ O F = ˘ O F . The lattice M ⊆ Λ − ⊗ ˘ O E is generated by e and Π e .With respect to this new basis, h is now given by the matrix (cid:18) x π y (cid:19) . Since all entries of the matrix are integral, we have M ⊆ M ] . But this already implies M ] = M , because they both have index 1 in Λ − ⊗ ˘ O E . Thus M ∈ N naive E ( k ) and i Λ − is well-defined. (cid:3) Remark 3.8. (1) Recall from Proposition 2.4 that the isomorphism type of a Π i -modular lattice Λ ⊆ C only depends on its norm ideal Nm(Λ) = h{ h ( x, x ) | x ∈ Λ }i = π ‘ O F ⊆ F . In the case that Λ = Λ or Λ − is unimodular or Π − -modular, ‘ can be anyinteger such that | | ≥ | π | ‘ ≥ | | . In particular, there are always at least two possiblevalues for ‘ . Recall from Lemma 2.5, that Λ is hyperbolic if and only if Nm(Λ) = 2 O F .(2) The intersection behaviour of the projective lines in N naive E ( k ) can be deduced fromProposition 2.7. In particular, for a given unimodular lattice Λ ⊆ C with Nm(Λ ) ⊆ π O F , there are q + 1 lines intersecting in M = Λ ⊗ ˘ O E . If Nm(Λ ) = O F , the lattice M = Λ ⊗ ˘ O E is only contained in one projective line. On the other hand, a projectiveline P (Λ − / ΠΛ − )( k ) ⊆ N naive E ( k ) contains q + 1 points corresponding to unimodularlattices in C . By Lemma 3.7 (1), these are exactly the k -rational points of P (Λ − / ΠΛ − ).(3) If we restrict the union at the right hand side of (3.8) to hyperbolic Π − -modularlattices Λ − ⊆ C ( i.e. , Nm(Λ − ) = 2 O F , see Lemma 2.5), we obtain a canonical subset N E ( k ) ⊆ N naive E ( k ) and there is a description of N E as a pro-representable functoron Nilp ˘O F (see below). We will see later (Theorem 3.16) that N E is isomorphic tothe Drinfeld moduli space M Dr , described in [3, I.3]. In particular, the underlyingtopological space of N E is connected. (The induced topology on the projective lines isthe Zariski topology, see Proposition 3.9.) Moreover, each projective line in N E ( k ) has q + 1 intersection points and there are 2 projective lines intersecting in each such point(see also Proposition 2.7).We fix such an intersection point P ∈ N E ( k ). Now going back to N naive E ( k ), there are q − P ∈ N naive E ( k ) that correspond to non-hyperboliclattices in C (see Proposition 2.7). Each of these additional lines contains P as its only“hyperbolic” intersection point, all other intersection points on this line and the lineitself correspond to unimodular resp. Π − -modular lattices Λ ⊆ C of norm Nm(Λ) = ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 15 (2 /π ) O F (whereas all hyperbolic lattices occuring have the norm ideal 2 O F , see Lemma2.5). Assume P (Λ / ΠΛ)( k ) ⊆ N naive E ( k ) is such a line and let P ∈ P (Λ / ΠΛ)( k ) be anintersection point, where P = P . There are again q more lines going through P (always q +1 in total) that correspond to lattices with norm ideal Nm(Λ) = (2 /π ) O F , and theselines again have more intersection points and so on. This goes on until we reach lines P (Λ / ΠΛ )( k ) with Nm(Λ ) = O F . Each of these lines contains q points that correspondto unimodular lattices Λ ⊆ C with Nm(Λ ) = O F . Such a lattice is only contained inone Π − -modular lattice (see part 4 of Proposition 2.7). Hence, these points are onlycontained in one projective line, namely P (Λ / ΠΛ )( k ).In other words, each intersection point P ∈ N E ( k ) has a “tail”, consisting of finitelymany projective lines, which is the connected component of P in ( N naive E ( k ) \ N E ( k )) ∪{ P } . Figure 1 shows a drawing of ( N naive E ) red for the cases F = Q (on the left handside) and F | Q a ramified quadratic extension (on the right hand side). The “tails” areindicated by dashed lines. (a) e = 1, f = 1. (b) e = 2, f = 1. Figure 1.
The reduced locus of N naive E for E | F of type (R-P) where F | Q has ramification index e and inertia degree f . Solid lines are givenby subschemes N E, Λ for hyperbolic lattices Λ.Fix a Π − -modular lattice Λ = Λ − ⊆ C . Let X +Λ be the formal O F -module overSpec k associated to the Dieudonné lattice M = Λ ⊗ ˘ O E ⊆ N . It comes with a canonicalquasi-isogeny % +Λ : X −→ X +Λ of F -height 1. We define a subfunctor N E, Λ ⊆ N naive E by mapping S ∈ Nilp ˘O F to N E, Λ ( S ) = { ( X, ι, λ, % ) ∈ N naive E ( S ) | ( % +Λ × S ) ◦ % is an isogeny } . (3.9)Note that the condition of (3.9) is closed, cf. [17, Prop. 2.9]. Hence N E, Λ is representableby a closed formal subscheme of N naive E . On geometric points, we have a bijection N E, Λ ( k ) ∼ −→ P (Λ / ΠΛ)( k ) , (3.10)as a consequence of Lemma 3.7 (1). Proposition 3.9.
The reduced locus of N naive E is given by ( N naive E ) red = [ Λ ⊆ C N E, Λ , where Λ runs over all Π − -modular lattices in C . For each Λ , there is an isomorphismof reduced schemes N E, Λ ∼ −→ P (Λ / ΠΛ) , inducing the map (3.10) on k -valued points.Proof. The embedding [ Λ ⊆ C ( N E, Λ ) red , −→ ( N naive E ) red (3.11)is closed, because each embedding N E, Λ ⊆ N naive E is closed and, locally on ( N naive E ) red ,the left hand side is always only a finite union of ( N E, Λ ) red . It follows already that(3.11) is an isomorphism, since it is a bijection on k -valued points (see the equations(3.8) and (3.10)) and ( N naive E ) red is reduced by definition and locally of finite type overSpec k by Remark 3.5 (2).For the second part of the proposition, we follow the proof presented in [11, 4.2]. Fixa Π − -modular lattice Λ ⊆ C and let M = Λ ⊗ ˘ O E ⊆ N , as above. Now X +Λ is theformal O F -module associated to M , but we also get a formal O F -module X − Λ associatedto the dual M ] = Π M of M . This comes with a natural isogenynat Λ : X − Λ −→ X +Λ and a quasi-isogeny % − Λ : X − Λ → X of F -height 1. For ( X, ι, λ, % ) ∈ N naive E ( S ) where S ∈ Nilp ˘O F , we consider the composition % − Λ ,X = % − ◦ ( % − Λ × S ) : ( X − Λ × S ) −→ X. By [11, Lemma 4.2], this composition is an isogeny if and only if ( % +Λ × S ) ◦ % is an isogeny,or, in other words, if and only if ( X, ι, λ, % ) ∈ N E, Λ ( S ). Let D X − Λ ( S ) be the (relative)Grothendieck-Messing crystal of X − Λ evaluated at S ( cf. [2, Def. 3.24] or [1, 5.2]). Thisis a locally free O S -module of rank 4, isomorphic to Λ /π Λ ⊗ O F O S . The kernel of D (nat Λ )( S ) is given by (Λ / ΠΛ) ⊗ O F O S , locally a direct summand of rank 2 of D X − Λ ( S ).For any ( X, ι, λ, % ) ∈ N E, Λ ( S ), the kernel of % − Λ ,X is contained in ker(nat Λ ). It followsfrom [20, Cor. 4.7] (see also [11, Prop. 4.6]) that ker D ( % − Λ ,X )( S ) is locally a directsummand of rank 1 of (Λ / ΠΛ) ⊗ O F O S . This induces a map N E, Λ ( S ) −→ P (Λ / ΠΛ)( S ) , functorial in S , and the arguments of [20, 4.7] show that it is an isomorphism. (Oneeasily checks that their results indeed carry over to the relative setting over O F .) (cid:3) Construction of the closed formal subscheme N E ⊆ N naive E . We now use aresult from section 5. By Theorem 5.2 and Remark 5.1 (2), there exists a principalpolarization e λ X : X → X ∨ on ( X , ι X , λ X ), unique up to a scalar in O × E , such that theinduced Rosati involution is the identity on O E . Furthermore, for any ( X, ι, λ, % ) ∈N naive E ( S ), the pullback e λ = % ∗ ( e λ X ) is a principal polarization on X .The next proposition is crucial for the construction of N E . Recall the notion of a hyperbolic lattice from Proposition 2.4 and the subsequent discussion. Proposition 3.10.
It is possible to choose ( X , ι X , λ X ) and e λ X such that λ X , = 12 ( λ X + e λ X ) ∈ Hom( X , X ∨ ) . Fix such a choice and let ( X, ι, λ, % ) ∈ N naive E ( k ) . Then, ( λ + e λ ) ∈ Hom(
X, X ∨ ) if andonly if ( X, ι, λ, % ) ∈ N E, Λ ( k ) for some hyperbolic lattice Λ ⊆ C .Proof. The polarization e λ X on X induces an alternating form ( , ) on the rational Dieu-donné module N = M X ⊗ ˘ O F ˘ F . For all x, y ∈ N , the form ( , ) satisfies the equations( F x, y ) = ( x, V y ) σ , (Π x, y ) = ( x, Π y ) . ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 17
It induces an ˘ E -alternating form b on N via b ( x, y ) = δ ((Π x, y ) + Π( x, y )) , where δ ∈ ˘ O F is a unit generating the unramified quadratic extension of F , chosen suchthat δ σ = − δ and δ ∈ ˘ O F , see page 7. On the other hand, we can describe ( , ) interms of b , ( x, y ) = Tr ˘ E | ˘ F (cid:18) δ · b ( x, y ) (cid:19) . (3.12)The form b is invariant under τ = Π V − , since b ( τ x, τ y ) = b ( − F Π − x, Π V − y ) = b (Π − x, Π y ) σ = b ( x, y ) σ . Hence b defines an E -linear alternating form on C = N τ , which we again denote by b .Denote by h , i the alternating form on M X induced by the polarization λ X and let h be the corresponding hermitian form, see (3.2). On N X , we define the alternating form h , i by h x, y i = 12 ( h x, y i + ( x, y )) . This form is integral on M X if and only if λ X , = ( λ X + e λ X ) is a polarization on X .We choose ( X , ι X , λ X ) such that it corresponds to a unimodular hyperbolic latticeΛ ⊆ ( C, h ) under the identifications of (3.7) and Lemma 3.6. There exists a basis( e , e ) of Λ such that h b = (cid:18) (cid:19) , b b = (cid:18) u − u (cid:19) , (3.13)for some u ∈ E × . Since e λ X is principal, the alternating form b is perfect on Λ , thus u ∈ O × E . After rescaling e λ X , we may assume that u = 1. We now have12 ( h ( x, y ) + b ( x, y )) ∈ O E , for all x, y ∈ Λ . Thus ( h + b ) is integral on M X = Λ ⊗ O E ˘ O E . This implies that h x, y i = 12 ( h x, y i + ( x, y )) = 12 Tr ˘ E | ˘ F (cid:18) · h ( x.y ) + 12Π δ · b ( x, y ) (cid:19) = Tr ˘ E | ˘ F (cid:18)
14Π ( h ( x, y ) + b ( x, y )) (cid:19) + Tr ˘ E | ˘ F (cid:18) − δ δ · b ( x, y ) (cid:19) ∈ ˘ O F , for all x, y ∈ M X . Indeed, in the definition of b , the unit δ has been chosen such that δ ∈ ˘ O F , so the second summand is in ˘ O F . The first summand is integral, since ( h + b ) is integral. It follows that λ X , = ( λ X + e λ X ) is a polarization on X .Let ( X, ι, λ, % ) ∈ N naive E ( k ) and assume that λ = ( λ + e λ ) = % ∗ ( λ X , ) is a polarizationon X . Then h , i is integral on the Dieudonné module M ⊆ N of X . By the abovecalculation, this is equivalent to ( h + b ) being integral on M . In particular, this impliesthat h ( x, x ) = h ( x, x ) + b ( x, x ) ∈ O F , for all x ∈ M . Let Λ = ( M + τ ( M )) τ . Then h ( x, x ) ∈ O F for all x ∈ Λ, henceNm(Λ) ⊆ O F . By Lemma 2.5 and the bound of norm ideals, we have Nm(Λ) = 2 O F and Λ is a hyperbolic lattice. It follows that ( X, ι, λ, % ) ∈ N E, Λ ( k ) for some hyperbolicΠ − -modular lattice Λ ⊆ C . Indeed, if M τ (cid:40) Λ then Λ is Π − -modular and Λ = Λ. If M τ = Λ then it is contained in some Π − -modular hyperbolic lattice Λ by Proposition2.7.Conversely, assume that ( X, ι, λ, % ) ∈ N E, Λ ( k ) for some hyperbolic lattice Λ ⊆ C .It suffices to show that ( h + b ) is integral on Λ. Indeed, it follows that ( h + b ) is integral on the Dieudonné module M . Thus h , i is integral on M and this is equivalentto λ = ( λ + e λ ) ∈ Hom(
X, X ∨ ).Let Λ ⊆ C be the Π − -modular lattice generated by e and Π − e , where ( e , e ) isthe basis of the lattice Λ corresponding to the framing object ( X , ι X , λ X ). By (3.13), h and b have the following form with respect to the basis ( e , Π − e ), h b = (cid:18) − Π − Π − (cid:19) , b b = (cid:18) Π − − Π − (cid:19) . In particular, Λ is hyperbolic and ( h + b ) is integral on Λ . By Proposition 2.4,there exists an automorphism g ∈ SU(
C, h ) mapping Λ onto Λ . Since det g = 1, thealternating form b is invariant under g . It follows that ( h + b ) is also integral on Λ. (cid:3) From now on, we assume ( X , ι X , λ X ) and e λ X chosen in a way such that λ X , = 12 ( λ X + e λ X ) ∈ Hom( X , X ∨ ) . Note that this determines the polarization e λ X up to a scalar in 1 + 2 O E . If we replace e λ X by e λ X = e λ X ◦ ι X (1 + 2 u ) for some u ∈ O E , then λ X , = λ X , + e λ X ◦ ι X ( u ).We can now formulate the straightening condition. Definition 3.11.
Let S ∈ Nilp ˘O F . An object ( X, ι, λ, % ) ∈ N naive E ( S ) satisfies the straightening condition if λ ∈ Hom(
X, X ∨ ) , (3.14)where λ = ( λ + e λ ) = % ∗ ( λ X , ).This definition is clearly independent of the choice of the polarization e λ X . We define N E as the functor that maps S ∈ Nilp ˘O F to the set of all tuples ( X, ι, λ, % ) ∈ N naive E ( S )that satisfy the straightening condition. By [17, Prop. 2.9], N E is representable by aclosed formal subscheme of N naive E . Remark 3.12.
The reduced locus of N E can be written as( N E ) red = [ Λ ⊆ C N E, Λ ’ [ Λ ⊆ C P (Λ / ΠΛ) , where we take the unions over all hyperbolic Π − -modular lattices Λ ⊆ C . By Propo-sition 2.7 and Lemma 3.7, each projective line contains q + 1 points corresponding tounimodular lattices and there are two lines intersecting in each such point. Recall fromRemark 3.8 (1) that there exist non-hyperbolic Π − -modular lattices Λ ⊆ C , thus wehave N E ( k ) = N naive E ( k ), and in particular ( N E ) red = ( N naive E ) red . Remark 3.13.
As has been pointed out to the author by A. Genestier, the straighteningcondition is not trivial on the rigid-analytic generic fiber of N naive E . However, we canshow that it is open and closed. Since a proper study of the generic fiber would go beyondthe scope of this paper, we restrain ourselves to indications rather than complete proofs.Let C be an algebraically closed extension of F and O C its ring of integers. Takea point x = ( X, ι, λ, % ) ∈ N naive E ( O C ) and consider its 2-adic Tate module T ( x ). It isa free O E -module of rank 2 and λ endows T ( x ) with a perfect (non-split) hermitianform h . If x ∈ N E ( O C ), then the straightening condition implies that ( T ( x ) , h ) is alattice with minimal norm Nm( T ( x )) in the vector space V ( x ) = T ( x ) ⊗ O E E (seeProposition 2.4 and [9]). But V ( x ) also contains selfdual lattices with non-minimalnorm ideal. Let Λ ⊆ V ( x ) be such a lattice with Nm(Λ) = Nm( T ( x )). Let Λ be theintersection of T ( x ) and Λ in V ( x ). The inclusions Λ , → Λ and Λ , → T ( x ) define Calling this lattice “hyperbolic” doesn’t make much sense here since it is anisotropic.
ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 19 canonically a formal O F -module Y with T ( Y ) = Λ and a quasi-isogeny ϕ : X → Y .By inheriting all data, Y becomes a point in N naive E ( O C ) that does not satisfy thestraightening condition.To see that the straightening condition is open and closed on the generic fiber, considerthe universal formal O F -module X = ( X , ι X , λ X ) over N naive E and let T ( X ) be its Tatemodule. Then T ( X ) is a locally constant sheaf over N naive , rig E with respect to the étaletopology. The polarization λ X defines a hermitian form h on T ( X ). Since T ( X ) isa locally constant sheaf, the norm ideal Nm( T ( X )) with respect to h (see Proposition2.4) is locally constant as well. Hence the locus where Nm( T ( X )) is minimal is openand closed in N naive , rig E . But this is exactly N rig E ⊆ N naive , rig E .3.3. The isomorphism to the Drinfeld moduli problem.
We now recall the Drin-feld moduli problem M Dr on Nilp ˘O F . Let B be the quaternion division algebra over F and O B its ring of integers. Let S ∈ Nilp ˘O F . Then M Dr ( S ) is the set of equivalenceclasses of objects ( X, ι B , % ) where • X is a formal O F -module over S of dimension 2 and height 4, • ι B : O B → End( X ) is an action of O B on X satisfying the special condition, i.e. ,Lie X is, locally on S , a free O S ⊗ O F O (2) F -module of rank 1, where O (2) F ⊆ O B is anyembedding of the unramified quadratic extension of O F into O B ( cf. [3]), • % : X × S S → X × Spec k S is an O B -linear quasi-isogeny of height 0 to a fixed framingobject ( X , ι X ) ∈ M Dr ( k ).Such a framing object exists and is unique up to isogeny. By a proposition of Drinfeld, cf. [3, p. 138], there always exist polarizations on these objects, as follows: Proposition 3.14 (Drinfeld) . Let Π ∈ O B a uniformizer with Π ∈ O F and let b b be the standard involution of B . Then b b ∗ = Π b Π − is another involution on B . (1) There exists a principal polarization λ X : X → X ∨ on X with associated Rosatiinvolution b b ∗ . It is unique up to a scalar in O × F . (2) Let λ X as in (1) . For ( X, ι B , % ) ∈ M Dr ( S ) , there exists a unique principal polariza-tion λ : X −→ X ∨ with Rosati involution b b ∗ such that % ∗ ( λ X ) = λ on S . We now relate M Dr and N E . For this, we fix an embedding E , → B . Any choice ofa uniformizer Π ∈ O E with Π ∈ O F induces the same involution b b ∗ = Π b Π − on B . For the framing object ( X , ι X ) of M Dr , let λ X be a polarization associated to thisinvolution by Proposition 3.14 (1). Denote by ι X ,E the restriction of ι X to O E ⊆ O B .For any object ( X, ι B , % ) ∈ M Dr ( S ), let λ be the polarization with Rosati involution b b ∗ that satisfies % ∗ ( λ X ) = λ , see Proposition 3.14 (2). Let ι E be the restriction of ι B to O E . Lemma 3.15. ( X , ι X ,E , λ X ) is a framing object for N naive E . Furthermore, the map ( X, ι B , % ) ( X, ι E , λ, % ) induces a closed immersion of formal schemes η : M Dr , −→ N naive E . Proof.
There are two things to check: that QIsog( X , ι X , λ X ) contains SU( C, h ) as a closedsubgroup and that ι E satisfies the Kottwitz condition. Indeed, once these two assertionshold, we can take ( X , ι X ,E , λ X ) as a framing object for N naive E and the morphism η iswell-defined. For any S ∈ Nilp ˘O F , the map η ( S ) is injective, because ( X, ι B , % ) and ( X , ι B , % ) ∈ M Dr ( S ) map to the same point in N naive E ( S ) under η if and only if thequasi-isogeny % ◦ % on S lifts to an isomorphism on S , i.e. , if and only if ( X, ι B , % ) and( X , ι B , % ) define the same point in M Dr ( S ). The functor F : S ( X, ι, λ, % ) ∈ N naive E ( S ) | ι extends to an O B -action } is pro-representable by a closed formal subscheme of N naive E by [17, Prop. 2.9]. Now,the formal subscheme η ( M Dr ) ⊆ F is given by the special condition. But the specialcondition is open and closed (see [18, p. 7]), thus η is a closed embedding.It remains to show the two assertions from the beginning of this proof. We first checkthe condition on QIsog( X , ι X , λ X ). Let G ( X ,ι X ) be the group of O B -linear quasi-isogenies ϕ : ( X , ι X ) → ( X , ι X ) of height 0 such that the induced homomorphism of Dieudonnémodules has determinant 1. Then we have (non-canonical) isomorphisms G ( X ,ι X ) ’ SL ,F and SL ,F ’ SU(
C, h ), since h is split. The uniqueness of the polarization λ X (upto a scalar in O × F ) implies that G ( X ,ι X ) ⊆ QIsog( X , ι X , λ X ). This is a closed embedding oflinear algebraic groups over F , since a quasi-isogeny ϕ ∈ QIsog( X , ι X , λ X ) lies in G ( X ,ι X ) if and only if it is O B -linear and has determinant 1, and these are closed conditions onQIsog( X , ι X , λ X ).Finally, the special condition implies the Kottwitz condition for any element b ∈ O B (see [18, Prop. 5.8]), i.e. , the characteristic polynomial for the action of ι ( b ) on Lie X ischar(Lie X, T | ι ( b )) = ( T − b )( T − b ) , where the right hand side is a polynomial in O S [ T ] via the structure homomorphism O F , → ˘ O F → O S . From this, the second assertion follows. (cid:3) Let O (2) F ⊆ O B be an embedding such that conjugation with Π induces the non-trivial Galois action on O (2) F , as in Lemma 2.3 (1). Fix a generator γ = δ of O (2) F with δ ∈ O × F . On ( X , ι X ), the principal polarization e λ X given by e λ X = λ X ◦ ι X ( δ )has a Rosati involution that induces the identity on O E . For any ( X, ι B , % ) ∈ M Dr ( S ),we set e λ = % ∗ ( e λ X ) = λ ◦ ι B ( δ ). The tuple ( X, ι E , λ, % ) = η ( X, ι B , % ) satisfies thestraightening condition (3.14), since λ = 12 ( λ + e λ ) = λ ◦ ι B ( γ ) ∈ Hom(
X, X ∨ ) . In particular, the tuple ( X , ι X ,E , λ X ) is a framing object of N E and η induces a naturaltransformation η : M Dr , −→ N E . (3.15)Note that this map does not depend on the above choices, as N E is a closed formalsubscheme of N naive E . Theorem 3.16. η : M Dr → N E is an isomorphism of formal schemes. We will first prove this on k -valued points: Lemma 3.17. η induces a bijection η ( k ) : M Dr ( k ) → N E ( k ) .Proof. We can identify the k -valued points of M Dr with a subset M Dr ( k ) ⊆ N naive E ( k ).The rational Dieudonné-module N of X is equipped with an action of B . Fix an em-bedding F (2) , → B as in Lemma 2.3 (1). This induces a Z / N = N ⊕ N of N , where N = { x ∈ N | ι ( a ) x = ax for all a ∈ F (2) } ,N = { x ∈ N | ι ( a ) x = σ ( a ) x for all a ∈ F (2) } , ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 21 for a fixed embedding F (2) , → ˘ F . The operators V and F have degree 1 with respectto this decomposition. Recall that λ has Rosati involution b Π b Π − on O B whichrestricts to the identity on O (2) F . The subspaces N and N are therefore orthogonalwith respect to h , i .Under the identification (3.7), a lattice M ∈ M Dr ( k ) respects this decomposition, i.e. , M = M ⊕ M with M i = M ∩ N i . Furthermore it satisfies the special condition:dim M / V M = dim M / V M = 1 . We already know that M Dr ( k ) ⊆ N E ( k ), so let us assume M ∈ N E ( k ). We want toshow that M ∈ M Dr ( k ), i.e. , that the lattice M is stable under the action of O B on N and satisfies the special condition. It is stable under the O B -action if and only if M = M ⊕ M for M i = M ∩ N i . Let y ∈ M and y = y + y with y i ∈ N i . For any x ∈ M , we have h x, y i = h x, y i + h x, y i ∈ ˘ O F . (3.16)We can assume that λ X , = λ X ◦ ι B ( γ ) with γ ∈ O (2) F under our fixed embedding F (2) , → B . Recall that γ σ = 1 − γ from page 7. Let h , i be the alternating form on M induced by λ X , . Then, h x, y i = γ · h x, y i + (1 − γ ) · h x, y i ∈ ˘ O F . (3.17)From the equations (3.16) and (3.17), it follows that h x, y i and h x, y i lie in ˘ O F . Since x ∈ M was arbitrary and M = M ∨ , this gives y , y ∈ M . Hence M respects thedecomposition of N and is stable under the action of O B .It remains to show that M satisfies the special condition: The alternating form h , i is perfect on M , thus the restrictions to M and M are perfect as well. If M is notspecial, we have M i = V M i +1 for some i ∈ { , } . But then, h , i cannot be perfect on M i . In fact, for any x, y ∈ M i +1 , h V x, V y i σ = h FV x, y i = π · h x, y i ∈ π ˘ O F . Thus M is indeed special, i.e. , M ∈ M Dr ( k ), and this finishes the proof of the lemma. (cid:3) Proof of Theorem 3.16.
We already know that η is a closed embedding η : M Dr , −→ N E . Let ( X , ι X ) be the framing object of M Dr and choose an embedding O (2) F ⊆ O B and agenerator γ of O (2) F as in Lemma 2.3 (1). We take ( X , ι X ,E , λ X ) as a framing object for N E and set e λ X = λ X ◦ ι X ( δ ).Let ( X, ι, λ, % ) ∈ N E ( S ) and e λ = % ∗ ( e λ X ). We have % − ◦ ι X ( γ ) ◦ % = % − ◦ λ − X ◦ λ X , ◦ % = λ − ◦ λ ∈ End( X ) , where λ X , = ( λ X + e λ X ) and λ = ( λ + e λ ). Since O B = O F [Π , γ ], this induces an O B -action ι B on X and makes % an O B -linear quasi-isogeny. We have to check that( X, ι B , % ) satisfies the special condition.Recall that the special condition is open and closed (see [18, p. 7]), so η is an openand closed embedding. Furthermore, η ( k ) is bijective and the reduced loci ( M Dr ) red and ( N E ) red are locally of finite type over Spec k . Hence η indcues an isomorphism onreduced subschemes. But any open and closed embedding of formal schemes, that is anisomorphism on the reduced subschemes, is already an isomorphism. (cid:3) The moduli problem in the case (R-U)
Let E | F be a quadratic extension of type (R-U), generated by a uniformizer Π sat-isfying an Eisenstein equation of the form Π − t Π + π = 0 where t ∈ O F and π | t | O F and O E be the rings of integers of F and E . We have O E = O F [Π]. As in thecase (R-P), let k be the common residue field, k an algebraic closure, ˘ F the completionof the maximal unramified extension with ring of integers ˘ O F = W O F ( k ) and σ the liftof the Frobenius in Gal( k | k ) to Gal( ˘ O F | O F ).4.1. The naive moduli problem.
Let S ∈ Nilp ˘O F . Consider tuples ( X, ι, λ ), where • X is a formal O F -module over S of dimension 2 and height 4. • ι : O E → End( X ) is an action of O E on X satisfying the Kottwitz condition : Thecharacteristic polynomial of ι ( α ) for some α ∈ O E is given bychar(Lie X, T | ι ( α )) = ( T − α )( T − α ) . Here α α is the Galois conjugation of E | F and the right hand side is a polynomial in O S [ T ] via the structure morphism O F , → ˘ O F → O S . • λ : X → X ∨ is a polarization on X with kernel ker λ = X [Π], where X [Π] is thekernel of ι (Π). Further we demand that the Rosati involution of λ satisfies ι ( α ) ∗ = ι ( α )for all α ∈ O E .We define quasi-isogenies ϕ : ( X, ι, λ ) → ( X , ι , λ ) and the group QIsog( X, ι, λ ) asin Definition 3.1.
Proposition 4.1.
Up to isogeny, there exists exactly one such tuple ( X , ι X , λ X ) over S =Spec k under the condition that the group QIsog( X , ι X , λ X ) contains a closed subgroupisomorphic to SU(
C, h ) for a -dimensional E -vector space C with split E | F -hermitianform h . Remark 4.2.
As in the case (R-P), we have QIsog( X , ι X , λ X ) ∼ = U( C, h ) for ( X , ι X , λ X )as in the Proposition. Proof of Proposition 4.1.
We first show uniqueness of the object. Let (
X, ι, λ ) / Spec k be a tuple as in the proposition and consider its rational Dieudonné-module N X . Thisis a 4-dimensional vector space over ˘ F equipped with an action of E and an alternatingform h , i such that h x, Π y i = h Π x, y i (4.1)for all x, y ∈ N X . Let ˘ E = ˘ F ⊗ F E . We can see N X as 2-dimensional vector space over˘ E with a hermitian form h given by h ( x, y ) = h Π x, y i − Π h x, y i . (4.2)Let F and V be the σ -linear Frobenius and the σ − -linear Verschiebung on N X . Wehave FV = VF = π and, since h , i comes from a polarization, h F x, y i = h x, V y i σ . Consider the σ -linear operator τ = Π V − = F Π − . The hermitian form h is invariantunder τ : h ( τ x, τ y ) = h ( F Π − x, Π V − y ) = h ( F x, V − y ) = h ( x, y ) σ . From the condition on QIsog( X , ι X , λ X ) it follows that N X is isotypical of slope andthus the slopes of τ are all zero. Let C = N τX . This is a 2-dimensional vector spaceover E with N X = C ⊗ E ˘ E and h induces an E | F -hermitian form on C . A priori,there are two possibilities for ( C, h ), either h is split or non-split. The group U( C, h )of automorphisms is isomorphic to QIsog( X , ι X , λ X ). But the unitary groups for h splitand h non-split are not isomorphic and do not contain each other as a closed subgroup.Thus the condition on QIsog( X , ι X , λ X ) implies that h is split. ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
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Assume we are given two different objects (
X, ι, λ ) and ( X , ι , λ ) as in the propo-sition. Then there is an isomorphism between the spaces ( C, h ) and ( C , h ) extendingto an isomorphism of N X and N X respecting all structure. This corresponds to aquasi-isogeny ϕ : ( X, ι, λ ) → ( X , ι , λ ).Now we prove the existence of ( X , ι X , λ X ). We start with a Π-modular lattice Λ in a 2-dimensional vector space ( C, h ) over E with split hermitian form. Then M = Λ ⊗ O E ˘ O E is an ˘ O E -lattice in N = C ⊗ E ˘ E . The σ -linear operator τ = 1 ⊗ σ on N has slopes areall 0. We can extend h to N such that h ( τ x, τ y ) = h ( x, y ) σ , for all x, y ∈ N . The operators F and V are given by the equations τ = Π V − = F Π − .Finally, the alternating form h , i is defined via h x, y i = Tr ˘ E | ˘ F (cid:18) tϑ · h ( x, y ) (cid:19) , for x, y ∈ N . The lattice M ⊆ N is the Dieudonné module of the object ( X , ι X , λ X ). Weleave it to the reader to check that this is indeed an object as considered above. (cid:3) We fix such an object ( X , ι X , λ X ) over Spec k from the proposition. We define thefunctor N naive E on Nilp ˘O F as in Definition 3.4. Remark 4.3. N naive E is pro-representable by a formal scheme, formally locally of finitetype over Spf ˘ O F , cf. [17, Thm. 3.25].We now study the k -valued points of the space N naive E . Let N = N X be the rationalDieudonné-module of ( X , ι X , λ X ). This is a 4-dimensional vector space over ˘ F , equippedwith an action of E , with two operators F and V and an alternating form h , i .Let ( X, ι, λ, % ) ∈ N naive E ( k ). This corresponds to an ˘ O F -lattice M = M X ⊆ N whichis stable under the actions of F , V and O E . The condition on the kernel of λ impliesthat M = Π M ∨ for M ∨ = { x ∈ N | h x, y i ∈ ˘ O F for all y ∈ M } . The alternating form h , i induces an ˘ E | ˘ F -hermitian form h on N , seen as 2-dimensionalvector space over ˘ E (see equation (4.2)): h ( x, y ) = h Π x, y i − Π h x, y i . We can recover the form h , i from h via h x, y i = Tr ˘ E | ˘ F (cid:18) tϑ · h ( x, y ) (cid:19) . (4.3)Since the inverse different of E | F is D − E | F = t O E (see Lemma 2.2), this implies that M is Π-modular with respect to h , as ˘ O E -lattice in N . We denote the dual of M withrespect to h by M ] . There is a natural bijection N naive E ( k ) = { ˘ O E -lattices M ⊆ N | M = Π M ] , π M ⊆ V M ⊆ M } . (4.4)Recall that τ = Π V − is a σ -linear operator on N with slopes all 0. Further C = N τ isa 2-dimensional E -vector space with hermitian form h . Lemma 4.4.
Let M ∈ N naive E ( k ) . Then: (1) M + τ ( M ) is τ -stable. (2) Either M is τ -stable and Λ = M τ ⊆ C is Π -modular with respect to h , or M is not τ -stable and then Λ = ( M + τ ( M )) τ ⊆ C is unimodular. The proof is the same as that of [11, Lemma 3.2]. We identify N with C ⊗ E ˘ E . Forany τ -stable lattice M ∈ N naive E ( k ), we have M = Λ ⊗ O E ˘ O E . If M ∈ N naive E ( k ) is not τ -stable, there is an inclusion M ⊆ Λ ⊗ O E ˘ O E of index 1. Recall from Proposition 2.4that the isomorphism class of a Π-modular or unimodular lattice Λ ⊆ C is determinedby the norm ideal Nm(Λ) = h{ h ( x, x ) | x ∈ Λ }i . There are always at least two types of unimodular lattices. However, not all of themappear in the description of N naive E ( k ). Lemma 4.5. (1)
Let Λ ⊆ C be a unimodular lattice with Nm(Λ) ⊆ π O F . There is aninjection i Λ : P (Λ / ΠΛ)( k ) , −→ N naive E ( k ) , that maps a line ‘ ⊆ Λ / ΠΛ ⊗ k k to its inverse image under the canonical projection Λ ⊗ O E ˘ O E −→ Λ / ΠΛ ⊗ k k. The k -valued points P (Λ / ΠΛ)( k ) ⊆ P (Λ / ΠΛ)( k ) are mapped to τ -invariant Dieudonnémodules M ⊆ Λ ⊗ O E ˘ O E under this embedding. (2) Identify P (Λ / ΠΛ)( k ) with its image under i Λ . The set N naive E ( k ) can be written as N naive E ( k ) = [ Λ ⊆ C P (Λ / ΠΛ)( k ) , where the union is taken over all lattices Λ ⊆ C with Nm(Λ) ⊆ π O F .Proof. Let Λ ⊆ C be a unimodular lattice. For any line ‘ ∈ P (Λ / ΠΛ)( k ), denote itspreimage in Λ ⊗ ˘ O E by M . The inclusion M ⊆ Λ ⊗ ˘ O E has index 1 and M is an˘ O E -lattice with Π(Λ ⊗ ˘ O E ) ⊆ M . Furthermore Λ ⊗ ˘ O E is τ -invariant by construction,hence Π(Λ ⊗ ˘ O E ) = V (Λ ⊗ ˘ O E ) = F (Λ ⊗ ˘ O E ). It follows that M is stable under theactions of F and V . Thus M ∈ N naive E ( k ) if and only if M = Π M ] . The hermitian form h induces a symmetric form s on Λ / ΠΛ. Now M is Π-modular if and only if it is thepreimage of an isotropic line ‘ ⊆ Λ / ΠΛ ⊗ k . Note that s is also anti-symmetric since weare in characteristic 2.We first consider the case Nm(Λ) ⊆ π O F . We can find a basis of Λ such that h hasthe form H Λ = (cid:18) x (cid:19) , x ∈ π O F , see (2.4). It follows that the induced form s is even alternating (because x ≡ π ).Hence any line in Λ / ΠΛ ⊗ k is isotropic. This implies that i Λ is well-defined, provingpart 1 of the Lemma.Now assume that Nm(Λ) = O F . There is a basis ( e , e ) of Λ such that h is repre-sented by H Λ = (cid:18) (cid:19) . The induced form s is given by the same matrix and ‘ = k · e is the only isotropic linein Λ / ΠΛ. Since ‘ is already defined over k , the corresponding lattice M ∈ N naive E ( k ) isof the form M = Λ ⊗ ˘ O E for a Π-modular lattice Λ ⊆ Λ. But, by Proposition 2.8, anyΠ-modular lattice in C is contained in a unimodular lattice Λ with Nm(Λ ) ⊆ π O F .It follows that we can write N naive E ( k ) as a union N naive E ( k ) = [ Λ ⊆ C P (Λ / ΠΛ)( k ) , where the union is taken over all unimodular lattices Λ ⊆ C with Nm(Λ) ⊆ π O F . Thisshows the second part of the Lemma. (cid:3) ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
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Remark 4.6.
We can use Proposition 2.8 to describe the intersection behaviour of theprojective lines in N naive E ( k ). A τ -invariant point M ∈ N naive E ( k ) corresponds to theΠ-modular lattice Λ = M τ ⊆ C . If Nm(Λ ) ⊆ π O F , there are q + 1 lines goingthrough M . If Nm(Λ ) = π O F , the point M is contained in one or 2 lines, dependingon whether Λ is hyperbolic or not, see part (3) and (4) of Proposition 2.8. The formercase ( i.e. , Λ is hyperbolic) appears if and only if π O F = Nm(Λ ) = tO F (see Lemma2.5). This happens only for a specific type of (R-U) extension E | F , see page 7. We referto Remark 4.8, Remark 4.11 and Section 4.4 for a further discussion of this special case.On the other hand, each projective line in N naive E ( k ) contains q + 1 τ -invariant points.Such a τ -invariant point M is an intersection point of 2 or more projective lines if andonly if | t | = | π | or Λ = M τ ⊆ C has a norm ideal satisfying Nm(Λ ) ⊆ π O F . (a) e = 2, f = 1, v ( t ) = 2. (b) e = 2, f = 1, v ( t ) = 1. Figure 2.
The reduced locus of N naive E for an (R-U) extension E | F where e and f are the ramification index and the inertia degree of F | Q and v ( t ) is the π -adic valuation of t . We always have 1 ≤ v ( t ) ≤ e .The solid lines lie in N E ⊆ N naive E .Let Λ ⊆ C as in Lemma 4.5. We denote by X +Λ the formal O F -module correspondingto the Dieudonné module M = Λ ⊗ ˘ O E . There is a canonical quasi-isogeny % +Λ : X −→ X +Λ of F -height 1. For S ∈ Nilp ˘O F , we define N E, Λ ( S ) = { ( X, ι, λ, % ) ∈ N naive E ( S ) | ( % +Λ × S ) ◦ % is an isogeny } . By [17, Prop. 2.9], the functor N E, Λ is representable by a closed formal subscheme of N naive E . On geometric points, we have N E, Λ ( k ) ∼ −→ P (Λ / ΠΛ)( k ) , (4.5)as follows from Lemma 4.5 (1). Proposition 4.7.
The reduced locus of N naive E is a union ( N naive E ) red = [ Λ ⊆ C N E, Λ where Λ runs over all unimodular lattices in C with Nm(Λ) ⊆ π O F . For each Λ , thereexists an isomorphism N E, Λ ∼ −→ P (Λ / ΠΛ) , inducing the bijection (4.5) on k -valued points. The proof is analogous to that of Proposition 3.9.
Remark 4.8.
Similar to Remark 3.8 (3), we let ( N E ) red ⊆ ( N naive E ) red be the union ofall projective lines N E, Λ corresponding to hyperbolic unimodular lattices Λ ⊆ C . Later,we will define N E as a functor on Nilp ˘O F and show that N E ’ M Dr , where M Dr is theDrinfeld moduli problem (see Theorem 4.14, a description of the formal scheme M Dr can be found in [3, I.3]). In particular, ( N E ) red is connected and each projective linein ( N E ) red has q + 1 intersection points and there are 2 lines intersecting in each suchpoint.It might happen that ( N E ) red = ( N naive E ) red (see, for example, Figure 2( b )), if there areno non-hyperbolic unimodular lattices Λ ⊆ C with Nm(Λ) ⊆ π O F . In fact, this is thecase if and only if | t | = | π | , see Proposition 2.4 and Lemma 2.5. (Note however thatwe still have N E = N naive E , see Remark 4.11 and Section 4.4.)Assume | t | 6 = | π | and let P ∈ N E ( k ) be an intersection point. Then, as in the casewhere E | F is of type (R-P) (compare Remark 3.8 (3)), the connected component of P in(( N naive E ) red \ ( N E ) red ) ∪ { P } consists of a finite union of projective lines (correspondingto non-hyperbolic lattices, by definition of ( N E ) red ). In Figure 2( a ), these componentsare indicated by dashed lines (they consist of just one projective line in that case).4.2. The straightening condition.
As in the case (R-P), see section 3.2, we use theresults of section 5 to define the straightening condition on N naive E . By Theorem 5.2 andRemark 5.1 (2), there exists a principal polarization e λ X on the framing object ( X , ι X , λ X )such that the Rosati involution is the identity on O E . We set e λ X = e λ X ◦ ι X (Π), which isagain a polarization on X with the Rosati involution inducing the identity on O E , butwith kernel ker e λ X = X [Π]. This polarization is unique up to a scalar in O × E , i.e. , anytwo polarizations e λ X and e λ X with these properties satisfy e λ X = e λ X ◦ ι ( α ) , for some α ∈ O × E . For any ( X, ι, λ, % ) ∈ N naive E ( S ), e λ = % ∗ ( e λ X ) = % ∗ ( e λ X ) ◦ ι (Π)is a polarization on X with kernel ker e λ = X [Π], see Theorem 5.2 (2).Recall that a unimodular or Π-modular lattice Λ ⊆ C is called hyperbolic if thereexists a basis ( e , e ) of Λ such that, with respect to this basis, h has the form (cid:18) Π i Π i (cid:19) , for i = 0 resp. 1. By Lemma 2.5, this is the case if and only if Nm(Λ) = tO F . Proposition 4.9.
For a suitable choice of ( X , ι X , λ X ) and e λ X , the quasi-polarization λ X , = 1 t ( λ X + e λ X ) is a polarization on X . Let ( X, ι, λ, % ) ∈ N naive E ( k ) and e λ = % ∗ ( e λ X ) . Then λ = t ( λ + e λ ) is a polarization if and only if ( X, ι, λ, % ) ∈ N E, Λ ( k ) for a hyperbolic unimodular lattice Λ ⊆ C .Proof. On the rational Dieudonné module N = M X ⊗ ˘ O F ˘ F , denote by h , i , ( , ) and h , i the alternating forms induced by λ X , e λ X and λ X , , respectively. The form h , i is integralon M X if and only if λ X , is a polarization on X . We have( F x, y ) = ( x, V y ) σ , (Π x, y ) = ( x, Π y ) , h x, y i = 1 t ( h x, y i + ( x, y )) , ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 27 for all x, y ∈ N . The form ( , ) induces an ˘ E -bilinear alternating form b on N by theformula b ( x, y ) = c ((Π x, y ) − Π( x, y )) . (4.6)Here, c is a unit in ˘ O E such that c · σ ( c ) − = ΠΠ − . Since ΠΠ = t − ΠΠ ∈ t Π ˘ O E , wecan even choose c ∈ t Π − ˘ O E . The dual of M with respect to this form is again M ] = Π − M , since ( x, y ) = Tr ˘ E | ˘ F (cid:18) tϑc · b ( x, y ) (cid:19) , and the inverse different of E | F is given by D − E | F = t − O E , see Lemma 2.2. Now b isinvariant under the σ -linear operator τ = Π V − = F Π − , because b ( τ x, τ y ) = b ( F Π − x, Π V − y ) = cσ ( c ) · b (Π − x, Π y ) σ = b ( x, y ) σ . Hence b defines an E -linear alternating form on C .We choose the framing object ( X , ι X , λ X ) such that M X is τ -invariant (see Lemma 4.4)and such that Λ = M τ X is hyperbolic. We can find a basis ( e , e ) of Λ such that h b = (cid:18) ΠΠ (cid:19) , b b = (cid:18) u − u (cid:19) , for some u ∈ E × . Since e λ X has the same kernel as λ X , we have u = Π u for some unit u ∈ O × E . We can choose e λ X such that u = 1 and u = Π. Now t ( h ( x, y ) + b ( x, y )) isintegral for all x, y ∈ Λ . Hence t ( h ( x, y ) + b ( x, y )) is also integral for all x, y ∈ M X .For all x, y ∈ M X , we have h x, y i = 1 t ( h x, y i + ( x, y )) = 1 t Tr ˘ E | ˘ F (cid:18) tϑ · h ( x, y ) + 1 tϑc · b ( x, y ) (cid:19) = Tr ˘ E | ˘ F (cid:18) t ϑ · ( h ( x, y ) + b ( x, y )) (cid:19) + Tr ˘ E | ˘ F (cid:18) − ct ϑc · b ( x, y ) (cid:19) . The first summand is integral since t ( h ( x, y ) + b ( x, y )) is integral. The second summandis integral since 1 − c is divisible by t Π − and b ( x, y ) lies in Π ˘ O E . It follows that thesecond summand above is integral as well. Hence h , i is integral on M X and this impliesthat λ X , is a polarization on X .Now let ( X, ι, λ, % ) ∈ N naive E ( k ) and denote by M ⊆ N its Dieudonné module. Assumethat λ = t − ( λ + e λ ) is a polarization on X . Then h , i is integral on M . But this isequivalent to t − ( h ( x, y ) + b ( x, y )) being integral for all x, y ∈ M . For x = y , we have h ( x, x ) = h ( x, x ) + b ( x, x ) ∈ t ˘ O F . Let Λ ⊆ C be the unimodular or Π-modular lattice given by Λ = M τ resp. Λ =( M + τ ( M )) τ , see Lemma 4.4. Then h ( x, x ) ∈ tO F for all x ∈ Λ. Thus Nm(Λ) ⊆ tO F and, by minimality, this implies that Nm(Λ) = tO F and Λ is hyperbolic (see Lemma2.5). Hence, in either case, the point corresponding to ( X, ι, λ, % ) lies in N E, Λ for ahyperbolic lattice Λ .Conversely, assume that ( X, ι, λ, % ) ∈ N E, Λ ( k ) for some hyperbolic lattice Λ ⊆ C . Wewant to show that λ is a polarization on X . This follows if h , i is integral on M , orequivalently, if t − ( h ( x, y ) + b ( x, y )) is integral on M . For this, it is enough to show that t − ( h ( x, y ) + b ( x, y )) is integral on Λ. Let Λ ⊆ C be the unimodular lattice generatedby Π − e and e , where ( e , e ) is the basis of the Π-modular lattice Λ = M X . Withrespect to the basis (Π − e , e ), we have h b = (cid:18) (cid:19) , b b = (cid:18) − (cid:19) . In particular, Λ is a hyperbolic lattice and t − ( h + b ) is integral on Λ . By Proposition2.4, there exists an element g ∈ SU(
C, h ) with g Λ = Λ . Since det g = 1, the alternatingform b is invariant under g . Thus t − ( h + b ) is also integral on Λ. (cid:3) From now on, we assume that ( X , ι X , λ X ) and e λ X are chosen in a way such that λ X , = 1 t ( λ X + e λ X ) ∈ Hom( X , X ∨ ) . Definition 4.10.
A tuple (
X, ι, λ, % ) ∈ N naive E ( S ) satisfies the straightening condition if λ = 1 t ( λ + e λ ) ∈ Hom(
X, X ∨ ) . (4.7)This condition is independent of the choice of e λ X . In fact, we can only change e λ X by a scalar of the form 1 + t Π − u , u ∈ O E . But if e λ X = e λ X ◦ ι (1 + t Π − u ), then λ X , = λ X , + e λ X ◦ ι (Π − u ) = λ X , + e λ X ◦ ι ( u ) and λ = λ + % ∗ ( e λ X ) ◦ ι ( u ). Clearly, λ is a polarization if and only if λ is one.For S ∈ Nilp ˘O F , let N E ( S ) be the set of all tuples ( X, ι, λ, % ) ∈ N naive E ( S ) that satisfythe straightening condition. By [17, Prop. 2.9], the functor N E is representable by aclosed formal subscheme of N naive E . Remark 4.11.
The reduced locus of N E is given by( N E ) red = [ Λ ⊆ C N E, Λ ’ [ Λ ⊆ C P (Λ / ΠΛ) , where the union goes over all hyperbolic unimodular lattices Λ ⊆ C . Note that, de-pending on the form of the (R-U) extension E | F , it may happen that all unimodularlattices are hyperbolic (when | t | = | π | ) and in that case, we have ( N E ) red = ( N naive E ) red .However, the equality does not extend to an isomorphism between N E and N naive E . Thiswill be discussed in section 4.4.4.3. The main theorem for the case (R-U).
As in the case (R-P), we want toestablish a connection to the Drinfeld moduli problem. Therefore, fix an embeddingof E into the quaternion division algebra B . Let ( X , ι X ) be the framing object of theDrinfeld problem. We want to construct a polarization λ X on X with ker λ X = X [Π]and Rosati involution given by b ϑb ϑ − on B . Here b b denotes the standardinvolution on B .By Lemma 2.3 (2), there exists an embedding E , → B of a ramified quadraticextension E | F of type (R-P), such that Π ϑ = − ϑ Π for a prime element Π ∈ E .From Proposition 3.14 (1) we get a principal polarization λ X on X with associated Rosatiinvolution b Π b Π − . If we assume fixed choices of E and Π , this is unique up toa scalar in O × F . We define λ X = λ X ◦ ι X (Π ϑ ) . Since λ X is a principal polarization and Π ϑ and Π have the same valuation in O B , wehave ker λ X = X [Π]. The Rosati involution of λ X is b ϑb ϑ − . On the other hand, anypolarization on X satisfying these two conditions can be constructed in this way (usingthe same choices for E and Π ). Hence: Lemma 4.12. (1)
There exists a polarization λ X : X → X ∨ , unique up to a scalar in O × F , with ker λ X = X [Π] and associated Rosati involution b ϑb ϑ − . (2) Fix λ X as in (1) and let ( X, ι B , % ) ∈ M Dr ( S ) . There exists a unique polarization λ on X with ker λ = X [Π] and Rosati involution b ϑb ϑ − such that % ∗ ( λ X ) = λ on S = S × Spf ˘ O F k . ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
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Note also that the involution b ϑb ϑ − does not depend on the choice of ϑ ∈ E .We write ι X ,E for the restriction of ι X to E ⊆ B and, in the same manner, we write ι E for the restriction of ι B to E for any ( X, ι B , % ) ∈ M Dr ( S ). Fix a polarization λ X of X as in Lemma 4.12 (1). Accordingly for a tuple ( X, ι B , % ) ∈ M Dr ( S ), let λ be thepolarization given by Lemma 4.12 (2). Lemma 4.13.
The tuple ( X , ι X ,E , λ X ) is a framing object of N naive E . Moreover, the map ( X, ι B , % ) ( X, ι E , λ, % ) induces a closed embedding of formal schemes η : M Dr , −→ N naive E . Proof.
We follow the same argument as in the proof of Lemma 3.15. Again it is enough tocheck that QIsog( X , ι X , λ X ) contains SU( C, h ) as a closed subgroup and that ι E satisfiesthe Kottwitz condition.By [18, Prop. 5.8], the special condition on ι B implies the Kottwitz condition for ι E .It remains to show that SU( C, h ) ⊆ QIsog( X , ι X , λ X ). But the group G ( X ,ι X ) of automor-phisms of determinant 1 of ( X , ι X ) is isomorphic to SL ,F and G ( X ,ι X ) ⊆ QIsog( X , ι X , λ X )is a Zariski-closed subgroup by the same argument as in Lemma 3.15. Hence the state-ment follows from the exceptional isomorphism SL ,F ’ SU(
C, h ). (cid:3) As a next step, we want to show that this already induces a closed embedding η : M Dr , −→ N E . (4.8)Let e E , → B an embedding of a ramified quadratic extension e E | F of type (R-U) as inLemma 2.3 (2). On the framing object ( X , ι X ) of M Dr , we define a polarization e λ X via e λ X = λ X ◦ ι X ( e ϑ ) , where e ϑ is a unit in e E of the form e ϑ = 1 + ( t /π ) · u , see Lemma 2.3 (2). The Rosatiinvolution of e λ X induces the identity on O E and we have λ X , = 1 t ( λ X + e λ X ) = 1 t · λ X ◦ ι B (1 + e ϑ ) = λ X ◦ ι B ( e Π /π )= λ X ◦ ι B (Π − γ ) ∈ Hom( X , X ∨ ) , using the notation of Lemma 2.3 (2). For ( X, ι B , % ) ∈ M Dr ( S ), we set e λ = λ ◦ ι B ( e ϑ ).By the same calculation, we have λ = t ( λ + e λ ) ∈ Hom(
X, X ∨ ). Thus the tuple( X, ι E , λ, % ) = η ( X, ι B , % ) satisfies the straightening condition. Hence we get a closedembedding of formal schemes η : M Dr → N E which is independent of the choice of e E . Theorem 4.14. η : M Dr → N E is an isomorphism of formal schemes. We first check this for k -valued points: Lemma 4.15. η induces a bijection η ( k ) : M Dr ( k ) → N E ( k ) .Proof. We only have to show surjectivity and we will use for this the Dieudonné theorydescription of N naive E ( k ), see (4.4). The rational Dieudonné-module N = N X of X nowcarries additionally an action of B . The embedding F (2) , → B given by γ Π · e Π π , (4.9)(see Lemma 2.3 (2)) induces a Z / N = N ⊕ N . Here, N = { x ∈ N | ι ( a ) x = ax for all a ∈ F (2) } ,N = { x ∈ N | ι ( a ) x = σ ( a ) x for all a ∈ F (2) } , for a fixed embedding F (2) , → ˘ F . The operators F and V have degree 1 with respect tothis grading. The principal polarization λ X , = 1 t ( λ X + e λ X ) = λ X ◦ ι X (Π − γ )induces an alternating form h , i on N that satisfies h x, y i = h x, ι (Π − γ ) · y i , for all x, y ∈ N . Let M ∈ N E ( k ) ⊆ N naive E ( k ) be an ˘ O F -lattice in N . We claim that M ∈ M Dr ( k ). For this, it is necessary that M is stable under the action of O (2) F (since O B = O F [Π , γ ] = O (2) F [Π], see Lemma 2.3 (2)) or equivalently, that M respects thegrading of N , i.e. , M = M ⊕ M for M i = M ∩ N i . Furthermore M has to satisfy the special condition: dim M / V M = dim M / V M = 1 . We first show that M = M ⊕ M . Let y = y + y ∈ M with y i ∈ N i . Since M = Π M ∨ ,we have h x, ι (Π) − y i = h x, ι (Π) − y i + h x, ι (Π) − y i ∈ ˘ O F , for all x ∈ M . Together with h x, y i = h x, y i + h x, y i = h x, ι ( e Π /π ) y i + h x, ι ( e Π /π ) y i = γ · h x, ι (Π − ) y i + (1 − γ ) · h x, ι (Π − ) y i ∈ ˘ O F , this implies that h x, ι (Π − ) y i and h x, ι (Π − ) y i lie in ˘ O F for all x ∈ M . Hence, y , y ∈ M and this means that M respects the grading. It follows that M is stableunder the action of O B .In order to show that M is special, note that h V x, V y i σ = h FV x, y i = π · h x, y i ∈ π ˘ O F , for all x, y ∈ M . The form h , i comes from a principal polarization, so it induces aperfect form on M . Now it is enough to show that also the restrictions of h , i to M and M are perfect. Indeed, if M was not special, we would have M i = V M i +1 for some i and this would contradict h , i being perfect on M i . We prove that h , i is perfect on M i by showing h M , M i ⊆ π ˘ O F .Let x ∈ M and y ∈ M . Then, h x, y i = (1 − γ ) · h x, ι (Π) − y i , h x, y i = −h y, x i = − γ · h y, ι (Π) − x i = γ · h x, ι (Π) − y i . We take the difference of these two equations. From Π ≡ Π mod π , it follows that h x, ι (Π) − y i ≡ π and thus also h x, y i ≡ π . The form h , i is henceperfect on M and M and the special condition follows. This finishes the proof ofLemma 4.15. (cid:3) Proof of Theorem 4.14.
Let ( X , ι X ) be a framing object for M Dr and let further η ( X , ι X ) = ( X , ι X ,E , λ X )be the corresponding framing object for N E . We fix an embedding F (2) , → B as inLemma 2.3 (2). For S ∈ Nilp ˘O F , let ( X, ι, λ, % ) ∈ N E ( S ) and e λ = % ∗ ( e λ X ). We have % − ◦ ι X ( γ ) ◦ % = % − ◦ ι X (Π) ◦ λ − X ◦ λ X , ◦ % = ι (Π) ◦ λ − ◦ λ ∈ End( X ) , ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 31 for λ = t − ( λ + e λ ), since ker λ = X [Π]. But O B = O F [Π , γ ] (see Lemma 2.3 (2)), sothis already induces an O B -action ι B on X . It remains to show that ( X, ι B , % ) satisfiesthe special condition (see the discussion before Proposition 3.14 for a definition).The special condition is open and closed (see [18, p. 7]) and η is bijective on k -points.Hence η induces an isomorphism on reduced subschemes( η ) red : ( M Dr ) red ∼ −→ ( N E ) red , because ( M Dr ) red and ( N E ) red are locally of finite type over Spec k . It follows that η : M Dr → N E is an isomorphism. (cid:3) Deformation theory of intersection points.
In this section, we will study thedeformation rings of certain geometric points in N naive E with the goal of proving that N E ⊆ N naive E is a strict inclusion even in the case | t | = | π | . In contrast to the non-2-adiccase, we are not able to use the theory of local models (see [15] for a survey) since thereis in general no normal form for the lattices Λ ⊆ C , see Proposition 2.4 and [17, Thm.3.16]. Thus we will take the more direct approach of studying the deformations of afixed point (
X, ι, λ, % ) ∈ N naive E ( k ) and using the theory of Grothendieck-Messing ([13]).Let Λ ⊆ C be a Π-modular hyperbolic lattice. By Lemma 4.5, there is a uniquepoint x = ( X, ι, λ, % ) ∈ N naive E ( k ) with a τ -stable Dieudonné module M ⊆ C ⊗ E ˘ E and M τ = Λ. Since Λ is hyperbolic, x satisfies the straightening condition, i.e. , x ∈ N E ( k ).(In Figure 2, x would lie on the intersection of two solid lines.)Let b O N naive E ,x be the formal completion of the local ring at x . It represents thefollowing deformation functor Def x . For an artinian ˘ O F -algebra R with residue field k ,we have Def x ( R ) = { ( Y, ι Y , λ Y ) /R | Y k ∼ = X } , where ( Y, ι Y , λ Y ) satisfies the usual conditions (see section 4.1) and the isomorphism Y k ∼ = X is actually an isomorphism of tuples ( Y k , ι Y , λ Y ) ∼ = ( X, ι, λ ) as in Definition 3.1.Now assume the quotient map R → k is an O F -pd-thickening (cf. [1]). For example,this is the case when m = 0 for the maximal ideal m of R . Then, by Grothendieck-Messing theory (see [13] and [1]), we get an explicit description of Def x ( R ) in terms ofliftings of the Hodge filtration:The (relative) Dieudonné crystal D X ( R ) of X evaluated at R is naturally isomorphicto the free R -module Λ ⊗ O F R and this isomorphism is equivariant under the action of O E induced by ι and respects the perfect form Φ = h , i ◦ (1 , Π − ) induced by λ ◦ ι (Π − ).The Hodge-filtration of X is given by F X = V · D X ( k ) ∼ = Π · (Λ ⊗ O F k ) ⊆ Λ ⊗ O F k .A point Y ∈ Def x ( R ) now corresponds, via Grothendieck-Messing, to a direct sum-mand F Y ⊆ Λ ⊗ O F R of rank 2 lifting F X , stable under the O E -action and totallyisotropic with respect to Φ. Furthermore, it has to satisfy the Kottwitz condition (seesection 4.1): For the action of α ∈ O E on Lie Y = (Λ ⊗ O F R ) / F Y , we havechar(Lie Y, T | ι ( α )) = ( T − α )( T − α ) . Let us now fix an O E -basis ( e , e ) of Λ and let us write everything in terms of the O F -basis ( e , e , Π e , Π e ). Since Λ is hyperbolic, we can fix ( e , e ) such that h isrepresented by the matrix h b = (cid:18) ΠΠ (cid:19) , It is possible define a local model for the non-naive spaces N E (also in the case (R-P)) and establisha local model diagram as in [17, 3.27]. The local model is then isomorphic to the local model of theDrinfeld moduli problem. This will be part of a future paper of the author. and then Φ = Tr E | F tϑ h ( · , Π − · ) b = t/π − − t /π t . An R -basis ( v , v ) of F Y can now be chosen such that( v v ) = y y y y , with y ij ∈ R . As an easy calculation shows, the conditions on F Y above are nowequivalent to the following conditions on the y ij : y + y = t,y y − y y = π ,t ( ty π + 2) = y ( ty π + 2) = y ( ty π + 2) = y ( ty π + 2) = 0 . Let T be the closed subscheme of Spec O F [ y , y , y , y ] given by these equations.Let T y be the formal completion of the localization at the ideal generated by the y ij and π . Then we have Def x ( R ) ∼ = T y ( R ) for any O F -pd-thickening R → k . In particular,the first infinitesimal neighborhoods of Def x and T y coincide. The first infinitesimalneighborhood of T y is given by Spec O F [ y ij ] / (( y ij ) , y + y − t, π ), hence T y hasKrull dimension 3 and so has Def x . However, M Dr is regular of dimension 2, cf. [3].Thus, Proposition 4.16. N naive E = M Dr , even when | t | = | π | . Indeed, dim b O N naive E ,x = dim Def x = 3 > b O N E ,x .5. A theorem on the existence of polarizations
In this section, we will prove the existence of the polarization e λ for any ( X, ι, λ, % ) ∈N naive E ( S ) as claimed in the sections 3.2 and 4.2 in both the cases (R-P) and (R-U). Infact, we will show more generally that e λ exists even for the points of a larger modulispace M E where we forget about the polarization λ .We start with the definition of the moduli space M E . Let F | Q p be a finite extension(not necessarily p = 2) and let E | F be a quadratic extension (not necessarily ramified).We denote by O F and O E the rings of integers, by k the residue field of O F and by k the algebraic closure of k . Furthermore, ˘ F is the completion of the maximal unramifiedextension of F and ˘ O F its ring of integers. Let B be the quaternion division algebraover F and O B the ring of integers.If E | F is unramified, we fix a common uniformizer π ∈ O F ⊆ O E . If E | F is ramifiedand p >
2, we choose a uniformizer Π ∈ O E such that π = Π ∈ O F . If E | F is ramifiedand p = 2, we use the notations of section 2 for the cases (R-P) and (R-U).For S ∈ Nilp ˘O F , let M E ( S ) be the set of isomorphism classes of tuples ( X, ι E , % ) over S . Here, X is a formal O F -module of dimension 2 and height 4 and ι E is an action of O E on X satisfying the Kottwitz condition for the signature (1 , i.e. , the characteristicpolynomial for the action of ι E ( α ) on Lie( X ) ischar(Lie X, T | ι ( α )) = ( T − α )( T − α ) , (5.1) ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 33 for any α ∈ O E , compare the definition of N naive E in the sections 3 and 4. The last entry % is an O E -linear quasi-isogeny % : X × S S −→ X × Spec k S, of height 0 to the framing object ( X , ι X ,E ) defined over Spec k . The framing object for M E is the Drinfeld framing object ( X , ι X ,B ) where we restrict the O B -action to O E for an arbitrary embedding O E , → O B . The special condition on ( X , ι X ,B ) implies theKottwitz condition for any α ∈ O E by [18, Prop. 5.8]. Remark 5.1. (1) Up to isogeny, there is more than one pair (
X, ι E ) over Spec k satisfy-ing the conditions above. Indeed, let N X be the rational Dieudonné module of ( X, ι E ).This is a 4-dimensional ˘ F -vector space with an action of O E . The Frobenius F on N X commutes with the action of O E . For a suitable choice of a basis of N X , it may be ofeither of the following two forms, F = π π σ or F = π π σ. This follows from the classification of isocrystals, see for example [17, p. 3]. In theleft case, F is isoclinic of slope 1 / O F -modules of dimension 1 and height 2 ( cf. [3, p.136-137]).(2) Let p = 2 and E | F ramified of type (R-P) or (R-U). We can identify the framingobjects ( X , ι X ,E ) for N naive E , M Dr and M E by Lemma 3.14 and Lemma 4.13. In this way,we obtain a forgetful morphism N naive E → M E . This is a closed embedding, since theexistence of a polarization λ for ( X, ι E , % ) ∈ M E ( S ) is a closed condition by [17, Prop.2.9].By [17, Thm. 3.25], M E is pro-representable by a formal scheme over Spf ˘ O F . Wewill prove the following theorem in this section. Theorem 5.2. (1)
There exists a principal polarization e λ X on ( X , ι X ,E ) such that theRosati involution induces the identity on O E , i.e. , ι ( α ) ∗ = ι ( α ) for all α ∈ O E . Thispolarization is unique up to a scalar in O × E , that is, for any two polarizations e λ X and e λ X of this form, there exists an element α ∈ O × E such that e λ X = e λ X ◦ ι X ,E ( α ) . (2) Fix e λ X as in part (1) . For any S ∈ Nilp ˘O F and ( X, ι E , % ) ∈ M E ( S ) , there exists aunique principal polarization e λ on X such that the Rosati involution induces the identityon O E and such that e λ = % ∗ ( e λ X ) . Remark 5.3. (1) We will see later that this theorem describes a natural isomorphismbetween M E and another space M E, pol which solves the moduli problem for tuples( X, ι E , e λ, % ) where e λ is a principal polarization with Rosati involution the identity on O E . This is an RZ-space for the symplectic group GSp ( E ) and thus the theorem givesus another geometric realization of an exceptional isomorphism of reductive groups, inthis case GSp ( E ) ∼ = GL ( E ).Since there is no such isomorphism in higher dimensions, the theorem does not gen-eralize to these cases and a different approach is needed to formulate the straighteningcondition.(2) With the Theorem 5.2 established, one can give an easier proof of the isomorphism N E ∼ −→ M Dr for the cases where E | F is unramified or E | F is ramified and p >
2, which is the main theorem of [11]. Indeed, the main part of the proof in loc. cit. consists of thePropositions 2.1 and 3.1, which claim the existence of a certain principal polarization λ X for any point ( X, ι, λ, % ) ∈ N E ( S ). But there is a canonical closed embedding N E , → M E and under this embedding, λ X is just the polarization e λ of Theorem 5.2,for a suitable choice of e λ X on the framing object. More explicitly, using the notation onpage 2 of loc. cit., we take e λ X = λ X ◦ ι − X (Π) = λ X ◦ ι X ( − δ ) in the unramified case and e λ X = λ X ◦ ι X ( ζ − ) in the ramified case.We will split the proof of this theorem into several lemmata. As a first step, we useDieudonné theory to prove the statement for all geometric points. Lemma 5.4.
Part (1) of theorem holds. Furthermore, for a fixed polarization e λ X on ( X , ι X ,E ) and for any ( X, ι E , % ) ∈ M E ( k ) , the pullback e λ = % ∗ ( e λ X ) is a polarization on X .Proof. This follows almost immediately from the theory of affine Deligne-Lusztig vari-eties (see, for example, [5]) since we are comparing the geometric points of RZ-spacesfor the isomorphic groups GL ( E ) and GSp ( E ).It is also possible to check this via a more direct computation using Dieudonné theory,as we will indicate briefly. Proceeding very similarly to Proposition 3.2 or Proposition4.1 (cf. [11] in the unramified case), we can associate to X a lattice Λ in the 2-dimensional E -vector space C (the Frobenius invariant points of the (rational) Dieudonné module).The choice of a principal polarization on X with trivial Rosati involution correspondsnow exactly to a choice of perfect alternating form on Λ. It immediately follows thatsuch a polarization exists and that it is unique up to a scalar in O × E .For the second part, let X ∈ M E ( k ) and M ⊆ C ⊗ E ˘ E be its Dieudonné module.Since % has height 0, we have[ M : M ∩ (Λ ⊗ E ˘ E )] = [(Λ ⊗ E ˘ E ) : M ∩ (Λ ⊗ E ˘ E )] , and one easily checks that a perfect alternating form b on Λ is also perfect on M . (cid:3) In the following, we fix a polarization e λ X on ( X , ι X ,E ) as in Theorem 5.2 (1). Let( X, ι E , % ) ∈ M E ( S ) for S ∈ Nilp ˘O F and consider the pullback e λ = % ∗ ( e λ X ). In general,this is only a quasi-polarization. It suffices to show that e λ is a polarization on X . Indeed,since % is O E -linear and of height 0, this is then automatically a principal polarizationon X such that the Rosati involution is the identity on O E .Define a subfunctor M E, pol ⊆ M E by M E, pol ( S ) = { ( X, ι E , % ) ∈ M E ( S ) | e λ = % ∗ ( e λ X ) is a polarization on X } . This is a closed formal subscheme by [17, Prop. 2.9]. Moreover, Lemma 5.4 shows that M E, pol ( k ) = M E ( k ). Remark 5.5.
Equivalently, we can describe M E, pol as follows. For S ∈ Nilp ˘O F , wedefine M E, pol ( S ) to be the set of equivalence classes of tuples ( X, ι E , e λ, % ) where • X is a formal O F -module over S of height 4 and dimension 2, • ι E is an action of O E on X that satisfies the Kottwitz condition in (5.1) and • e λ is a principal polarization on X such that the Rosati involution induces the identityon O E . • Furthermore, we fix a framing object ( X , ι X ,E , e λ X ) over Spec k , where ( X , ι X ,E ) is theframing object for M E and e λ X is a polarization as in Theorem 5.2 (1). Then % is an O E -linear quasi-isogeny % : X × S S −→ X × Spec k S, ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
1) 35 of height 0 such that, locally on S , the (quasi-)polarizations % ∗ ( e λ X ) and e λ on X only differby a scalar in O × E , i.e. , there exists an element α ∈ O × E such that % ∗ ( e λ X ) = e λ ◦ ι E ( α ).Two tuples ( X, ι E , e λ, % ) and ( X , ι E , e λ , % ) are equivalent if there exists an O E -linearisomorphism ϕ : X ∼ −→ X such that ϕ ∗ ( e λ ) and e λ only differ by a scalar in O × E .In this way, we gave a definition for M E, pol by introducing extra data on points of themoduli space M E , instead of extra conditions. It is now clear, that M E, pol describes amoduli problem for p -divisible groups of (PEL) type. It is easily checked that the twodescriptions of M E, pol give rise to the same moduli space.Theorem 5.2 now holds if and only if M E, pol = M E . This equality is a consequenceof the following statement. Lemma 5.6.
For any point x = ( X, ι E , % ) ∈ M E, pol ( k ) , the embedding M E, pol , → M E induces an isomorphism of completed local rings b O M E, pol ,x ∼ = b O M E ,x . For the proof of this Lemma, we use the theory of local models, cf. [17, Chap. 3]. Wepostpone the proof of this lemma to the end of this section and we first introduce thelocal models M loc E and M loc E, pol for M E and M E, pol .Let C be a 4-dimensional F -vector space with an action of E and let Λ ⊆ C be an O F -lattice that is stable under the action of O E . Furthermore, let ( , ) be an F -bilinearalternating form on C with ( αx, y ) = ( x, αy ) , (5.2)for all α ∈ E and x, y ∈ C and such that Λ is unimodular with respect to ( , ). It iseasily checked that ( , ) is unique up to an isomorphism of C that commutes with the E -action and that maps Λ to itself.For an O F -algebra R , let M loc E ( R ) be the set of all direct summands F ⊆ Λ ⊗ O F R of rank 2 that are O E -linear and satisfy the Kottwitz condition . That means, for all α ∈ O E , the action of α on the quotient (Λ ⊗ O F R ) / F has the characteristic polynomialchar(Lie X, T | α ) = ( T − α )( T − α ) . The subset M loc E, pol ( R ) ⊆ M loc E ( R ) consists of all direct summands F ∈ M loc E ( R ) that arein addition totally isotropic with respect to ( , ) on Λ ⊗ O F R .The functor M loc E is representable by a closed subscheme of Gr(2 , Λ) O F , the Grassma-nian of rank 2 direct summands of Λ, and M loc E, pol is representable by a closed subschemeof M loc E . In particular, both M loc E and M loc E, pol are projective schemes over Spec O F .These local models have already been studied by Deligne and Pappas. In particular,we have: Proposition 5.7 ([6]) . M loc E, pol = M loc E . In other words, for an O F -algebra R , any directsummand F ∈ M loc E ( R ) is totally isotropic with respect to ( , ) . The moduli spaces M E and M E, pol are related to the local models M loc E and M loc E, pol via local model diagrams, cf. [17, Chap. 3]. Let M large E be the functor that maps ascheme S ∈ Nilp ˘O F to the set of isomorphism classes of tuples ( X, ι E , % ; γ ). Here,( X, ι E , % ) ∈ M E ( S ) , and γ is an O E -linear isomorphism γ : D X ( S ) ∼ −→ Λ ⊗ O F O S . On the left hand side, D X ( S ) denotes the (relative) Grothendieck-Messing crystal of X evaluated at S , cf. [1, 5.2]. Let b M loc E be the π -adic completion of M loc E ⊗ O F ˘ O F . Then there is a local modeldiagram: M large Ef (cid:123) (cid:123) g (cid:35) (cid:35) M E b M loc E The morphism f on the left hand side is the projection ( X, ι E , % ; γ ) ( X, ι E , % ). Themorphism g on the right hand side maps ( X, ι E , % ; γ ) ∈ M large E ( S ) to F = ker(Λ ⊗ O F O S γ − −−→ D X ( S ) −→ Lie X ) ⊆ Λ ⊗ O F O S . By [17, Thm. 3.11], the morphism f is smooth and surjective. The morphism g isformally smooth by Grothendieck-Messing theory, see [13, V.1.6], resp. [1, Chap. 5.2]for the relative setting ( i.e. , when O F = Z p ).We also have a local model diagram for the space M E, pol . We define M large E, pol as thefiber product M large E, pol = M E, pol × M E M large E . Then M large E, pol is closed formal subschemeof M large E with the following moduli description. A point ( X, ι E , % ; γ ) ∈ M large E ( S ) liesin M large E, pol ( S ) if e λ = % ∗ ( e λ X ) is a principal polarization on X . In that case, e λ induces analternating form ( , ) X on D X ( S ) which, under the isomorphism γ , is equal to the form( , ) on Λ ⊗ O F O S , up to a unit in O E ⊗ O F O S .The local model diagram for M E, pol now looks as follows. M large E, pol f pol (cid:122) (cid:122) g pol (cid:35) (cid:35) M E, pol b M loc E, pol (5.3)Here, b M loc E, pol is the π -adic completion of M loc E, pol ⊗ O F ˘ O F and f pol and g pol are the restric-tions of the morphisms f and g above. Again, g pol is formally smooth by Grothendieck-Messing theory and f pol is smooth and surjective by construction.We can now finish the proof of Lemma 5.6. Proof of Lemma 5.6.
We have the following commutative diagram. M E, pol M large E, pol f pol (cid:111) (cid:111) g pol (cid:47) (cid:47) b M loc E, pol M E M large Ef (cid:111) (cid:111) g (cid:47) (cid:47) b M loc E (cid:95)(cid:127) (cid:15) (cid:15) (cid:95)(cid:127) (cid:15) (cid:15) (5.4)The equality on the right hand side follows from Proposition 5.7. The other verticalarrows are closed embeddings.Let x ∈ M E, pol ( k ). By [17, Prop. 3.33], there exists an étale neighbourhood U of x in M E and section s : U → M large E such that g ◦ s is formally étale. Similarly, U pol = U × M E M E, pol and s pol is the base change of s to U pol . Then the composition g pol ◦ s pol is also formally étale. This formally étale maps induce isomorphism of localrings b O M E ,x ∼ −→ b O b M loc E ,x and b O M E, pol ,x ∼ −→ b O b M loc E, pol ,x , x = s ( g ( x )). By Proposition5.7, we have b O b M loc E ,x = b O b M loc E, pol ,x and since this identification commutes with g ◦ s (resp. g pol ◦ s pol ), we get the desired isomorphism b O M E, pol ,x ∼ = b O M E ,x . (cid:3) ONSTRUCTION OF A RZ-SPACES FOR 2-ADIC RAMIFIED GU(1 ,
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