Regular integral models for Shimura varieties of orthogonal type
aa r X i v : . [ m a t h . N T ] F e b REGULAR INTEGRAL MODELS FOR SHIMURA VARIETIESOF ORTHOGONAL TYPE
G. PAPPAS AND I. ZACHOS
Abstract.
We consider Shimura varieties for orthogonal or spin groups acting onhermitian symmetric domains of type IV. We give regular p -adic integral models forthese varieties over odd primes p at which the level subgroup is the connected stabilizerof a vertex lattice in the orthogonal space. Our construction is obtained by combiningresults of Kisin and the first author with an explicit presentation and resolution of acorresponding local model. Contents
1. Introduction 12. Preliminaries: quadratic forms, lattices and spinors 53. Quadrics and linked quadrics 94. Local models and variants 135. Equations and a resolution for the local model 176. Resolution and the linked quadric 227. Application to Shimura varieties 30References 331.
Introduction
V, Q ) of dimension d with real signature ( d − , d ≥
7, these Shimura varieties are not of PEL typeand so they cannot be given directly as moduli spaces of abelian varieties with polarization,endomorphisms and level structure. However, they are always either of Hodge or of abeliantype. So, they can still be constructed using a relation to Siegel moduli spaces, i.e. tomoduli spaces of polarized abelian varieties.Shimura varieties have canonical models over the “reflex” number field. In the caseswe consider here the reflex field is the field of rational numbers Q . They are also expectedto afford reasonable integral models. However, the behavior of these depends very muchon the “level subgroup”. Here, we consider level subgroups determined by the choice of a lattice Λ ⊂ V on which the quadratic form takes integral values, i.e. for whichΛ ⊂ Λ ∨ where Λ ∨ is the dual lattice. In fact, we study p -adic integral models and their reductionover odd primes p at which the p -power part of the discriminant module Λ ∨ / Λ is anni-hilated by p . The level subgroup at p is parahoric in the sense of Bruhat-Tits and thisallows us to apply the results of the first author with Kisin [10] and construct a p -adicintegral model with controlled singularities. We then build on this and, by using work ofthe second author in [23], we study and resolve the singularities. This leads to regularmodels for these Shimura varieties over the p -adic integers Z p . The models we constructhave very simple local structure: Their fibers over p are divisors with normal crossings,with multiplicities one or two, and with no more than three branches intersecting at apoint. We expect that our construction will find applications to the study of arithmeticintersections of special cycles and Kudla’s program. (See [1] for an important applicationof integral models of spin/orthogonal Shimura varieties to number theory.)1.2. Let us give some details. For technical reasons, it is simpler to discuss Shimuravarieties for the group G = GSpin( V, Q ) of spinor similitudes. We take X to be thecorresponding hermitian symmetric domain of type IV (see § p be an odd primeand choose a Z p -lattice Λ ⊂ V ⊗ Q Q p with p Λ ∨ ⊂ Λ ⊂ Λ ∨ . The lattice defines theparahoric subgroup K p = { g ∈ GSpin( V ⊗ Q Q p ) | g Λ g − = Λ , η ( g ) ∈ Z × p } which we fix below. (Here, η : GSpin( V ⊗ Q Q p ) → Q × p is the spinor similitude, and for v ∈ V ⊗ Q Q p , gvg − is defined using the Clifford algebra, see § § K p of the prime-to- p finite adelic points G ( A pf )of G and set K = K p K p . The Shimura variety Sh K ( G, X ) with complex pointsSh K ( G, X )( C ) = G ( Q ) \ X × G ( A f ) /K is of Hodge type and has a canonical model over the reflex field Q . The following is aspecial case of [10, Theorem 4.2.7], see also [15, Theorem 3.7]. Theorem 1.2.1.
There is a scheme S K ( G, X ) , flat over Spec ( Z p ) , with S K ( G, X ) ⊗ Z p Q p = Sh K ( G, X ) ⊗ Q Q p , and which supports a “local model diagram” (1.2.2) f S K ( G, X ) S K ( G, X ) M loc (Λ) π K q K such that: a) π K is a G -torsor for the parahoric group scheme G that corresponds to K p , b) q K is smooth and G -equivariant. In the above, G is the smooth connected “Bruhat-Tits” group scheme over Spec ( Z p )such that • G ⊗ Z p Q p = G ⊗ Q Q p , • G ( Z p ) = K p , NTEGRAL MODELS FOR SHIMURA VARIETIES 3 and M loc (Λ) is the “local model” as defined by the first author and X. Zhu [19] (followingprevious work by Rapoport-Zink [20] and others).The integral model S K ( G, X ) satisfies several additional properties, see [10] and §
7. Infact, it is “canonical” in the sense of [16]. Set δ = length Z p (Λ ∨ / Λ) and δ ∗ = min( δ, d − δ ).When δ ∗ = 0, then Λ = Λ ∨ or Λ = π Λ ∨ , the parahoric group K p is hyperspecial, and bothM loc (Λ) and S K ( G, X ) are smooth over Z p . This is the case of good reduction studiedin general by Kisin in [9]. In what follows, we will exclude the case δ ∗ = 0, which, forthe goals of this paper, holds no interest. When δ ∗ = 1, both M loc (Λ) and S K ( G, X ) areregular, as established by Madapusi Pera [12] (see also [8, 12.7.2]).Note that, since G is smooth, properties (a) and (b) imply that every point of S K ( G, X )has an ´etale neighborhood which is also ´etale over the local model M loc (Λ). The localmodel M loc (Λ) is a flat projective scheme over Spec ( Z p ) with G -action. It is a Z p -modelof the compact dual X ∨ of the hermitian domain X . Here, the compact dual X ∨ is thequadric hypersuface Q ( V ) in P d − which parametrizes isotropic lines L ⊂ V . Hence, thelocal model M loc (Λ) is a distinguished Z p -model of the quadric Q ( V ) obtained from thelattice Λ. It is a “ p -adic degeneration” of the quadric hypersurface Q ( V ).Note that P (Λ) is a distinguished Z p -model of the projective space P ( V ). One couldhope that M loc (Λ) is the flat closure of Q ( V Q p ) ⊂ P ( V Q p ) in P (Λ) but this is not true, unless δ ∗ = 0 or δ ∗ = 1. The next naive guess is that M loc (Λ) is isomorphic to the flat closureof Q ( V Q p ) ⊂ P ( V Q p ) × P ( V Q p ) in P (Λ) × P (Λ ∨ ) but this is also not correct (see below).Nevertheless, the relation of the local model M loc (Λ) with this last flat closure, which wedenote by Q (Λ , Λ ∨ ) and call a “linked quadric”, plays an important role in our discussion.Let us mention here that, local models for Shimura varieties of PEL type have beenstudied using connections to linked Grassmannians and other classical algebraic varieties(see [18]). The current paper extends this to the first non-trivial non-PEL type case. Theplot now is thicker because the Hodge type Shimura cocharacter of the orthogonal groupis no longer minuscule in the standard representation. So, one should not really expect astraightforward embedding of the local model in the standard linked Grassmannian. Tounderstand M loc (Λ) one needs, either to use a spin representation (which has very highdimension), or apply directly the definition of M loc (Λ) ([19]) via Beilinson-Drinfeld affineGrassmannians. The latter route was taken by the second author in his thesis [23] toobtain an explicit description of a dense open affine chart as we explain below.Note that the G -action of M loc (Λ) has only a finite number of orbits (parametrized bythe µ -admissible set) and a unique closed orbit given by a distinguished point ∗ in thespecial fiber of M loc (Λ), which we call the “worst point”.Set Z = ( z ij ) ∈ Mat δ × ( d − δ ) . Denote by D δ × ( d − δ ) = { Z | ∧ Z = 0 } ⊂ Mat δ × ( d − δ ) thedeterminantal subscheme (a Segre cone) of the affine space of matrices Z over Spec ( Z p )given by the vanishing of all 2 × Z . For simplicity, we assume that the quadraticform on V ⊗ Q Q p is split, or quasi-split. (We can always reduce to this case after basechanging to Q p .). The following result is essentially shown in [23], but is stated there ina different form: Theorem 1.2.3.
There exists an open affine chart U ⊂ M loc (Λ) which contains the worstpoint ∗ and which is isomorphic to the closed subscheme of the determinantal scheme D δ × ( d − δ ) given by the quadratic equation X ≤ i ≤ δ, ≤ j ≤ d − δ z i d − δ +1 − j z δ +1 − i j = − p. G. PAPPAS AND I. ZACHOS
Observe that the open subscheme U intersects every G -orbit in M loc (Λ), since it containsthe unique closed G -orbit ∗ . Hence, U “captures all the singularities” of M loc (Λ) andhence also of S K ( G, X ). When δ ∗ = 1, it is an open affine in the quadric Q (Λ). The case δ ∗ = 1 was studied in [12] and [8]. For δ ∗ ≥
2, the schemes M loc (Λ) and U are harder tounderstand. Let us add that, for δ ∗ = 0, none of these models are smooth or semi-stable(as also follows from the result of [8]). The level subgroup which is the stabilizer of a pairof lattices (Λ , Λ ) with π Λ ⊂ Λ ⊂ Λ and Λ ∨ = Λ , Λ ∨ = p − Λ , gives a semi-stablemodel, as was first shown by Faltings [6], but it is not of the type considered here.1.3. We can now consider the blow-up of M loc (Λ) at the worst point ∗ . This gives a G -birational projective morphism r bl : M bl (Λ) −→ M loc (Λ) . Using the explicit description of U above, we show: Theorem 1.3.1.
The scheme M bl (Λ) is regular and has special fiber a divisor with nor-mal crossings. In fact, M bl (Λ) is covered by open subschemes which are smooth over Spec ( Z p [ u, x, y ] / ( u xy − p )) when δ ∗ ≥ , or over Spec ( Z p [ u, x ] / ( u x − p )) when δ ∗ = 1 . We quickly see that the corresponding blow-up S reg K ( G, X ) of the integral model S K ( G, X ) inherits the same nice properties as M bl (Λ). In fact, there is a local modeldiagram for S reg K ( G, X ) similar to (7.2.3) but with M loc (Λ) replaced by M bl (Λ). See The-orem 7.2.1 for the precise statement about the model S reg K ( G, X ); this theorem is themain result of the paper. The construction of S reg K ( G, X ) from r bl and the local modeldiagram (7.2.3) is an example of a “linear modification” in the sense of [14].1.4. To understand the full picture, we also give two alternative descriptions of theresolution r bl : M bl (Λ) → M loc (Λ):The first is partially moduli-theoretic and is inspired by a classical idea: The determi-nantal scheme D n × m = { A ∈ Mat n × m | ∧ A = 0 } can be resolved by considering e D n × m = { ( A, L ) ∈ Mat n × m × P m − | ∧ A = 0 , Im( A ) ⊂ L } which maps birationally to D n × m by the forgetful ( A, L ) A . This leads to the definitionof a G -equivariant birational morphism r : M (Λ) −→ M loc (Λ)which we then show agrees with the blow-up r bl via a G -equivariant isomorphism(1.4.1) M (Λ) ≃ M bl (Λ) . The second description relates M loc (Λ) to the classical linked quadric Q (Λ , Λ ∨ ). Weshow that there is a blow-up e Q (Λ , Λ ∨ ) −→ Q (Λ , Λ ∨ ) along an irreducible component ofthe special fiber of Q (Λ , Λ ∨ ) (this component is a divisor which is not Cartier) and a G -equivariant isomorphism(1.4.2) e Q (Λ , Λ ∨ ) ≃ M bl (Λ) . The proofs of the two isomorphisms (1.4.1) and (1.4.2) are intertwined. As a result, wehave a diagram of G -equivariant birational projective morphisms(1.4.3) M bl (Λ)M loc (Λ) Q (Λ , Λ ∨ ) r bl ρ NTEGRAL MODELS FOR SHIMURA VARIETIES 5 and we can pass from the linked quadric Q (Λ , Λ ∨ ) to M loc (Λ) as follows: We first blow-upalong an irreducible component Z (a non-Cartier divisor, if δ ∗ ≥
2) of the special fiberof Q (Λ , Λ ∨ ). Then we blow the proper transform P δ − × P d − δ − = e Z ⊂ e Q (Λ , Λ ∨ ) ofanother irreducible component Z down to the point ∗ , to obtain M loc (Λ).Let us mention here that families of degenerating quadrics, like the schemes M loc (Λ)and Q (Λ , Λ ∨ ), are objects of the old theory of complete quadrics (see [4] and the referencesthere). An interesting problem is to reinterpret our constructions within the frameworkof this classical theory.1.5. In the above, we only discussed Shimura varieties for G = GSpin( V ). However, theresults also apply to G = SO( V ) (see § F/ Q ( G F ), where G F = GSpin( V F ), or G = SO( V F ), for a quadratic space V F over the number field F , provided F is unramifiedover p . In that case, the local models over p are given as d -fold products of the local modelsabove and we can construct integral models of Shimura varieties by using the productsof the corresponding resolutions. Similarly, we can apply these results to obtain regular(formal) models of the corresponding Rapoport-Zink spaces constructed in [7, § Acknowledgement.
We thank M. Rapoport for useful suggestions.2.
Preliminaries: quadratic forms, lattices and spinors
Quadratic forms and lattices.
Let us fix an odd prime p and consider a finite fieldextension F/ Q p . Let O = O F be the ring of integers and let π be a uniformizer of O . Wewill denote by k = O / ( π ) the residue field and by ¯ k an algebraic closure of k . Denote alsoby ˘ F the completion of the maximal unramified extension of F in an algebraic closure ¯ F and by ˘ O the integers of ˘ F . We can assume ¯ k = ˘ O / ( π ).Let V be an F -vector space of dimension d and fix h , i : V × V → F, a non-degenerate symmetric F -bilinear form. Set Q ( v ) = 12 h v, v i , for the associated quadratic form Q : V → F . We assume that d ≥ ⊂ V is an O -lattice, we setΛ ∨ = { v ∈ V | h v, a i ∈ O , ∀ a ∈ Λ } for the dual of Λ. In what follows, we will fix a “vertex lattice” Λ in V . By definition thisis an O -lattice Λ such that π Λ ∨ ⊂ Λ ⊂ Λ ∨ . Set δ (Λ) := length O (Λ ∨ / Λ) = dim k (Λ ∨ / Λ), δ ∗ = min( δ, d − δ ). We will often assume δ (Λ) ≤ d/
2. Indeed, we can always replace the form by a multiple and exchange the rolesof Λ and Λ ∨ to come to this situation. Then δ ∗ = δ .The form h , i induces symmetric O -bilinear forms h , i : Λ × Λ → O , π h , i : Λ ∨ × Λ ∨ → O . We also obtain perfect symmetric k -bilinear forms h , i : Λ /π Λ ∨ × Λ /π Λ ∨ → k, h , i : Λ ∨ / Λ × Λ ∨ / Λ → k, given by h x, y i = h x, y i mod ( π ), h x, y i = π h x, y i mod ( π ). G. PAPPAS AND I. ZACHOS
Normal forms.
As in [20, Appendix, Lemma A.25], we can writeΛ = M ⊕ N, Λ ∨ = M ⊕ π − N, where M , N are free O F -submodules such that h M, N i = 0 and with the property that theform h , i is perfect on M and π − h , i is perfect on N . In fact, by loc. cit. Prop. A.21,after a finite unramified base change F ′ /F we can find an O F ′ -basis { e i } of Λ ⊗ O F O F ′ ,i.e. write Λ O F ′ = Λ ⊗ O F O F ′ = d M i =1 O F ′ · e i such that exactly one of the following cases occurs:(1) d = 2 n , δ (Λ) = 2 r :Λ O F ′ = ( e , . . . , e n − r ) ⊕ ( e n − r +1 , . . . , e n + r ) ⊕ ( e n + r +1 , . . . , e d ) ,M = ( e , . . . , e n − r ) ⊕ ( e n + r +1 , . . . , e d ) , N = ( e n − r +1 , . . . , e n + r ) , with h e i , e d +1 − j i = δ ij , for i < n − r + 1 , or n + r < i, h e i , e d +1 − j i = πδ ij , for n − r + 1 ≤ i ≤ n + r. (2) d = 2 n , δ (Λ) = 2 r + 1:Λ O F ′ = ( e , . . . , e n − r − ) ⊕ ( e n − r , . . . , e n − ) ⊕ ( e n , e n +1 ) ⊕ ( e n +2 , . . . , e n + r +1 ) ⊕ ( e n + r +2 , . . . , e d ) ,M = ( e , . . . , e n − r − ) ⊕ ( e n +1 ) ⊕ ( e n + r +2 , . . . , e d ) ,N = ( e n − r , . . . , e n − ) ⊕ ( e n ) ⊕ ( e n +2 , . . . , e n + r +1 ) , with h e i , e d +1 − j i = δ ij , for i < n − r, or n + r + 1 < i, h e i , e d +1 − j i = πδ ij , for n − r ≤ i ≤ n + r + 1 , i = n, i = n + 1 , h e n , e n i = π, h e n +1 , e n +1 i = 1 , h e n , e n +1 i = 0 . (3) d = 2 n + 1, δ (Λ) = 2 r :Λ O F ′ = ( e , . . . , e n − r ) ⊕ ( e n − r +1 , . . . , e n ) ⊕ ( e n +1 ) ⊕ ( e n +2 , . . . , e n + r +1 ) ⊕ ( e n + r +2 , . . . , e d ) ,M = ( e , . . . , e n − r ) ⊕ ( e n +1 ) ⊕ ( e n + r +2 , . . . , e d ) ,N = ( e n − r +1 , . . . , e n ) ⊕ ( e n +2 , . . . , e n + r +1 ) , with h e i , e d +1 − j i = δ ij , for i < n + 1 − r, or i = n + 1 , or n + 1 + r < i, h e i , e d +1 − j i = πδ ij , for n + 1 − r ≤ i ≤ n + 1 + r, and i = n + 1 . (4) d = 2 n + 1, δ (Λ) = 2 r + 1:Λ O F ′ = ( e , . . . , e n − r ) ⊕ ( e n − r +1 , . . . , e n + r +1 ) ⊕ ( e n + r +2 , . . . , e d ) ,M = ( e , . . . , e n − r ) ⊕ ( e n + r +2 , . . . , e d ) , N = ( e n − r +1 , . . . , e n + r +1 ) , with h e i , e d +1 − j i = δ ij , for i < n + 1 − r, or n + 1 + r < i, h e i , e d +1 − j i = πδ ij , for n + 1 − r ≤ i ≤ n + 1 + r. NTEGRAL MODELS FOR SHIMURA VARIETIES 7
In all the above, the parentheses give a short-hand notation for the O F ′ -lattice gener-ated by the included vectors. For simplicity, we omit the notation of the base change of M , N . In all cases, we will denote by S the (symmetric) matrix with entries h e i , e j i where { e i } is the basis above. We can then write S = S + πS where S , S both have entries only 0 or 1. For example, in case (1) we write: S := ( n − r ) (2 r ) ( n − r ) , S := ( n − r ) (2 r ) ( n − r ) . Spinor groups.
The Clifford algebra of V is a Z / Z -graded F -algebra denoted C ( V ) = C + ( V ) ⊕ C − ( V ) . It is a vector space of rank 2 d over F , generated as an algebra by the image of a canonicalinjection V ֒ → C − ( V ) satisfying v · v = Q ( v ). The canonical involution on C ( V ) is the F -linear endomorphism c c ∗ characterized by ( v · · · v m ) ∗ = v m · · · v , for v , . . . , v m ∈ V .For an F -algebra R , the tensor product V R = V ⊗ F R is a nondegenerate quadraticspace over R with Clifford algebra C ( V R ) = C ( V ) ⊗ F R . The spinor similitude group G = GSpin( V ) is the reductive group over F with R -pointsGSpin( V )( R ) = { g ∈ C + ( V R ) × : gV R g − = V R , g ∗ g ∈ R × } , and the spinor similitude η : GSpin( V ) → G m is the character η ( g ) = g ∗ g .The conjugation action of G on C ( V ) leaves invariant the F -submodule V , and thisaction of G on V is denoted g · v = gvg − . There is a short exact sequence of reductivegroup schemes(2.3.1) 1 → G m → G g g · −−−→ SO( V ) → F , and the restriction of η to the central G m is z z .The spinor group Spin( V ) is the kernel of η and there is a short exact sequence ofreductive groups(2.3.2) 1 → Spin( V ) → G = GSpin( V ) η −→ G m → . We have G der = Spin( V ) which is simply connected. Hence, π ( G ) = π ( G m ) ≃ Z with trivial Galois action. In particular, we have π ( G ) I ≃ Z for the coinvariants of theaction by the inertia group I = Gal( ¯ F / ˘ F ). Recall, π ( G ) I is the target of the Kottwitzhomomorphism κ : G ( ˘ F ) → π ( G ) I . Hence, in our case, the Kottwitz homomorphism is κ : GSpin( V )( ˘ F ) → Z . In fact, by the definition of κ and (2.3.2), we can see that(2.3.3) κ ( g ) = val( η ( g )) , i.e. κ is given by the valuation of the spinor similitude. G. PAPPAS AND I. ZACHOS
Orthogonal parahoric groups.
We first consider the special orthogonal groupSO( V ). The self-dual lattice chain · · · ⊂ π Λ ∨ ⊂ Λ ⊂ Λ ∨ ⊂ π − Λ ⊂ · · · gives a point x Λ of the Bruhat-Tits building B (SO( V ) , F ) (see for example [3]). Let usconsider the subgroup K = K Λ = { g ∈ SO( V ) | g (Λ) = Λ } ⊂ SO( V )which preserves the lattice. Then every element g ∈ K also preserves the dual lattice, g (Λ ∨ ) = Λ ∨ , and so it induces elements ¯ g ∈ GL(Λ /π Λ ∨ ), ¯ g ∈ GL(Λ ∨ / Λ), which liein the corresponding orthogonal groups for the forms h , i , h , i . We can see that g (¯ g , ¯ g ) gives a group homomorphism K → O(Λ /π Λ ∨ ) × O(Λ ∨ / Λ) . Composing with det × det : O(Λ /π Λ ∨ ) × O(Λ ∨ / Λ) → {± } × {± } gives ε : K → {± } × {± } . The corresponding parahoric subgroup K ◦ ⊂ K is the kernelof ε . (See [3] and [22, Example 3.12]. The map ε can also be related to the Kottwitzhomomorphism for SO( V ) which is given by the spinor norm.)Set i : Λ → Λ ∨ , j : π Λ ∨ → Λ for the natural O -linear inclusions. We have i · j = π .Let R be an O -algebra. For simplicity, set Λ R = Λ ⊗ O R , Λ ∨ R = Λ ∨ ⊗ O R . We identifyΛ ∨ = Hom R (Λ R , R )using the form h , i R = h , i ⊗ O R . Consider the group scheme G = G Λ over Spec ( O )which has R -valued points given by g ∈ GL(Λ R ) for which(2.4.1) π Λ ∨ R j R −→ Λ R i R −→ Λ ∨ R πg ∨ ↓ g ↓ ↓ g ∨ π Λ ∨ R j R −→ Λ R i R −→ Λ ∨ R commutes, and with det( g ) = 1, det( g ∨ ) = 1. In the above, g ∨ denotes the R -linear mapwhich is R -dual to g : Λ R → Λ R .As we can see, using Appendix [20], the group G is smooth and has O -points given by K . The homomorphism ε above is the composition K = G ( O ) → G ( k ) → {± } × {± } and the kernel gives the neutral component G ◦ of G . As in [3], G = G Λ , G ◦ = G ◦ Λ , are theBruhat-Tits group schemes that correspond to x Λ , or, in other words, to K , resp. K ◦ .2.5. Spinor parahoric groups.
Consider the (extended) Bruhat-Tits building B e ( G, F )for G = GSpin( V ) over F . The central exact sequence (2.3.1) induces by [2, (4.2.15)], or[11, Theorem 2.1.8], a canonical G ( F )-equivariant map B e ( G, F ) → B e (SO( V ) , F ) = B (SO( V ) , F )which lifts an identification B ( G, F ) ∼ −→ B (SO( V ) , F )between the (classical) Bruhat-Tits buildings. Consider now x ∈ B e ( G, F ) which weassume maps to the point x Λ ∈ B ( G, F ) ≃ B (SO( V ) , F ) defined by a vertex latticeΛ ⊂ V . Then, the stabilizer of x is the subgroup G ( F ) x = { g ∈ GSpin( V )( F ) | g Λ g − = Λ , η ( g ) ∈ O × } . NTEGRAL MODELS FOR SHIMURA VARIETIES 9
By (2.3.3), G ( ˘ F ) x ⊂ ker( κ ). Hence, by [17, Appendix by Haines-Rapoport], the corre-sponding Bruhat-Tits group scheme G x is connected, i.e. G x = G ◦ x , and it is the corre-sponding parahoric group scheme over O for GSpin( V ) over F . By [10, Prop. 1.1.4], wehave an exact sequence(2.5.1) 1 → G m → G x → G ◦ Λ → O ) which extends (2.3.1). Remark 2.5.2.
Assume that we are in one of the cases of § B (GSpin( V ) , F ) ∼ −→ B (SO( V ) , F ) and the description of the building for B (SO( V ) , F )given in [3], we see that each maximal compact subgroup of GSpin( V ) is of the form G ( F ) x for x = x Λ given, as above, by some vertex lattice Λ with δ ∗ (Λ) = 2. As wementioned above, when δ ∗ (Λ) = 0, G ( F ) x is hyperspecial. When δ ∗ (Λ) = 1, G ( F ) x isspecial. The vertex lattices Λ with δ ∗ (Λ) = 2, i.e. δ (Λ) = 2 or δ (Λ) = d −
2, give parahoricsubgroups which are not maximal. Indeed, when δ (Λ) = 2, there are exactly two self-duallattices Λ , Λ ′ , that fit in an oriflamme configuration:Λ ⊂ ⊂ Λ Λ ′ ⊂ ⊂ Λ ∨ Then G ( F ) x is the intersection of the two hyperspecial subgroups GSpin(Λ ), GSpin(Λ ′ ).If δ (Λ) = d −
2, then there are exactly two lattices Λ and Λ ′ with Λ ∨ = π Λ , Λ ′∨ = π Λ ′ ,that fit in a similar oriflamme configuration between Λ ∨ and π − Λ. The group G ( F ) x isagain the intersection of the two hyperspecial subgroups which stabilize Λ and Λ ′ .3. Quadrics and linked quadrics
Quadrics.
Suppose that R is an O -algebra and that ( W, h , i ) is a free rank dR -module with a symmetric R -bilinear form h , i : W × W → R. We denote by Q ( W, h , i ), or simply by Q ( W ) ⊂ P ( W ) ≃ P d − R , when the form is under-stood, the closed subscheme parametrizing isotropic R -lines L (i.e. locally R -free directsummands of W of rank 1) with h L, L i = 0.3.2. Linked quadrics.
Suppose now that V and Λ ⊂ V are as in § δ = length O (Λ ∨ / Λ) ≤ d/ . Consider the O -scheme P (Λ , Λ ∨ ) parametrizing linked isotropic lines( L, L ′ ) ∈ Q (Λ) × O Q (Λ ∨ ) ⊂ P (Λ) × O P (Λ ∨ )where the “linked” condition for an R -valued point ( L, L ′ ) is that j R ( πL ′ ) ⊂ L, i R ( L ) ⊂ L ′ , i.e. we have a commutative diagram π Λ ∨ R j R −→ Λ R i R −→ Λ ∨ R ∪ ∪ ∪ πL ′ −→ L −→ L ′ . (Of course, isotropic means that we require h L, L i R = 0, π h L ′ , L ′ i R = 0.) There is anatural isomorphism P (Λ , Λ ∨ ) ⊗ O F ∼ = Q ( V ) , since, for an F -algebra R , we have Λ R = Λ ∨ R = V R .We set Q (Λ , Λ ∨ ) for the (flat) reduced closure of the generic fiber Q ( V ) in the O -scheme P (Λ , Λ ∨ ). By (2.4.1), the group scheme G acts on P (Λ , Λ ∨ ) and also on theclosure Q (Λ , Λ ∨ ). Theorem 3.2.1.
The scheme Q (Λ , Λ ∨ ) is normal and is a relative local complete inter-section, flat and projective over Spec ( O ) . Its generic fiber is the smooth quadric Q ( V ) in P ( V ) . The scheme is regular if and only if δ ≤ .a) Assume δ = 0 . Then Q (Λ , Λ ∨ ) = Q (Λ) is a smooth quadric hypersurface in P d − O .b) Assume δ ≥ . The reduced special fiber is the union of reduced codimension subschemes Z , Z and Z , which have the following properties: The subschemes Z , Z are projective space bundles over the quadric hypersurfaces Q (Λ /π Λ ∨ , h , i ) ⊂ P (Λ /π Λ ∨ ) and Q (Λ ∨ / Λ , h , i ) ⊂ P (Λ ∨ / Λ) respectively. Z ≃ P (Λ ∨ / Λ) × P (Λ /π Λ ∨ ) and we have Z ∩ Z ≃ Q (Λ /π Λ ∨ , h , i ) × P (Λ ∨ / Λ) ,Z ∩ Z ≃ P (Λ /π Λ ∨ ) × Q (Λ ∨ / Λ , h , i ) , for the scheme theoretic intersections. div ( π ) = 2 · ( Z ) + ( Z ) + ( Z ) as Weil divisors on Q (Λ , Λ ∨ ) .If δ > , then Z , Z and Z are smooth and irreducible. If δ = 2 , then Z , Z aresmooth and irreducible but Z is either irreducible or the disjoint union of two smoothirreducible components. Finally, if δ = 1 , then Q (Λ ∨ / Λ , h , i ) and Z are empty and Z , Z are smooth and irreducible.Proof. Part (a) is straightfoward and so, in what follows, we assume that δ >
0. We startby giving a description of Q (Λ , Λ ∨ ). WriteΛ = M ⊕ N, Λ ∨ = M ⊕ π − N with M , N free O -submodules of Λ such that h M, N i = 0 and with the form h , i perfecton M and π − h , i perfect on N . We have rank O ( N ) = δ , rank O ( M ) = d − δ . Note that¯ M := M ⊗ O k = Λ /π Λ ∨ , ¯ N := N ⊗ O k ≃ Λ ∨ / Λ . Set h , i : M × M → O , h m, m ′ i = h m, m ′ i . h , i : N × N → O , h n, n ′ i = π − h n, n ′ i . Now write L = ( f ) = ( m + n ) , L ′ = ( f ′ ) = ( m ′ + π − n ′ )with m , m ′ ∈ M R and n , n ′ ∈ N R . The linking conditions are m + n = u ( m ′ + π − n ′ ) , πm ′ + n ′ = π ( m ′ + π − n ′ ) = v ( m + n ) , for some u , v ∈ R . These give m = um ′ , n ′ = vn , uv = π .The isotropic conditions are h f, f i = h m, m i + π h n, n i = 0 , π h f ′ , f ′ i = π h m ′ , m ′ i + h n ′ , n ′ i = 0 . These translate to, respectively, u h m ′ , m ′ i + uv h n, n i = 0 , uv h m ′ , m ′ i + v h n, n i = 0 . NTEGRAL MODELS FOR SHIMURA VARIETIES 11
In the flat closure, they amount to a single equation: u h m ′ , m ′ i + v h n, n i = 0 . Now consider the affine scheme over Spec ( O ) given as the product X := A ( M ) × O A ( N ) × O Spec ( O [ u, v ] / ( uv − π )) = Spec ( O [ x, y, u, v ] / ( uv − π )) . (We identify A ( M ), A ( N ) with Spec ( O [ x ]), Spec ( O [ y ]), after picking a basis of M , N respectively.) Let X = X − ( T ∪ T ∪ T ) be the open subscheme of X which is thecomplement of the union of the three subschemes T , T , T , defined respectively by theideals ( x, v ) , ( y, u ) , ( x, y ) . The scheme X supports an action of the torus G m × G m given by( λ, µ ) · ( x, y, u, v ) = ( λx, µy, λ − µu, λµ − v ) . The subscheme Y of X defined by the equation(3.2.2) u · h x, x i + v · h y, y i = 0 , is stable under the torus action. The quotient Y / ( G m × G m )is flat over O and we can see, by the above, that it is isomorphic to Q (Λ , Λ ∨ ).Let Z , Z , Z be the subschemes of Q (Λ , Λ ∨ ) (all in the special fiber) given by theclosures of u = 0, v = 0, h y, y i = 0, of v = 0, u = 0, h x, x i = 0, and of u = v = 0,respectively: • Z is given by the quotient of the subscheme h y, y i = 0 in the complement[ A ( ¯ M ) × k ( A ( ¯ N ) − { } ) × k A ] − (0 × ( A ( ¯ N ) − { } ) × G m × G m . Here, ( λ, µ ) acts by( λ, µ ) · ( x, y, v ) = ( λx, µy, λµ − v ) . This is a projective space bundle over the G m -quotient of h y, y i = 0 in A ( ¯ N ) −{ } ;this last quotient is the projective quadric hypersurface Q (Λ ∨ / Λ , h , i ). • Z is similarly a projective space bundle over the projective quadric hypersurface Q (Λ /π Λ ∨ , h , i ). • Z is the quotient( A ( ¯ M ) − { } ) × k ( A ( ¯ N ) − { } ) / ( G m × G m ) , so it is a product of projective spaces P (Λ ∨ / Λ) × P (Λ /π Λ ∨ ).The statement in (2) about the intersections also follows. We still have to show that Q (Λ , Λ ∨ ) is a normal relative local complete intersection and that the identity of Weildivisors in (3) is true.Consider a ¯ k -point of Q (Λ , Λ ∨ ) obtained from a ¯ k -valued point of Y for which x = 0.Since Y ⊂ X , we necessarily have v = 0 and hence u = 0. Then we also have y = 0 andthe point lies in Z but not on Z . The equation (3.2.2) amounts to just h y, y i = 0. Sincewe are considering a non-zero point y , the scheme Q (Λ , Λ ∨ ) k is smooth there. The sameargument works for a point for which y = 0; this lies on Z but not on Z .It remains to deal with points for which x = 0 and y = 0. Consider the correspondingopen subscheme Q (Λ , Λ ∨ ) = [( A ( M ) − { } ) × O ( A ( N ) − { } ) × Spec ( O [ u, v ] / ( uv − π ))] / ( G m × G m ) of Q (Λ , Λ ∨ ). We will obtain an explicit description of Q (Λ , Λ ∨ ):Recall, we can choose bases of M and N that give coordinates x = ( x , . . . , x d − δ ) on A ( M ) and y = ( y , . . . , y δ ) on A ( N ). Denote by Q ( x , . . . , x d − δ ), Q ( y , . . . , y δ ) thequadratic forms given as h x, x i / h y, y i / Q (Λ , Λ ∨ ) is covered by the open affines V i,j with equations uv = π, uQ ( x , . . . , x d − δ ) + vQ ( y , . . . , y δ ) = 0 , x i = 1 , y j = 1 . For simplicity, we will just consider the case i = 1, j = 1, and set V = V .As also later, it helps to consider the simpler “basic” scheme B = Spec ( O [ u, v, S, T ] / ( uv − π, uS + vT )There is a morphism f : V → B given by S Q , T Q . We can see that f is smooth.We can now check the desired properties for B :The scheme B has relative dimension 2 over Spec ( O ) and is a relative complete inter-section. The radical of ( uS + vT, uv ) is ( uS, vT, uv ) and so the reduced special fiber of B has 3 smooth irreducible components given by the prime ideals ( u, v ), ( S, v ), (
T, u ). Thecomponent given by ( u, v ) is not reduced in the special fiber; the corresponding primaryideal is ( u , uv, v , uS + vT ). The other components are reduced. We can see that B isregular in codimension 1 and it easily follows by Serre’s criterion that B is normal.It now follows that we have a similar picture for the special fiber of V and that V is anormal relative complete intersection of relative dimension d −
2. The result for Q (Λ , Λ ∨ )and then also Q (Λ , Λ ∨ ) follows. Remark 3.2.3.
Note that in the proof above, the case δ = 1 is special. Indeed, then V is given by uv = π, uQ (1 , x , . . . , x d − δ ) + v = 0 , i.e. − u Q (1 , x , . . . , x d − δ ) = π. In this case, V and hence Q (Λ , Λ ∨ ) is regular and its special fiber is a divisor with normalcrossings. This result appears in [8, 12.7.2]: There, Q (Λ , Λ ∨ ) is denoted as P (Λ) fl and isidentified with the blow-up of the singular quadric Q (Λ) of isotropic lines in Λ R at theunique singular point of its special fiber Q (Λ) k (see loc. cit. Lemma 12.8).3.3.
A Blow-up.
Now let us consider the blow-up β i : e Q ( i ) (Λ , Λ ∨ ) → Q (Λ , Λ ∨ )along the (reduced) subscheme Z i , i = 1, 2. We first observe that when δ = 1, by Remark3.2.3, Z is locally principal and so e Q (1) (Λ , Λ ∨ ) = Q (Λ , Λ ∨ ). Proposition 3.3.1. a) There is a canonical isomorphism e Q (1) (Λ , Λ ∨ ) ∼ = e Q (2) (Λ , Λ ∨ ) . Byabusing notation, we will denote both these schemes by e Q (Λ , Λ ∨ ) .b) The scheme e Q (Λ , Λ ∨ ) is regular. Its special fiber is a divisor with (non-reduced)normal crossings and the multiplicity of each component is one or two. In fact, e Q (Λ , Λ ∨ ) is covered by open affine subschemes which are smooth over Spec ( O [ u, x, y ] / ( u xy − p )) when δ ≥ , or over Spec ( O [ u, x ] / ( u x − p )) when δ = 1 .c) The reduced special fiber of e Q (Λ , Λ ∨ ) is a union of smooth irreducible components.It has three irreducible components if δ > , four or three if δ = 2 , and two if δ = 1 . NTEGRAL MODELS FOR SHIMURA VARIETIES 13
Proof.
By Remark 3.2.3 and the above, these statements hold when δ = 1. Assume δ ≥ e B of B = Spec ( O [ u, v, S, T ] / ( uv − π, uS + vT )along ( S, v ). It is a closed subscheme of { Sy − vx = 0 | ( S, T, u, v ) × ( x ; y ) } ⊂ B × O P O , where P O has projective coordinates ( x ; y ). This is covered by two open affine subschemes:I) y = 1. Then S = vx and we obtain uv = π , v ( ux + T ) = 0, so uv = π , T = − ux .This open is isomorphic to Spec ( O [ u, v, x ] / ( uv − π )) . II) x = 1. Then v = Sy and we obtain: uSy = π , uS + SyT = 0 , so uSy = π , u + yT = 0. Hence, ST y = π . This open is isomorphic toSpec ( O [ S, T, y ] / ( ST y + π )) . Notice that the ideal (
T, u ) becomes principal on both of these charts. On (I) we have T = − ux , so ( T, u ) = ( u ). On (II) we have u = − yT , so ( T, u ) = ( T ). Using the universalproperty of the blow-up and symmetry gives that the blow-ups of ( S, v ) and (
T, u ) areisomorphic.By the proof of Theorem 3.2.1, we see that the open affine subscheme
V ⊂ Q (Λ , Λ ∨ )supports a smooth morphism f : V → B = Spec ( O [ u, v, S, T ] / ( uv − π, uS + vT ); S Q , T Q . Note that, by the same proof, Q (Λ , Λ ∨ ) is smooth at the points of Q (Λ , Λ ∨ ) −V and the di-visors Z and Z are principal at these points. Therefore, the blow-ups q i : e Q ( i ) (Λ , Λ ∨ ) →Q (Λ , Λ ∨ ) are isomorphisms locally over Q (Λ , Λ ∨ ) − V . To show our result is enough toconsider the blow-ups of V at Z ∩ V and at Z ∩ V . Since Z ∩ V is given by the ideal( Q , u ), the blow-up e V of V along Z ∩ V is obtained as the fiber product e V ≃ V × B e B . The morphism e V → e B is smooth and (b) and (c) now follows from our discussion aboveand Theorem 3.2.1 and its proof. Part (a), i.e. e Q (1) (Λ , Λ ∨ ) ≃ e Q (2) (Λ , Λ ∨ ), follows from e V ≃ e V since, by the observation above, the blow-ups of B at ( S, v ) and (
T, u ) areisomorphic. 4.
Local models and variants
Local models after [19] . We briefly recall some of the constructions in [19].Let G be a connected reductive group over F . Assume that G splits over a tamely ram-ified extension of F . Let { µ } be the conjugacy class of a minuscule geometric cocharacter µ : G m ¯ F → G ¯ F . Let K be a parahoric subgroup of G ( F ), which is the connected stabilizerof some point x in the (extended) Bruhat-Tits building B e ( G, F ) of G ( F ). Define E tobe the extension of F which is the field of definition of the conjugacy class { µ } . Theconstruction of the local model M loc K ( G, { µ } ) is done as follows:First, give an affine group scheme G which is smooth over Spec ( O F [ t ]) and which,among other properties, satisfies:(1) The base change of G by Spec ( O F ) → Spec ( O F [ t ]) = A O F given by t → π is theBruhat-Tits group scheme G which corresponds to K (see [2]).(2) The group scheme G ⊗ O F [ t ] O F [ t, t − ] is reductive. Next, consider the global (“Beilinson-Drinfeld”) affine GrassmannianAff G → A O F given by G , which is an ind-projective ind-scheme. The base change t → π , gives anequivariant isomorphism Aff G ∼ −→ Aff G × A O F Spec ( F )where Aff G is the affine Grassmannian of G ; this is the ind-projective ind-scheme overSpec ( F ) that represents the fpqc sheaf associated to R → G ( R (( t ))) /G ( R [[ t ]]) , where R is an F -algebra (see also [17]). The cocharacter µ gives an ¯ F (( t ))-valued point µ ( t ) of G . This gives a ¯ F -point [ µ ( t )] = µ ( t ) G ( ¯ F [[ t ]]) of Aff G . Since µ is minuscule and { µ } is defined over the reflex field E the orbit G ( ¯ F [[ t ]])[ µ ( t )] ⊂ Aff G ( ¯ F ) , is equal to the set of ¯ F -points of a closed subvariety X µ of Aff G,E = Aff G ⊗ F E. Definition 4.1.1.
The local model M loc K ( G, { µ } ) is the flat projective scheme over Spec ( O E )with G ⊗ O F O E -action given by the reduced Zariski closure of the image of X µ ⊂ Aff G,E ∼ −→ Aff G × A O F Spec ( E )in the ind-scheme Aff G × A O F Spec ( O E ).These “PZ” local models of [19] are independent of choices in their construction ([8,Theorem 2.7]) and have the following property (see [8, Prop. 2.14] and its proof). Proposition 4.1.2. If F ′ /F is a finite unramified extension, then (equivariantly) M loc K ( G, { µ } ) ⊗ O E O E ′ ∼ −→ M loc K ′ ( G ⊗ F F ′ , { µ ⊗ F F ′ } ) . Note that here the reflex field E ′ of ( G ⊗ F F ′ , { µ ⊗ F F ′ } ) is the join of E and F ′ . Also, K ′ is the parahoric subgroup of G ⊗ F F ′ with K = K ′ ∩ G . (cid:3) Remark 4.1.3.
Let us note that the PZ local models are not well behaved when p dividesthe order of the algebraic fundamental group π ( G der ). To correct this defect, one needsto adjust the definition by using a z -extension of G , as in [8, 2.6]. The resulting variant isbetter behaved: It satisfies a property of invariance under central extensions and shouldagree with the local model conjectured to exist in [21, 21.4], see [8, Conjecture 2.16]. Inthis paper, we only consider groups G with adjoint group G ad ≃ (SO( V )) ad . In this case, | π ( G der ) | is a power of 2. Since we assume that p is odd, the local models of [8] coincidewith the PZ local models given as above. In particular, the central extension invarianceproperty of [8, Prop. 2.14] holds in the cases we consider.4.2. Lattices over O [ u ] and orthogonal local models. We now concentrate our at-tention to G = SO( V ), where V is an F -vector space of dimension d ≥ F -bilinear form h , i . We consider the minuscule coweight µ : G m → SO( V ) to be given by µ ( t ) = diag( t − , , . . . , , t ), defined over E = F . We take K to be the parahoric subgroup of SO( V ) which is the connected stabilizer of a vertexlattice Λ ⊂ V , with δ (Λ) = length O F (Λ ∨ / Λ).For simplicity, we set O = O F . We extend the data of the vector space V with itssymmetric bilinear form h , i from F to O [ u, u − ] by following the procedure in [19, 5.2,5.3]. This is simpler to explain when we are in one of the 4 cases of § NTEGRAL MODELS FOR SHIMURA VARIETIES 15 assume after an unramified extension). Then, we define V = ⊕ di =1 O [ u, u − ] e i and let h , i : V × V → O [ u, u − ] to be a symmetric O [ u, u − ]-bilinear form such that the valueof h e i , e j i is the same as the one for V but with π is replaced by u . Similarly, we define µ : G m → SO( V ) as above by using the { e i } basis for V . We set L = ⊕ di =1 O [ u ] · e i ⊂ V . From the above, we see that the base change of ( V , L , h , i ) from O [ u, u − ] to F given by u π is ( V, Λ , h , i ).Let us now consider the local model M loc (Λ) = M loc K (SO( V ) , { µ } ) where K is theparahoric stabilizer K ◦ Λ of Λ.As in [19], we consider the smooth, affine group scheme G over O [ u ] given by g ∈ SO( V )that also preserve L and L ∨ . The base change of G by u π gives the Bruhat-Tits groupscheme G = G Λ of SO( V ) which is the stabilizer of the lattice chain Λ ⊂ Λ ∨ ⊂ π − Λas in § G → Spec ( O [ u ]) represents the functor that sends the O [ u ]-algebra R , given by u r , to theset of finitely generated projective R [ u ]-submodules L of V ⊗ O R [( u − r ) − ] = ⊕ i R [ u, u − , ( u − r ) − ] e i which are such that ( u − r ) N L R ⊂ L ⊂ ( u − r ) − N L R for some N > L / ( u − r ) N L R , ( u − r ) − N L R / L both R -projective, and which satisfy L δ ⊂ L ∨ d − δ ⊂ u − L with all successive quotients R -finite projective of the indicated rank. Here, we set L R = L ⊗ O R .As in [8, 2.7.2], or [23], we see that the local model M loc = M loc (Λ) is a subfunctor ofAff G ⊗ O [ u ] O (with the base change given by u π ). Definition 4.2.1.
We set M naive to be the functor (a subfunctor of Aff G ⊗ O [ u ] O ), whichsends an O -algebra R to the set of R [ u ]-modules L that satisfy:(1) L is a finitely generated projective R [ u ]-submodule of L ⊗ O [ u ] R [ u, u − , ( u − π ) − ].(2) L δ ⊂ L ∨ d − δ ⊂ u − L , with all successive quotients R -finite projective and of theindicated rank,(3) ( u − π ) L R ⊂ L ⊂ ( u − π ) − L R , with the quotients L / ( u − π ) L R , ( u − π ) − L R / L ,both R -finite projective of rank d ,(4) ( u − π ) L ∨ R ⊂ L ∨ ⊂ ( u − π ) − L ∨ R , with L ∨ / ( u − π ) L ∨ R , ( u − π ) − L ∨ R / L ∨ , both R -finite projective of rank d ,(5) Consider the R -linear mapΦ : L / ( u − π ) L R → ( u − π ) − L R / L R = Λ R given by the inclusion in (3). This is an R -map between two finitely generatedprojective R -modules of rank d . We require that Φ has R -rank ≤
1, i.e. that ∧ Φ = 0 . (6) Similarly, we considerΨ : L ∨ / ( u − π ) L ∨ R → ( u − π ) − L ∨ R / L ∨ R = Λ ∨ R , given by the inclusion in (4), and require that ∧ Ψ = 0 . Proposition 4.2.2.
The functor M naive is represented by a closed projective subscheme M naive = M naive (Λ) of Aff G ⊗ O [ u ] O with G -action. There is a G -equivariant closed im-mersion i : M loc ֒ → M naive . Proof.
The proof of the first part (representability) is standard. The second part followsfrom the construction of M loc . See [8, 12.7.2], or [23], for more details.
Remark 4.2.3.
The immersion i is not an isomorphism, i.e. the conditions (1)-(6) arenecessary but not always sufficient for L to correspond to an R -valued point of M loc .In fact, the generic fibers M naive ⊗ O F and M loc ⊗ O F are not equal. Indeed, M naive ( F )contains the additional F -point L = L F which is not in the orbit X µ = Q ( V ) of µ inAff G ⊗ O [ u ] F = Aff G . We can easily see that the reduced locus of M naive ⊗ O F decomposesinto the (disjoint) union of Q ( V ) = M loc ⊗ O F with this point. By its definition, M loc isthe (reduced) Zariski closure of M ⊗ O F = Q ( V ) in M naive .However, M naive is not very different from M loc : We will see in § naive is thepush-out of M loc and Spec ( O ), “glued” at the point ∗ = Spec ( k ). In particular, whenwe just regard the underlying topological spaces, the image of i only misses the isolatedpoint of the generic fiber of M naive that corresponds to L = L F . The scheme M naive is O -flat but has non-reduced special fiber with the non-reduced locus supported at ∗ .4.3. Spinor local models.
Recall the constructions and considerations of § § G = GSpin( V ) and recall the central extension1 → G m → GSpin( V ) α −→ SO( V ) → . Define the cocharacter e µ : G m → G by µ ( t ) = t − f f d + f d f where f , f d are part of a basis ( f i ) i of V such that h f , f d i = 1, h f , f i = h f d , f d i = 0,and the arithmetic on the right hand side takes place in the Clifford algebra C ( V ). Underthe representation α : G → SO( V ), we have( α · e µ ( t )) · f i = e µ ( t ) f i e µ ( t ) − = t − f i if i = 1 f i if 2 ≤ i < dtf i if i = d. Hence, α · e µ : G m → SO( V ) is given by the minuscule µ ( t ) = diag( t − , , . . . , , t ).Now let x ∈ B e ( G, F ) be a point in the extended Bruhat-Tits building of G ( F ) suchthat x maps to x Λ under α ∗ : B e ( G, F ) → B (SO( V ) , F ), where Λ is a vertex lattice of V .Then the corresponding parahoric group of G is given by K = G x ( O ) = { g ∈ GSpin( V )( F ) | g Λ g − = Λ , η ( g ) ∈ O × } . By Remark 4.1.3 and [8, Proposition 12.4], there is an equivariant isomorphism betweenlocal models(4.3.1) M loc K ( G, { e µ } ) ∼ −→ M loc K ◦ Λ (SO( V ) , { µ } ) = M loc (Λ)where K ◦ Λ is the parahoric stabilizer of Λ in SO( V ). NTEGRAL MODELS FOR SHIMURA VARIETIES 17
The equivariance here is meant in the following sense: The natural action of G x = G ◦ x on M loc K ( G, { e µ } ) factors via the quotient G x / G m ≃ G ◦ Λ (see (2.5.1)) and, under theisomorphism (4.3.1), agrees with the action of the corresponding parahoric group scheme G ◦ Λ of SO( V ) on M loc (Λ).Hence, all our results on M loc (Λ), are really also about local models for the spin simil-itude groups. 5. Equations and a resolution for the local model
An affine chart of the local model following [23] . We continue with the samenotations, see especially § U naive and U of M naive and M loc , respectively. We will work over ˘ O but, for simplicity, sometimesomit the base change from the notation.We start with U naive . For an R -valued point L of M naive we set¯ L := Im( L ) ⊂ ( u − π ) − L R / ( u − π ) L R which determines L uniquely. Recall we are fixing a basis L = ⊕ di =1 ˘ O [ u ] · e i which corresponds to the basis ( e i ) of Λ = L / ( u − π ) L . We can describe an affine chartof M naive as follows: We set¯ L = { v + X ( u − π ) − v | v = d X i =1 v i e i } where X is a variable d × d matrix with entries in R . This naturally produces an R -basis e i + X ( u − π ) − e i for ¯ L , i.e. an R -module isomorphism R d ≃ ¯ L . Then the mapΦ : L / ( u − π ) L R = ¯ L ֒ → ( u − π ) − L R / ( u − π ) L R → ( u − π ) − L R / L R of § X .Similarly, we can write¯ L ∨ = { v + Y ( u − π ) − v | v = d X i =1 v i e ∨ i } . where Y is a variable d × d matrix with entries in R . As above, the mapΨ : L ∨ / ( u − π ) L ∨ R = ¯ L ∨ ֒ → ( u − π ) − L ∨ R / ( u − π ) L ∨ R → ( u − π ) − L ∨ R / L ∨ R of § Y .a) The condition that L ∨ is the dual of L , gives that we have h ¯ L , ¯ L ∨ i = 0 under the R -base change of the pairing h , i : ( u − π ) − L / ( u − π ) L × ( u − π ) − L ∨ / ( u − π ) L ∨ → ( u − π ) − ˘ O [ u ] / ˘ O [ u ] . Hence, on this affine chart h v + X ( u − π ) − v, w + Y ( u − π ) − w i = 0 . This is equivalent to(5.1.1) Y + X t = 0 , X t · Y = 0since h Xv, w i = h w, X t w i . Under ( u − π ) − L / ( u − π ) L ∼ −−→ L R / ( u − π ) L R given by multiplication by ( u − π ), the module ¯ L can be identified with ¯ L = Im(( u − π ) : L → L R / ( u − π ) L R ) used in [23]. b) The conditions ∧ Φ = 0, ∧ Ψ = 0, immediately translate to(5.1.2) ∧ X = 0 , ∧ Y = 0 . c) As in [23], we see that the condition L ⊂ L ∨ amounts to(5.1.3) X t S X − πSX = 0 , X t S X + 2 SX = 0 . Similarly, the condition L ∨ ⊂ u − L amounts to(5.1.4) Y t S Y + 2( S Y + πS Y ) = 0 , Y t S Y − π ( S Y + πS Y ) = 0 . In the above, S , S , are the matrices with S = S + πS = ( h e i , e j i ) i,j as in § L and ¯ L ∨ should be u -stable. This translates to X = 0 , Y = 0 . However, these are implied by the equations (5.1.1) above.Denote by J m the unit antidiagonal matrix of size m , J m := . . . . We now set X = ( x ij ), Y = ( y ij ), and denote by I naive the ideal of the polynomial ring˘ O [( x ij ) , ( y ij )] in 2 d variables, which is generated by the entries of the above relations(5.1.1), (5.1.2), (5.1.3), (5.1.4). Also denote by U naive the corresponding scheme overSpec ( ˘ O ) given by pairs of d × d matrices X , Y which satisfy these relations (for moredetails see Section 3 in [23]). Set U naive = Spec ( ˘ O [ X, Y ] / I naive )which is an open subscheme of M naive by Proposition 4.2.2. We also set U = U naive ∩ M loc = Spec ( ˘ O [ X, Y ] / I ) , with I naive ⊂ I . The scheme U is an open affine subscheme of M loc . Denote by ˘ O ( U naive ),resp. ˘ O ( U ), the affine coordinate rings of U naive , resp. U , over ˘ O .Following [23], we distinguish two cases:I) The integers d and δ have the same parity (cases (1) and (4) of § X into blocks as follows. We write(5.1.5) X = D C D B A B D C D , where A is of size δ × δ , D , D , D , D , are of size ( n − r ) × ( n − r ). (Recall, d = 2 n , δ = 2 r , or d = 2 n + 1, δ = 2 r + 1.) We setT( B | B ) = Tr( B J n − r B t J δ ) . II) The integers d and δ have different parity (cases (2) and (3) of § X in 9 blocks as above, but the recipe for the dimensionsof these blocks is somewhat different. In order to define the submatrices A , B i , C j , D j ,giving the block decomposition of X we set: r ′ = (cid:26) r if δ = 2 rr + 1 if δ = 2 r + 1 . NTEGRAL MODELS FOR SHIMURA VARIETIES 19
Then we write the matrix X as before, with blocks A of size ( δ + 1) × ( δ + 1), D , D , D , D of size ( n − r ′ ) × ( n − r ′ ).We denote by A ′ the δ × δ matrix which is obtained from A by erasing the part that isin the ( n + 1)-row and ( n + 1)-column of X . Similarly we denote by B ′ , B ′ the δ × ( n − r ′ )matrices which are obtained from B , B by erasing the part that is on the ( n + 1)-rowof X . Lastly, we denote by E the ( r ′ + 1)-column of A and E ′ the ( r ′ + 1)-column of A with the ( n + 1)-entry erased. (Recall, d = 2 n , δ = 2 r + 1, or d = 2 n + 1, δ = 2 r .) We setT( B ′ | E ′ | B ′ ) = Tr(( B ′ J n − r ′ ( B ′ ) t + 12 E ′ ( E ′ ) t ) J δ ) . Finally, for simplicity, we set Z = ( [ B | B ] , if d ≡ δ mod 2 , [ B ′ | E ′ | B ′ ] , if d δ mod 2 . Then Z = ( z ij ) ∈ Mat δ × ( d − δ ) , in both cases. Theorem 5.1.6. ( [23] ) The inclusion ˘ O [ Z ] → ˘ O [ X, Y ] induces isomorphisms ˘ O [ Z ] / ( ∧ Z, T( Z ) + 2 π ) ∼ −→ ˘ O ( U ) , (5.1.7) ˘ O [ Z ] / ( ∧ Z, (T( Z ) + 2 π ) · Z ) ∼ −→ ˘ O ( U naive ) . (5.1.8)5.2. This is essentially contained in loc. cit. but, for completeness, we will also give theargument below. Before we do that, we discuss some corollaries:A) By an explicit calculation, we find thatT( Z ) = 12 X ≤ i ≤ δ, ≤ j ≤ d − δ z i d − δ +1 − j z δ +1 − i j . (The same expression is valid in both cases, of same (I), or different (II), parity.) Theresult stated in the introduction follows:Denote by D δ × ( d − δ ) = { Z | ∧ Z = 0 } ⊂ Mat δ × ( d − δ ) the “determinantal” subscheme ofthe affine space of matrices Z over Spec ( ˘ O ). Theorem 5.2.1.
The affine chart U ⊂ M loc is isomorphic to the closed subscheme D T of the determinantal scheme D δ × ( d − δ ) which is defined by the quadratic equation X ≤ i ≤ δ, ≤ j ≤ d − δ z i d − δ +1 − j z δ +1 − i j = − π. Remark 5.2.2.
As in [23], one can see, by using the classical result that the determinantalscheme is Cohen-Macaulay, that D T above is ˘ O -flat, Cohen-Macaulay and of relativedimension d −
2. This observation is used in the proof of Theorem 5.1.6: It is needed toestablish that the ˘ O -algebra giving U in the statement above is indeed ˘ O -flat. It easilyfollows that M loc is also Cohen-Macaulay.B) By Theorem 5.1.6,(5.2.3) ˘ O ( U ) = ˘ O ( U naive ) / (T( Z ) + 2 π ) , where we slightly abuse notation by denoting by T( Z ) + 2 π the image of this elementin the quotient ring ˘ O ( U naive ). We can easily check that the annihilator of the elementT( Z ) + 2 π in the coordinate ring ˘ O ( U naive ) is the ideal ( Z ) generated by the entries of Z .We obtain an exact sequence0 → ˘ O ( U naive ) → ˘ O ( U ) × ˘ O −→ k → , where the second map is the difference of the reductions modulo the maximal ideals( Z ) + ( π ) and ( π ). Hence, ˘ O ( U naive ) = ˘ O ( U ) × k ˘ O where, on the right hand side, we have the fibered product of rings. This exhibits U naive as the push-out Spec (¯ k ) Spec ( O ) U U naive . X = Y =0 i It easily follows that M naive is also a push-out:(5.2.4) Spec ( k ) Spec ( O )M loc M naive . ∗ L = L i . We now give the proof of Theorem 5.1.6. Proof.
This is a variation of the work in [23]. We explain a simpler version of the argument.(We will also omit some of the explicit calculations.)We start with the case (I) of same parity d ≡ δ mod 2.First, we prove that ˘ O ( U naive ), resp. ˘ O ( U ), are quotient rings of ˘ O [ B , B ]:Let R be the ideal generated by (the entries of) the elements (1)-(8):(1) D + J n − r B t J δ B , (2) D + J n − r B t J δ B , (3) D + J n − r B t J δ B , (4) D + J n − r B t J δ B , (5) C + J n − r B t J δ A, (6) C + J n − r B t J δ A ,(7) A − B J n − r B t J δ ,(8) Y + X t .We observe that R ⊂ I naive . Indeed, the first six relations (1)-(6) are implied from the relation X t S X − πSX ∈ I naive (5.1.3), relation (7) from (5.1.1) and (5.1.4), and (8) from (5.1.1).Hence, the relations (1)-(8) express each block A , D i , C j of X and Y , in terms of B and B , modulo I naive . It follows that ˘ O ( U naive ) and, therefore, also ˘ O ( U ), is a quotientof ˘ O [ B , B ]. In fact, since I naive contains all the 2 × X , it follows that˘ O ( U naive ) and ˘ O ( U ) are also quotients of ˘ O [ B , B ] / ( ∧ [ B | B ]). This last ring is theaffine coordinate ring of the cone over the Segre embedding of P δ − × P d − δ − over ˘ O andso it is integral and flat of relative dimension d − O .We now continue to uncover additional relations in I naive . As is observed in [23], thecondition ∧ X = 0 together with the fact that the blocks B , A , and B , all share thesame rows of X , easily give(5.2.5) AB = Tr( A ) B , AB = Tr( A ) B . By [23, Lemma 4.4], the relation ∧ X = 0 implies that B t J n − r B is symmetric, so,modulo I naive , we have B t J n − r B = B t J n − r B . NTEGRAL MODELS FOR SHIMURA VARIETIES 21
By looking at the appropriate blocks, we see that the relations (5.1.3) imply A t J δ B + 2 πJ δ B ∈ I naive , A t J δ B + 2 πJ δ B ∈ I naive . Now A − B J n − r B t J δ ∈ I naive (relation (7) above), so, modulo I naive , A t J δ B + 2 πJ δ B = ( B J n − r B t J δ ) t J δ B + 2 πJ δ B = J δ B J n − r B t J δ B + 2 πJ δ B = J δ AB + 2 πJ δ B = J δ (Tr( A ) + 2 π ) B , where the last identity uses (5.2.5). This implies that (Tr( A ) + 2 π ) B ∈ I naive . Similarly,starting from A t J δ B + 2 πJ δ B ∈ I naive , we obtain (Tr( A ) + 2 π ) B ∈ I naive . Since T( B | B ) = Tr( A ) modulo I naive , this gives that˘ O ( U naive ) is a quotient of ˘ O [ B , B ] / ( ∧ [ B | B ] , (T( B | B ) + 2 π )[ B | B ]).Observe that B = B = 0 gives X = Y = 0, which corresponds to L = L . This pointdoes not belong to the generic fiber of the ˘ O -flat M loc . Hence, we find that ˘ O ( U ) is aquotient of A = ˘ O [ B , B ] / ( ∧ [ B | B ] , T( B | B ) + 2 π ). The ring A is the coordinate ringof a hypersurface in the integral Cohen-Macaulay ˘ O [ B , B ] / ( ∧ [ B | B ] , T( B | B ) + 2 π ));we can easily see as in [23] that A is integral of dimension d − O -flat. Since U alsoshares both these properties and ˘ O ( U ) is a quotient of A , it follows that ˘ O ( U ) = A , aswe wanted. The result for ˘ O ( U naive ) also quickly follows: Indeed, ˘ O ( U naive ) is a quotientof A + := ˘ O [ B , B ] / ( ∧ [ B | B ] , (T( B | B ) + 2 π )[ B | B ]) . However, this quotient has to be big enough to also allow both ˘ O ( U ) = A and ˘ O = O [ B , B ] / ([ B | B ]) to appear as quotients. The corresponding spectrum has to supporta morphism from the push-out of U and Spec ( ˘ O ), glued at the point X = Y = π = 0.But, as in § A + , so˘ O ( U naive ) = A + = ˘ O [ B , B ] / ( ∧ [ B | B ] , (T( B | B ) + 2 π )[ B | B ]) , as we wanted.The case (II) of different parity is similar: The role of A is now played by A ′ and therelation (7) above is replaced by A ′ − ( B ′ J n − r ′ ( B ′ ) t + 12 E ′ ( E ′ ) t ) J δ ∈ I naive which gives Tr( A ′ ) = T( B ′ | E ′ | B ′ ) modulo I naive . See also the proof of [23, Theorem 3.3]for more details.5.3. The blow-up of the local model M loc . Let r bl : M bl (Λ) → M loc (Λ)be the blow-up of M loc (Λ) at the closed point ∗ of its special fiber that corresponds to L = L . We will now show Theorem 1.3.1 of the introduction: Proof.
By Theorem 5.2.1, it is enough to show the conclusion of the theorem for theblow-up e D T of D T at the (maximal) ideal given by ( z ij ). For simplicity, we write D forthe determinantal scheme D δ × ( d − δ ) over Spec ( ˘ O ). This is the affine cone over the Segreembedding ( P δ − × P d − δ − ) ˘ O ֒ → P δ ( d − δ ) − O . Also, we set T = 12 X ≤ i ≤ δ, ≤ j ≤ d − δ z i d − δ +1 − j z δ +1 − i j . Let us consider the blow-up e D −→ D of the determinantal scheme over Spec ( ˘ O ) along the vertex of the cone, i.e. along thesubscheme defined by the ideal ( z ij ). Then, the blow-up e D T is isomorphic to the stricttransform of the hypersurface D T ⊂ D given by T + 2 π = 0. Let V s,t be the open affinechart of e D over which the image of z st generates the pull-back of the ideal ( z ij ). Then V s,t = Spec ( ˘ O [( u i,j ) ≤ i ≤ δ, ≤ j ≤ d − δ ] / (( u i,j − u s,j u i,t ) i,j , u s,t − . The intersection V s,t ∩ e D T is obtained by substituting z ij = u i,j z st and u i,j = u s,j u i,t , forall i , j , in the equation T = − π . This amounts to setting z ij = u s,j u i,t z st and gives 4 π + z st ( X ≤ i ≤ δ, ≤ j ≤ d − δ u s,d − δ +1 − j u i,t u s,j u δ +1 − i,t ) = 0 . This is(5.3.1) 4 π + z st ( δ X i =1 u i,t u δ +1 − i,t )( d − δ X j =1 u s,j u s,d − δ +1 − j ) = 0 . Note that, since u s,t = 1, the two sums in the line above are S = u δ +1 − s,t + X i = s u i,t u δ +1 − i,t , S = u s,d − δ +1 − t + X j = t u s,j u s,d − δ +1 − j . (If δ ∗ = 1, i.e. δ = 1 or d − δ = 1, then one of the sums is equal to u s,t = 1.) If δ ∗ ≥ u z st , x
7→ − S / y S /
2, defines a smooth morphism V s,t ∩ e D T −→ Spec ( ˘ O [ u, x, y ] / ( u xy − π )) . If δ ∗ = 1, we similarly obtain a smooth morphism to Spec ( ˘ O [ u, x ] / ( u x − π )).6. Resolution and the linked quadric
Here, we relate the blow-up M bl (Λ) of the local model with the linked quadric Q (Λ , Λ ∨ )by introducing a third auxiliary scheme M (Λ).6.1. A resolution via additional lines.
We first define a scheme M naive = M naive (Λ)over Spec ( O ) with G -action and a G -equivariant morphism r : M naive −→ M naive .We give M naive as a functor on O -algebras as follows. Definition 6.1.1.
The functor M naive associates to an O -algebra R the set of triples( W + , W − , L ), where1) L is a finitely generated projective R [ u ]-module which gives a point of M naive , i.e.satisfies conditions (1)-(6) of § NTEGRAL MODELS FOR SHIMURA VARIETIES 23 W + , W − , are two finitely generated projective R [ u ]-modules such that( u − π ) L R ⊂ W + ⊂ ⊂ L R L R ⊂ ⊂ W − ⊂ ( u − π ) − L R , with all the quotients L R /W + , L /W + , W − / L R , W − / L , finitely generated projec-tive R -modules of rank 1.We can see that the forgetful morphism r : M naive −→ M naive given by r ( W + , W − , L ) = L is representable by a projective morphism. Indeed, M naive is naturally a closed subschemeof P (Λ) × P (Λ ∨ ) × M naive and r is given by the projection. Since M naive is projective byProposition 4.2.2, M naive is also represented by a projective scheme over Spec ( O ).This map r is an isomorphism over the open locus M ⊂ M naive where L 6 = L . Indeed,over M we have W − = L ∩ L , W + = L + L , and the data ( W + , W − , L ) are uniquelydetermined by L .Set M = r − (M ) which, by the above, is isomorphic to M via r . For the genericfibers we have M ⊗ O F = M ⊗ O F = Q ( V ) . Definition 6.1.2.
The scheme M = M (Λ) is the (reduced) Zariski closure of the genericfiber Q ( V ) = M ⊗ O F in M naive .The restriction of the morphism r to M factors through M loc ⊂ M naive and gives aprojective birational G -equivariant morphism r : M −→ M loc . We can identify M (Λ) with both the blow-up M bl (Λ) of the local model and the blow-up e Q (Λ , Λ ∨ ) of the linked quadric Q (Λ , Λ ∨ ): Theorem 6.1.3.
There are G -equivariant isomorphisms M bl (Λ) ≃ M (Λ) ≃ e Q (Λ , Λ ∨ ) . In fact, we obtain a diagram of G -equivariant birational projective morphisms(6.1.4) M bl (Λ) M (Λ)M loc (Λ) Q (Λ , Λ ∨ ) . r bl ∼ βr and the isomorphism M bl (Λ) ≃ M (Λ) makes the diagram commute.We now give the proof of Theorem 6.1.3.6.2. Comparing resolutions.
We first define a morphism ρ : M −→ Q (Λ , Λ ∨ )which we then show identifies ρ with the blow-up β : e Q (Λ , Λ ∨ ) = e Q i (Λ , Λ ∨ ) −→ Q (Λ , Λ ∨ ) . Let us consider an R -point of M given by ( W + , W − , L ). Taking duals gives( u − π ) L ∨ R ⊂ W ∨− ⊂ ⊂ L ∨ R L ∨ ⊂ ⊂ W ∨ + ⊂ ( u − π ) − L ∨ R . We now set(0) ⊂ ¯ W + 1 ⊂ L R / ( u − π ) L R = Λ R , (0) ⊂ ¯ W − ⊂ ( u − π ) − L R / L R = Λ R , (0) ⊂ ¯ W ∨− ⊂ L ∨ R / ( u − π ) L ∨ R = Λ ∨ R , (0) ⊂ ¯ W ∨ + ⊂ ( u − π ) − L ∨ R / L ∨ R = Λ ∨ R , where we denote by bar the images of the submodules W + , W − , W ∨ + , W ∨− , in the corre-sponding quotients.There is a morphism ρ : M −→ P (Λ) × O P (Λ ∨ )given by ( W + , W − , L ) ( ¯ W − , ¯ W ∨ + ). Recall that the generic fiber M ⊗ O F is isomorphicto the quadric of isotropic lines in the quadratic space V . Since M is by definition, flatover Spec ( O ), and Q (Λ , Λ ∨ ) is, also by definition, the flat closure of the same quadric in Q (Λ) × O Q (Λ ∨ ) ⊂ P (Λ) × O P (Λ ∨ ), the morphism ρ factors through Q (Λ , Λ ∨ ) to give ρ : M −→ Q (Λ , Λ ∨ ) . We will use ρ to identify M with the blow-up e Q (Λ , Λ ∨ ) of Q (Λ , Λ ∨ ).6.3. Proof of the main comparison.
We continue with the proof of Theorem 6.1.3.We have already given morphisms r : M → M loc and ρ : M → Q (Λ , Λ ∨ ). We would liketo show that these induce identifications with the blow-up r bl : M bl → M loc . Using theuniversal property of the blow-up, we see that it is enough to prove this statement afterbase changing to O F ′ , where F ′ /F is an unramified extension, or even to ˘ O . This allowsus to assume that we are in one of the 4 cases listed in § M is a closed subscheme of P (Λ) × P (Λ ∨ ) × M loc . This can also be seen asfollows: Start with ( x ; . . . ; x d ) × ( y ; . . . ; y d ) ∈ P (Λ) × P (Λ ∨ ) . Take W = W + by giving ¯ W + ⊂ Λ R to be the perpendicular of the line ( y ; . . . ; y d ) ∈ P (Λ ∨ ) under the perfect pairing Λ R × Λ ∨ R → R . Similarly, we take W ′ = W ∨− by giving¯ W ′ ⊂ Λ ∨ R to be the perpendicular of the line ( x ; . . . ; x d ) ∈ P (Λ) under the perfect pairingΛ R × Λ ∨ R → R . Set e ∨ i for the dual basis of e i ∈ Λ so that h e i , e ∨ j i = δ ij . Then we have ¯ W ∨ + = ( X i y i e ∨ i ) ⊂ Λ ∨ R , ¯ W − = ( X i x i e i ) ⊂ Λ R . The pair ( W + , W − ) is part of a triple ( W + , W − , L ) that corresponds to a point of M naive ,when there is L ∈ M naive ( R ), such that the image of L in ( u − π ) − L R / L R = Λ R , i.e. theimage of Φ, is contained in ¯ W − = ( P i x i e i ), and the image of L ∨ in ( u − π ) − L ∨ R / L ∨ R = Λ ∨ R is contained in ¯ W ∨ + = ( P i y i e ∨ i ).Let us now consider the inverse image e U := r − ( U ) under r : M −→ M loc . NTEGRAL MODELS FOR SHIMURA VARIETIES 25
Recall U = U naive ∩ M loc . Over the affine chart U naive , Φ is given by the matrix X . Hence,the first condition translates to(6.3.1) X = ( x , . . . , x d ) t · ( λ , . . . , λ d ) = (cid:0) λ x t |· · · | λ d x t (cid:1) = λ x . . . λ d x ... . . . ... λ x d . . . λ d x d for some ( λ , . . . , λ d ) ∈ R d . Similarly, the second condition translates to(6.3.2) Y = ( y , . . . , y d ) t · ( µ , . . . , µ d ) = (cid:0) µ y t |· · · | µ d y t (cid:1) = µ y . . . µ d y ... . . . ... µ y d . . . µ d y d for some ( µ , . . . , µ d ) ∈ R d . In particular, it follows that the equations (6.3.1) and (6.3.2)are true over e U .To continue with our calculation, it is convenient to recall the decompositions Λ = M ⊕ N , Λ ∨ = M ⊕ π − N as in § M , N listed there. Recall thatwe assume that we are in one of the 4 cases of § c , to be theset of 1 ≤ i ≤ d with e i ∈ N , resp. e i ∈ M , where { e i } is the basis listed there. Then { , . . . , d } = ∆ c ⊔ ∆ and δ , c = d − δ . For x = ( x i ) ≤ i ≤ d ∈ A (Λ) we write x = x + x , where x ∈ A ( M ), x ∈ A ( N ). Let w = ( w j ) j ∈ ∆ c be a point of A ( M ), resp. z = ( z i ) i ∈ ∆ a point of A ( N ). As in the proof of Theorem 3.2.1, we set Q ( w ) = h w, w i , Q ( z ) = h z, z i π . Let e U s,t , where x s = 1 and y t = 1, be the affine patches that cover e U . Using theequation Y + X t = 0 we obtain the following relations:(6.3.3) λ t = − µ s , λ i = λ t y i , and µ j = − λ t x j for 1 ≤ i, j ≤ d. We can now determine e U s,t . Proposition 6.3.4.
We have e U s,t ≃ Spec O [ λ t , ( x i ) ≤ i ≤ d , ( y j ) ≤ j ≤ d ]( λ t Q ( x ) Q ( y ) + π ) + K s,t ! where K s,t = (cid:16) x s − , y t − , ( x i + λ t Q ( x ) y d +1 − i ) i ∈ ∆ c , ( y j − λ t Q ( y ) x d +1 − j ) j ∈ ∆ (cid:17) . Before we give the proof, we note that this implies that the charts e U s,t with s ∈ ∆, t ∈ ∆ c , cover e U . Indeed, if x i = 0 on e U s,t , for all i ∈ ∆, then also Q ( x ) = 0. Hence,since x i + λ t Q ( x ) y d +1 − i ∈ K s,t , we obtain x i ∈ K s,t for all i ∈ ∆ c also, which is acontradiction. Hence, we also have x i = 0 for some i ∈ ∆. A similar argument gives y j = 0 for some j ∈ ∆ c . If s ∈ ∆ and t ∈ ∆ c , then(6.3.5) e U s,t ≃ Spec O [ λ t , ( x i ) i ∈ ∆ , ( y j ) j ∈ ∆ c ]( λ t Q ( x ) Q ( y ) + π, x s − , y t − ! . Proof.
We will assume d and δ have the same parity and leave the very similar (butnotationally more involved) case of different parity to the reader. Set l = ( d − δ ) / n − r .Recall the decomposition X = ( x , . . . , x d ) t · ( λ , . . . , λ d ) = λ x . . . λ d x ... . . . ... λ x d . . . λ d x d = D C D B A B D C D . The block decomposition corresponds to separating a vector x into 3 parts x = x ⊕ x = x − ⊕ x ⊕ x , in spaces of dimension l , δ , l , respectively, with x − = ( x , . . . , x l ) , x = ( x l +1 , . . . , x l + δ ) , x = ( x l + δ +1 , . . . , x d ) . Denote by x ∗ , resp. x ∗ ± , the result of reversing the order of the coordinates in the vector x , resp. x ± . We have A = x t · λ = λ l +1 x l +1 . . . λ l + δ x l +1 ... . . . ... λ l +1 x l + δ . . . λ l + δ x l + δ ,B = x t · λ − = λ x l +1 . . . λ l x l +1 ... . . . ... λ x l + δ . . . λ l x l + δ , B = x t · λ = λ l + δ +1 x l +1 . . . λ d x l +1 ... . . . ... λ l + δ +1 x l + δ . . . λ d x l + δ , . Similarly, D = x t − · λ − , etc. We have B J l B t J δ = x t · λ · J l · ( x t · λ − ) t · J δ (6.3.6) = x t · ( λ · J l · λ t − ) · x · J δ = x t · x ∗ · Q ( λ ) , since Q ( λ ) = λ · J l · λ t − and x · J δ = x ∗ . The relation Tr( B J l B t J δ ) + 2 π = 0, thatholds over U by Theorem 5.1.6, translates to(6.3.7) Q ( x ) · Q ( λ ) + π = 0 . Using the relations (6.3.3) we obtain:(6.3.8) λ t Q ( y ) Q ( x ) + π = 0 . Notice that this last equation implies that λ t is not a zero divisor in the coordinate ringof the O -flat scheme e U s,t . Since, by (6.3.3), λ i = λ t y i , the relation A = B J n − r B t J δ , and(6.3.6) gives λ t · x t · y = x t · x ∗ · λ t Q ( y ) . Since λ t is not a zero divisor, we obtain x t · y = x t · x ∗ · λ t Q ( y )or, after taking transpose,(6.3.9) ( y − λ t Q ( y ) · x ∗ ) t · x = 0 . By a similar calculation as in the proof of (6.3.6), we obtain12 J l B t J δ B = ( λ ∗ ) t · λ − · Q ( x ) . NTEGRAL MODELS FOR SHIMURA VARIETIES 27
Hence, the relation D + J l B t J δ B = 0 which holds over U ⊂ M loc (see the proof ofTheorem 5.1.6) amounts to x t − · λ − = λ t · x t − · y − = − λ t · ( y ∗ ) t · y − · Q ( x ) , and over the O -flat e U s,t to(6.3.10) x t − · y − = − ( y ∗ ) t · y − · λ t Q ( x ) . Equivalently, this is(6.3.11) ( x − + λ t · Q ( x ) · y ∗ ) t · y − = 0 . Similarly, from C = − J l B t J δ A , D = − J l B t J δ B , we obtain(6.3.12) ( x − + λ t · Q ( x ) · y ∗ ) t · y = 0 , (6.3.13) ( x − + λ t · Q ( x ) · y ∗ ) t · y = 0 . Since y t = 1, these, all together, amount to(6.3.14) x − + λ t · Q ( x ) · y ∗ = 0 . We now examine the relations C = − J l B t J δ A , D = − J l B t J δ B , D = − J l B t J δ B .In a similar fashion, we find that these, all together, amount to(6.3.15) x + λ t · Q ( x ) · y ∗ − = 0 . Combining the relations (6.3.14) and (6.3.15) we obtain the desired equations x i + λ t Q ( x ) y d +1 − i = 0 , ∀ i ∈ ∆ c . Finally, (6.3.9) amounts to x i ( y j − λ t Q ( y ) x d +1 − j ) = 0 , ∀ i, j ∈ ∆ . By (6.3.8), Q ( x ) is not a zero divisor in the coordinate ring of the O -flat scheme e U s,t .Since Q ( x ) belongs to the ideal ( x i ) i ∈ ∆ , we see that over e U s,t , we also have y j − λ t Q ( y ) x d +1 − j = 0 , ∀ j ∈ ∆ . Assembling the above, we see that all the generators of the ideal K s,t vanish on e U s,t . It isnow easy to see, by comparing dimensions, that e U s,t is as in the statement.We now continue on to show that there is a G -equivariant isomorphism M ≃ e Q (Λ , Λ ∨ ).First, we show that ρ : M → Q (Λ , Λ ∨ ) factors through the blow-up M e Q (Λ , Λ ∨ ) Q (Λ , Λ ∨ ) . ρ β It is enough, by the universal property of the blow-up, to show that the pull-back ρ ∗ ( Z )of Z ⊂ Q (Λ , Λ ∨ ) to M is a Cartier divisor. Recall that Z is locally defined by the ideal( u, Q ( x )), where u is the “linking multiplier”, defined up to unit by i R ( L ) = uL ′ , where L and L ′ are the universal isotropic lines over Q (Λ , Λ ∨ ). We will show that over eachaffine chart e U s,t ⊂ M the linking multiplier is, in fact, a multiple of Q ( x ). Hence, thepull-back of the ideal ( u, Q ( x )) is principal, as desired. In Proposition 6.3.4 above, we are using the dual basis e ∨ i of Λ ∨ , which by definition,is given by h e i , e ∨ j i = δ ij . We have e ∨ i = e d +1 − i for i ∈ S ( M ) , and e ∨ j = π − e d +1 − j for i ∈ S ( N ) . With this basis, the “linking” i R : Λ R → Λ ∨ R is given by the symmetric matrix S = ( h e i , e j i ) i,j . For example, in the case d and δ havethe same parity, we have i R ( x − , x , x ) = ( x − , πx , x ) . Hence, in view of the elements generating K s,t , we see that Proposition 6.3.4 implies thatover e U s,t ⊂ M we have i R ( x ) = u · y, j R ( πy ) = v · x, with(6.3.16) u = − Q ( x ) λ t , v = Q ( y ) λ t . This establishes that the pullback of the ideal ( u, Q ( x )) is principal over e U s,t andso, locally principal over e U . Therefore, the restriction ρ | e U : e U → Q (Λ , Λ ∨ ) factorsthrough the blow-up e Q (Λ , Λ ∨ ) → Q (Λ , Λ ∨ ). The result for ρ easily follows since ρ is G -equivariant. Indeed, the G -translates of U cover M loc . Hence, the G -translates of theopen e U = r − ( U ) ⊂ M cover M . This, combined with the above, implies that ρ factorsthrough e ρ : M → e Q (Λ , Λ ∨ ).It remains to show that e ρ : M → e Q (Λ , Λ ∨ ) is an isomorphism. This is easily obtainedby using the description of the affine charts given in Proposition 6.3.4, and the discussionabove together with G -equivariance: The map e ρ is birational. From Proposition 6.3.4and the usual G -equivariance argument, M is a regular scheme and is projective and flatover Spec ( O ) of relative dimension d −
2. The same is true for e Q (Λ , Λ ∨ ) by Proposition3.3.1. In fact, by comparing the explicit description of the affine charts e U s,t ⊂ M givenby Proposition 6.3.4, with that of the affine charts for e Q (Λ , Λ ∨ ) given in the proof ofProposition 3.3.1, we can easily see, using (6.3.16) above, that e ρ gives a bijection on ¯ k -points. Hence, the morphism e ρ is birational quasi-finite and then, by using Zariski’s maintheorem, an isomorphism. This concludes the proof of the existence of a G -isomorphism M ≃ e Q (Λ , Λ ∨ ).It remains to give a G -equivariant isomorphism M ∼ −→ M bl . Again, we first show that r : M → M loc factors through the blow-up r bl : M bl → M loc . By using G -equivariancewe see it is enough to show that the pull-back of the ideal generated by the entries ofthe matrix X becomes principal over r − ( U ) = e U ⊂ M . This follows immediately fromthe equations in the proof of Proposition 6.3.4. The isomorphism M ∼ −→ M bl also followsby a similar argument. In fact, after unravelling the various identifications of coordinatesystems, we can see that the description of the affine chart e U s,t in (6.3.5) matches thedescription of the corresponding affine chart V s,t ∩ U bl of the blow up U bl given by (5.3.1).Hence, M → M bl restricts to an isomorphism e U s,t ∼ −→ V s,t ∩ U bl . This concludes the proofof Theorem 6.1.3. (cid:3) NTEGRAL MODELS FOR SHIMURA VARIETIES 29
Additional properties.
We now give some further properties:
Proposition 6.4.1. a) The exceptional locus of r : M = M bl −→ M loc is r − ( ∗ ) = P (Λ ∨ / Λ) × k P (Λ /π Λ ∨ ) ≃ P δ − × k P d − δ − . b) The morphism ρ : M → Q (Λ , Λ ∨ ) is an isomorphism over the complement of theintersection Z ∩ Z ∩ Z = Q (Λ ∨ / Λ) × k Q (Λ /π Λ ∨ ) ⊂ Q (Λ , Λ ∨ ) . Over this intersection ρ is a P -bundle.Proof. Let us describe the inverse image r − ( ∗ ) ⊂ M : We easily see that r − ( ∗ ) is aclosed subscheme of { ( x ; . . . ; x d ) , ( y ; . . . ; y d ) } = P (Λ) k × k P (Λ ∨ ) k . Over the intersection e U s,t ∩ r − ( ∗ ) we have x s = 1 and X = x t · λ = 0. This gives λ = 0and in particular λ t = 0. Using the equations for e U s,t given by Proposition 6.3.4, we seethat e U s,t ∩ r − ( ∗ ) is defined by x i = 0 for all i ∈ ∆ c , x s = 1, and y j = 0, for all j ∈ ∆, y t = 1. Therefore, the inverse image r − ( ∗ ) is P ( N ) k × k P ( M ) k = P (Λ ∨ / Λ) × P (Λ /π Λ ∨ ) ≃ P δ − × k P d − δ − . This proves (a). (Alternatively, we could have used the description of r as a blow-upfrom Theorem 6.1.3 and its proof.) Part (b) follows from the description of the blow-up e Q (Λ , Λ ∨ ) → Q (Λ , Λ ∨ ) in Proposition 3.3.1 and its proof, and Theorem 6.1.3 whichidentifies ρ with this blow-up. Remark 6.4.2. a) We can now explain the birational map Q (Λ , Λ ∨ ) M loc as follows: We first perform the blow-up e Q (Λ , Λ ∨ ) of Q (Λ , Λ ∨ ). Then to obtain M loc , wecontract a subscheme e Z , isomorphic to P δ − × P d − δ − (which is in fact an irreduciblecomponent of the special fiber of the blow-up) to a point. This subscheme is the stricttransform of the component Z ≃ P δ − × P d − δ − of the special fiber of Q (Λ , Λ ∨ ).b) The cases δ = 0 and δ = 1 are different.i) When δ = 0, Λ = Λ ∨ and M loc is the smooth quadric Q (Λ). ThenM loc = e Q (Λ , Λ ∨ ) = Q (Λ , Λ ∨ ) . ii) When δ = 1, we also have M loc ≃ Q (Λ) (see [8, Prop. 12.7]). This case wasalso discussed in detail, but from slightly different perspectives, in [8, 12.7.2] and in [12,Prop. 2.16]. Now, M loc is not smooth over Spec ( O ) but only regular; the special fiberhas an isolated singular point. In this case, Q (Λ , Λ ∨ ) (which is denoted by P (Λ) fl in[8, 12.7.2]) is a blow-up of Q (Λ) at this singular point. We have e Q (Λ , Λ ∨ ) = Q (Λ , Λ ∨ ), ρ : M → Q (Λ , Λ ∨ ) is an isomorphism, and r : M = M bl = e Q (Λ , Λ ∨ ) = Q (Λ , Λ ∨ ) −→ M loc = Q (Λ)can be identified with the blow-up of M loc = Q (Λ) at its singular point, discussed above.In particular, M = M bl is isomorphic to P (Λ) fl of loc. cit. Application to Shimura varieties
Spin and orthogonal Shimura data.
We now discuss some Shimura varieties towhich we can apply these results. We start with an odd prime p and an orthogonalquadratic space V over Q of dimension d ≥ d − , § C ( V ) is endowed with a Z / Z -grading C ( V ) = C + ( V ) ⊕ C − ( V ) and a canonical involution c c ∗ . The group of spinorsimilitudes G = GSpin( V ) is the reductive group over Q defined by G ( R ) = { g ∈ C + ( V R ) × | gV R g − = V R , g ∗ g ∈ R × } for any Q -algebra R . The spinor similitude η : G → G m is defined by η ( g ) = g ∗ g , andthere is a representation α : G → SO( V ) defined by g · v = gvg − .Consider the hermitian symmetric domain X = { z ∈ V C : h z, z i = 0 , h z, ¯ z i < } / C × of dimension d −
2. The group G ( R ) acts on X through G → SO( V ), and the action ofany g ∈ G ( R ) with η G ( g ) < X .Now write z ∈ X as z = u + iv with u, v ∈ V R . Then, the subspace Span R { u, v } isa negative definite plane in V R , oriented by the ordered orthogonal basis u, v . There arenatural R -algebra maps C ∼ −→ C + (Span R { u, v } ) → C + ( V R ) . The first is determined by i uv p Q ( u ) Q ( v ) , and the second is induced by the inclusion Span R { u, v } ⊂ V R . The above compositionrestricts to an injection h z : C × → G ( R ), which arises from a morphism h z : S → G R ofreal algebraic groups. Here S = Res C / R G m is Deligne’s torus. The construction z h z realizes X ⊂ Hom( S , G R ) as a G ( R )-conjugacy class. The pair ( G, X ) is a Shimura datumof Hodge type.Using the conventions of [5], the Hodge structure on V determined by h z is(7.1.1) V (1 , − C = C z, V (0 , C = ( C z + C ¯ z ) ⊥ , V ( − , C = C ¯ z. This implies that the Shimura cocharacter µ z : G m C → G C obtained from { h z } isconjugate to ˜ µ : G m C → G C given by˜ µ ( t ) = t − f f d + f d f where f , f d are part of a basis ( f i ) i of V C such that h f , f d i = 1, h f , f i = h f d , f d i = 0,and the arithmetic on the right hand side takes place in the Clifford algebra C ( V C ). Thisagrees with the cocharacter considered in § § α · e µ : G m C → SO( V ) C is given by µ ( t ) = diag( t − , , . . . , , t ).Observe that the action of G ( R ) on X factors through SO( V )( R ) via α , and we alsoobtain a Shimura datum (SO( V ) , X ) which is now of abelian type. The correspondingShimura cocharacter is conjugate to µ ( t ) = diag( t − , , . . . , , t ). NTEGRAL MODELS FOR SHIMURA VARIETIES 31
Spinor integral models.
We continue with the notations and assumptions of theprevious paragraph. In particular, we take G = GSpin( V ) and X the G ( R )-conjugacyclass of { h z } : S → G R above that define the spin similitude Shimura datum ( G, X ).In addition, we choose a vertex lattice Λ ⊂ V ⊗ Q Q p with π Λ ∨ ⊂ Λ ⊂ Λ ∨ and δ =length Z p (Λ ∨ / Λ), δ ∗ = min( δ, d − δ ), and assume δ ∗ ≥
1. This defines the parahoricsubgroup K p = { g ∈ GSpin( V ⊗ Q Q p ) | g Λ g − = Λ , η ( g ) ∈ Z × p } which we fix below. Choose also a sufficiently small compact open subgroup K p of theprime-to- p finite adelic points G ( A pf ) of G and set K = K p K p . The Shimura varietySh K ( G, X ) with complex pointsSh K ( G, X )( C ) = G ( Q ) \ X × G ( A f ) /K is of Hodge type and has a canonical model over the reflex field Q . Theorem 7.2.1.
For every K p as above, there is a scheme S reg K ( G, X ) , flat over Spec ( Z p ) ,with S reg K ( G, X ) ⊗ Z p Q p = Sh K ( G, X ) ⊗ Q Q p , and which supports a “local model diagram” (7.2.2) f S reg K ( G, X ) S reg K ( G, X ) M bl (Λ) π reg K q reg K such that: a) π reg K is a G -torsor for the parahoric group scheme G that corresponds to K p , b) q reg K is smooth and G -equivariant. c) S reg K ( G, X ) is regular and has special fiber which is a divisor with normal cross-ings. The multiplicity of each irreducible component of the special fiber is ei-ther one or two and the components of multiplicity two are each isomorphic to P δ − × P d − δ − over ¯ F p . In fact, S reg K ( G, X ) can be covered, in the ´etale topology,by schemes which are smooth over Spec ( Z p [ u, x, y ] / ( u xy − p )) when δ ∗ ≥ , orover Spec ( Z p [ u, x ] / ( u x − p )) when δ ∗ = 1 .In addition, we have: The schemes { S reg K ( G, X ) } K p , for variable K p , support correspondences that ex-tend the standard prime-to- p Hecke correspondences on { Sh K ( G, X ) } K p . Thesecorrespondences extend to the local model diagrams above (acting trivially on M bl (Λ) ). The projective limit S reg K p ( G, X ) = lim ←− K p S K p K p ( G, X ) satisfies the “dvr extension property”: For every dvr R of mixed characteristic (0 , p ) we have: S reg K p ( G, X )( R ) = Sh K p ( G, X )( R [1 /p ]) . Note that (a) and (b) together amount to the existence of a smooth morphism¯ q K : S reg K ( G, X ) → [ G\ M bl (Λ)]where the target is the quotient algebraic stack. Proof.
By [10, Theorem 4.2.7], there are schemes S K ( G, X ) which satisfy similar proper-ties, excluding (c), but with M bl (Λ) replaced by the PZ local model M loc (Λ). (Note thatall the assumptions of [10, Theorem 4.2.7] are satisfied: ( G, X ) is of Hodge type, p is odd,the group G splits over a tamely ramified extension of Q p , and, by the discussion in § G x is connected.) In particular, we have(7.2.3) f S K ( G, X ) S K ( G, X ) M loc (Λ) π K q K with π K a G -torsor and q K smooth and G -equivariant. We set f S reg K ( G, X ) = f S K ( G, X ) × M loc (Λ) M bl (Λ)which carries a diagonal G -action. Since r : M bl (Λ) −→ M loc (Λ) is given by a blow-up, isprojective, and we can see ([14, § π reg K : f S reg K ( G, X ) −→ S reg K ( G, X ) := G\ f S reg K ( G, X )is represented by a scheme and gives a G -torsor. (This is an example of a “linear mod-ification”, see [14, § S reg K ( G, X ) is the blow-up of S K ( G, X ) at the subscheme of closed points that corre-spond to ∗ ∈ M loc (Λ) under the local model diagram (7.2.3). This set of points is thediscrete Kottwitz-Rapoport stratum of the special fiber of S K ( G, X ). The projectiongives a smooth G -morphism q reg K : f S reg K ( G, X ) −→ M bl (Λ)which completes the local model diagram. Property (c) follows from Theorem 6.1.3,Proposition 3.3.1 and its proof, Proposition 6.4.1, and properties (a) and (b) which implythat S reg K ( G, X ) and M bl (Λ) are locally isomorphic for the ´etale topology. The rest of theproperties in the statement follow from the corresponding properties for S K ( G, X ) andthe construction.
Remark 7.2.4. a) As we see in the proof, S reg K ( G, X ) is the blow-up of the discreteKottwitz-Rapoport stratum of S K ( G, X ). The geometric fibers of the blow-up morphism S reg K ( G, X ) → S K ( G, X )over points in this stratum are each isomorphic to P δ − × P d − δ − . These fibers are exactlythe components of multiplicity two in the geometric special fiber of S reg K ( G, X ).b) When δ ∗ = 1, M loc (Λ) is already regular. Hence, in this case S K ( G, X ) is alsoregular: This integral model has appeared, via a different construction, in [12].7.3.
Orthogonal integral models.
Let us mention that a result exactly like Theorem7.2.1 can be obtained for the Shimura varieties associated to the Shimura data (SO( V ) , X )and the parahoric subgroup given by the connected stabilizer K p = { g ∈ SO( V ⊗ Q Q p ) | g Λ = Λ , ε ( g ) = 1 } by combining the previous results with [10, Theorem 4.6.23]: Note that (SO( V ) , X ) is ofabelian type. The corresponding integral model S K (SO( V ) , X ) is obtained as a quotientof the integral model S K (GSpin( V ) , X ) by a finite group action ([10, § S reg K (SO( V ) , X ) with M bl (Λ) as its local model byfollowing the argument in the proof of Theorem 7.2.1 above. NTEGRAL MODELS FOR SHIMURA VARIETIES 33
Rapoport-Zink spaces.
Finally, we observe that our result can be applied to con-struct regular formal models of certain related Rapoport-Zink spaces. Our discussionwill be brief, since passing from integral models of Shimura varieties to correspondingRapoport-Zink formal schemes (which can be thought of as integral models of local
Shimura varieties) is, for the most part, routine. See for example [8, §
4] for anotherinstance of this parallel treatment.We consider a local Shimura datum (GSpin( V ) , b, { µ } ), with V over Q p and µ asabove and fix the level subgroup K to be the stabilizer of a vertex lattice Λ. Then a“Rapoport-Zink formal scheme” M b := M (GSpin( V ) ,µ,b,K ) over Spf( Z p ) is constructed in[7, § b is basic or GSpin( V ) is residually split. (By work of R. Zhou[24, Proposition 6.5], this assumption implies that Axiom (A) of [7, 5.3] is satisfied so theconstruction in loc. cit. applies, but it should not be necessary. The group is residuallysplit when ( V, h , i ) affords a basis as in one of the four cases of § b is basic. Then the formal scheme M b uniformizes the formal completionof the integral model S K ( G, X ) of a corresponding Shimura variety Sh K ( G, X ) along thebasic locus of its special fiber, see [7], [13].By its construction, M b supports a local model diagram(7.4.1) g M b M b \ M loc (Λ) ˆ π ˆ q of formal schemes over Spf( Z p ). (In this, ˆ π is a G -torsor, ˆ q is formally smooth, and \ M loc (Λ)denotes the formal p -adic completion of M loc (Λ), see [13] for details.) Our results nowimply that the blow-up M b, reg of M b along the discrete stratum is regular and has thesame ´etale local structure as described for S reg K ( G, X ) in Theorem 7.2.1 (c). In fact, wecan see that M b, reg can be used to uniformize the completion of S reg K ( G, X ) along itsbasic locus and that it affords a diagram(7.4.2) f M b, reg M b, reg \ M bl (Λ) ˆ π reg ˆ q reg with similar properties as above. References [1] F. Andreatta, E. Goren, B. Howard, K. Madapusi Pera,
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Dept. of Mathematics, Michigan State Univ., E. Lansing, MI 48824, USA
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