ON 2-ADIC LOCAL AND INTEGRAL MODELS OF SHIMURA VARIETIES By Jie Yang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics—Doctor of Philosophy 2025 ABSTRACT This thesis is about integral models of Shimura varieties with emphasis on the reduction at the prime p = 2. In the first part of the thesis, we construct local models for wildly ramified unitary similitude groups of odd dimension n ≥ 3 with special parahoric level structure and signature (n − 1, 1). We first give a lattice-theoretic description for parahoric subgroups using Bruhat- Tits theory in residue characteristic two, and apply them to define local models following the lead of Rapoport-Zink [RZ96] and Pappas-Rapoport [PR09]. In our case, there are two conjugacy classes of special parahoric subgroups. We show that the local models are smooth for the one class and normal, Cohen-Macaulay for the other class. We also prove that they represent the v-sheaf local models of Scholze-Weinstein. Under some additional assumptions, we obtain an explicit moduli interpretation of the local models. The second part of the thesis focuses on constructing integral models over p = 2 for some Shimura varieties of abelian type with parahoric level structure, extending the previous work of Kim-Madapusi [KM16] and Kisin, Pappas, and Zhou [KP18; KZ24; KPZ24]. For Shimura varieties of Hodge type, we show that our integral models are canonical in the sense of Pappas-Rapoport [PR24]. Copyright by JIE YANG 2025 ACKNOWLEDGEMENTS Completing this thesis has been a long and challenging journey, and I could not have done it without the support, guidance, and encouragement of so many people along the way. I would like to take this opportunity to thank everyone who helped make this thesis possible. First and foremost, I would like to express my profound gratitude to my advisor, Professor G. Pappas, for his unwavering support, patience, and guidance throughout my doctoral studies. His mentorship and constant encouragement have been invaluable in shaping this thesis and deeply influenced my growth as a mathematician. I would also like to extend my sincere thanks to professors R. Kulkarni, A. Levin, I. Rapinchuk, P. Wake. Their courses and our discussion opened my eyes to many new ideas and inspired me to explore further. To my peers at MSU, S. Bhutani, P. ˇCoupek, C. Guan, K. Huang, B. Kong, A. Oswal, P. Qi, A. Roy, J. Ruiter, A. Sathyanarayana, E. Sgallova, Y. Shen, Z. Xiao, S. Xu, thanks for creating a stimulating and supportive environment. The countless discussions, both formal and informal, have been a source of inspiration and motivation. Your friendship has been a source of joy and comfort. I also want to thank my roommate L. Shen for these years of friendship and support. My heartfelt thanks go to my family for their unconditional love and encouragement throughout this journey. Thank you for always being there for me. Your constant support has been my anchor during the most challenging times. This dissertation represents not only my efforts but also the collective support, guidance, and encouragement of many individuals. To everyone who has been part of my academic and personal growth, thank you from the bottom of my heart! iv TABLE OF CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 2 BRUHAT-TITS THEORY FOR UNITARY GROUPS IN RESIDUE CHARACTERISTIC TWO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notations 2.2 Bruhat-Tits buildings in terms of norms . . . . . . . . . . . . . . . . . . 2.3 Bruhat-Tits buildings in terms of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Parahoric subgroups and lattices CHAPTER 3 WILDLY RAMIFIED ODD UNITARY LOCAL MODELS . . . . 3.1 Quadratic extensions of 2-adic fields . . . . . . . . . . . . . . . . . . . . . 3.2 Hermitian quadratic modules and parahoric group schemes . . . . . . . . 3.3 Construction of the unitary local models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Comparison with the v-sheaf local models CHAPTER 4 THE CASE I = {0} . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The case I = {0} and (R-U) . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The case I = {0} and (R-P) . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 5 THE CASE I = {m} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The case I = {m} and (R-U) 5.2 The case I = {m} and (R-P) . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 15 18 18 18 25 29 32 32 35 39 42 46 46 73 79 79 86 CHAPTER 6 NORMAL FORMS OF HERMITIAN QUADRATIC MODULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Hermitian quadratic modules of type Λm . . . . . . . . . . . . . . . . . . 6.2 Hermitian quadratic modules of type Λ0 93 93 . . . . . . . . . . . . . . . . . . 104 CHAPTER 7 2-ADIC INTEGRAL MODELS OF SHIMURA VARIETIES . . . 110 7.1 p-divisible groups and Lau’s classification . . . . . . . . . . . . . . . . . . 110 7.2 Deformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Integral models of Shimura varieties of abelian type . . . . . . . . . . . . 135 7.3 . . . . . . . . . 155 7.4 Bruhat-Tits group schemes and tame Galois fixed points BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 v CHAPTER 1 INTRODUCTION 1.1 Background Shimura varieties, first reformulated in a modern framework by Deligne in his seminal papers [Del71; Del79], are higher dimensional generalizations of modular curves and play a central role in number theory. Let p be a prime number. Let Af denote the ring of finite ad`eles over Q, and Ap f denote the ring of prime-to-p finite ad`eles over Q. Let (G, X) be a Shimura datum, i.e., G is a reductive group over Q, X is a G(R)-conjugacy class of an algebraic group homomorphism h : S := ResC/RGm → GR, and (G, X) satisfies Deligne’s axioms ([Del79, (2.1.1.1)-(2.1.1.3)]). It follows from these axioms that each connected component of X is a hermitian symmetric domain. For a sufficiently small open compact subgroup K ⊂ G(Af ), the associated Shimura variety is the double coset space ShK(G, X) := G(Q)\X × G(Af )/K, which naturally carries the structure of a complex analytic space induced by X. By work of Baily and Borel, ShK(G, X) is in fact a quasi-projective smooth projective variety over C. Due to Shimura, Deligne, Borovoi, Milne, and others, the system Sh(G, X) := lim ←− K ShK(G, X) has a canonical model defined over a number field E, known as the reflex field, which only depends on the Shimura datum (G, X). The simplest Shimura varieties are the modular curves, which are given by the Shimura datum (GL2, H±). Here H± = C − R, the union of upper and lower half planes. Let v|p be a place of E and E be the completion of E at v. One active area of interest in the study of Shimura varieties is the construction of integral models. These are schemes over OE with generic fiber ShK(G, X)E. Integral models are useful in computing the Hasse-Weil 1 zeta function of a Shimura variety in terms of automorphic L-functions, which is part of the Langlands program. The construction of integral models is also a starting point for Kudla’s program relating special cycles on Shimura varieties and derivatives of Eisenstein series and L-functions. Let G be a Bruhat-Tits stabilizer group scheme (see §3.4) over Zp for GQp with neutral p := G◦(Zp). Suppose K◦ ⊂ G(Af ) is of the form component G◦. Set Kp := G(Zp) and K◦ K◦ = K◦ pKp, where Kp ⊂ G(Ap f ) is a sufficiently small open compact subgroup. When (G, X) is of PEL type, the corresponding Shimura varieties ShK(G, X) are (essentially) moduli spaces of abelian varieties with polarization, endomorphisms and level structure. Integral models of these Shimura varieties are studied in [Kot92, §5] and [RZ96, Chapter 6]. More generally, let (G, X) be a Shimura datum of abelian type, a large class that includes almost all cases where G is a classical group. Shimura varieties of abelian type are closely related to those of Hodge type, which can described as moduli spaces of abelian varieties equipped with families of Hodge tensors. If Kp is hyperspecial (which implies that GQp extends to a reductive group scheme G over Zp such that G(Zp) = Kp), Kisin [Kis10] and Kim-Madapusi [KM16] (when p = 2) constructed (smooth) canonical integral models over OE of ShK◦(G, X), which are uniquely characterized by Milne’s extension property. If p > 2, Kisin, Pappas and Zhou [KZ24; KPZ24], following earlier work of Kisin-Pappas [KP18], constructed normal flat integral models over OE of ShK◦(G, X) with arbitrary parahoric level structure. Using Scholze’s theory of p-adic shtukas, Pappas-Rapoport [PR24] and Daniels [Dan23] made the following conjecture about the existence of the canonical integral model of ShK◦(G, X) with parahoric level structure for any Shimura datum (G, X). Conjecture 1.1.1 ([PR24, Conjecture 4.2.2],[Dan23, Conjecture 4.5]). There exists a unique system {SK◦}Kp of normal flat schemes over OE, extending {ShK◦(G, X)}Kp and equipped with a p-adic shtuka satisfying the axioms in loc. cit.. By [PR24, Theorem 1.3.2], Conjecture 1.1.1 holds when (G, X) is of Hodge type and K◦ p is a stabilizer parahoric subgroup (i.e., Kp = K◦ p). Assuming the existence of SK◦ 2 as in Conjecture 1.1.1, Pappas and Rapoport also conjectured (at least when G satisfies the blanket assumption in [PR24, §4.1]) that SK◦ fits into a scheme-theoretic local model diagram. Specifically, there should exist a diagram of OE-schemes SK◦ π(cid:111) (cid:102)SK◦ q (cid:47) Mloc G◦,µh , where µh denotes the geometric cocharacter of GQp corresponding to the Hodge cocharacter attached to (G, X), the OE-scheme Mloc G◦,µh denotes the scheme local model used in [PR24, §4.9.2] (see also Theorem 3.4.4), π is a G◦-torsor, and q is G◦-equivariant and smooth of relative dimension dim G, such that the compatibility conditions in [PR24, Definition 4.9.1] are satisfied. For the current status of Conjecture 1.1.1, we refer readers to Daniels-van Hoften-Kim-Zhang [DvHKZ24] and Daniels-Youcis [DY24], which build upon the work of Kisin, Pappas and Zhou [KP18; KZ24; KPZ24]. Local models are certain flat projective schemes over the p-adic integers which are ex- pected to model the singularities of the integral models of Shimura varieties. Rapoport and Zink studied local models for Shimura varieties of PEL type with parahoric level structure at p in [RZ96]. Their local models were later called naive local models, since they are not always flat if the corresponding reductive group is ramified at p as pointed out in [Pap00, §4]. The construction of the naive local models relies on the lattice-theoretic description of parahoric subgroups, which is significantly more involved if p = 2 and the group is ramified. A more general approach is given in [PZ13] (see also a variant in [HPR20]) which constructs (flat) local models attached to a local model triple (G, {µ} , G), where G is a tamely ramified connected reductive group over a p-adic field L, {µ} is a geometric conjugacy class of cochar- acters of G with reflex field E, and G is a parahoric group scheme over OL with generic fiber G. Subsequent works [Lev16; Lou23; FHLR22] allow us to define local models for all triples (G, {µ} , G) excluding the case that p = 2 and Gad contains, as an ˘L-factor, a wildly ramified unitary group of odd dimension. Here ˘L denotes the completion of the maximal unramified extension of L in a fixed algebraic closure of L. These constructions a priori depend on certain auxiliary choices. 3 (cid:111) (cid:47) Another construction of local models is proposed in [SW20] using v-sheaves. The advan- tage is that this approach is canonical (without any auxiliary choices) and applies to arbi- trary triples (G, {µ} , G), even for wildly ramified groups G and p = 2. It has been proven in [AGLR22; GL24] that when {µ} is minuscule, the v-sheaf local models are representable by flat normal projective schemes Mloc G,µ over OE with reduced special fibers. Roughly, the local model Mloc G,µ is constructed as the weak normalization of certain orbit closure inside a Beilinson-Drinfeld type affine Grassmannian, extending the construction of Pappas and Zhu in [PZ13]. Excluding the case that p = 2 and Gad contains, as an ˘L-factor, a wildly ramified unitary group of odd dimension, one can show that the corresponding scheme local models are Cohen-Macaulay with Frobenius split special fibers. We refer the readers to [FHLR22, Remark 2.2] for some explanation on this exceptional case. A key aspect of understanding the special fibers of local models is their identification with a union of (semi-normalizations of) Schubert varieties in affine flag varieties. It is worth noting that the theory of local models also has applications in the study of Galois deformation rings, leading to strong re- sults in modularity lifting theorems, Breuil-M´ezard conjecture, etc. See for example [Kis09; LLHLM23]. In the present thesis, we study the local and integral models of Shimura varieties over p = 2. Now we explain the main results of our work. 1.2 Main results 1.2.1 2-adic local models The first part of the thesis focuses on the 2-adic local models for unitary similitude groups of odd dimension n ≥ 3 with special parahoric level structure when the signature is (n−1, 1). Let F0/Q2 be a finite extension and F be a (wildly) ramified quadratic extension of F0. For any x ∈ F , we write x for the Galois conjugate of x in F . We can pick uniformizers π ∈ F and π0 ∈ F0 such that F/F0 falls into one of the following two distinct cases (see §3.1): 4 √ θ), where θ is a unit in OF0. The uniformizer π satisfies an Eisenstein (R-U) F = F0( equation where t = π + π ∈ OF0 satisfies π0|t|2. We have θ = 1 − 2π/t and θ = 1 − 4π0/t2. π2 − tπ + π0 = 0, √ (R-P) F = F0( √ π0), where π2 + π0 = 0. Let (V, h) be a hermitian space, where V is an F -vector space of dimension n = 2m+1 ≥ 3 and h : V × V → F is a non-degenerate hermitian form. In this Introduction, we will assume that h is split, i.e., there exists an F -basis (ei)1≤i≤n of V such that h(ei, ej) = δi,n+1−j for 1 ≤ i, j ≤ n. Let G := GU(V, h) denote the unitary similitude group over F0 attached to (V, h). Our first result is the lattice-theoretic description of parahoric subgroups of G(F0). Theorem 1.2.1 (Proposition 2.4.1). Let I be a non-empty subset of {0, 1, . . . , m}. Define Λi := OF ⟨π−1e1, . . . , π−1ei, ei+1, . . . , em+1, λem+2, . . . , λen⟩, for 0 ≤ i ≤ m, where λ = π/t in the (R-U) case and λ = 1/2 in the (R-P) case. Then the subgroup PI := {g ∈ G(F0) | gΛi = Λi, for i ∈ I} is a parahoric subgroup of G(F0). Furthermore, any parahoric subgroup of G(F0) is conjugate to PI for a unique I ⊂ {0, 1, . . . , m}. The conjugacy classes of special parahoric subgroups correspond to the sets I = {0} and {m}. The proof of Theorem 1.2.1 is based on Bruhat-Tits theory in (residue) characteristic two. Note that in our case, parahoric subgroups of G(F0) no longer correspond to self-dual lattice chains, which causes difficulties in the study of local models. Given a special parahoric subgroup of G(F0) corresponding to I = {0} or {m}, we define in §3.3 the naive local model Mnaive I of signature (n − 1, 1), which is an analogue of the naive unitary local model considered in [RZ96]. To explain the construction, we start with a crucial 5 but simple observation on the structure of the lattices Λi in Theorem 1.2.1. Set   t ε := in the (R-U) case,  2 in the (R-P) case. (1.2.1) The hermitian form h defines a symmetric F0-bilinear form s(−, −) : V × V → F0 and a quadratic form q : V → F0 via s(x, y) := ε−1 TrF/F0 h(x, y) and q(x) := 1 2 s(x, x), for x, y ∈ V . (1.2.2) Set L := ε−1OF0, which is an invertible OF0-module. Then for 0 ≤ i ≤ m, the forms in (1.2.2) induce the L -valued forms s : Λi × Λi −→ L and q : Λi −→ L . (1.2.3) The triple (Λi, q, L ) is an L -valued hermitian quadratic module over OF0 in the sense of Definition 3.2.1, which roughly means that the quadratic form q is compatible with the OF -action. For I = {0} or {m}, denote ΛI := Λ0 or Λm respectively. Let Λs I := {x ∈ V | s(x, ΛI) ⊂ OF0} be the dual lattice of ΛI with respect to the pairing s in (1.2.2). Then we have a perfect OF0-bilinear pairing ΛI × Λs I −→ OF0 (1.2.4) induced by the symmetric pairing in (1.2.2), and an inclusion of lattices ΛI (cid:44)→ αΛs I, where α :=   π/ε  1/ε if I = {0}, if I = {m}. We define the naive unitary local model Mnaive I to be the functor Mnaive I : (Sch/OF )op −→ Sets which sends an OF -scheme S to the set of OS-modules F such that 6 (1) (π-stability condition) F is an OF ⊗OF0 OS-submodule of ΛI ⊗OF0 OS and as an OS- module, it is a locally direct summand of rank n. (2) (Kottwitz condition) The action of π ⊗ 1 ∈ OF ⊗OF0 OS on F has characteristic polynomial det(T − π ⊗ 1 | F) = (T − π)(T − π)n−1. (3) Let F ⊥ be the orthogonal complement of F in Λs I ⊗OF0 OS with respect to the perfect pairing (ΛI ⊗OF0 OS) × (Λs I ⊗OF0 OS) → OS induced by the perfect pairing in (1.2.4). We require that the map ΛI ⊗OF0 αΛs I sends F to αF ⊥, where αF ⊥ denotes the image OS induced by ΛI (cid:44)→ αΛs OS → I ⊗OF0 of F ⊥ under the isomorphism α : Λs I ⊗OF0 OS ∼−→ αΛs I ⊗OF0 OS. (4) F is totally isotropic with respect to the pairing s : (ΛI ⊗OF0 OS) × (ΛI ⊗OF0 OS) → L ⊗OF0 OS induced by s in (1.2.3), i.e., s(F, F) = 0 in L ⊗OF0 OS. The moduli functor Mnaive I is representable by a closed OF -subscheme of the Grassmannian Gr(n, ΛI)OF . It turns out that Mnaive I is not flat over OF . We define, as in [PR09], the local model Mloc I to be the flat closure of the generic fiber in Mnaive I . By construction, we have a closed immersion Mloc I (cid:44)→ Mnaive I of projective schemes over OF whose generic fibers are isomorphic to the (n − 1)-dimensional projective space over F . We have the following results on further geometric properties of the scheme Mloc I . 7 Theorem 1.2.2. (1) If I = {0}, then Mloc {0} is flat projective of relative dimension n − 1 over OF , normal and Cohen-Macaulay with geometrically integral special fiber. More- over, Mloc {0} is smooth over OF on the complement of a single closed point. (2) If I = {m}, then Mloc {m} is smooth projective of relative dimension n − 1 over OF with geometrically integral special fiber. Let us explain the strategy of the proof of Theorem 1.2.2 in greater detail. For I = {0} or {m}, let HI denote the group scheme1 of similitude automorphisms of the hermitian quadratic module (Λm, q, L ) (resp. (Λ0, q, L , ϕ)), see Definition 3.2.2 and 3.2.3. Then HI acts naturally on Mnaive I , and hence on Mloc I . Let k denote the algebraic closure of the residue field of F . Using the results in Chapter 6, we can show that the (geometric) special fiber Mloc I ⊗OF k has two orbits under the action of HI ⊗OF0 k. One of the orbits consists of just one closed point. We call it the worst point of the local model. Using this, we are reduced to proving that there is an open affine subscheme of Mloc I containing the worst point and satisfying the geometric properties (normality, Cohen-Macaulayness, etc) as stated in Theorem 1.2.2. To get the desired open affine subscheme of Mloc I , we introduce a refinement MI, as a closed subfunctor, of the moduli functor Mnaive I such that Mloc I ⊂ MI ⊂ Mnaive I . It turns out that the underlying topological space of MI is equal to that of Mloc I . For a matrix A, we will write OF [A] for the polynomial ring over OF whose variables are entries of the matrix A. Viewing MI as a closed subscheme of the Grassmannian Gr(n, ΛI)OF , we can find an open affine subscheme UI of MI which contains the worst point and which is isomorphic to a closed subscheme of Spec OF [Z], where Z is an n × n matrix, such that the worst point is defined by Z = 0 and π = 0. Then we explicitly write down the affine coordinate ring of 1In Chapter 6, we prove that HI is smooth over OF0 and isomorphic to the parahoric group scheme attached to ΛI . 8 UI defined by matrix identities. From this, we obtain the affine coordinate ring of UI ∩ Mloc I by calculating the flat closure of UI. Theorem 1.2.3. Let Y (resp. X) be a 2m × 2m (resp. 2m × 1) matrix with variables as entries. Let H2m denote the 2m × 2m anti-diagonal unit matrix. There is an open affine subscheme Uloc I of Mloc I which contains the worst point and satisfies the following properties. (1) If I = {0}, then Uloc {0} is isomorphic to Spec (cid:16) Spec (cid:16) OF [Y |X] tr(H2mY ) ∧2(Y |X), Y − Y t, ( π π 2 OF [Y |X] ∧2(Y |X), Y − Y t, ( tr(H2mY ) 2 √ + π θ)Y + XX t (cid:17) , in the (R-U) case, − π)Y + XX t (cid:17) , in the (R-P) case. (We remark that under the relation Y − Y t = 0, the polynomial tr(H2mY ), which is the sum of the anti-diagonal entries of Y , is indeed divisible by 2 in OF [Y ].) (2) If I = {m}, then Uloc {m} is isomorphic to Spec (cid:16) OF [Y |X] ∧2(Y |X), Y − Y t, ( tr(H2mY ) t + √ θ)Y + XX t (cid:17), in the (R-U) case, Spec OF [X], in the (R-P) case. Using the above result, we reduce the proof of Theorem 1.2.2 to a purely commutative algebra problem. We need to show that the affine coordinate rings in Theorem 1.2.3 satisfy the geometric properties stated in Theorem 1.2.2. The hardest part is to show the Cohen- Macaulayness when I = {0}, where we use a converse version of the miracle flatness theorem. We refer to Lemma 4.1.16 for more details. We can also relate Mloc I to the v-sheaf local models considered in [SW20, §21.4] (see also §3.4). By the results in [AGLR22; FHLR22; GL24] (see Theorem 3.4.4), we already know that the v-sheaf local models in our case are representable by normal projective flat OF -schemes MI (denoted by Mloc G,µ in §3.4). 9 Theorem 1.2.4 (Theorem 3.4.5). The local model Mloc I is isomorphic to MI. As a corollary, our result gives a very explicit construction of MI and a more elementary proof of the representability of the v-sheaf local models in our setting. Remark 1.2.5. If F/F0 is of type (R-P), the arguments in [AGLR22] (see the paragraph after Theorem 1.1 in loc. cit. ) also imply that MI is Cohen-Macaulay. However, our methods can also deal with the (R-U) case, and we are able to give explicit local affine coordinate rings. It should be pointed out that it could be useful to provide an explicit moduli interpreta- tion of Mloc I . As a by-product of our analysis of Uloc I (see Lemma 4.1.13), we obtain such a description in a special case. Theorem 1.2.6. Suppose F/F0 is of type (R-U) and assume that the valuations of t and π0 are equal2. Then Mloc {0} represents the functor (Sch/OF )op −→ Sets which sends an OF -scheme S to the set of OS-modules F such that 3 LM1 (π-stability condition) F is an OF ⊗OF0 OS-submodule of Λ0 ⊗OF0 OS and as an OS- module, it is a locally direct summand of rank n. LM2 (Kottwitz condition) The action of π ⊗ 1 ∈ OF ⊗OF0 OS on F has characteristic poly- nomial det(T − π ⊗ 1 | F) = (T − π)(T − π)n−1. LM3 Let F ⊥ be the orthogonal complement in Λs 0 ⊗OF0 OS of F with respect to the perfect pairing (Λ0 ⊗OF0 OS) × (Λs 0 ⊗OF0 OS) → OS 2This holds if F0 is unramified over Q2, see some more discussion in Remark 4.1.14. 3As in [Smi15, Lemma 5.2, Remark 5.4], the conditions LM2 and LM5 are in fact implied by LM6. 10 induced by the perfect pairing in (1.2.4). We require that the map Λ0 ⊗OF0 π t Λs F ⊥ under the isomorphism π I sends F to π ∼−→ π OS induced by Λ0 (cid:44)→ π t F ⊥, where π t Λs t Λs t : Λs 0 ⊗OF0 0 ⊗OF0 0 ⊗OF0 OS. OS t F ⊥ denotes the image of OS → LM4 (Hyperbolicity condition) The quadratic form q : Λ0 ⊗OF0 OS → L ⊗OF0 OS induced by q : Λ0 → L satisfies q(F) = 0. LM5 (Wedge condition) The action of π ⊗ 1 − 1 ⊗ π ∈ OF ⊗OF0 OS on F satisfies ∧2(π ⊗ 1 − 1 ⊗ π | F) = 0. LM6 (Strengthened spin condition) The line ∧nF ⊂ W (Λ0) ⊗OF OS is contained in Im (cid:0)W (Λ0)n−1,1 −1 ⊗OF OS → W (Λ0) ⊗OF OS (cid:1) . (See §4.1.1.1 for the explanation of the notation in this condition.) 1.2.2 2-adic integral models The second part of the thesis focuses on the 2-adic models of Shimura varieties. Assume p = 2 and that (G, X) is a Shimura datum of abelian type. Let v|p be a place of E and E be the completion of E at v. Denote by OE,(v) the localization of OE at v. Denote by kE the residue field of E and by k the algebraic closure of kE. We will construct 2-adic integral models over OE,(v) for ShK◦(G, X) under one of the following assumptions: (A) (Gad, X ad) has no factor of type DH, GQp is unramified, and K◦ p is contained in some hyperspecial subgroup; (B) G = GU(n−1, 1) is the unitary similitude group over Q of signature (n−1, 1) for some odd integer n ≥ 3, GQp is (wildly) ramified, and K◦ p is a special parahoric subgroup. Theorem 1.2.7. Assume that either (A) or (B) holds. (1) The E-scheme ShK◦ p(G, X) := lim ←− Kp ShK◦ pKp(G, X) 11 admits a G(Ap f )-equivariant extension to a flat normal OE,(v)-scheme SK◦ p(G, X). Any sufficiently small Kp ⊂ G(Ap f ) acts freely on SK◦ p(G, X), and the quotient SK◦(G, X) := SKp(G, X)/Kp is a flat normal OE,(v)-scheme extending ShK◦(G, X). (2) For any discrete valuation ring R of mixed characteristic 0 and p, the map SK◦ p(G, X)(R) → SK◦ p(G, X)(R[1/p]) is a bijection. (3) There exists a diagram of OE-schemes (cid:102)S ad K◦ p π q SK◦ p(G, X)OE Mloc G◦,µh , where π is a G(Ap f )-equivariant Gad◦ Zp -equivariant, and for any suf- ficiently small Kp ⊂ G(Ap f ), the map (cid:102)S ad K◦ p induced by q is smooth of Zp -torsor, q is Gad◦ /Kp → Mloc G◦,µh relative dimension dim Gad. (4) If κ is a finite extension of kE and y ∈ SK◦ p(G, X)(κ), then there exists z ∈ Mloc G◦,µh (κ) such that we have an isomorphism of henselizations Oh S K◦ p (G,X),y ≃ Oh Mloc G◦,µh ,z. Here in (3), Gad◦ defined by G◦ using the map B(GQp, Qp) → B(Gad Zp denotes the parahoric group scheme over Zp with generic fiber Gad Qp, Qp, Qp) between extended Bruhat-Tits buildings, see §7.3.2. The proof of Theorem 1.2.7 will be given in §7.3.2.2 and §7.3.3.3. Remark 1.2.8. (1) When K◦ p is hyperspecial, Theorem 1.2.7 has been proved by Kim- Madapusi [KM16]. In loc. cit., (Gad, X ad) is allowed to have a factor of type DH. 12 (cid:121) (cid:121) (cid:35) (cid:35) (2) We expect that the results of van Hoften [vHof24] and Gleason-Lim-Xu [GLX22] can be extended to the 2-adic models constructed in this thesis. Let us give two interesting cases in which Theorem 1.2.7 can be applied to obtain integral models over Z(2) for ShK◦ 2K2(G, X) when K◦ 2 is a parahoric subgroup contained in some hyperspecial subgroup. Let F be a totally real number field which is unramified at primes over 2. (i) G = ResF/QGSpin(V, Q), where GSpin(V, Q) is the spin similitude group over F attached to a quadratic space (V, Q) of signature (n, 2) at each real place (assume GSpin(V, Q) is unramified over Fv, v|2) and X is (a product of) the space of oriented negative definite planes; (ii) G = ResF/Q GU, where GU is the unitary similitude group over F that is unramified over Fv, v|2. We note that this case is also known by [RSZ21, Appendix A]. As in [KP18, Corolary 0.3], Theorem 1.2.7 implies the following. Corollary 1.2.9. With the assumptions as in Theorem 1.2.7, the special fiber SK◦ p(G, X) ⊗ kE is reduced, and the strict henselizations of the local rings on SK◦ p(G, X) ⊗ kE have irre- ducible components which are normal and Cohen-Macaulay. If K◦ p is associated to a point x ∈ B(GQp, Qp) which is a special vertex in B(GQp, Qur p ), then the special fiber SK◦ p(G, X) ⊗ kE is normal and Cohen-Macaulay. We now explain the idea to prove Theorem 1.2.7. The overall strategy follows that of [KP18] and [KPZ24]. As in loc. cit., the crucial case is when (G, X) is of Hodge type. A key step in this case involves identifying the formal neighborhood of SK(G, X) with that of the local model Mloc G,µh . For p > 2, this identification is obtained in [KP18; KPZ24] by constructing a versal deformation of p-divisible groups (equipped with a family of crystalline tensors) over the formal neighborhood of the local model. The construction of this versal deformation uses Zink’s theory of Dieudonn´e displays that classify p-divisible groups. For 13 p = 2, we modify Zink’s theory by using Lau’s results from [Lau14], and obtain a similar deformation theory for 2-divisible groups. A technical requirement arising in this step is that we need to find a Hodge embedding ι : (G, X) (cid:44)→ (GSp(V, ψ), S±), where V is a Q-vector space of dimension 2g equipped with a perfect alternating pairing ψ, such that ιQp extends to a very good integral Hodge embedding (G, µh) (cid:44)→ (GL(Λ), µg), where Λ ⊂ VQp is a self-dual Zp-lattice with respect to ψ. The concept of very good integral Hodge embeddings was introduced in [KPZ24, §5.2] for p > 2, refining the notion of good integral Hodge embeddings in [KZ24, Definition 3.1.6]. We generalize the concept to the case p = 2 (see Definition 7.2.13). Roughly speaking, a good integral integral Hodge embedding is an integral Hodge embedding (cid:101)ι : (G, µh) (cid:44)→ (GL(Λ), µg) extending ιQp such that (cid:101)ι induces a closed immersion Mloc G,µh (cid:44)→ Mloc GL(Λ),µg ⊗Zp OE = Gr(g, Λ) ⊗Zp OE of local models, where Gr(g, Λ) denotes the Grassmannian of rank g subspaces of Λ. The key idea behind very good Hodge embeddings is that certain collection of tensors (sα) in the tensor algebra Λ⊗, cutting out G in GL(Λ), should satisfy a “horizontal” condition under the natural connection isomorphism. We refer to §7.2.2 for more details. For a good integral Hodge embedding (cid:101)ι, Kisin-Pappas-Zhou proved in [KPZ24, Proposition 5.3.1, Lemma 5.3.2] that this horizontality condition is satisfied in the following two cases (including for p = 2): (1) For any x ∈ Mloc G,µh (k), the image of the natural map (cid:8)f ∈ Mloc G,µh (k[[t]]) | f mod(t) = x(cid:9) → TxMloc G,µh spans, as a k-vector space, the tangent space TxMloc G,µh . 14 (2) The tensors (sα) ⊂ Λ⊗ are in Λ ⊗Zp Λ∨. Using this, they can produce sufficiently many very good Hodge embeddings when p > 2. When p = 2, it is in general difficult to find a very good integral Hodge embedding (cid:101)ι for a Shimura datum of Hodge type. In the present thesis, we establish the existence of very good Hodge embeddings under the assumption (A) or (B). For Case (A), by applying [KPZ24, Proposition 5.3.1, Lemma 5.3.2], we are reduced to presenting the stabilizer group scheme G as (ResOF /ZpH)Γ, where F/Qp is a tame Galois extension with Galois group Γ and H is a reductive group over OF . For Case (B), we directly prove that the tangent space of the local model Mloc G,µh at any closed point is spanned by formal curves (see Lemma 7.3.17), using the explicit description of the (local) coordinate rings of the unitary local models in the first part of the thesis. 1.3 Organization We now give an overview of the thesis. In Chapter 2, we discuss Bruhat-Tits theory for (odd) unitary groups in residue charac- teristic two. In particular, we describe the maxi-minorant norms (norme maximinorante in French) used in [BT87] in terms of graded lattice chains, and thus obtain a lattice-theoretic description of the Bruhat-Tits buildings of unitary groups. As a corollary, we deduce Theo- rem 1.2.1. In Chapter 3, we first discuss some basic facts about quadratic extensions of 2-adic fields. Then we equip the lattices Λi in Theorem 1.2.1 with the structure of hermitian quadratic modules. Using this, we define the naive local models Mnaive I and local models Mloc I . In §3.4, we review the Beilinson-Drinfeld Grassmannian (in mixed characteristic) and v-sheaf local models of Scholze-Weinstein. Assuming Theorem 1.2.2, we show that the local models in Theorem 1.2.2 represent the v-sheaf local models, thereby proving Theorem 1.2.4. In Chapter 4 and 5, we prove Theorem 1.2.2, 1.2.3 and 1.2.6. We address the (R-U) and (R-P) case separately, although the techniques are very similar. In each chapter, we introduce the refinement MI of Mnaive I by imposing certain linear algebraic conditions and 15 then explicitly write down the local affine coordinate rings. We then obtain Theorem 1.2.3 by computing the flat closure of these affine coordinate rings. Utilizing the group action on local models, we finish the proof of Theorem 1.2.2 and Theorem 1.2.6. In Chapter 6, we show that, under certain conditions, hermitian quadratic modules ´etale locally have a normal form up to similitude. Along the way, we prove in Theorem 6.1.13 and Theorem 6.2.8 that the similitude automorphism group scheme of Λm (resp. (Λ0, ϕ)) is affine smooth over OF0 and is isomorphic to the parahoric group scheme attached to Λm (resp. Λ0). The results in this chapter are used in Chapter 4 and 5. In Chapter 7, we construct 2-adic integral models of Shimura varieties of abelian type and prove Theorem 1.2.7. Very often we will refer the readers to corresponding arguments in [KP18; KPZ24] that are similar or can be directly extended to the case p = 2 without repeating the proofs. In §7.1, we review Lau’s results in [Lau14], which generalizes Zink’s theory of Dieudonn´e displays so that we can classify 2-divisible groups over 2-adic rings (see Theorem 7.1.14). A new feature of the theory of Dieudonn´e displays in the case p = 2 is the modified Verschiebung map for the Zink ring (see Lemma 7.1.2). In §7.1.2, we construct the natural “connection isomorphisms” for Dieudonn´e pairs when p = 2 (see Lemma 7.1.13), generalizing [KPZ24, Lemma 5.1.3] for p > 2. In §7.1.4, we compare Lau’s classification of p-divisible groups with Breuil-Kisin’s classification. This comparison is later used in §7.3.1.2 to construct (GW , µy)- adapted deformations of p-divisible groups in the sense of Definition 7.2.17. In §7.2, we apply Lau’s theory to construct a versal deformation of p-divisible groups, extending results from [KP18, §3] to the case p = 2. We also generalize the concept of very good Hodge embeddings, introduced in [KPZ24], to p = 2. This is used to construct versal deformations of p-divisible groups with crystalline tensors (see Proposition 7.2.16). In Proposition 7.2.18, we establish a criterion for determining when a deformation is (GW , µy)- adapted, extending [Zho20, Proposition 4.7] to p = 2. In §7.3, we apply results in §7.2 to construct 2-adic integral models of Shimura varieties 16 of abelian type under certain assumptions (see Theorem 7.3.9). The overall strategy follows that of [KP18; KPZ24]. We first treat the case of Shimura varieties of Hodge type and then extend to Shimura varieties of abelian type by finding suitable Hodge type lifts while closely following [KP18]. In §7.3.2.2 and §7.3.3.3, we complete the proof of Theorem 1.2.7 by verifying that the assumptions in Theorem 7.3.9 are satisfied in Case (A) or (B). In §7.4, we show that, for an unramified group G over a 2-adic field F , if a stabilizer group scheme G satisfies G(OF ) ⊂ H for some hyperspecial subgroup H of G(F ), then G can be written as the tame Galois fixed points of the Weil restriction of scalars of a reductive group scheme. This result is used in the construction of very good integral Hodge embeddings in Case (A). 17 CHAPTER 2 BRUHAT-TITS THEORY FOR UNITARY GROUPS IN RESIDUE CHARACTERISTIC TWO In this chapter, we discuss Bruhat-Tits theory for (odd) unitary groups in residue charac- teristic two. In particular, we describe the maxi-minorant norms (norme maximinorante in French) used in [BT87] in terms of graded lattice chains, and thus obtain a lattice-theoretic description of the Bruhat-Tits buildings of unitary groups. As a corollary, we deduce Theo- rem 1.2.1 in the Introduction. 2.1 Notations Let F0 be a finite extension of Q2. Let ω : F0 → Z ∪ {+∞} denote the normalized valuation on F0. Let F/F0 be a (wildly totally) ramified quadratic extension. The valuation ω uniquely extends to a valuation on F , which is still denoted by ω. Denote by σ the nontrivial element in Gal(F/F0). For x ∈ F , we will write xσ or x for the Galois conjugate of x in F . Let OF (resp. OF0) be the ring of integers of F (resp. F0) with uniformizer π (resp. π0). We assume NF/F0(π) = π0. Let k be the common residue field of F and F0. Let V be an F -vector space of dimension n = 2m + 1 ≥ 3 with a non-degenerate hermitian form h : V × V → F . We assume that there exists an F -basis (ei)1≤i≤n of V such that h(ei, ej) = δi,n+1−j for 1 ≤ i, j ≤ n. In this case, we will say that the hermitian form h is split, or (V, h) is a split hermitian space. (We remark that all results in Chapter 2 have analogous (simpler) counterparts when F0 is a finite extension of Qp for p > 2, see Remark 2.2.6 and 2.2.10.) 2.2 Bruhat-Tits buildings in terms of norms In this section, we would like to recall the description of Bruhat-Tits buildings of odd dimensional (quasi-split) unitary groups in residue characteristic two in terms of norms. The standard reference is [BT87]. There is a summary (in English) in [Lem09, §1]. See also [Tit79, Example 1.15, 2.10]. Let G := U(V, h) denote the unitary group over F0 attached to (V, h). Then there is an 18 embedding of (enlarged) buildings B(G, F0) (cid:44)→ B(GLF (V ), F ). Definition 2.2.1. A norm on V is a map α : V → R ∪ {+∞} such that for x, y ∈ V and λ ∈ F , we have α(x + y) ≥ inf {α(x), α(y)} , α(λx) = ω(λ) + α(x), and x = 0 ⇔ α(x) = +∞. Example 2.2.2. (1) Let V be a one dimensional F -vector space. Then any norm α on V is uniquely determined by its value of a non-zero element in V : for any 0 ̸= x ∈ V and λ ∈ F , we have α(λx) = ω(λ) + α(x). (2) Let V1 and V2 be two finite dimensional F -vector spaces. Let αi be a norm on Vi for i = 1, 2. The direct sum of α1 and α2 is defined as a norm α1 ⊕α2 : V1 ⊕V2 → R∪{+∞} via (α1 ⊕ α2)(x1 + x2) := inf {α1(x1), α2(x2)} , for xi ∈ Vi. Proposition 2.2.3 ([KP23, 15.1.11]). Let α be a norm on V . Then there exists a basis (ei)1≤i≤n of V and n real numbers ci for 1 ≤ i ≤ n such that n (cid:88) α( i=1 xiei) = inf 1≤i≤n {ω(xi) − ci} . In this case, we say (ei)1≤i≤n is a splitting basis of α, or α is split by (ei)1≤i≤n. Denote by N the set of all norms on V . Then N carries a natural GLF (V )(F )-action via (gα)(x) := α(g−1x), for g ∈ GLF (V )(F ) and x ∈ V. (2.2.1) For each F -basis (ei)1≤i≤n of V , we have a corresponding maximal F -split torus T of GLF (V ) whose F -points are diagonal matrices with respect to the basis (ei)1≤i≤n. The 19 cocharacter group X∗(T ) has a Z-basis (µi)1≤i≤n, where µi : Gm,F → T is a cocharacter characterized by µi(t)ej = t−δij ej, for t ∈ F × and 1 ≤ i, j ≤ n, (2.2.2) where δij is the Kronecker symbol. Fixing an origin, we may identify the apartment A ⊂ B(GLF (V ), F ) corresponding to T with X∗(T )R. Proposition 2.2.4 ([BT84b, 2.8, 2.11]). The map A = X∗(T )R −→ N (cid:32) n n (cid:88) ciµi (cid:55)→ (cid:88) i=1 i=1 xiei (cid:55)→ inf 1≤i≤n {ω(xi) − ci} , (cid:33) (2.2.3) where ci ∈ R, xi ∈ F and (cid:80)n i=1 xiei ∈ V , extends uniquely to an isomorphism of GLF (V )-sets B(GLF (V ), F ) ∼−→ N . Moreover, the image of X∗(T )R in N is the set of norms on V admitting (ei)1≤i≤n as a splitting basis. By Proposition 2.2.4, we can identify the building B(GLF (V ), F ) with the set N of norms on V . Next we will describe the image of the inclusion B(G, F0) (cid:44)→ B(GLF (V ), F ) = N in terms of maxi-minorant norms (norme maximinorante in French). Set Fσ := {λ − λσ | λ ∈ F }. Then Fσ is an F0-subspace of F and we denote by F/Fσ the quotient space. We can associate the hermitian form h with a map q : V → F/Fσ, called the pseudo-quadratic form in [BT87], defined by q(x) := 1 2 h(x, x) + Fσ, for x ∈ V. The valuation ω induces a quotient norm ω on the F0-vector space F/Fσ: ω(λ + Fσ) := sup {ω(λ + µ − µσ) | µ ∈ F } , for λ ∈ F . 20 Definition 2.2.5. Let α be a norm on V . We say α minorizes (minores in French) (h, q) if for all x, y ∈ V , α(x) + α(y) ≤ ω(h(x, y)) and α(x) ≤ 1 2 ω(q(x)). Following the terminology of [KP23, Remark 15.2.12], we say α is maxi-minorant (maximi- norante in French) for (h, q) if α minorizes (h, q) and α is maximal for this property. Denote by Nmm (⊂ N ) the set of maxi-minorant norms for (h, q) on V . One can easily check that Nmm carries a G(F0)-action via (2.2.1). Here we view G(F0) as a subgroup of GLF (V ). Remark 2.2.6. Let α be a norm on V . Set α∨(x) := inf y∈V {ω(h(x, y)) − α(y)} , for x ∈ V . Then α∨ is also a norm on V , called the dual norm of α. We say α is self-dual if α = α∨. If F has odd residue characteristic, then by [BT87, 2.16], the norm α ∈ Nmm if and only if α is self-dual. Note that for x ∈ V , we have q(x) = 1 2 h(x, x) + Fσ = { 1 2 = {λh(x, x) | λ ∈ F, λ + λσ = 1} ∈ F/Fσ. h(x, x) + µ − µσ | µ ∈ F } Therefore, Set ω(q(x)) = sup {ω(λh(x, x)) | λ ∈ F, λ + λσ = 1} = ω(h(x, x)) + sup {ω(λ) | λ ∈ F, λ + λσ = 1} . δ := sup {ω(λ) | λ ∈ F, λ + λσ = 1} . (2.2.4) We obtain that α minores (h, q) if and only if for x, y ∈ V , we have α(x) + α(y) ≤ ω(h(x, y)) and α(x) ≤ 1 2 ω(h(x, x)) + 1 2 δ. 21 Definition 2.2.7. Let (V, h) be a (split) hermitian F -vector space of dimension n as in §2.1. (1) A Witt decomposition of V is a decomposition V = V− ⊕ V0 ⊕ V+ such that V− and V+ are two maximal isotropic subspaces of V , and V0 is the orthogonal complement of V− ⊕ V+ with respect to h. As we assume h is split, we have dimF V− = dimF V+ = m and dimF V0 = 1. (2) For any F -basis (ei)1≤i≤n of V , we put V− := spanF {e1, . . . , em} , V0 := spanF {em+1} , V+ := spanF {em+2, . . . , en} . We say (ei)1≤i≤n induces a Witt decomposition of V if V− ⊕ V0 ⊕ V+ is a Witt decom- position of V and h(ei, ej) = δi,n+1−j for 1 ≤ i, j ≤ n. Let (ei)1≤i≤n be a basis of V inducing a Witt decomposition. Such a basis defines a maximal F0-split torus S of G whose F0-points are given by    g ∈ G(F0) ⊂ GLF (V )(F ) gei = xiei and xixn+1−i = xm+1 = 1   for some xi ∈ F0 and 1 ≤ i ≤ n  . The centralizer of S in G ⊗F0 F ≃ GLF (V ) is T . For m + 2 ≤ i ≤ n, let λi : Gm,F0 → S be the cocharacter of S defined by λi(t)ei = t−1ei, λi(t)en+1−i = ten+1−i, and λi(t)ej = ej for t ∈ F × 0 and j ̸= i, n + 1 − i. (2.2.5) Then the set (λi)m+2≤i≤n forms a Z-basis of X∗(S). Fixing an origin, we may identify the apartment A(G, S) of B(G, F0) corresponding to S with X∗(S)R. Then we have the following proposition. Proposition 2.2.8. The map X∗(S)R −→ Nmm n (cid:88) ciλi (cid:55)→ (cid:32) n (cid:88) i=m+2 i=1 xiei (cid:55)→ inf{ω(xi) − ci, ω(xm+1) + 1 2 δ | 1 ≤ i ≤ n and i ̸= m + 1} , (2.2.6) (cid:33) 22 where ci := −cn+1−i if 1 ≤ i ≤ m, extends uniquely to an isomorphism of G(F0)-sets B(G, F0) → Nmm. The image of X∗(S)R in Nmm is the set of maxi-minorant norms admitting (ei)1≤i≤n as a splitting basis. Moreover, a norm α ∈ Nmm is special, i.e., α corresponds to a special point in B(G, F0), if and only if there is a basis (fi)1≤i≤n of V inducing a Witt decomposition and a constant C ∈ 1 4 n (cid:88) α( i=1 Z such that for xi ∈ F , we have xifi) = inf{ω(xi) − C, ω(xj) + C, ω(xm+1) + 1 2 δ | 1 ≤ i < m + 1 and m + 1 < j ≤ n}. Proof. See [BT87, 2.9, 2.12] and [Tit79, Example 2.10]. Corollary 2.2.9. Let α ∈ N . Then α ∈ Nmm if and only if there exists a basis (fi)1≤i≤n of V inducing a Witt decomposition V = V− ⊕ V0 ⊕ V+ such that α = α± ⊕ α0, where α± is a self-dual norm on V− ⊕ V+ split by the basis (fi)i̸=m+1, and α0 is the unique norm on V0 with α(fm+1) = 1 2δ. Proof. (⇒) We can view X∗(S)R as a subset of Nmm via the map (2.2.6). Using the G(F0)- action, we may assume α lies in X∗(S)R, say α = (cid:80)n i=m+2 ciλi ∈ X∗(S)R for ci ∈ R. Then we take (fi) to be (ei), which induces a Witt decomposition V = V− ⊕ V0 ⊕ V+. Define the norm α± on V− ⊕ V+ by V− ⊕ V+ −→ R ∪ {+∞} (cid:88) 1≤i≤n,i̸=m+1 xifi (cid:55)→ inf {ω(xi) − ci | 1 ≤ i ≤ n and i ̸= m + 1} , (2.2.7) where we define ci := −cn+1−i for 1 ≤ i ≤ m. Clearly α± is split by (fi)i̸=m+1. As h(fi, fn+1−j) = δij and ci = −cn+1−i for 1 ≤ i, j ≤ n, we deduce that α± is self-dual by [KP23, Remark 15.2.7]. Moreover, from the expression of (2.2.6), we immediately see that α decomposes as α = α± ⊕ α0. 23 (⇐) Under the assumptions, there exist n real numbers ci for 1 ≤ i ≤ n such that cn+1−i = −ci and α± is given by the norm as in (2.2.7). Let S′ be the maximal F0-split torus in G corresponding to the basis (fi)1≤i≤n. Let (λ′ as in (2.2.5). Then α is the norm corresponding to the point (cid:80)n i)m+2≤i≤n be a Z-basis of X∗(S′) defined i=m+2 ciλ′ i ∈ X∗(S′)R via a similar map as in (2.2.6). In particular, α ∈ Nmm. Remark 2.2.10. Assume F has odd residue characteristic. Then δ = 0, and hence α0 is self- dual. Then the norm α± ⊕ α0 as in the Corollary 2.2.9 is self-dual. When F has odd residue characteristic, any self-dual norm admits a splitting basis inducing a Witt decomposition of V , see for example [KP23, Proposition 15.2.10]. Then we see again that α ∈ Nmm if and only α is self-dual. Remark 2.2.11. We can define a “twisted” Galois action of Gal(F/F0) on GLF (V )(F ) as follows: for g ∈ GLF (V )(F ), define σ(g) to be the element satisfying h(g−1x, y) = h(x, σ(g)y), for x, y ∈ V . Then we have G(F0) = GLF (V )(F )σ=1, the set of fixed points of σ. This twisted Galois action induces an involution on N = B(GLF (V ), F ) = B(G ⊗F0 F, F ), which is still denoted by σ. Next we give an explicit description of this involution. Let (ei)1≤i≤n be a basis inducing a Witt decomposition V = V− ⊕ V0 ⊕ V+. Let T be the induced maximal torus of GLF (V ). Let A(T ) ⊂ B(GLF (V ), F ) be the apartment corresponding to T . We can identify A(T ) with X∗(T )R through the injection (cf. (2.2.3)) X∗(T )R −→ N n (cid:88) i=1 ciµi (cid:55)→    n (cid:88) i=1 xiei (cid:55)→ inf   ω(xm+1) − cm+1 + 1 2δ, ω(xi) − ci for 1 ≤ i ≤ n and i ̸= m + 1    ,    where µi is defined as in (2.2.2), xi ∈ F and (cid:80)n i=1 xiei ∈ V . As G is quasi-split, we can pick a σ-stable point as the origin such that the twisted σ-action on A(T ) is transported by the twisted σ-action on X∗(T )R. For α ∈ N , there is a g ∈ GLF (V )(F ) such that gα ∈ X∗(T )R, 24 since GLF (V )(F ) acts transitively on the apartments of N . Then gα = α1 ⊕ (α0 + C), where α1 is a norm on V− ⊕ V+ admitting (ei)i̸=m+1 as a splitting basis, α0 is the norm on V0 as in the Corollary 2.2.9, and C ∈ R is a certain constant. The twisted σ-action on X∗(T )R implies that σ(α1 ⊕ (α0 + C)) = α∨ 1 ⊕ (α0 − C). Hence, we see that σ acts on α as σ(α) = σ(g−1) (α∨ 1 ⊕ (α0 − C)) . For α ∈ Nmm = B(G, F0), we may take g ∈ G(F0) and C = 0. Thus, we get an inclusion B(G, F0) (cid:44)→ B(GLF (V ), F )σ=1. The inclusion is strict: any norm of the form α1 ⊕α0, where α1 is a self-dual norm on V− ⊕V+ but not split by any basis of V−⊕V+ inducing a Witt decomposition, lies in B(GLF (V ), F )σ=1 but not in B(G, F0). Such a norm can only exist when the residue characteristic of F is two. For an explicit example, see Example 2.3.7. 2.3 Bruhat-Tits buildings in terms of lattices In this section, we will translate the results in §2.2 into the language of lattices, which is more useful in the theory of local models. Definition 2.3.1. Let V be a finite dimensional F -vector space. (1) A lattice L in V is a finitely generated OF -submodule of V such that L ⊗OF F = V . (2) A (periodic) lattice chain of V is a non-empty set L• of lattices in V such that lattices in L• are totally ordered with respect to the inclusion relation, and λL ∈ L• for λ ∈ F × and L ∈ L•. (3) A graded lattice chain is a pair (L•, c), where L• is a lattice chain of V and c : L• → R is a strictly decreasing function such that for any λ ∈ F and L ∈ L•, we have c(λL) = ω(λ) + c(L). 25 The function c is called a grading of L•. (4) An F -basis (ei)1≤i≤n of V is called adapted to a graded lattice chain (L•, c) of V if for every L ∈ L•, there exist x1, . . . , xn ∈ F such that (xiei)1≤i≤n is an OF -basis of L. In this case, we also say (L•, c) is adapted to the basis (ei)1≤i≤n. Remark 2.3.2. Since L• is stable under homothety, the set L• is determined by a finite number of lattices satisfying πL0 ⊊ Lr−1 ⊊ Lr−2 ⊊ · · · ⊊ L1 ⊊ L0. We say (L0, L1, . . . , Lr−1) is a segment of L•, and the integer r is the rank of L•. Denote by GLC the set of graded lattice chains of V . There is a GLF (V )(F )-action on GLC: for (L•, c) ∈ GLC and g ∈ GLF (V )(F ), define g(L•, c) := (gL•, gc), where gL• consists of lattices of the form gL for L ∈ L•, and (gc)(gL) := c(L) for L ∈ L•. Lemma 2.3.3. (1) There is a one-to-one correspondence between N and GLC. More pre- cisely, given α ∈ N , we can associate a graded lattice chain (Lα, cα), where Lα is the set of following lattices Lα,r = {x ∈ V | α(x) ≥ r} , for r ∈ R, and the grading cα is defined by cα(Lα,r) = inf x∈Lα,r α(x). Conversely, given a graded lattice chain (L•, c) ∈ GLC, we can associate a norm α(L•,c)(x) := sup {c(L) | x ∈ L and L ∈ L•} . We say the norm α and the graded lattice chain (Lα, cα) in the above bijection corre- spond to each other. (2) The bijection in (1) is GLF (V )(F )-equivariant. 26 (3) Let (ei)1≤i≤n be a basis of V . Let (L•, c) be the graded lattice chain corresponding to a norm α via (1). Then (ei)1≤i≤n is adapted to (L•, c) if and only if (ei)1≤i≤n is a splitting basis of α. Proof. The proof of (1) and (3) can be found in [KP23, Proposition 15.1.21]. The assertion in (2) can be checked by direct computation. Using the above lemma, we can easily extend operations like direct sums or duality on norms to graded lattice chains. Lemma 2.3.4. (1) Let V and V ′ be two finite dimensional F -vector spaces. Let α and α′ be two norms on V and V ′ respectively. Let (L•, c) and (L′ •, c′) be graded lattice chains corresponding to α and α′ respectively. Then the graded lattice chain (L•, c) ⊕ (L′ •, c′) corresponding to α ⊕ α′ is a pair (L• ⊕ L′ •, c ⊕ c′), where L• ⊕ L′ • is the set of lattices of the form Lα,r ⊕ Lα′,r for r ∈ R, and (c ⊕ c′)(Lα,r ⊕ Lα′,r) := inf {c(Lα,r), c′(Lα′,r)} . (2) Let (L•, c) be the graded lattice chain corresponding to a norm α on V . Then the dual norm α∨ corresponds to the graded lattice chain (L∨ • , c∨), where L∨ • is the set of the lattices of the form L∨ := {x ∈ V | h(x, L) ∈ OF } for L ∈ L•, and c∨(L∨) := −c(L−) − 1, where L− is the smallest member of L• that properly contains L. Proof. The proof of (1) is straightforward. The proof of (2) can be found in [KP23, Fact 15.2.18]. We say (L•, c) is self-dual if (L•, c) = (L∨ • , c∨). Proposition 2.3.5. Let (L•, c) ∈ GLC. Then (L•, c) corresponds to a norm in Nmm if and only if there exists a basis (fi)1≤i≤n of V inducing a Witt decomposition V = V− ⊕ V0 ⊕ V+ 27 and (L•, c) decomposes as (L± • , c±) ⊕ (L0 •, c0), such that (L± • , c±) is a self-dual graded lattice chain of V− ⊕ V+ adapted to the basis (fi)i̸=m+1, and (L0 •, c0) is the graded lattice chain corresponding to the norm α0 on V0. Proof. This is a translation of Corollary 2.2.9 in view of the previous two lemmas. Remark 2.3.6. Let (L± • , c±) be a self-dual graded lattice chain adapted to the basis (fi)i̸=m+1 as in Proposition 2.3.5. Then for any L ∈ L± • , there exist xi ∈ F for i ̸= m + 1 such that (xifi)i̸=m+1 forms an OF -basis of L. As h(fi, fj) = δi,n+1−j, we see that L is isomorphic to an orthogonal sum of “hyperbolic planes” of the form H(i) (i ∈ Z). Here H(i) denotes a lattice in a two dimensional hermitian F -vector space (W, h) such that H(i) is OF ⟨x, y⟩ spanned by some x, y ∈ W with h(x, x) = h(y, y) = 0 and h(x, y) = πi. A lattice in W which is isomorphic to H(i) for some i ∈ Z is also called a hyperbolic lattice in the sense of [Kir17, §2]. For any lattice K in W , define the norm ideal n(K) of K to be the ideal in OF0 generated by h(x, x) for x ∈ K. Let K ∨ denote the dual lattice of K with respect to the hermitian form h on W . Then by [Kir17, §2] (see also [Jac62, Proposition 9.2 (a)]), any lattice K ⊂ W satisfying K = πiK ∨ (that is, K is πi-modular) and n(K) = n(H(i)) is isomorphic to H(i). Example 2.3.7. Let F0 = Q2 and F = Q2( √ 3). Pick uniformizers π = √ 3 − 1 ∈ F and π0 = −2 ∈ F0 so that π2 + 2π − 2 = 0. We have δ = sup {ω(λ) | λ ∈ F, λ + λσ = 1} = ω( π 2 ) = − 1 2 . Let (V, h) be a 3-dimensional (split) hermitian F -vector space. Let (ei)1≤i≤3 be a basis of V inducing a Witt decomposition V = V− ⊕ V0 ⊕ V+. Denote V± := V− ⊕ V+ = F ⟨e1, e3⟩. Set f1 := π−1(e1 + e3), f2 := e2, f3 := π−1(e1 − e3). Then L1 := OF ⟨f1, f3⟩ is a self-dual lattice in (V±, h). By [Jac62, Equation (9.1)], the self-dual hyperbolic plane H(0) in V± has norm ideal 2OF0. On the other hand, we have 28 n(L1) = OF0 by direct computation. In particular, the self-dual lattice L1 in (V±, h) is not isomorphic to H(0), and hence L1 is not adapted to any basis of V± induing a Witt decomposition. Now define L := L1 ⊕ OF f2. Then the graded lattice chain (L•, c), where L• := {πiL}i∈Z and c(πiL) := i 2 + δ 2 = i 2 − 1 4, defines a norm α : V −→ R ∪ {+∞} 3 (cid:88) i=1 xifi (cid:55)→ inf 1≤i≤3 {ω(xi) − 1 4 }. Then we see α lies in the fixed point set B(GLF (V ), F )σ=1 = N σ=1, but does not lie in Nmm. 2.4 Parahoric subgroups and lattices Let us keep the notations as in §2.2. In particular, the set (ei)1≤i≤n denotes a basis of V inducing a Witt decomposition V = V− ⊕ V0 ⊕ V+ and S denotes the corresponding maximal F0-split torus of G = U(V, h). Denote by (ai)m+2≤i≤n ∈ X ∗(S) the dual basis of (λi)m+2≤i≤n ∈ X∗(S). By the calculations in [Tit79, Example 1.15], the relative root system Φ = Φ(G, S) is {±ai ± aj | m + 2 ≤ i, j ≤ n, i ̸= j} ∪ {±ai, ±2ai | m + 2 ≤ i ≤ n} , and the affine root system Φa is {±ai ± aj + ∪{±ai + 1 2 δ + 1 2 1 2 Z | m + 2 ≤ i, j ≤ n, i ̸= j} Z | m + 2 ≤ i ≤ n} ∪ {±2ai + 1 2 + δ + Z | m + 2 ≤ i ≤ n}. Here δ is defined as in (2.2.4). These affine roots endow X∗(S)R with a simplicial structure. Following [Tit79, Example 3.11], we pick a chamber defined by the inequalities 1 2 δ < am+2 < · · · < an < 1 2 δ + 1 4 . 29 Then we obtain m + 1 vertices v0, . . . , vm in X∗(S)R such that for 0 ≤ i ≤ m, aj(vi) =    1 2δ if m + 2 ≤ j ≤ n − i, 1 2δ + 1 4 if n − i < j ≤ n. Now each vi defines a (maxi-minorant) norm, and hence a graded lattice chain, by Proposition 2.2.8 and Lemma 2.3.3. Let λ ∈ F be an element satisfying ω(λ) = δ. We shall see an explicit expression of λ in Lemma 3.2.4. Define Λi := OF ⟨π−1e1, . . . , π−1ei, ei+1, . . . , em+1, λem+2, . . . , λen⟩, (2.4.1) Λ′ i = OF ⟨e1, . . . , em, em+1, λem+2, . . . , λen−i, λπen+1−i, . . . , λπen⟩. Then the graded lattice chain corresponding to vi is of rank 2 and has a segment πΛi ⊂ Λ′ i ⊂ Λi. Let (cid:101)G = GU(V, h) be the unitary similitude group attached to the hermitian space (V, h). Let I be a non-empty subset of {0, 1, . . . , m}. Define (cid:110) PI := g ∈ (cid:101)G(F0) | gΛi = Λi, for i ∈ I (cid:111) . As in [PR09, 1.2.3], the Kottwitz map restricted to PI is trivial. In particular, we obtain that the (maximal) parahoric subgroup of (cid:101)G(F0) is the stabilizer of vi in (cid:101)G(F0), which also equals the stabilizer of Λi in (cid:101)G(F0) (as the stabilizer of Λ′ i is larger). More generally, we have the following proposition. Proposition 2.4.1. Denote (cid:101)G = GU(V, h). The subgroup PI is a parahoric subgroup of (cid:101)G(F0). Any parahoric subgroup of (cid:101)G(F0) is conjugate to a subgroup PI for a unique I ⊂ {0, 1, . . . , m}. The conjugacy classes of special parahoric subgroups correspond to the sets I = {0} and {m}. Proof. The results are similar to those in [PR08, §4] and [PR09, 1.2.3]. The first two as- sertions follow from the observation that (cid:101)G(F0) acts transitively on the chambers in the 30 building, and each I determines a (unique) facet in a chamber. The last assertion follows from the explicit expressions of the vertices vi and Proposition 2.2.8. 31 CHAPTER 3 WILDLY RAMIFIED ODD UNITARY LOCAL MODELS In this chapter, we construct local models for unitary similitude groups of odd dimension n ≥ 3 with special parahoric level structure when the signature is (n − 1, 1). 3.1 Quadratic extensions of 2-adic fields We start with some basic facts about quadratic extensions of 2-adic fields. The readers can find more details in [Jac62, §5] and [OMe00, §63]. Proposition 3.1.1. Let E be a finite extension of Q2 of degree d with ring of integer OE. Let e (resp. f ) be the ramification degree (resp. residue degree) of the field extension E/Q2. Note that d = ef . (1) The map sending a to E( √ a) defines a bijection between E×/(E×)2 and the set of isomorphism classes of field extensions of E of degree at most two. Furthermore, the cardinality of E×/(E×)2 is 22+d. In particular, we have 22+d − 1 quadratic extensions of E. (2) Let U be the unit group of OE and ϖ be a uniformizer of OE. For i ≥ 1, let Ui := 1 + ϖiOE be a subgroup of U . Then Ui is contained in U 2 for i ≥ 2e + 1 and the quotient U2e/(U2e ∩ U 2) has two elements corresponding to the trivial extension and the unramified quadratic extension of E. Note that U2e = 1 + 4OE. (3) Any non-trivial element in E×/(E×)2 has a representative of the following three forms: (i) a unit in U2e − U2e+1 (elements in U2e but not in U2e+1), (ii) a prime element in E, (iii) a unit in U2i−1 − U2i for some 1 ≤ i ≤ e. The corresponding quadratic extensions in (ii) and (iii) are ramified. Following [Jac62, §5], we will say the (ramified) quadratic extensions in (ii) and (iii) are of type (R-P) 32 and (R-U) respectively. There are 21+d quadratic extensions of E of type (R-P) and 21+d − 2 quadratic extensions of E of type (R-U). √ (4) Let E( θ)/E be a quadratic extension of type (R-U) for some unit θ ∈ U2i−1 − U2i for some 1 ≤ i ≤ e. Then there exists a prime π in E( √ θ) and a prime π0 in E satisfying π2 − tπ + π0 = 0 for some t ∈ OE with ord(t) = e + 1 − i, where ord denotes the normalized valuation on E. Proof. (1) The bijection is well-known from Kummer theory. The formula for the cardinality can be found in [OMe00, 63:9]. (2) See [OMe00, 63:1, 63:3]. (3) See [OMe00, 63:2]. The number of quadratic extensions of type (R-U) or (R-P) follows from the cardinality formula of E×/(E×)2 in (1). (4) Let ϖ be any prime in E. By assumption, θ = 1 + ϖ2i−1u for some unit u. Set √ π := √ θ 1 − ϖi−1 ∈ E( θ). Let π be the Galois conjugate of π. Then π + π = 2 ϖi−1 and ππ = −ϖu. Now take π0 to be −ϖu and t to be 2 ϖi−1 . Then t ∈ OE, as ord(t) = e + 1 − i ≥ 1, and π satisfies In particular, π is a prime element in E( √ θ). π2 − tπ + π0 = 0. Example 3.1.2. The (ramified) quadratic extension Q2( √ 3)/Q2 is of type (R-U), while √ Q2( 2)/Q2 is a quadratic extension of type (R-P). 33 Let us return to the setting in §2.1. By Proposition 3.1.1, we can find uniformizers π ∈ F and π0 ∈ F0 such that the quadratic extension F/F0 falls into one of the following two distinct cases1: (R-U) F = F0( √ θ), where θ is a unit in OF0. The uniformizer π satisfies π2 − tπ + π0 = 0. Here t ∈ OF0 with π0|t|2 and ω(t) depends only on F . We have √ θ = 1 − 2π t and θ = 1 − 4π0 t2 . (R-P) F = F0( √ π0), where π2 + π0 = 0. Lemma 3.1.3. Let F, F0, π and π0 be as above. (1) Suppose F/F0 is of type (R-U). Then the inverse different of F/F0 is 1 t OF . (2) Suppose F/F0 is of type (R-P). Then the inverse different of F/F0 is 1 2π OF . Proof. As π satisfies an Eisenstein polynomial f , by [Ser13, Chapter III, §6, Corollary 2] and [Ser13, Chapter I, §6, Proposition 18], we obtain that OF = OF0[π] and the inverse different of F/F0 is given by More precisely, δ−1 F/F0 = 1 f ′(π) OF . (1) when F/F0 is of type (R-U), then f (T ) = T 2 − tT + π0 and δ−1 F/F0 as t|2. = 1 2π−tOF = 1 t OF , (2) when F/F0 is of type (R-P), then f (T ) = T 2 + π0 and δ−1 F/F0 = 1 2π OF . 1When F0/Q2 is an unramified finite extension, there is a description in [Cho16, §2A] of these two cases in terms of the ramification groups of Gal(F/F0). 34 3.2 Hermitian quadratic modules and parahoric group schemes In this section, we define hermitian quadratic modules following [Ans18, §9] and relate them to parahoric group schemes. Let R be an OF0-algebra. The non-trivial Galois involution on OF extends to a map OF ⊗OF0 R → OF ⊗OF0 R, x ⊗ r (cid:55)→ x ⊗ r for x ∈ OF and r ∈ R. We will also denote the map by a (cid:55)→ a for a ∈ OF ⊗OF0 R. The norm map on OF induces the map NF/F0 : OF ⊗OF0 R → R, a (cid:55)→ aa. Definition 3.2.1 ([Ans18, Definition 9.1]). Let R be an OF0-algebra. Let d ≥ 1 be an integer. Consider a triple (M, q, L ), where M is a locally free OF ⊗OF0 d, L is an invertible R-module, and q : M → L is an L -valued quadratic form. Let R-module of rank f : M × M → L denote the symmetric R-bilinear form sending (x, y) ∈ M × M to f (x, y) := q(x + y) − q(x) − q(y) ∈ L . We say the triple (M, q, L ) is a hermitian quadratic module of rank d over R if for any a ∈ OF ⊗OF0 R and any x, y ∈ M , we have q(ax) = NF/F0(a)q(x) and f (ax, y) = f (x, ay). (3.2.1) A quadratic form q : M → L satisfying (3.2.1) is called an L -valued hermitian quadratic form on M . Definition 3.2.2. Let (M1, q1, L1) and (M2, q2, L2) be two hermitian quadratic modules over an OF0-algebra R. A similitude isomorphism or simply similitude between (Mi, qi, Li) ∼−→ M2 is an isomorphism of for i = 1, 2 is a pair (φ, γ) of isomorphisms, where φ : M1 OF ⊗OF0 R-modules and γ : L1 ∼−→ L2 is an isomorphism of R-modules such that q2(φ(m1)) = γ(q1(m1)), for any m1 ∈ M1. 35 We will write Sim ((M1, q1, L1), (M2, q2, L2)) , or simply Sim (M1, M2), (3.2.2) for the functor over R which sends an R-algebra S to the set Sim(M1 ⊗R S, M2 ⊗R S) of similitude isomorphisms between (Mi ⊗R S, qi ⊗R S, Li ⊗R S) for i = 1, 2. In the case (M1, q1, L1) = (M2, q2, L2), we will write Sim(M1, q1, L1), or simply Sim(M1), (3.2.3) for Sim ((M1, q1, L1), (M2, q2, L2)). This is in fact a group functor, and represented by an affine group scheme of finite type over R. Definition 3.2.3. Let R be an OF0-algebra. Denote by CR the category of quadruples (M, q, L , ϕ) such that (M, q, L ) is a hermitian quadratic module over R and ϕ is an R- bilinear form ϕ : M × M → L such that for x, y ∈ M , we have ϕ(x, πy) = q(x + y) − q(x) − q(y), ϕ(πx, y) = ϕ(x, πy), ϕ(x, y) = ϕ (cid:19) y, x , ϕ(x, x) = (cid:18) π π t π0 q(x). (3.2.4) Here t := π + π. In particular, t = 0 if F/F0 is of type (R-P). We will say an object (M, q, L , ϕ) ∈ CR is a hermitian quadratic module with ϕ, or simply a hermitian quadratic module. Let (Mi, qi, Li, ϕi) ∈ CR for i = 1, 2. A similitude isomorphism preserving ϕ between (Mi, qi, Li, ϕi) is a pair (φ, γ) of isomorphisms such that (φ, γ) is a similitude between (Mi, qi, Li), and for m1, m′ 1 ∈ M1, we have ϕ2(φ(m1), φ(m′ 1)) = γ(ϕ1(m1, m′ 1)). We will use a similar notation as in (3.2.2) and (3.2.3) to denote the functor of similitudes preserving ϕ between two hermitian quadratic modules in CR. Recall that we defined in §2.4 lattices Λi for 0 ≤ i ≤ m via Λi = OF ⟨π−1e1, . . . , π−1ei, ei+1, . . . , em+1, λem+2, . . . , λen⟩, 36 where λ is an element in F such that ω(λ) = δ = sup x∈F {ω(x) | x + x = 1} . The stabilizer of Λi is a maximal parahoric subgroup of GU(V, h). We sometimes call these lattices Λi standard lattices. A more explicit expression of λ is given as follows. Lemma 3.2.4. (1) Suppose F/F0 is of type (R-U). Then we may take λ = π t . (2) Suppose F/F0 is of type (R-P). Then we may take λ = 1 2. Proof. (1) By construction, we have ω(λ) ≥ ω( π t ) > ω( 1 2). Write λ = a + b √ θ ∈ F for some a, b ∈ F0. Then λ = a − b If ω( 1 2) ̸= ω(b √ θ), then √ θ. Since λ + λ = 1, we get a = 1 2 and ω(λ) = ω( √ θ). + b 1 2 ω(λ) = min{ω( √ ), ω(b 1 2 θ)} ≤ ω( 1 2 ), which is a contradiction. Therefore, we may assume ω(b) = ω(b write b = 1 2u for some unit u in OF0. Then √ θ) = ω( 1 2). Then we can ω(λ) = ω( 1 2 + 1 2 u(1 − 2π t )) = ω(( 1 2 + u) − π t u). Since ω(π) = 1/2, we have ω( 1 2 + u) ̸= ω( π t u). It implies that ω(λ) = min{ω( 1 2 + u), ω( π t )} ≤ ω( π t ) Thus, we have ω(λ) = ω( π t ). (2) By construction, we have ω(λ) ≥ ω( 1 2). Write λ = a + bπ ∈ F for some a, b ∈ F0. Then λ = a − bπ. Since λ + λ = 1, we have a = 1 2. As ω( 1 2) is even and ω(bπ) is odd, they cannot be equal. We get ω(λ) = ω( 1 2 + bπ) = min{ω( 1 2 ), ω(bπ)} ≤ ω( 1 2 ). Thus, we have ω(λ) = ω( 1 2). 37 Set ε :=    t in the (R-U) case, 2 in the (R-P) case. The hermitian form h defines a symmetric F0-bilinear form s(−, −) : V × V → F0 and a quadratic form q : V → F0 via s(x, y) := ε−1 TrF/F0 h(x, y) and q(x) := 1 2 s(x, x), for x, y ∈ V . Set L := ε−1OF0, which is an invertible OF0-module. Then for 0 ≤ i ≤ m, we obtain induced forms s : Λi × Λi −→ L and q : Λi −→ L . (3.2.5) It is straightforward to verify the following lemma. Lemma 3.2.5. (1) For 0 ≤ i ≤ m, the triple (Λi, q, L ) forms an L -valued hermitian quadratic module of rank n over OF0 in the sense of Definition 3.2.1. (2) Define ϕ : Λ0 × Λ0 → ε−1OF0, (x, y) (cid:55)→ ε−1 TrF/F0 h(x, π−1y). Then (Λ0, q, L , ϕ) is a hermitian quadratic module with ϕ. Now we state two theorems on hermitian quadratic modules. The proofs will be given in Chapter 6. Theorem 3.2.6. The functor Sim(Λm) (resp. Sim(Λ0, ϕ)) is representable by an affine smooth group scheme over OF0 with generic fiber GU(V, h). Moreover, the scheme Sim(Λ) (resp. Sim(Λ0, ϕ)) is isomorphic to the parahoric group scheme attached to Λm (resp. Λ0). Proof. See Theorem 6.1.13 and 6.2.8, Corollary 6.1.14 and 6.2.9. 38 Theorem 3.2.7 (Theorem 6.1.12, 6.2.7). Let R be an OF0-algebra. Let (M, q, L ) (resp. (N, q, L , ϕ)) be a hermitian quadratic module over R of rank n. Assume that (M, q, L ) (resp. (N, q, L , ϕ)) is of type Λm (resp. Λ0) in the sense of Definition 6.1.8 (resp. Defi- nition 6.2.4). Then the hermitian quadratic module (M, q, L ) is ´etale locally isomorphic to (Λm, q, ε−1OF0) ⊗OF0 R (resp. (Λ0, q, ε−1OF0, ϕ) ⊗OF0 R) up to similitude. 3.3 Construction of the unitary local models 3.3.1 Naive local models Let I = {0} or {m}. Then I corresponds to a special parahoric subgroup of GU(V, h) by Proposition 2.4.1. Let ΛI denote the corresponding lattice, which is either Λ0 or Λm. Set Λh I := {x ∈ V | h(x, ΛI) ⊂ OF } , Λs I := {x ∈ V | s(x, ΛI) ⊂ OF0} . The symmetric pairing s on V induces a perfect OF0-bilinear pairing ΛI × Λs I → OF0, which is still denotes by s(−, −). By Lemma 3.1.3, one can check that Λs =    Λh in the (R-U) case, π−1Λh in the (R-P) case. Note that (3.3.1) (3.3.2) Λh 0 = OF ⟨λ −1 Λh m = OF ⟨λ −1 e1, . . . , λ e1, . . . , λ −1 −1 em, em+1, em+2, . . . , en⟩, em, em+1, πem+2, . . . , πen⟩. Using (3.3.2) and Lemma 3.2.4, we have Λs 0 (cid:44)→ Λ0 (cid:44)→ π t Λs 0, in the (R-U) case, πΛs 0 (cid:44)→ Λ0 (cid:44)→ π 2 Λs 0, in the (R-P) case, and Λs m (cid:44)→ Λm (cid:44)→ 1 t Λs m, in the (R-U) case, πΛs m (cid:44)→ Λm (cid:44)→ 1 2 Λs m, in the (R-P) case. 39 In summary, we have an inclusion of lattices ΛI (cid:44)→ αΛs I, where α :=   π/ε  1/ε if I = {0}, if I = {m}. We define the naive unitary local model of type I (and of signature (n − 1, 1)) as follows. Definition 3.3.1. Let Mnaive I be the functor Mnaive I : (Sch/OF )op −→ Sets which sends an OF -scheme S to the set of OS-modules F such that (1) F is an OF ⊗OF0 OS-submodule of ΛI ⊗OF0 OS and as an OS-module, it is a locally direct summand of rank n. (2) (Kottwitz condition) The action of π ⊗ 1 ∈ OF ⊗OF0 OS on F has characteristic polynomial det(T − π ⊗ 1 | F) = (T − π)(T − π)n−1. (3) Let F ⊥ be the orthogonal complement of F in Λs I ⊗OF0 OS with respect to the perfect pairing (ΛI ⊗OF0 OS) × (Λs I ⊗OF0 OS) → OS induced by (3.3.1). We require that the map ΛI ⊗OF0 the inclusion ΛI (cid:44)→ αΛs I sends F to αF ⊥, where αF ⊥ denotes the image of F ⊥ under OS → αΛs I ⊗OF0 OS induced by the isomorphism α : Λs I ⊗OF0 OS ∼−→ αΛs I ⊗OF0 OS. (4) F is totally isotropic with respect to the form (ΛI ⊗OF0 induced by s in (3.2.5), i.e., s(F, F) = 0 in L ⊗OF0 OS. OS)×(ΛI ⊗OF0 OS) → L ⊗OF0 OS Lemma 3.3.2. The functor Mnaive I is representable by a projective scheme over OF and the generic fiber is isomorphic to the (n − 1)-dimensional projective space Pn−1 F over F . 40 Proof. This is similar to [PR09, 1.5.3]. The representability follows by identifying Mnaive I with a closed subscheme of the Grassmannian Gr(n, ΛI)OF classifying locally direct summands of rank n in ΛI. As π ⊗ 1 is a semisimple operator on V ⊗F0 F , we have V ⊗F0 F = Vπ ⊕ Vπ, where Vπ (resp. Vπ) denotes the π-eigenspace (resp. π-eigenspace) of π ⊗1. Both eigenspaces Vπ and Vπ are n-dimensional F -vector spaces. We claim that Vπ is totally isotropic for the induced symmetric pairing, which is still denoted by s(−, −), on V ⊗F0 F . Indeed, for any x, y ∈ Vπ, we have (π ⊗ 1)x = πx and (π ⊗ 1)y = πy. Then s(x, y) = π−2s(πx, πy) = π−2s ((π ⊗ 1)x, (π ⊗ 1)y) = (π0/π2)s(x, y). So s(x, y) = 0. Similarly, we obtain that Vπ is also totally isotropic. It implies that the induced pairing s(−, −) : Vπ × Vπ → F (3.3.3) is perfect. Let Pn−1 F be the projective space associated with Vπ. For any F -algebra R, define φ : Mnaive I (R) −→ Pn−1 F (R), F (cid:55)→ ker(π ⊗ 1 − 1 ⊗ π | F). By the Kottwitz condition for F, this is a well-defined map. Conversely, let G ∈ Pn−1 F (R), i.e., G is a direct summand of rank one of Vπ ⊗F R. The perfect pairing (3.3.3) gives a (unique) direct summand G′ of rank n − 1 of Vπ ⊗F R such that s(G, G′) = 0. Set F := G ⊕ G′ ⊂ V ⊗F0 R. Then by our construction, we have F ∈ Mnaive I (R). This process defines an inverse map of φ. In particular, φ is bijective, and hence the generic fiber of Mnaive I is isomorphic to Pn−1 F . Similar arguments as in [Pap00, Proposition 3.8] on the dimension of the special fiber of Mnaive I show that Mnaive I is not flat over OF . 41 3.3.2 Local models Definition 3.3.3. The local model Mloc I is defined to be the (flat) Zariski closure of the generic fiber of Mnaive I in Mnaive I . By construction, the scheme Mloc I is a flat projective scheme of (relative) dimension n − 1 over OF . In Chapter 4-5, we will prove Theorem 1.2.2-1.2.6 in the Introduction. The proof of Theorem 1.2.2 and 1.2.3 will be divided into four cases, depending on the index set I and the ramification types of F/F0, see §4-5.2. In the course of the proof, we also establish Theorem 1.2.6. 3.4 Comparison with the v-sheaf local models In this section, assuming Theorem 1.2.2 and 1.2.3, we relate the local model Mloc I for I = {0} or {m} to the v-sheaf local models considered in [SW20, §21.4] and [AGLR22]. We give a proof of Theorem 1.2.4. We first briefly introduce the v-sheaf local models in the sense of Scholze-Weinstein. Let G be any connected reductive group over a complete discretely valued field L/Qp, where p is any prime. Let B(G, L) denote the associated (extended) Bruhat-Tits building, which carries an action of G(L). For x ∈ B(G, L), the associated Bruhat-Tits stabilizer group scheme Gx, in the sense of [BT84a], is a smooth affine group scheme over OL such that the generic fiber of Gx is G and Gx(OL) is the stabilizer subgroup of x in G(L). By definition, the neutral component G◦ x is the parahoric group scheme associated to x. Recall that a smooth affine group scheme G over OL is quasi-parahoric if the neutral component of G is a parahoric group scheme and G◦ x(O ˘L) ⊂ G(O ˘L) ⊂ Gx(O ˘L) for some Bruhat-Tits stabilizer group scheme Gx. Here ˘L denotes the completion of the maximal unramified extension of L in the algebraic closure Q p of Qp. Definition 3.4.1. A local model triple over L is a triple (G, {µ} , G), where G is a connected reductive group over L, {µ} is the G(L)-conjugacy class of a minuscule cocharacter µ : G m,L → GL, and G is a quasi-parahoric group scheme for G. 42 We will often write (G, µ) (resp. (G, µ)) for (G, {µ} , G) (resp. (G, {µ})). A morphism of local model triples (G, µ) → (G′, µ′) is a group scheme homomorphism G → G′ taking {µ} to {µ′}. Let (G, {µ} , G) be a local model triple over L. Denote by E the reflex field of {µ}. Then we can form the Beilinson-Drinfeld Grassmannian GrG, which is a v-sheaf over OL. We have the following properties. Theorem 3.4.2. (1) The structure morphism GrG −→ Spd OL is ind-proper and ind- representable in spatial diamonds. The generic fiber of GrG can be naturally identified with the B+ dR-affine Grassmannian GrG. (2) If G (cid:44)→ H is a closed immersion of parahoric group schemes, then the induced morphism GrG → GrH is a closed immersion. Proof. See [SW20, Proposition 20.3.6, Proposition 20.5.4, Theorem 21.2.1], or [AGLR22, Theorem 4.9, Lemma 4.10]. Recall that the B+ dR-affine Grassmannian GrG is a union of (open) Schubert diamonds Gr◦ G,{µ} indexed by geometric conjugacy classes {µ} of cocharacters of G. Let GrG,{µ} denote the v-closure of Gr◦ G,{µ}. If {µ} is minuscule with reflex field E, then GrG,{µ} is representable by a projective scheme over E (see [SW20, Proposition 19.4.2]). More precisely, GrG,{µ} is the associated diamond of the flag variety FℓG,{µ} := G/P{µ}, where P{µ} := {g ∈ G | lim t→∞ µ(t)gµ(t)−1 exists}. is the parabolic subgroup associated to {µ}. Sometimes, we will write µ for {µ} for simplicity. Definition 3.4.3. Let GrG,OE be the base change of GrG. The v-sheaf local model Mv G,µ is the v-closure of GrG,µ inside GrG,OE . Recall that given a scheme X proper over OE, there is a functorially associated v-sheaf X ♢ over Spd OE. For details of the definition, we refer to [AGLR22, §2.2]. We have the following representability result of the v-sheaf local models. 43 Theorem 3.4.4 (Scholze-Weinstein Conjecture). Assume {µ} is minuscule. Then there exists a unique (up to unique isomorphism) flat, projective and normal OE-scheme Mloc G,µ with a closed immersion Mloc♢ G,µ (cid:44)→ GrG ⊗OLOE prolonging Fℓ♢ G,µ ∼−→ GrG,µ ⊂ GrG ⊗LE. In particular, Mloc♢ G,µ = Mv G,µ. Proof. See [AGLR22, Theorem 1.1] and [GL24, Corollary 1.4]. We also have Mloc G,µ = Mloc G◦,µ by [SW20, Proposition 21.4.3]. By functoriality, any mor- phism (G, µ) → (G′, µ′) of local model triples induces a natural morphism Mloc G,µ → Mloc G′,µ′ of local models. Now we return to the situation in §2.1. In particular, we let G denote the unitary similitude group GU(V, h) over F0 attached to a split hermitian F/F0-vector space (V, h) of dimension n = 2m+1 ≥ 3, and there is an F -basis (ei)1≤i≤n of V such that h(ei, ej) = δi,n+1−j for 1 ≤ i, j ≤ n. Let G be the (special) parahoric group scheme corresponding to the index set I = {0} or {m}. Let T be the maximal torus of G consisting of diagonal matrices with respect to the basis (ei)1≤i≤n. Under the isomorphism GF ≃ GLn,F × Gm,F , we can identify X∗(T ) with Zn × Z. Let µ := µn−1,1 ∈ X∗(T ) be the (minuscule) cocharacter corresponding to (1, 0(n−1), 1) ∈ Zn × Z. We write 0(n−1) for a list of n − 1 copies of 0. Then the reflex field E of {µ} equals F . Let Mloc denote the local model Mloc I for I = {0} or {m} constructed in §3.3.2. Theorem 3.4.5. The scheme Mloc is isomorphic to Mloc G,µ in Theorem 3.4.4. Proof. We have shown that the scheme Mloc is normal, flat and projective over OF . By the uniqueness part of Theorem 3.4.4, it suffices to show that Mv G,µ = Mloc,♢. 44 By our concrete description of G in Corollary 6.1.14 and 6.2.9, we have a closed immersion G (cid:44)→ GL(Λ) ≃ GL2n (3.4.1) over OF0, prolonging the closed immersion G (cid:44)→ GLF0(V ) ≃ GL2n,F0, where Λ is either Λ0 or Λm depending on what G is. Let T ′ be the maximal torus of GL2n,F0 consisting of diagonal matrices. Then the map G (cid:44)→ GLF0(V ) transports {µn−1,1} to the geometric conjugacy class {µn} of cocharacters of T ′. Here, µn corresponds to (1(n), 0(n)) ∈ X∗(T ′) ≃ Z2n. By Theorem 3.4.2 (2), the closed immersion (3.4.1) induces a closed immersion Mv G,µ (cid:44)→ Mv GL2n,µn ⊗OF0 OF = Gr(n, 2n)♢ OF , and we may identify Mv G,µ with the v-closure of Fℓ♢ G,µ inside Gr(n, 2n)♢ OF . By Lemma 3.3.2, we can identify the generic fiber Mloc ⊗OF F with Pn−1 F ≃ FℓG,µ, and there exists a closed immersion FℓG,µ (cid:44)→ FℓGL2n,µn,F = Gr(n, 2n)F induced by the embedding G (cid:44)→ GLF0(V ). By our construction of Mloc, the scheme Mloc is the Zariski closure of FℓG,{µ} along FℓG,µ (cid:44)→ FℓGL2n,µn,F (cid:44)→ Gr(n, 2n)OF . Applying the diamond functor, we see that Mloc,♢ is the v-closure of Fℓ♢ . Hence, we G,µ inside Gr(n, 2n)♢ OF have Mv G,µ = Mloc,♢. This completes the proof of Theorem 1.2.4. Remark 3.4.6. The proof of the above theorem also gives another proof of the representability of the v-sheaf local model Mv G,µ in our setting. 45 CHAPTER 4 THE CASE I = {0} 4.1 The case I = {0} and (R-U) In this section, we will prove Theorem 1.2.2 in the case when I = {0} and the quadratic extension F/F0 is of (R-U) type. In particular, we have π2 − tπ + π0 = 0, where t ∈ OF0 with π0|t|2. Consider the following ordered OF0-basis of Λ0 and Λs 0: Λ0 : π t em+2, . . . , π t Λs 0 : em+2, . . . , en, en, e1, . . . , em, em+1, π0 t em+2, . . . , π0 t en, πe1, . . . , πem, πem+1, (4.1.1) t π e1, . . . , t π em, em+1, πem+2, . . . , πen, te1, . . . , tem, πem+1. (4.1.2) 4.1.1 A refinement of Mnaive {0} in the (R-U) case In this subsection, we will propose a refinement of the functor Mnaive {0} . We first recall the “strengthened spin condition” raised by Smithling in [Smi15]. 4.1.1.1 The strengthened spin condition Take g1, . . . , g2n to be the ordered F -basis e1 ⊗ 1 − πe1 ⊗ π−1, . . . , en ⊗ 1 − πen ⊗ π−1, πe1 ⊗ π t − e1 ⊗ π0 t , . . . , πen ⊗ π t − en ⊗ π0 t of V ⊗F0 F . Then with respect to the basis (gi)1≤i≤2n, the symmetric pairing s(−, −) ⊗F0 F on V ⊗F0 F is represented by the 2n × 2n matrix anti-diag(θ, . . . , θ). Recall θ = 1 − 4π0 t2 . One can easily check that • (gi)1≤i≤n is a basis for Vπ (the π-eigenspace of the operator π ⊗ 1 acting on V ⊗F0 F ), • (gi)n+1≤i≤2n is a basis for Vπ (the π-eigenspace of the operator π ⊗1 acting on V ⊗F0 F ). Take f1, . . . , f2n to be the ordered OF -basis e1 ⊗ 1, . . . , em+1 ⊗ 1, π t em+2 ⊗ 1, . . . , π t en ⊗ 1, 46 πe1 ⊗ 1, . . . , πem+1 ⊗ 1, π0 t em+2 ⊗ 1, . . . , π0 t en ⊗ 1 OF . This is the base change of the basis in (4.1.1), but in different order. We of Λ0 ⊗OF0 have (g1, . . . , g2n) = (f1, . . . , f2n)          Im+1 0 − 1 π Im+1 0 t π Im 0 − π0 t Im+1 0 0 −πIm π t Im+1 0 π2 π0 Im          . (4.1.3) 0 − t π2 Im 0 As in [Smi15], we use the following convenient notations: • For an integer i, we write i∨ := n + 1 − i, i∗ := 2n + 1 − i. For S ⊂ {1, . . . , 2n} of cardinality n, we write S∗ := {i∗ | i ∈ S} , S⊥ := {1, . . . , 2n} \S∗. Let σS be the permutation on {1, . . . , 2n} sending {1, . . . , n} to S in increasing order and sending {n + 1, . . . , 2n} to {1, . . . , 2n} \S in increasing order. Denote by sgn(σS) ∈ {±1} the sign of σS. • Set W := ∧n(V ⊗F0 F ). For S = {i1 < · · · < in} ⊂ {1, . . . , 2n} of cardinality n, we write eS := fi1 ∧ · · · ∧ fin ∈ W, similarly, gS := gi1 ∧ · · · ∧ gin ∈ W. Note that (eS){#S=n} (or (gS){#S=n}) is an F -basis of W . • Set W±1 := spanF {gS ± sgn(σS)gS⊥ | S ⊂ {1, . . . , 2n} and #S = n} . 47 This is a sub F -vector space of W . For any OF -lattice Λ in V ⊗F0 F , set W (Λ) := ∧n (cid:0)Λ ⊗OF0 OF (cid:1) , W (Λ)±1 := W±1 ∩ W (Λ). Then W (Λ) (resp. W (Λ)±1) is an OF -lattice in W (resp. W±1). • Set W n−1,1 := (cid:0)∧n−1Vπ (cid:1) ⊗F (Vπ), W n−1,1 ±1 := W n−1,1 ∩ W±1, W (Λ)n−1,1 ±1 := W n−1,1 ±1 ∩ W (Λ). Then the strengthened spin condition states that For any OF -algebra R and F ∈ Mnaive {0} (R), the line ∧nF ⊂ W (Λ0) ⊗OF R is contained in the space Im (cid:0)W (Λ0)n−1,1 −1 ⊗OF R → W (Λ0) ⊗OF R(cid:1) . 4.1.1.2 The definition of the refinement Definition 4.1.1. Let M{0} be the functor M{0} : (Sch/OF )op −→ Sets which sends an OF -scheme S to the set of OS-modules F such that LM1 (π-stability condition) F is an OF ⊗OF0 OS-submodule of Λ0 ⊗OF0 OS and as an OS- module, it is a locally direct summand of rank n. LM2 (Kottwitz condition) The action of π ⊗ 1 ∈ OF ⊗OF0 OS on F has characteristic polynomial det(T − π ⊗ 1 | F) = (T − π)(T − π)n−1. LM3 Let F ⊥ be the orthogonal complement in Λs 0 ⊗OF0 OS of F with respect to the perfect pairing s(−, −) : (Λ0 ⊗OF0 OS) × (Λs 0 ⊗OF0 OS) → OS. 48 t Λs t F ⊥ denotes the image of F ⊥ under the isomorphism π OS induced by Λ0 (cid:44)→ π t Λs t : Λs OS → ( π 0) ⊗OF0 0 sends F to ∼−→ OS 0 ⊗OF0 We require the map Λ0 ⊗OF0 t F ⊥, where π π t Λs 0 ⊗OF0 OS. π LM4 (Hyperbolicity condition) The quadratic form q : Λ0 ⊗OF0 OS → L ⊗OF0 OS induced by q : Λ0 → L satisfies q(F) = 0. LM5 (Wedge condition) The action of π ⊗ 1 − 1 ⊗ π ∈ OF ⊗OF0 OS satisfies ∧2(π ⊗ 1 − 1 ⊗ π | F) = 0. LM6 (Strengthened spin condition) The line ∧nF ⊂ W (Λ0) ⊗OF OS is contained in Im (cid:0)W (Λ0)n−1,1 −1 ⊗OF OS → W (Λ0) ⊗OF OS (cid:1) . Then M{0} is representable by a projective OF -scheme, which is a closed subscheme of Mnaive {0} . Note that over the generic fiber of M{0}, the quadratic form q is determined by s via q(x) = 1 2s(x, x). So, over the generic fiber, the hyperbolicity condition LM4 is implied by the Condition (3) in Mnaive {0} . Similarly as in [PR09, 1.5] and [Smi15, 2.5], we can deduce that the rest of the conditions of M{0} do not affect the generic fiber of Mnaive {0} , and hence M{0} and Mnaive {0} have the same generic fiber. Hence, we have closed immersions Mloc {0} ⊂ M{0} ⊂ Mnaive {0} of projective schemes over OF , where all schemes have the same generic fiber. 4.1.2 An affine chart U{0} around the worst point Set F0 := (π ⊗ 1)(Λ0 ⊗OF0 k). Then we can check that F0 ∈ M{0}(k). We call it the worst point of M{0}. With respect to the basis (4.1.1), the standard affine chart around F0 in Gr(n, Λ0)OF is (cid:1). We denote by U{0} the intersection of M{0} with the OF -scheme of 2n × n matrices (cid:0) X In 49 the standard affine chart in Gr(n, Λ0)OF . The worst point F0 of M{0} is contained in U{0} and corresponds to the closed point defined by X = 0 and π = 0. The conditions LM1-6 yield the defining equations for U{0}. We will analyze each condition in detail. A reader who is only interested in the affine coordinate ring of U{0} may proceed directly to Proposition 4.1.10. 4.1.2.1 Condition LM1 Let R be an OF -algebra. With respect to the basis (4.1.1), the operator π ⊗ 1 acts on Λ0 ⊗OF0 R via the matrix    0 −π0In In tIn    . Then the π-stability condition LM1 on F means there exists an n × n matrix P ∈ Mn(R) such that         0 −π0In In tIn     X In   =   X In   P. We obtain P = X + tIn and X 2 + tX + π0In = 0. 4.1.2.2 Condition LM2 We have already shown that π ⊗ 1 acts on F via X + tIn. Then the Kottwitz condition LM2 translates to Equivalently, Note that det(T − (X + tIn)) = (T − π)(T − π)n−1. det(T − (X + πIn)) = (T + π − π)T n−1. det(T − (X + πIn)) = n (cid:88) i=0 (−1)i tr(∧i(X + πIn))T n−i. Then by comparing the coefficients of T n−i, the Kottwitz condition LM2 becomes tr(X + πIn) = π − π, tr (cid:0)∧i(X + πIn)(cid:1) = 0, for i ≥ 2. (4.1.4) 50 4.1.2.3 Condition LM3 With respect to the bases (4.1.1) and (4.1.2), the perfect pairing s(−, −) : (Λ0 ⊗OF0 R) × (Λs 0 ⊗OF0 R) → R and the map Λ0 ⊗OF0 R → π t Λs 0 ⊗OF0 R are represented respectively by the matrices          S = 2 t H2m 0 H2m 2 t H2m 0 2π0 0 0 t H2m 0 1 0          0 1 0 2π0 t and N =                 Im 0 0 0 −Im 0 0 0 0 0 0 0 0 0 t π0 Im 0 0 t π0 0 0 0 Im 0 0                 0 −tIm 0 0 0 0 −t 0 , t2−π0 π0 Im 0 0 t2 π0 where H2m denotes the 2m × 2m anti-diagonal unit matrix, and Im denotes the m × m identity matrix. Then the Condition LM3 translates to    X In  t       S  N   X In     = 0, or equivalently,  t      X In                 0 t2−2π0 tπ0 Hm 0 2 t Hm 0 0 Hm 0 0 0 Hm 0 0 0 t π0 0 0 2 Write 0 Hm 0 0 2π0 t Hm t2−3π0 π0 Hm 0 0 t2−2π0 t Hm 0 0                 0 0 t2−2π0 π0 0 0 t    = 0.    X In (4.1.5) 0       X =       , A B E C D F G H x 51 where A, B, C, D ∈ Mm(R), E, F ∈ Mm×1(R), G, H ∈ M1×m(R) and x ∈ R. Then Equation (4.1.5) translates to C tHmA + C tHmB + t2 − 2π0 tπ0 t2 − 2π0 tπ0 AtHmC + AtHmD + t π0 t π0 GtG + HmC + C tHm = 0, (LM3-1) GtH + HmD + t2 − 3π0 π0 AtHm + t2 − 2π0 t Hm = 0, 2 t 2 t 2 t 2 t 2 t 2 t 2 t 2 t 2 t C tHmE + DtHmA + DtHmB + DtHmE + F tHmA + F tHmB + F tHmE + t2 − 2π0 tπ0 t2 − 2π0 tπ0 t2 − 2π0 tπ0 t2 − 2π0 tπ0 t2 − 2π0 tπ0 t2 − 2π0 tπ0 t2 − π0 tπ0 AtHmF + Gtx + HmF + t2 − 2π0 π0 Gt = 0, (LM3-2) (LM3-3) BtHmC + H tG + HmA + DtHm + 2π0 t Hm = 0, (LM3-4) BtHmD + H tH + HmB + BtHm = 0, (LM3-5) t2 − 3π0 π0 t2 − 2π0 π0 BtHmF + xH t + HmE + H t = 0, EtHmC + xG + 2G + F tHm = 0, EtHmD + xH + 2H + EtHm = 0, EtHmF + x2 + 2x + x + t = 0. t2 − 3π0 π0 t2 − 2π0 π0 (LM3-6) (LM3-7) (LM3-8) (LM3-9) t π0 t π0 t π0 t π0 t π0 t π0 t π0 4.1.2.4 Condition LM4 Recall L = t−1OF0. With respect to the basis (4.1.1), the induced (L ⊗OF0 R)-valued symmetric pairing on Λ0 ⊗OF0 R is represented by the matrix Hm 0 0 t2−2π0 t Hm 0 Hm 0 0                 S1 = 0 0 0 2π0 t Hm 2 0 0 2π0 t Hm 0 t2−2π0 t Hm 0 0 0 0 π0Hm t 0                 . 0 0 t 0 0 2π0 0 0 π0Hm 0 0 Convention: Throughout the rest of the thesis, we often encounter a matrix M = (Mij) ∈ Mℓ×ℓ(R) whose diagonal entries are of the form Mii = 2aii for some aii ∈ R. We then use 52 1 2Mii to denote aii. When we refer to “half of the diagonal of M ”, we mean the row matrix consisting of the entries 1 2Mii for 1 ≤ i ≤ ℓ. The Condition LM4 translates to    X In  t     t     S1   X In   = 0 and half of the diagonal of   X In   S1   X In   equals zero. One can check that the diagonal entries of (cid:0) X In (cid:1)tS1 (cid:0) X In (cid:1) are indeed divisible by 2 in R. Equivalently, we obtain the following equations. C tHmA + AtHmC + 2GtG + C tHmB + AtHmD + 2GtH + AtHm + π0Hm = 0, (LM4-2) C tHmE + AtHmF + 2xGt + HmF + tGt = 0, DtHmA + BtHmC + 2H tG + HmA + DtHm + π0Hm = 0, DtHmB + BtHmD + 2H tH + HmB + BtHm = 0, C tHm = 0, 2π0 t t2 − 2π0 t 2π0 t t2 − 2π0 t HmC + HmD + 2π0 t 2π0 t 2π0 t t2 − 2π0 t t2 − 2π0 t t2 − 2π0 t 2π0 t t2 − 2π0 t DtHmE + BtHmF + 2xH t + HmE + tH t = 0, F tHmA + EtHmC + 2xG + tG + F tHm = 0, F tHmB + EtHmD + 2xH + tH + EtHm = 0, F tHmE + EtHmF + 2x2 + 2tx + 2π0 = 0, (LM4-1) (LM4-3) (LM4-4) (LM4-5) (LM4-6) (LM4-7) (LM4-8) (LM4-9) half of the diagonal of matrices in LM4-1,5,9 equals 0. (LM4-10) 4.1.2.5 Condition LM5 We already know from §4.1.2.1 that π ⊗ 1 acts as right multiplication by X + tIn on F. Thus, the wedge condition LM5 on F translates to ∧2(X + πIn) = 0. 4.1.2.6 Condition LM6 We will use the same notations as in §4.1.1.1. To find the equations induced by the strengthened spin condition LM6 on F, we need to determine an OF -basis of W (Λ0)n−1,1 −1 . 53 Definition 4.1.2. Let S ⊂ {1, . . . , 2n} be a subset of cardinality n. (1) We say S is of type (n − 1, 1) if #(S ∩ {1, . . . , n}) = n − 1 and #(S ∩ {n + 1, . . . , 2n}) = 1. Such S necessarily has the form {1, . . . , (cid:98)j, . . . , n, n + i} for some i, j ∈ {1, . . . , n}. (2) Let S be of type (n − 1, 1). Denote by iS the unique element in S ∩ {n + 1, . . . , 2n}. Define S ≼ S⊥ if iS ≤ iS⊥. Set B := {S ⊂ {1, . . . , 2n} | #S = n} , Bn−1,1 := {S ∈ B | S is of type (n − 1, 1)} , B0 := (cid:8)S ∈ Bn−1,1 | S ≼ S⊥(cid:9) . By construction, the F -vector space W (Λ0)n−1,1 −1 ⊗OF F equals W n−1,1 −1 , which is an F -subspace of W . Lemma 4.1.3. (1) The set {eS | S ∈ B} (resp. {gS | S ∈ B}) is an F -basis of W . (2) For S ∈ B, denote hS := gS − sgn(σS)gS⊥. The set {hS | S ∈ B0} is an F -basis of W n−1,1 −1 . Proof. (1) As W = ∧n(V ⊗F0 F ) by definition, the statement is a standard fact about the wedge product of vector spaces. (2) By [Smi15, Lemma 4.2], the F -space W n−1,1 −1 is spanned by the set {hS | S ∈ Bn−1,1}. These hS’s are not linearly independent over F . Indeed, for S ∈ Bn−1,1, we have hS⊥ = −sgn(σS)hS by using that (S⊥)⊥ = S and sgn(σS) = sgn(σS⊥) (by [Smi15, Lemma 2.8]). However, the set {hS | S ∈ B0} is F -linearly independent, since {gS | S ∈ B} is F -linearly independent. So the set {hS | S ∈ B0} is an F -basis of W n−1,1 −1 . 54 Definition 4.1.4. Let w = (cid:80) S∈B cSeS ∈ W . The worst term of w is defined to be W T (w) := (cid:88) cSeS, S∈B(w) where B(w) ⊂ B consists of elements S ∈ B such that ω(cS) ≤ ω(cT ) for all T ∈ B. √ Recall θ = 1 − 2π/t ∈ O× F . Using (4.1.3), we immediately obtain the following. Lemma 4.1.5. Let S ∈ Bn−1,1. Then exactly we have the following six cases. (1) If S = {1, . . . ,(cid:98)i, . . . , n, n + i} for some i ≤ m + 1, then W T (gS) = (−1)i−1 tm−1 π3m−1 e{n+1,...,2n}. (2) If S = {1, . . . ,(cid:98)i, . . . , n, n + i} for some m + 1 < i ≤ n, then W T (gS) = (−1)i−1 tm−1 π3m−3π0 e{n+1,...,2n}. (3) If S = {1, . . . , (cid:98)j, . . . , n, n + i} for some i, j ≤ m + 1 with i ̸= j, then √ W T (gS) = − θ tm π3m−1 e{i,n+1,...,(cid:91)n+j,...,2n}. (4) If S = {1, . . . , (cid:98)j, . . . , n, n + i} for some i ≤ m + 1 < j, then √ W T (gS) = − θ tm−1 π3m−2 e{i,n+1,...,(cid:91)n+j,...,2n}. (5) If S = {1, . . . , (cid:98)j, . . . , n, n + i} for some j ≤ m + 1 < i, then W T (gS) = − √ θ tm+1 π3m−2π0 e{i,n+1,...,(cid:91)n+j,...,2n}. (6) If S = {1, . . . , (cid:98)j, . . . , n, n + i} for some i, j > m + 1 with i ̸= j, then W T (gS) = − √ θ tm π3m−3π0 e{i,n+1,...,(cid:91)n+j,...,2n}. Definition 4.1.6. For S ∈ Bn−1,1, the weight vector wS ∈ Zn attached to S is defined to be an element of Zn such that the i-th coordinate of wS is #(S ∩ {i, n + i}). 55 Note that if S ∈ Bn−1,1, then S = {1, . . . , (cid:98)j, . . . , n, n + i} for some 1 ≤ i, j ≤ n. Moreover, we have sgn(σS) = (−1)i+j+1 (see [Smi15, Remark 4.3]) and S⊥ = {1, . . . , (cid:98)i∨, . . . , n, j∗}. Similar arguments in [Smi15, Lemma 4.10] imply the following lemma. Lemma 4.1.7. Let S ∈ B0. Then exactly we have the following nine cases. (1) S = {1, . . . , (cid:92)m + 1, . . . , n, n + m + 1}. Then S = S⊥, wS = (1, . . . , 1), and W T (hS) = W T (2gS) = (−1)m 2tm−1 π3m−1 e{n+1,...,2n}. (2) S = {1, . . . , (cid:98)i∨, . . . , n, n + i} for some i < m + 1. Then S = S⊥, wS ̸= (1, . . . , 1), and W T (hS) = W T (2gS) = − √ θ 2tm−1 π3m−2 e{i,n+1,..., (cid:98)i∗,...,2n}. (3) S = {1, . . . , (cid:98)i∨, . . . , n, n + i} for some i > m + 1. Then S = S⊥, wS ̸= (1, . . . , 1), and W T (hS) = W T (2gS) = − √ θ 2tm+1 π3m−2π0 e{i,n+1,..., (cid:98)i∗,...,2n}. (4) S = {1, . . . ,(cid:98)i, . . . , n, n + i} for some i < m + 1. Then S ̸= S⊥, wS = wS⊥ = (1, . . . , 1), and W T (hS) = W T (g{1,...,(cid:98)i,...,n,n+i} + g{1,..., (cid:98)i∨,...,n,i∗}) = (−1)i−1 tm π3m−2π0 e{n+1,...,2n}. (5) S = {1, . . . , (cid:98)j, . . . , n, n + i} for some i < j∨ < m + 1. Then S ̸= S⊥, wS, wS⊥ and (1, . . . , 1) are pairwise distinct and W T (hS) = W T (g{1,...,(cid:98)j,...,n,n+i} + (−1)i+jg{1,..., (cid:98)i∨,...,n,j∗}) √ √ tm−1 π3m−2 e{i,n+1,...,(cid:91)n+j,...,2n} + (−1)i+j+1 = − θ θ tm−1 π3m−2 e{j∨,n+1,..., (cid:98)i∗,...,2n}. (6) S = {1, . . . , (cid:92)m + 1, . . . , n, n + i} for some i < m + 1. Then S ̸= S⊥, wS, wS⊥ and (1, . . . , 1) are pairwise distinct and W T (hS) = W T (g{1,..., (cid:91)m+1,...,n,n+i} − (−1)m+ig{1,..., (cid:98)i∨,...,n+m+1}) = (−1)m+i √ θ tm−1 π3m−2 e{m+1,n+1,..., (cid:98)i∗,...,2n}. 56 (7) S = {1, . . . , (cid:98)j, . . . , n, n + i} for some i < m + 1 < j∨. Then S ̸= S⊥, wS, wS⊥ and (1, . . . , 1) are pairwise distinct and W T (hS) = W T (g{1,...,(cid:98)j,...,n,n+i} − (−1)i+j+1g{1,..., (cid:98)i∨,...,n,j∗}) √ = − θ tm π3m−1 e{i,n+1,...,(cid:91)n+j,...,2n} − (−1)i+j √ θ tm π3m−3π0 e{j∨,n+1,..., (cid:98)i∗,...,2n}. (8) S = {1, . . . , (cid:98)j, . . . , n, n + m + 1} for some j∨ > m + 1. Then S ̸= S⊥, wS, wS⊥ and (1, . . . , 1) are pairwise distinct and W T (hS) = W T (g{1,...,(cid:98)j,...,n,n+m+1} − (−1)m+j+1g{1,..., (cid:91)m+1,...,n,j∗}) √ θ = − tm π3m−1 e{m+1,n+1,...,(cid:91)n+j,...,2n}. (9) S = {1, . . . , (cid:98)j, . . . , n, n + i} for some j∨ > i > m + 1. Then S ̸= S⊥, wS, wS⊥ and (1, . . . , 1) are pairwise distinct and W T (hS) = W T (g{1,...,(cid:98)j,...,n,n+i} − (−1)i+j+1g{1,..., (cid:98)i∨,...,j∗}) √ √ e{i,n+1,...,(cid:91)n+j,...,2n} + (−1)i+j+1 = − θ tm+1 π3m−2π0 θ tm+1 π3m−2π0 e{j∨,n+1,..., (cid:98)i∗,...,2n}. Let w ∈ W n−1,1 −1 . Recall that {hS |S ∈ B0} is an F -basis of W n−1,1 −1 by Lemma 4.1.3. Write w = (cid:88) S∈B0 aShS = (cid:88) (cid:88) aShS, aS ∈ F. w∈Zn S∈B0 and wS = w Then as in the proof of [Smi15, Proposition 4.12], we have w ∈ W (Λ0)n−1,1 −1 ⇐⇒ (cid:88) S∈B0 and wS = w aShS ∈ W (Λ0)n−1,1 −1 , for each w ∈ Zn We have two distinct situations for w: Case 1: w ̸= (1, . . . , 1). Then there exists at most one S ∈ B0 such that wS = w. Case 2: w = (1, . . . , 1). Then S is necessarily of the form Si := (cid:110) 1, . . . ,(cid:98)i, . . . , n, n + i (cid:111) 57 for some 1 ≤ i ≤ m + 1. For any 1 ≤ i < m + 1, we have hSi = gSi + gSi∨ = (−1)ig1 ∧ · · · ∧ (cid:98)gi ∧ · · · ∧ (cid:99)gi∨ ∧ · · · ∧ gn ∧ (gi ∧ gi∗ − gi∨ ∧ gn+i) = (−1)ig1 ∧ · · · ∧ (cid:98)gi ∧ · · · ∧ (cid:99)gi∨ ∧ · · · ∧ gn t2 − π0 π0 fi ∧ fi∗ − 2fi∨ ∧ fn+i − ∧ (−tfi ∧ fi∨ + fn+i ∧ fi∗), t π0 and hSm+1 = 2gSm+1 = −2 · g1 ∧ · · · ∧ (cid:98)gi ∧ · · · ∧ (cid:99)gi∨ ∧ · · · ∧ gn ∧ gn+m+1 ∧ (gi ∧ gi∨) = −2 · g1 ∧ · · · ∧ (cid:98)gi ∧ · · · ∧ (cid:99)gi∨ ∧ · · · ∧ gn ∧ gn+m+1 ∧ ( t π fi ∧ fi∨ − t π2 fi ∧ fi∗ + t π2 fi∨ ∧ fn+i + t π3 fn+i ∧ fi∗). Define (cid:101)hSi :=    2πhSi + (−1)m+ithSm+1 if i ̸= m + 1, hSm+1 if i = m + 1. Then for 1 ≤ i < m + 1, terms of (cid:101)hSi do not contain (multiples of) W T (hSm+1) = (−1)m 2tm−1 π3m−1 e{n+1,...,2n}, and W T ((cid:101)hSi) = − √ θ 2tmπ0 π3m e{i,n+1,...,(cid:91)n+i,...,2n} − √ θ 2tm π3m−2 e{i∨,n+1,..., (cid:98)i∗,...,2n}. (4.1.6) For S with wS ̸= (1, . . . , 1), we set (cid:101)hS := hS. By Lemma 4.1.3, the set {(cid:101)hS | S ∈ B0} forms an F -basis of W n−1,1 −1 . Previous analysis on w together with similar arguments in [Smi15, Proposition 4.12] imply the following lemma. Lemma 4.1.8. For each S ∈ B0, pick bS ∈ F such that the worst term W T (bS(cid:101)hS) is a sum of terms of the form uT eT for some unit uT ∈ O× F and T ∈ B. Then the set {bS(cid:101)hS | S ∈ B0} forms an OF -basis of the OF -module W (Λ0)n−1,1 −1 . 58 For the matrix (cid:0) X In (cid:1) corresponding to F, denote by v ∈ ∧nF the wedge product of n- columns of the matrix in the order from left to right. Then the strengthened spin condition LM6 on F amounts to that v ∈ Im (cid:0)W (Λ0)n−1,1 −1 ⊗OF R → W (Λ0) ⊗OF R(cid:1) . Write v = (cid:80) S∈B aSeS for some aS ∈ R. By Lemma 4.1.8, we have v = (cid:88) S∈B aSeS = (cid:88) S∈B0 cSbS(cid:101)hS (4.1.7) for some cS ∈ R. By comparing the coefficients of both sides in Equation (4.1.7), we will obtain the defining equations of the condition LM6 on the chart U{0}. Recall X =       A B E C D F G H x       , where A, B, C, D ∈ Mm(R), E, F ∈ Mm×1(R), G, H ∈ M1×m(R) and x ∈ R. In the following, we use aij to denote the (i, j)-entry of the matrix A. We use similar notations for other block matrices in X. For 1 ≤ i < m + 1, comparing the coefficients of e{n+1,...,2n} and eSi = e{1,...,(cid:98)i,...,n,n+i} in (4.1.7), we obtain cSm+1(−1)mbSm+1 (cid:18) √ cSm+1bSm+1(−1)m+i 2tm−1 cSm+1bSm+1(−1)m+i 2tm−1 π3m−2 + cSibSi π3m−2 + cSibSi − θ (cid:18) − √ θ 2tm−1 π3m−1 = 1, (cid:19) 2tmπ0 π3m 2tm π3m−2 (cid:19) = (−1)1+idii, = (−1)1+iam+i−i,m+1−i. Hence, dii = √ π0 π2 am+1−i,m+1−i + t θ. (4.1.8) For 1 ≤ i, j < m + 1 and i ̸= j, by comparing the coefficients of e{1,...,(cid:98)j,...,n,n+i} and e{j∨,n+1,..., (cid:98)i∗,...,2n}, we obtain (cid:18) √ − c{1,...,(cid:98)j,...,n,n+i}b{1,...,(cid:98)j,...,n,n+i} 59 (cid:19) θ tm π3m−1 = (−1)1+jdij, c{1,...,(cid:98)j,...,n,n+i}b{1,...,(cid:98)j,...,n,n+i} (cid:18) (−1)1+i+j (cid:19) √ θ tm π3m−3π0 = (−1)1+iam+1−j,m+1−i. Hence, dij = π0 π2 am+1−j,m+1−i. (4.1.9) Combining (4.1.8) and (4.1.9), we obtain D = π0 π2 HmAtHm + t √ θIm. Here the matrix HmAtHm is the reflection of A over its anti-diagonal. Equivalently, D + πIm = π π Hm(A + πIm)tHm. (4.1.10) Similarly, we can obtain B = HmBtHm, C = HmC tHm, E = t π HmH t, F = t π HmGt. (4.1.11) Write (cid:101)H2m :=    0 Hm π π Hm 0     , X1 :=      . A B C D Combining (4.1.10) and (4.1.11), we have (cid:101)H2m (X1 + πI2m) = (X1 + πI2m)t (cid:101)H t 2m. (4.1.12) 4.1.2.7 A simplification of equations First we can see that under the wedge condition ∧2(X + πIn) = 0, the Kottwitz condition (4.1.4) becomes Next we claim that the equation tr(X + πIn) = π − π. (4.1.13) X 2 + tX + π0In = 0 (4.1.14) of Condition LM1 is implied by the Kottwitz condition LM2 and the wedge condition LM5. To justify the claim, we need an easy but useful lemma. 60 Lemma 4.1.9. Let X be an n × n matrix. Then X 2 ≡ (tr X)X modulo (∧2X). Proof. The (i, j)-entry of the matrix X 2 − tr(X)X is n (cid:88) k=1 XikXkj − n (cid:88) k=1 XkkXij = n (cid:88) k=1 (XikXkj − XkkXij) , which is a sum of 2-minors of X. By Lemma 4.1.9 and the wedge condition LM5, the equation (4.1.14) X 2 + tX + π0In = (X + πIn)2 + (t − 2π)(X + πIn) = 0 is equivalent to tr(X + πIn)(X + πIn) + (t − 2π)(X + πIn) = (tr(X + πIn) + π − π) (X + πIn) = 0, which is implied by the Kottwitz condition (4.1.13). Next, we examine the Condition LM3. For the equation (LM3-1), we have 2 t 2 t = = 2 t C tHmA + t2 − 2π0 tπ0 AtHmC + t π0 GtG + HmC + C tHm C tHm(A + πIm) + t2 − 2π0 tπ0 (A + πIm)tHmC − 2π t C tHm − t2 − 2π0 tπ0 πHmC + t π0 GtG + HmC + C tHm C tHm(A + πIm) + t2 − 2π0 tπ0 (A + πIm)tHmC + t π0 GtG + √ θC tHm + √ θHmC. π π A similar argument as in the proof of Lemma 4.1.9 implies that C tHm(A + πIm) ≡ (A + πIm)tHmC modulo (∧2(X + πIm)). Hence, the equation (LM3-1) gives the same restriction on U{0} as the equation t π0 (A + πIm)tHmC + t π0 √ GtG + θC tHm + √ π π θHmC = 0. By (4.1.11), we further obtain t π0 (A + πIm)tHmC + GtG + √ t π t π0 61 θHmC = 0, (4.1.15) (A + πIm)tHmC = (C(A + πIm))tHm. Again, as in Lemma 4.1.9, the matrix C(A + πIm) is equivalent to tr(A + πIm)C. Thus, the equation (4.1.15) is equivalent to t π0 tr(A + πIm)C tHm + t π0 GtG + t π √ θHmC = 0. Equivalently, (tr(A + πIm) + π √ (cid:17) θ)HmC + GtG = 0. (cid:16) t π0 (4.1.16) Similarly, under the wedge condition LM5 and the strengthened spin condition LM6, one can verify that the equation (LM3-2) can be simplified to (cid:16) (tr(A + πIm) + π √ θ)Hm(D + πIm) + GtH (cid:17) = 0; (4.1.17) t π0 the equation (LM3-3) is trivial; the equation (LM3-4) is equivalent to (LM3-2); the equa- tion (LM3-5) is equivalent to (cid:18) ( π π t π0 tr(A + πIm) + π √ θ)HmB + H tH (cid:19) = 0; (4.1.18) the rest of the equations are trivial. Set X1 :=    A B C D      , X2 :=   E F   , X3 := (cid:18) (cid:19) G H , X4 := x. Then X = (cid:0) X1 X2 X3 X4 (cid:1), and equations (4.1.16), (4.1.17), (4.1.18) translate to (tr(A + πIm) + π √ θ) (cid:101)H2m(X1 + πI2m) + X t 3X3 (cid:17) = 0. (cid:16) t π0 Using similar arguments, one can check that under the wedge condition LM5 and the strengthened spin condition LM6, equations (LM4-1) to (LM4-9) are implied by the Con- dition LM3, and the equation (LM4-10) is equivalent to the diagonal of (tr(A + πIm) + π √ θ) (cid:101)H2m(X1 + πI2m) + X t 3X3 equals 0. 62 Denote by OF [X] the polynomial ring over OF whose variables are entries of the matrix X. Then we can view the affine chart U{0} ⊂ M{0} as a closed subscheme of Spec OF [X]. In summary, we have shown the following. Proposition 4.1.10. The scheme U{0} is a closed subscheme1 of U′ {0} := Spec OF [X]/I, where I is the ideal of OF [X] generated by: tr(X + πIn) − π + π, ∧2(X + πIn), (cid:101)H2m (X1 + πI2m) − (X1 + πI2m)t (cid:101)H t 2m, E − t π HmH t, F − t π HmGt, t π0 the diagonal of (tr(A + πIm) + π Set (cid:16) (tr(A + πIm) + π √ θ) (cid:101)H2m(X1 + πI2m) + X t 3X3 (cid:17) , √ θ) (cid:101)H2m(X1 + πI2m) + X t 3X3. (cid:101)X1 := X1 + πI2m, (cid:101)A := A + πIm, (cid:101)X :=    .    (cid:101)X1 X3 As X2 and X4 are determined by X1 and X3 by relations in I, we obtain the following proposition. Proposition 4.1.11. The scheme U′ {0} = Spec OF [X]/I is isomorphic to Spec OF [ (cid:101)X]/(cid:101)I, where (cid:101)I is the ideal of OF [ (cid:101)X] generated by: ∧2 ( (cid:101)X), (cid:101)H2m (cid:101)X1 − (cid:101)X t 1 (cid:101)H t the diagonal of (tr( (cid:101)A) + π 2m, √ (cid:16) t π0 (tr( (cid:101)A) + π √ θ) (cid:101)H2m (cid:101)X1 + X t 3X3 (cid:17) , θ) (cid:101)H2m (cid:101)X1 + X t 3X3. Definition 4.1.12. Denote by Ufl {0} the closed subscheme of U′ {0} = Spec OF [ (cid:101)X]/I defined by the ideal (cid:101)I fl ⊂ OF [ (cid:101)X] that is generated by: ∧2 ( (cid:101)X), (cid:101)H2m (cid:101)X1 − (cid:101)X t 1 (cid:101)H t 2m, (tr( (cid:101)A) + π √ θ) (cid:101)H2m (cid:101)X1 + X t 3X3. Note that the ideal (cid:101)I fl contains (cid:101)I. 1In fact, we expect that U{0} = U′ {0}. This amounts to saying that the equations obtained by comparing coefficients of eS in (4.1.7) for S not of type (n − 1, 1) are implied by relations in I. 63 4.1.2.8 Geometric properties of U{0} and Ufl {0} In the following, we write Rfl for the ring OF [ (cid:101)X]/(cid:101)I fl and R for the ring OF [ (cid:101)X]/(cid:101)I. Lemma 4.1.13. If ω(π0) = ω(t), then R = Rfl. Proof. Note that ω(π0) = ω(t) if and only if t/π0 is a unit in OF . By comparing the lists of generators of (cid:101)I and (cid:101)I fl, we immediately see that (cid:101)I = (cid:101)I fl, and hence R = Rfl. Remark 4.1.14. Since π0|t|2, the condition ω(t) = ω(π0) clearly holds if F0/Q2 is unramified. More generally, by applying Proposition 3.1.1 (4) to F0, we have ω(t) = ω(π0) if and only if θ ∈ U2e−1 − U2e. Namely, given a quadratic extension F of F0 with a uniformizer π satisfying an Eisenstein equation π2 − tπ + π0 = 0, the condition ω(t) = ω(π0) holds if and only if F is of the form F0( √ θ) for some unit θ ∈ U2e−1 − U2e. We will count the number of such extensions F in the following. We have a short exact sequence 0 → U2e U 2 ∩ U2e → U2e−1 U 2 ∩ U2e−1 → U2e−1 U2e(U 2 ∩ U2e−1) → 0. (4.1.19) We claim that U 2 ∩U2e−1 ⊂ U2e. For any x ∈ U 2 ∩U2e−1, we can find a ∈ OF0 and u ∈ U such that x = 1 + π2e−1 0 a = u2. We want to show ω(a) ≥ 1. Set b = u − 1. Then b(b + 2) = π2e−1 0 a. If ω(b) < e = ω(2), then ω(b + 2) = ω(b) and ω(π2e−1 0 a) = ω (b(b + 2)) = 2ω(b). As 2e − 1 is odd, this forces ω(a) to be odd and in particular ω(a) ≥ 1. If ω(b) ≥ e, then ω(π2e−1 0 a) = ω (b(b + 2)) ≥ ω(b) + ω(2) ≥ 2e. Again we have ω(a) ≥ 1. This proves the claim. Then we have U2e(U 2 ∩ U2e−1) = U2e and by the short exact sequence (4.1.19), (cid:12) (cid:12) (cid:12) (cid:12) U2e−1 U 2 ∩ U2e−1 (cid:12) (cid:12) (cid:12) (cid:12) = (cid:12) (cid:12) (cid:12) (cid:12) U2e U 2 ∩ U2e (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) U2e−1 U2e (cid:12) (cid:12) (cid:12) (cid:12) = 2 · 2f = 21+f , 64 where f denotes the residue degree of F0/Q2. Note that there are two elements in U2e−1 U 2∩U2e−1 defining the trivial extension and the unramified quadratic extension of F0. Thus, we have 21+f − 2 ramified quadratic extensions of F0 of type (R-U) with ω(t) = ω(π0). By (4.1.10), we have So we can rewrite Rfl as Rfl = (cid:16) (cid:16) ∧2 (cid:17) (cid:101)X1 X3 tr( (cid:101)X1) = tr( (cid:101)A) + tr( (cid:101)D) = t π tr( (cid:101)A). OF [ (cid:16) (cid:17) ] (cid:101)X1 X3 , (cid:101)H2m (cid:101)X1 − (cid:101)X t 1 (cid:101)H t 2m, ( π t tr( (cid:101)X1) + π √ θ) (cid:101)H2m (cid:101)X1 + X t 3X3 (cid:17). Let Y := (cid:101)H2m (cid:101)X1. Then (cid:101)X1 = π π (cid:101)H2mY and Rfl ≃ (cid:16) ∧2 (cid:16) π π (cid:101)H2mY X3 (cid:17) X3 (cid:1)] OF [(cid:0) Y tπ tr( (cid:101)H2mY ) + π , Y − Y t, ( π2 √ θ)Y + X t 3X3 (cid:17) = (cid:16) ∧2(cid:0) Y X3 (cid:1), Y − Y t, ( π (cid:1)] OF [(cid:0) Y X3 2π tr(H2mY ) + π √ θ)Y + X t 3X3 (cid:17). For 1 ≤ i, j ≤ 2m, we denote by yij the (i, j)-entry of Y and by xi the (1, i)-entry of X3. Lemma 4.1.15. The scheme Ufl {0} is irreducible of Krull dimension n and smooth over OF on the complement of the worst point, which is the closed point defined by Y = X3 = π = 0. Proof. For 1 ≤ ℓ ≤ 2m, consider the principal open subscheme D(yℓℓ) of Ufl {0}, i.e., the locus where yℓℓ is invertible. Then one can easily verify that D(yℓℓ) is isomorphic to the closed subscheme of Spec OF [yij, xi | 1 ≤ i, j ≤ 2m] defined by the ideal generated by the relations yij = yji, yij = y−1 ℓℓ yℓiyℓj, xi = y−1 ℓℓ xℓyℓi, −x2 ℓ = ( yℓiyℓ,n−i) + π √ θyℓℓ. π π m (cid:88) i=1 Hence, the scheme D(yℓℓ) is isomorphic to Spec OF [xℓ, yℓ1, . . . , yℓℓ, . . . , yℓ,2m, y−1 ℓℓ ] ℓ + ( π (x2 θyℓℓ) i=1 yℓiyℓ,n−i) + π (cid:80)m √ π . 65 By the Jacobian criterion, D(yℓℓ) is smooth over OF of Krull dimension n. Note that the worst point is defined (set-theoretically) by the ideal generated by π and yℓℓ for 1 ≤ ℓ ≤ 2m. Since the generic fiber of Ufl {0} is smooth, we obtain that Ufl {0} is smooth over OF on the complement of the worst point. As the generic fiber and all D(yℓℓ), for 1 ≤ ℓ ≤ 2m, are irreducible, we conclude that Ufl {0} is irreducible. Lemma 4.1.16. The scheme Ufl {0} is Cohen-Macaulay. Proof. Let S denote the polynomial ring OF [yii | 1 ≤ i ≤ 2m]. Then we have an obvious ring homomorphism S → Rfl. By the wedge condition LM5 and Y = Y t, for 1 ≤ i, j ≤ 2m, we have y2 ij = yijyji = yiiyjj and x2 i = −( yiℓyi,n−ℓ) − π √ θyii π π m (cid:88) ℓ=1 In particular, we deduce that Rfl is integral (also of finite type) over S, and hence Rfl is a finitely generated S-module. Since S is a domain of the same Krull dimension as Rfl, the map S → Rfl is necessarily injective. By [Eis13, Corollary 18.17], to show Rfl is Cohen- Macaulay, it suffices to show that Rfl is a flat S-module. Equivalently, we need to show that the induced morphism ψ : Spec Rfl → Spec S ≃ A2m is flat. Let P0 be the closed point in Spec S corresponding to the maximal ideal m0 := (π, y11, . . . , y2m,2m). Then ψ maps the worst point of Spec Rfl to P0 and the preimage of Spec S[y−1 ℓℓ ] is the scheme D(yℓℓ) considered in the proof of Lemma 4.1.15. As D(yℓℓ) is smooth over OF , by miracle flatness (see [Eis13, Theorem 18.16 b.]), the restriction ψ|D(yℓℓ) is flat. Similarly, we obtain that ψ restricted to the generic fiber of Ufl {0} is flat. It remains to show that ψ is flat at the worst point, i.e., the localization map Sm0 → Rfl m0 is flat. The local ring Sm0 has residue field k. Let K denote the fraction field of Sm0. By an application of Nakayama’s lemma (see [Har13, Chapter II, Lemma 8.9]), we are reduced to show that dimK(Rfl m0 ⊗Sm0 K) = dimk(Rfl m0 ⊗Sm0 k). (4.1.20) 66 Note that K is the field F (y11, . . . , y2m,2m) of rational functions. By the following Lemma 4.1.17, we have the desired equality (4.1.20) of dimensions. Lemma 4.1.17. The K-vector space (resp. k-vector space) Rfl m0 ⊗Sm0 K (resp. Rfl m0 ⊗Sm0 k) has a K-basis (resp. k-basis) consisting of (images of ) monomials i yβ1 xα i1j1 yβ2 i2j2 · · · yβℓ iℓjℓ , where α, βi ∈ {0, 1}, 0 ≤ ℓ ≤ m, and 1 ≤ i < i1 < j1 < i2 < j2 < · · · < iℓ < jℓ ≤ 2m. Let S denote the set of these monomials. Then the cardinality #S equals 22m. In particular, dimK(Rfl m0 ⊗Sm0 K) = dimk(Rfl m0 ⊗Sm0 k) = 22m. (4.1.21) Proof. We first count the cardinality of S. For an integer 0 ≤ ℓ ≤ m, the number of monomials of the form xiyβ1 i1j1 yβ2 i2j2 · · · yβℓ iℓjℓ in S is the number of tuples (i, i1, j1, . . . , iℓ, jℓ) such that 1 ≤ i < i1 < j1 < i2 < j2 < · · · < iℓ < jℓ ≤ 2m. It is well-known that the number is (cid:0) 2m (cid:1) = 0 if ℓ = m. Similarly, the number of monomials of the form (cid:1). Hence, we obtain that (cid:1). Here, we set (cid:0) 2m in S is (cid:0)2m 2ℓ+1 2ℓ+1 yβ2 i2j2 yβ1 i1j1 · · · yβℓ iℓjℓ 2ℓ #S = m (cid:88) ℓ=0 (cid:19) (cid:18) 2m 2ℓ + 1 + m (cid:88) ℓ=0 (cid:19) (cid:18)2m 2ℓ = 2m (cid:88) i=0 (cid:19) (cid:18)2m i = 22m. yβ2 i2j2 · · · yβℓ iℓjℓ be a general monomial in Rfl m0 ⊗Sm0 K. As y2 ij = yijyji = yiiyjj Let xα i xα′ j yβ1 i1j1 in Rfl m0, we may assume βi for 1 ≤ i ≤ ℓ lies in {0, 1}. As −X t 3X3 = ( π 2π tr(H2mY ) + π √ θ)Y in Rfl m0, we see xixj can be expressed by entries in Y . Hence, we may assume α′ = 0 and α ∈ {0, 1}. We claim that the monomial xα i yi1j1yi2j2 · · · yiℓjℓ for α ∈ {0, 1} is generated by elements in S. By the wedge condition and Y = Y t, it is straightforward to check that the product xryijypq only depends on the indices {r, i, j, p, q}, namely, changing the order of indices gives the same product in Rfl m0. Since yii ∈ K, we may assume 1 ≤ i < i1 < j1 < i2 < j2 < · · · < iℓ < jℓ ≤ 2m, and hence we may assume 0 ≤ ℓ ≤ m. Thus, the K-vector 67 space Rfl m0 ⊗Sm0 K is generated by (images of) the elements in S. Now it suffices to show that these elements are K-linearly independent. m0 ⊗Sm0 K, the ring Rfl Note that the ring Rfl K corresponds to the generic point of Spec Rfl. Since y11 is invertible over Rfl m0 ⊗Sm0 m0 ⊗Sm0 K is in fact the function field of D(y11) in the proof of Lemma 4.1.15 (take ℓ = 1), and the field Rfl m0 ⊗Sm0 K = (cid:16) 12 − y11y22, . . . , y2 y2 1,2m − y11y2m,2m, x2 1 + ( π π i=1 y1iy1,n−i) + π K[y12, y13, . . . , y1,2m, x1] (cid:80)m (cid:17) . √ θy11 is a compositum of successive quadratic extensions. In particular, dimK(Rfl m0 ⊗Sm0 K) = 22m. As #S = 22m, elements in S are K-linearly independent, i.e., elements in S form a K-basis of Rfl K. m0 ⊗Sm0 Similar arguments (just note that now yii = 0 in k) as before imply that Rfl m0 ⊗Sm0 k is generated by (images of) elements in S. Hence, dimk(Rfl m0 ⊗Sm0 k) ≤ #S = dimK(Rfl m0 ⊗Sm0 K). On the other hand, by Nakayama’s lemma, we always have dimk(Rfl m0 ⊗Sm0 k) ≥ dimK(Rfl m0 ⊗Sm0 K). This completes the proof of the lemma. Corollary 4.1.18. The scheme Ufl {0} is normal and flat over OF . The geometric special fiber Ufl {0} ⊗OF k is reduced and irreducible. Proof. As Ufl {0} is smooth over OF on the complement of a closed point, and Cohen-Macaulay by Lemma 4.1.15 and 4.1.16, the normality of Ufl {0} follows from the Serre’s criterion for normality (see [Sta24, 031S]). By Lemma 4.1.15, the scheme Ufl {0} ⊗OF k is smooth over k on the complement of the worst point. The proof of Lemma 4.1.15 also implies that Ufl {0} ⊗OF k 68 is irreducible of dimension n − 1. As Ufl {0} is Cohen-Macaulay and Spec OF is regular, then Ufl {0} is flat over OF by the miracle flatness (see [Eis13, Theorem 18.16 b.]). Since Ufl {0} is Cohen-Macaulay and π is not a zero divisor (follows from the flatness), the scheme Ufl {0} ⊗OF k is also Cohen-Macaulay. Then Ufl {0} ⊗OF k is reduced by the Serre’s criterion for reducedness (see [Sta24, 031R]). Lemma 4.1.19. The schemes U{0} and Ufl {0} have the same underlying topological space. Proof. (1) Since Ufl {0} is flat over OF , the scheme Ufl {0} is the Zariski closure of its generic fiber. Then we have closed immersions Ufl {0} (cid:44)→ U{0} (cid:44)→ U′ {0} where all schemes have the same generic fiber. Then it suffices to prove that the special fibers of Ufl {0} and U′ {0} have the same underlying topological space. Since Ufl {0} ⊗OF k is reduced, we are reduced to show that I fl ⊗OF k is contained in the radical of I ⊗OF k. If ω(π0) = ω(t), then the assertion follows from Lemma 4.1.13. We may assume t/π0 is not a unit. In this case, we have I ⊗OF k = I fl ⊗OF k = (cid:18) ∧2(cid:0) Y X3 (cid:18) ∧2(cid:0) Y X3 (cid:1), Y − Y t, the diagonal of ( (cid:1), Y − Y t, tr(H2mY ) 2 Y + X t tr(H2mY ) 2 (cid:19) . 3X3 Y + X t 3X3) (cid:19) , Let M denote the matrix tr(H2mY ) 2 Y + X t 3X3. Then for 1 ≤ i, j ≤ 2m, the (i, j)-entry Mij of M is αyij + xixj, α := tr(H2mY )/2. Since char(k) = 2, we obtain M 2 ij = α2y2 ij + x2 i x2 j . Therefore, we have M 2 ij − MiiMjj = α2(y2 ij − yiiyjj) − αx2 i yjj − αx2 j yii = α2(y2 ij − yiiyjj) − x2 i Mjj − x2 j Mii + 2x2 i x2 j = α2(y2 ij − yiiyjj) − x2 i Mjj − x2 j Mii ∈ (cid:101)I ⊗OF k 69 In particular, any M 2 ij for 1 ≤ i, j ≤ 2m lies in (cid:101)I ⊗OF k. Hence, I fl ⊗OF k is contained in the radical of I ⊗OF k. This finishes the proof. In summary, we have proven the following. Proposition 4.1.20. (1) The scheme Ufl {0} is flat over OF of relative dimension n − 1. In particular, Ufl {0} is isomorphic to an open subscheme of the local model Mloc {0} containing the worst point. Furthermore, Ufl {0} is normal, Cohen-Macaulay, and smooth over OF on the complement of the worst point. The special fiber Ufl {0} ⊗OF k is (geometrically) reduced and irreducible. (2) U{0} and Ufl {0} have the same underlying topological space. (3) If ω(π0) = ω(t), then U{0} = Ufl {0}. 4.1.2.9 Global results Recall that (Λ0, q, L , ϕ) is a hermitian quadratic module with ϕ over OF0 by Lemma 3.2.5. Let H{0} := Sim((Λ0, q, L , ϕ)) be the group scheme over OF0 of similitudes preserving ϕ of (Λ0, q, L , ϕ). By Theorem 6.2.8, H{0} is an affine smooth group scheme over OF0. Lemma 4.1.21. The group scheme H{0} acts on Mnaive {0} and M{0}. Proof. It suffices to show the result for M{0}. Let R be an OF -algebra. Let g = (φ, γ) ∈ H{0}(R) be a similitude preserving ϕ. For F ∈ M{0}, we define gF := φ(F) ⊂ Λ0 ⊗OF0 We need to show that gF ∈ M{0}(R). It is clear that gF satisfies conditions LM1,2,4. R. Recall that ϕ : Λ0 × Λ0 → t−1OF0 is defined by (x, y) (cid:55)→ t−1 TrF/F0 h(x, π−1y). We also use ϕ to denote the base change to Λ0 ⊗OF0 R. Then we see that F satisfies LM3 if and only if ϕ(F, F) = 0. As g preserves ϕ, we have that ϕ(gF, gF) = γϕ(F, F) = 0. So gF satisfies LM3. As g is OF ⊗OF0 R-linear by definition, we obtain that (π ⊗ 1 − 1 ⊗ π) ◦ g = g ◦ (π ⊗ 1 − 1 ⊗ π). 70 By the functoriality of the wedge product of linear maps, we have ∧2(π ⊗ 1 − 1 ⊗ π | gF) = ∧2(g ◦ (π ⊗ 1 − 1 ⊗ π) | F) = ∧2(g) ◦ ∧2(π ⊗ 1 − 1 ⊗ π | F) = 0. Therefore, gF satisfies the wedge condition LM5. Since H{0} is smooth over OF0, using a similar argument of [RSZ18, Lemma 7.1], we can show that the R-submodule Im (cid:0)W (Λ0)n−1,1 −1 ⊗OF R → W (Λ0) ⊗OF R(cid:1) of W (Λ0) ⊗OF R is stable under the natural action of H{0}(R) on the space W (Λ0) ⊗OF R = ∧n(Λ0 ⊗OF0 R). It follows that gF satisfies the strengthened spin condition LM6. Lemma 4.1.22. Let k be the algebraic closure of the residue field k. Then M{0} ⊗OF k has two H{0} ⊗OF0 k-orbits, one of which consists of the worst point. Proof. By Lemma 4.1.21, the special fiber M{0} ⊗OF k has an action of H{0} ⊗OF0 F ∈ M{0}(k). In particular, the subspace F ⊂ (Λ0 ⊗OF0 space. The wedge condition in this case becomes ∧2(π ⊗ 1 | F) = 0. Therefore, the image k) is an n-dimensional k-vector k. Let (π ⊗ 1)F is at most one dimensional. We have the following two cases. Suppose (π ⊗ 1)F = 0. Then F = (π ⊗ 1)(Λ0 ⊗OF0 k), namely, F is the worst point. Suppose (π ⊗1)F is one-dimensional. Then there exists a vector v ∈ F such that (π ⊗1)v generates (π ⊗ 1)F. For simplicity, write π for π ⊗ 1. Recall the k-bilinear form ϕ(−, −) : (Λ0 ⊗OF0 k) × (Λ0 ⊗OF0 k) −→ L ⊗OF0 k (x, y) (cid:55)→ s(x, π−1y) = t−1 Tr h(x, π−1y), where π−1 is the induced isomorphism Λ0 ⊗OF0 L ⊗OF0 of Λ0 ⊗OF0 k with k by sending t−1 ⊗ 1 to 1. Denote by N := k⟨em+1, πem+1⟩ the submodule k. Then one can check that the radical of ϕ is contained in N . We claim that k ∼−→ (π−1Λ0) ⊗OF0 k. We can identify πv is not in N . Otherwise, after rescaling, we may assume v = em+1 ⊗ 1 + πv1 for some v1 ∈ Λ0 ⊗OF0 k. Then for the quadratic form q : Λ0 ⊗OF0 k −→ L ⊗OF0 k ≃ k, 71 we have q(v) = q(em+1 ⊗ 1 + πv1) = q(em+1 ⊗ 1) + s(em+1 ⊗ 1, πv1) + q(πv1). One can check that q(em+1 ⊗ 1) = 1 and s(em+1 ⊗ 1, πv1) = q(πv1) = 0. Hence q(v) ̸= 0. This contradicts the hyperbolicity condition LM4 that q(F) = 0. In particular, we obtain that πv is not in the radical of ϕ. Thus, we can find w ∈ Λ0 ⊗OF0 k such that ϕ(w, πv) ̸= 0 in k. By rescaling, we may assume ϕ(w, πv) = 1. Note that for a ∈ k, q(w + av) = q(w) + as(w, v) + a2q(v) = q(w) + aϕ(w, πv) + 0, since q(v) = 0, = q(w) + a. Replacing w by w − q(w)v, we may assume q(w) = 0. Put b := −ϕ(w, v). One can check that ϕ(w + bπw) = 0. Replacing w by w + bπw, we have q(w) = q(v) = 0, ϕ(w, v) = 0 and ϕ(w, πv) = 1. Denote W1 := ⟨v, πv, w, πw⟩, the k-subspace of Λ0 ⊗OF0 k generated by v, πv, w, πw. Then ϕ restricts to a perfect pairing on W1. Now we can write Λ0 ⊗OF0 k = W1 ⊕ W, (4.1.22) where W is the orthogonal complement of W1 with respect to ϕ whose dimension is 2n − 4 over k. Note that the Condition LM3 in Definition 4.1.1 of M{0} implies that ϕ(F, F) = 0, and hence F ∩ ⟨w, πw⟩ = 0. Since ⟨v, πv⟩ ⊂ F and ϕ(F, F) = 0, we obtain that the k- dimension of F ∩ W is n − 2 and F ∩ W is contained in πW = ker(π | W ). Therefore, F ∩ W = πW for dimension reasons. By (4.1.22), we have disc′(ϕ) = disc(ϕ|W1)disc′(ϕ|W ). Here, disc′(ϕ) is the divided discriminant in the sense of Definition 6.2.4, and we view it as an element in k by using a basis of Λ0 ⊗OF0 k. By Example 6.2.6, we have disc′(ϕ) ∈ k × . 72 Since ϕ is perfect on W1, we obtain that disc(ϕ|W1) ∈ k × , and hence disc′(ϕ|W ) ∈ k × . So W is a hermitian quadratic module of type Λ0 over k in the sense of Definition 6.2.4. Set v1 := v and vn := w. By applying Theorem 6.2.7 to W , we deduce that there is an OF ⊗OF0 k- basis {vi : 1 ≤ i ≤ n} of Λ0 ⊗OF0 k with the property that q(vm+1) generates R, q(vi) = 0, ϕ(vi, vj) = 0 and ϕ(vi, πvj) = δi,n+1−j for all 1 ≤ i < j ≤ n. With respect to this basis, we have F = ⟨v, πv⟩ ⊕ (F ∩ W ) = ⟨v, πv⟩ ⊕ (πW ) = ⟨v1, πv1, πvi, 2 ≤ i ≤ n − 1⟩. This shows that points F ∈ M{0}(k) with dimk πF = 1 are in the same H{0}(k)-orbit. As Ufl {0} is flat over OF , we may view Ufl {0} as an open subscheme of Mloc {0} containing the worst point. By Lemma 4.1.22, the H{0}-translation of Ufl {0} covers Mloc {0}. By Proposition 4.1.20, we have shown Theorem 1.2.6, and Theorem 1.2.2, 1.2.3 in the case I = {0} and (R-U). 4.2 The case I = {0} and (R-P) In this section, we consider the case when F/F0 is of (R-P) type. In particular, we have π2 + π0 = 0 and π = −π. Consider the following ordered OF0-basis of Λ0 and Λs 0: Λ0 : 1 2 em+2, . . . , 1 2 en, e1, . . . , em, em+1, π 2 em+2, . . . , π 2 en, πe1, . . . , πem, πem+1, (4.2.1) Λs 0 : π−1em+2, . . . , π−1en, 2 π e1, . . . , 2 π em, π−1em+1, em+2, . . . , en, 2e1, . . . , 2em, em+1. (4.2.2) Recall that (Λ0, q, L ) is a hermitian quadratic module for L = 1 2OF0. 4.2.1 A refinement of Mnaive {0} in the (R-P) case Definition 4.2.1. Let M{0} be the functor M{0} : (Sch/OF )op −→ Sets which sends an OF -scheme S to the set of OS-modules F such that 73 LM1 (π-stability condition) F is an OF ⊗OF0 OS-submodule of Λ0 ⊗OF0 OS and as an OS- module, it is a locally direct summand of rank n. LM2 (Kottwitz condition) The action of π ⊗ 1 ∈ OF ⊗OF0 OS on F has characteristic polynomial det(T − π ⊗ 1 | F) = (T − π)(T − π)n−1. LM3 Let F ⊥ be the orthogonal complement in Λs 0 ⊗OF0 OS of F with respect to the perfect pairing s(−, −) : (Λ0 ⊗OF0 OS) × (Λs 0 ⊗OF0 OS) → OS. We require the map Λ0 ⊗OF0 to π 2 Λs 2 F ⊥ is the image of F ⊥ under the isomorphism π 2 F ⊥, where π 0) ⊗OF0 OS → ( π OS induced by Λ0 (cid:44)→ π 2 : Λs 2 Λs 0 ⊗OF0 0 sends F ∼−→ OS π 2 Λs 0 ⊗OF0 OS. LM4 (Hyperbolicity condition) The quadratic form q : Λ0 ⊗OF0 OS → L ⊗OF0 OS induced by q : Λ0 → L satisfies q(F) = 0. LM5 (Wedge condition) The action of π ⊗ 1 − 1 ⊗ π ∈ OF ⊗OF0 OS on F satisfies ∧2(π ⊗ 1 − 1 ⊗ π | F) = 0. Then as in the (R-U) case, the functor M{0} is representable and we have closed immer- sions Mloc {0} ⊂ M{0} ⊂ Mnaive {0} of projective schemes over OF , where all schemes have the same generic fiber. 4.2.2 An affine chart U{0} around the worst point Set F0 := (π ⊗ 1)(Λ0 ⊗OF0 k). Then we can check that F0 ∈ M{0}(k). We call it the worst point of M{0}. 74 With respect to the basis (4.2.1), the standard affine chart around F0 in Gr(n, Λ0)OF is (cid:1). We denote by U{0} the intersection of M{0} with the OF -scheme of 2n × n matrices (cid:0) X In the standard affine chart in Gr(n, Λ0)OF . The worst point F0 of M{0} is contained in U{0} and corresponds to the closed point defined by X = 0 and π = 0. The conditions LM1-5 yield the defining equations for U{0}. We will analyze each condition as in the (R-U) case. A reader who is only interested in the affine coordinate ring of U{0} may proceed directly to Proposition 4.2.2. 4.2.2.1 Condition LM1 Let R be an OF -algebra. With respect to the basis (4.2.1), the operator π ⊗ 1 acts on Λ0 ⊗OF0 R via the matrix    0 −π0In In 0    . Then the π-stability condition LM1 on F means there exists an n × n matrix P ∈ Mn(R) such that       0 −π0In     X In   =   X In   P. 0   In We obtain P = X and X 2 + π0In = 0. 4.2.2.2 Condition LM2 We have already shown that π ⊗ 1 acts on F via right multiplication of X. Then as in the (R-U) case, the Kottwitz condition LM2 translates to tr(X + πIn) = π − π = 2π, tr (cid:0)∧i(X + πIn)(cid:1) = 0, for i ≥ 2. (4.2.3) 4.2.2.3 Condition LM3 With respect to the bases (4.2.1) and (4.2.2), the perfect pairing s(−, −) : (Λ0 ⊗OF0 R) × (Λs 0 ⊗OF0 R) → R 75 0 0 0 0 0 0 0          and the map Λ0 ⊗OF0  R → π 2 Λs 0 ⊗OF0 R are represented respectively by the matrices 0 H2m 0 0 0         S = −H2m 0 −1 I2m 0 2 0 0          and N =    1     0   0 0 0 0 0 I2m 0 0 0  0   0       2 . Then the Condition LM3 translates to    X In  t       S  N   X In     = 0, or equivalently, 0 H2m 0  t      X In 0 0 −H2m 0 −2 0 0 0    2     0   0    = 0.    X In (4.2.4) Write X =    X1 X2 X3 x    , where X1 ∈ M2m(R), X2 ∈ M2m×1(R), X3 ∈ M1×2m(R) and x ∈ R. Then (4.2.4) translates to    X t 1H2m − H2mX1 2X t 3 − H2mX2 X t 2H2m − 2X3 0    = 0. 4.2.2.4 Condition LM4 Recall L = 1 2OF0. With respect to the basis (4.2.1), the induced L ⊗OF0 R-valued symmetric pairing on Λ0 ⊗OF0 R is represented by the matrix H2m 0 2 0 0 S1 =          0 0 0 0 0 0          . 0 π0H2m 0 0 2π0 76 The Condition LM4 translates to   t     t     X In   S1   X In   = 0 and half of the diagonal of   X In   S1   X In   equals zero. One can check that the diagonal entries of (cid:0) X In (cid:1)tS1 (cid:0) X In (cid:1) are indeed divisible by 2 in R. Equivalently, we obtain    X t 1H2mX1 + 2X t 3X3 + π0H2m X t 1H2mX2 + 2xX t 3 X t 2H2mX1 + 2xX3 X t 2H2mX2 + 2x2 + 2π0    = 0, half of the diagonal of X t 1 2 (cid:0)X t 2H2mX2 + 2x2 + 2π0 1H2mX1 + 2X t (cid:1) = 0. 3X3 + π0H2m equals 0, 4.2.2.5 Condition LM5 As π ⊗ 1 acts as right multiplication by X on F, the wedge condition on F translates to ∧2(X + πIn) = 0. 4.2.2.6 A simplification of equations As in the (R-U) case, we can simplify the above equations and obtain the following proposition. Proposition 4.2.2. The scheme U{0} = Spec OF [X]/I, where I is the ideal generated by: tr(X + πIn) − 2π, ∧2(X + πIn), X t 1H2m − H2mX1, 2X t 3 − H2mX2, (tr(X1 + πI2m) − 2π)H2m(X1 + πI2m) + 2X t 3X3, half of the diagonal of (tr(X1 + πI2m) − 2π)H2m(X1 + πI2m) + 2X t 3X3. Set (cid:101)X1 := X1 + πI2m, (cid:101)X :=    .    (cid:101)X1 X3 Then we have the following proposition. 77 Proposition 4.2.3. The scheme U{0} is isomorphic to Spec OF [ (cid:101)X]/(cid:101)I, where (cid:101)I is the ideal in OF [ (cid:101)X] generated by: ∧2 ( (cid:101)X), H2m (cid:101)X1 − (cid:101)X t 1H2m, (tr( (cid:101)X1) − 2π)H2m (cid:101)X1 + 2X t 3X3, half of the diagonal of (tr( (cid:101)X1) − 2π)H2m (cid:101)X1 + 2X t 3X3. Definition 4.2.4. Denote by Ufl {0} the closed subscheme of U{0} = Spec OF [ (cid:101)X]/(cid:101)I defined by the ideal (cid:101)I fl ⊂ OF [ (cid:101)X] generated by: ∧2 ( (cid:101)X), H2m (cid:101)X1 − (cid:101)X t 1H2m, ( 1 2 tr( (cid:101)X1) − π)H2m (cid:101)X1 + X t 3X3. Note that tr( (cid:101)X1) is divisible by 2 by the relation H2m (cid:101)X1 = (cid:101)X t 1H2m. 4.2.2.7 Global results As in the (R-U) case, we can prove the following proposition. Proposition 4.2.5. (1) The scheme Ufl {0} is flat over OF of relative dimension n − 1. In particular, Ufl {0} is isomorphic to an open subscheme of Mloc {0} containing the worst point. Furthermore, Ufl {0} is normal, Cohen-Macaulay, and smooth over OF on the complement of the worst point. The special fiber Ufl {0} ⊗OF k is (geometrically) reduced and irreducible. (2) U{0} and Ufl {0} have the same underlying topological space. Similar arguments as in the proof of Lemma 4.1.22 imply that the special fiber M{0} ⊗OF k has only two H{0}(k)-orbits. Together with Proposition 4.2.5, we can deduce Theorem 1.2.2 and 1.2.3 in the case I = {0} and (R-P). 78 CHAPTER 5 THE CASE I = {m} 5.1 The case I = {m} and (R-U) In this section, we will prove Theorem 1.2.2 in the case when F/F0 is of (R-U) type and I = {m}. In particular, we have π2 − tπ + π0 = 0, where t ∈ OF0 with π0|t|2. Consider the following ordered OF0-basis of Λm and Λs m: Λm : π t em+2, . . . , π t en, π−1e1, . . . , π−1em, em+1, π0 t em+2, . . . , π0 t en, e1, . . . , em, πem+1, (5.1.1) Λs m : πem+2, . . . , πen, t π e1, . . . , t π em, em+1, π0em+2, . . . , π0en, te1, . . . , tem, πem+1. (5.1.2) Recall (Λm, q, L ) is a hermitian quadratic module for L = t−1OF0. 5.1.1 A refinement of Mnaive {m} in the (R-U) case Definition 5.1.1. Let M{m} be the functor M{m} : (Sch/OF )op −→ Sets which sends an OF -scheme S to the set of OS-modules F such that LM1 (π-stability condition) F is an OF ⊗OF0 OS-submodule of Λm ⊗OF0 OS and as an OS- module, it is a locally direct summand of rank n. LM2 (Kottwitz condition) The action of π ⊗ 1 ∈ OF ⊗OF0 OS on F has characteristic polynomial det(T − π ⊗ 1 | F) = (T − π)(T − π)n−1. LM3 Let F ⊥ be the orthogonal complement in Λs m ⊗OF0 OS of F with respect to the perfect pairing s(−, −) : (Λm ⊗OF0 OS) × (Λs m ⊗OF0 OS) → OS. 79 We require that the map Λm ⊗OF0 Λm (cid:44)→ t−1Λs m sends F to t−1F ⊥, where t−1F ⊥ is the image of F ⊥ under the isomor- OS → (t−1Λs m) ⊗OF0 OS induced by the inclusion phism t−1 : Λs m ⊗OF0 OS ∼−→ t−1Λs m ⊗OF0 OS. LM4 (Hyperbolicity condition) The quadratic form q : Λm ⊗OF0 OS → L ⊗OF0 OS induced by q : Λm → L satisfies q(F) = 0. LM5 (Wedge condition) The action of π ⊗ 1 − 1 ⊗ π ∈ OF ⊗OF0 OS on F satisfies ∧2(π ⊗ 1 − 1 ⊗ π | F) = 0. Then M{m} is representable and we have closed immersions Mloc {m} ⊂ M{m} ⊂ Mnaive {m} of projective schemes over OF , where all schemes have the same generic fiber. 5.1.2 An affine chart U{m} around the worst point Set F0 := (π ⊗ 1)(Λm ⊗OF0 k). Then we can check that F0 ∈ M{m}(k). We call it the worst point of M{m}. With respect to the basis (5.1.1), the standard affine chart around F0 in Gr(n, Λm)OF is (cid:1). We denote by U{m} the intersection of M{m} with the OF -scheme of 2n × n matrices (cid:0) X In the standard affine chart in Gr(n, Λm)OF . The worst point F0 of M{m} is contained in U{m} and corresponds to the point defined by X = 0 and π = 0. The conditions LM1-5 yield the defining equations for U{m}. We will analyze each condition as in the case I = {0}. A reader who is only interested in the affine coordinate ring of U{m} may proceed directly to Proposition 5.1.2. 80 5.1.2.1 Condition LM1 Let R be an OF -algebra. With respect to the basis (5.1.1), the operator π ⊗ 1 acts on Λm ⊗OF0 R via the matrix    0 −π0In In tIn    . Then the π-stability condition LM1 on F means there exists an n × n matrix P ∈ Mn(R) such that         0 −π0In In tIn     X In   =   X In   P. We obtain P = X + tIn and X 2 + tX + π0In = 0. 5.1.2.2 Condition LM2 We have already shown that π ⊗ 1 acts on F via right multiplication of X + tIn. Then the Kottwitz condition LM2 translates to tr(X + πIn) = π − π, tr (cid:0)∧i(X + πIn)(cid:1) = 0, for i ≥ 2. (5.1.3) 5.1.2.3 Condition LM3 With respect to the bases (5.1.1) and (5.1.2), the perfect pairing s(−, −) : (Λm ⊗OF0 R) × (Λs m ⊗OF0 R) → R and the map Λm ⊗OF0 R → 1 t Λs m ⊗OF0 R are represented respectively by the matrices          S = 2 t H2m 0 H2m 2 t H2m 0 2π0 0 0 t H2m 0 1 0 and N =          0 1 0 2π0 t 81          0 0 0 I2m 0 0 0 t  0   0       t . 0 I2m 0 0 0 Then the Condition LM3 translates to    X In  t       S  N   X In     = 0, or equivalently,  t      X In          2 t H2m 0 H2m 0 2 0 H2m 0 2π0 t H2m 0 t 0 0 t 0 2π0             = 0.    X In It amounts to the following equation. ( 2 t X t + In)      X + X t     H2m 0 0 t H2m 0 0 t    +    2π0 t H2m 0 0 2π0    = 0. Note that the π-stability condition LM1 on F implies 2 t (X t)2 + 2X t + 2π0 t In = 0, and hence ( 2 t X t + In)2 = (1 − 4π0 t2 )In = θIn. (5.1.4) (5.1.5) Multiplying 2 t X t + In on both sides of (5.1.5), we can obtain      X = X t    H2m 0 0 t H2m 0 0 t   .   Write X =    X1 X2 X3 x    , where X1 ∈ M2m(R), X2 ∈ M2m×1(R), X3 ∈ M1×2m(R) and x ∈ R. Equivalently, we obtain H2mX1 = X t 1H2m, H2mX2 = tX t 3. 82 5.1.2.4 Condition LM4 Recall L = 1 t OF0. With respect to the basis (5.1.1), the induced L ⊗OF0 R-valued symmetric pairing on Λm ⊗OF0 R is represented by the matrix S1 =          2 t H2m 0 H2m 0 2 0 H2m 0 2π0 t H2m 0 t 0 0 t 0 2π0          . (5.1.6) The Condition LM4 translates to    X In  t     t     S1   X In   = 0 and half of the diagonal of   X In   S1   X In   equals zero. Equivalently, we obtain 2 t    X t 1H2mX1 + 2X t 3X3 + H2mX1 + X t 1H2m + half of the diagonal of 2 t X t 2H2m 2H2mX1 + 2xX3 + tX3 + X t 2 t 2H2mX2 + 2x2 + 2tx + 2π0) = 0. 1H2mX1 + 2X t X t 1 2 ( 2 t X t 2π0 t H2m 2 t 3 + H2mX2 + tX t 3 X t 1H2mX3 + 2xX t 2 t X t 2H2mX2 + 2x2 + 2tx + 2π0   = 0,  3X3 + H2mX1 + X t 1H2m + 2π0 t H2m equals 0, 5.1.2.5 Condition LM5 As π ⊗ 1 acts as right multiplication by X + tIn on F, the wedge condition LM5 on F translates to ∧2(X + πIn) = 0. 5.1.2.6 A simplification of equations As in the case I = {0}, we can simplify the above equations and obtain the following. Proposition 5.1.2. The scheme U{m} = Spec OF [X]/I, where I is the ideal generated by: tr(X + πIn) − π + π, ∧2(X + πIn), X t 1H2m − H2mX1, tX t 3 − H2mX2, half of the diagonal of ( tr(X1 + πI2m) + 2 √ 2 t θ)H2m(X1 + πI2m) + 2X t 3X3. 83 Set (cid:101)X1 := X1 + πI2m, (cid:101)X :=    .    (cid:101)X1 X3 Then we have the following proposition. Proposition 5.1.3. The scheme U{m} is isomorphic to Spec OF [ (cid:101)X]/(cid:101)I, where (cid:101)I is the ideal generated by ∧2 ( (cid:101)X), H2m (cid:101)X1 − (cid:101)X t 1H2m, half of the diagonal of ( 2 t tr( (cid:101)X1) + 2 √ θ)H2m (cid:101)X1 + 2X t 3X3. Definition 5.1.4. Denote by Ufl {m} the closed subscheme of U{m} = Spec OF [ (cid:101)X]/(cid:101)I defined by the ideal (cid:101)I fl ⊂ OF [ (cid:101)X] generated by ∧2 ( (cid:101)X), H2m (cid:101)X1 − (cid:101)X t 1H2m, ( tr( (cid:101)X1) t + Note that (cid:101)I ⊂ (cid:101)I fl. 5.1.2.7 Global results √ θ)H2m (cid:101)X1 + X t 3X3. We first give the results for the schemes U{m} and Ufl {m}. Proposition 5.1.5. (1) Ufl {m} is smooth over OF of relative dimension n − 1. The special fiber is geometrically reduced and irreducible. (2) U{m} and Ufl {m} have the same underlying topological space. Proof. The proof of (2) is similar as that of Lemma 4.1.19. Now we prove the smoothness of Ufl {m}. We use the notation as in the proof of Lemma 4.1.15. In particular, Rfl = (cid:16) ∧2(cid:0) Y X3 (cid:1), Y − Y t, ( 1 (cid:1)] OF [(cid:0) Y X3 t tr(H2mY ) + √ θ)Y + X t 3X3 (cid:17) . Then one can similarly show that D(yℓℓ) for 1 ≤ ℓ ≤ 2m is smooth over OF . Let z := √ 1 t tr(H2mY ) + have in Rfl[z−1] that θ. Consider the principal open subscheme D(z) = Spec Rfl[z−1]. Then we Y = −z−1X t 3X3. 84 Thus, Y is determined by X3 and Rfl[z−1] ≃ OF [X3] is smooth over OF . Note that the scheme Ufl {m} is covered by D(z) and D(yℓℓ) for 1 ≤ ℓ ≤ 2m. Hence, we conclude that Ufl {m} is smooth over OF . The special fiber is geometrically reduced by the smoothness. It is geometrically irreducible because the geometric special fibers of D(z) and D(yℓℓ) for 1 ≤ ℓ ≤ 2m are irreducible. Recall (Λm, q, L ) is a hermitian quadratic module over OF0 for L = 1 t OF0. Let H{m} := Sim((Λm, q, L )) be the group scheme over OF0 of similitude automorphisms of (Λm, q, L ). By Theorem 6.1.13, H{m} is an affine smooth group scheme over OF0. As in Lemma 4.1.21, the group scheme H{m} acts on M{m}. Lemma 5.1.6. Let k be the algebraic closure of the residue field k. Then M{m} ⊗OF k has two H{m} ⊗OF0 k-orbits, one of which consists of the worst point. Proof. Let F ∈ M{m}(k). In particular, the subspace F ⊂ (Λm ⊗OF0 k-vector space. The wedge condition LM5 in this case becomes ∧2(π⊗1 | F) = 0. Therefore, k) is an n-dimensional the image (π ⊗ 1)F is at most one dimensional. We have the following two cases. Suppose (π ⊗ 1)F = 0. Then F = (π ⊗ 1)(Λm ⊗OF0 k), namely, F is the worst point. Suppose (π ⊗1)F is one-dimensional. Then there exists a vector v ∈ F such that (π ⊗1)v generates (π⊗1)F. For simplicity, write π for π⊗1. Let f : (Λm⊗OF0 k)×(Λm⊗OF0 k) → L ≃ k denote the associated symmetric pairing on Λm ⊗OF0 k. As in the proof of Lemma 4.1.22, we see that πv is not in the radical of the paring f , because q(v) = 0. Then we can find some w ∈ Λm ⊗OF0 k such that f (w, πv) ̸= 0 in k. By rescaling, we may assume that f (w, πv) = 1. Similar arguments in Lemma 4.1.22 imply that after some linear transformations, we may assume q(w) = q(v) = f (w, v) = 0 and f (w, πv) = 1. 85 Let W1 := ⟨v, πv, w, πw⟩. Then f restricts to a perfect symmetric pairing on W1. Now we can write Λm ⊗OF0 k = W1 ⊕ W, (5.1.7) where W is the orthogonal complement of W1 with respect to f whose dimension is 2n − 4 over k. Since q(F) = 0, we have F ∩ ⟨w, πw⟩ = 0. Hence, we obtain that the k-dimension of F ∩ W is n − 2 and F ∩ W ⊂ πW = ker(π | W ). Therefore, F ∩ W = πW for dimension reasons. Note that the space W carries a structure of hermitian quadratic module. By (5.1.7), we have disc′(q) = disc(q|W1)disc′(q|W ). Here disc′(q) is the divided discriminant in the sense of Definition 6.1.8, and we view it as an element in k by using a basis of Λm ⊗OF0 k. By Example 6.1.10, we have disc′(q) ∈ k × . Since ϕ is perfect on W1, we obtain that disc(ϕ|W1) ∈ k × , and hence disc′(q|W ) ∈ k × . In particular, W is a hermitian quadratic module of type Λm over k in the sense of Definition 6.1.8. Applying Theorem 6.1.12 to W and using similar arguments as in the proof of Lemma 4.1.22, we can conclude that points F ∈ M{m}(k) with dimk πF = 1 are in the same orbit under the action of H{m} ⊗OF0 k. As Ufl {m} is flat over OF , we may view Ufl {m} as an open subscheme of Mloc {m} containing the worst point. By Lemma 5.1.6, the H{m}-translation of Ufl {m} covers Mloc {m}. Together with Proposition 5.1.5, we have proven Theorem 1.2.2 and 1.2.3 in the case I = {m} and (R-U). 5.2 The case I = {m} and (R-P) In this section, we consider the case when F/F0 is of (R-P) type and I = {m}. In particular, we have π2 + π0 = 0 and π + π = 0. Consider the following ordered OF0-basis of Λm and Λs m: Λm : 1 2 em+2, . . . , 1 2 en, π−1e1, . . . , π−1em, em+1, π 2 em+2, . . . , π 2 en, e1, . . . , em, πem+1, (5.2.1) 86 Λs m : em+2, . . . , en, 2 π e1, . . . , 2 π em, π−1em+1, πem+2, . . . , πen, 2e1, . . . , 2em, em+1. (5.2.2) Recall (Λm, q, L ) is a hermitian quadratic module for L = 2−1OF0. 5.2.1 A refinement of Mnaive {m} in the (R-P) case Definition 5.2.1. Let M{m} be the functor M{m} : (Sch/OF )op −→ Sets which sends an OF -scheme S to the set of OS-modules F such that LM1 (π-stability condition) F is an OF ⊗OF0 OS-submodule of Λm ⊗OF0 OS and as an OS- module, it is a locally direct summand of rank n. LM2 (Kottwitz condition) The action of π ⊗ 1 ∈ OF ⊗OF0 OS on F has characteristic polynomial det(T − π ⊗ 1 | F) = (T − π)(T − π)n−1. LM3 Let F ⊥ be the orthogonal complement in Λs m ⊗OF0 OS of F with respect to the perfect pairing s(−, −) : (Λm ⊗OF0 OS) × (Λs m ⊗OF0 OS) → OS. We require the map Λm ⊗OF0 sends F to 2−1F ⊥, where 2−1F ⊥ denotes the image of F ⊥ under the isomorphism OS induced by Λm (cid:44)→ 2−1Λs m OS → (2−1Λs m) ⊗OF0 2−1 : Λs m ⊗OF0 OS ∼−→ 2−1Λs m ⊗OF0 OS. LM4 (Hyperbolicity condition) The quadratic form q : Λm ⊗OF0 OS → L ⊗OF0 OS induced by q : Λm → L satisfies q(F) = 0. LM5 (Wedge condition) The action of π ⊗ 1 − 1 ⊗ π ∈ OF ⊗OF0 OS satisfies ∧2(π ⊗ 1 − 1 ⊗ π | F) = 0. LM6 (Strengthened spin condition) The line ∧nF ⊂ W (Λm) ⊗OF OS is contained in Im (cid:0)W (Λm)n−1,1 −1 ⊗OF OS → W (Λm) ⊗OF OS (cid:1) . 87 Here we use similar notations as in §4.1.1.1. Then M{m} is representable and we have closed immersions Mloc {m} ⊂ M{m} ⊂ Mnaive {m} of projective schemes over OF , where all schemes have the same generic fiber. 5.2.2 An affine chart U{m} around the worst point Set F0 := (π ⊗ 1)(Λm ⊗OF0 k). Then we can check that F0 ∈ M{m}(k). We call it the worst point of M{m}. With respect to the basis (5.1.1), the standard affine chart around F0 in Gr(n, Λm)OF is (cid:1). We denote by U{m} the intersection of M{m} with the OF -scheme of 2n × n matrices (cid:0) X In the standard affine chart in Gr(n, Λm)OF . The worst point F0 of M{m} is contained in U{m} and corresponds to the closed point defined by X = 0 and π = 0. The conditions LM1-6 yield the defining equations for U{m}. We will analyze each condition as in the (R-U) case. A reader who is only interested in the affine coordinate ring of U{m} may proceed directly to Proposition 5.2.2. 5.2.2.1 Condition LM1 Let R be an OF -algebra. With respect to the basis (5.2.1), the operator π ⊗ 1 acts on Λm ⊗OF0 R via the matrix    0 −π0In In 0    . Then the π-stability condition LM1 on F means there exists an n × n matrix P ∈ Mn(R) such that       0 −π0In   In 0 We obtain P = X and X 2 + π0In = 0.     X In   =   X In   P. 88 5.2.2.2 Condition LM2 We have already shown that π ⊗ 1 acts on F via right multiplication by X. Then the Kottwitz condition LM2 translates to tr(X + πIn) = π − π = 2π, tr (cid:0)∧i(X + πIn)(cid:1) = 0, for i ≥ 2. (5.2.3) 5.2.2.3 Condition LM3 With respect to the bases (5.2.1) and (5.2.2), the perfect pairing s(−, −) : (Λm ⊗OF0 R) × (Λs m ⊗OF0 R) → R and the map Λm⊗OF0 → 1 2Λs m ⊗OF0 R are represented respectively by the matrices J2m 0    1     0   0 and N =          I2m 0 0 0 0 0 0 0 0 −2π0 0 I2m 2 0 0 0          , 0 0 0 S =          0 0 −J2m 0 0 0 0 −1    0 Hm −Hm 0   . where J2m := Then the Condition LM3 translates to    X In  t       S  N   X In     = 0, or equivalently,  t      X In          Write 0 0 0 J2m 2 −J2m 0 0 0          0 0 0 2π0     X In   = 0. (5.2.4) 0 0 0 X =    X1 X2 X3 x    , 89 where X1 ∈ M2m(R), X2 ∈ M2m×1(R), X3 ∈ M1×2m(R) and x ∈ R. The Equation (5.2.4) translates to    2X t 3X3 + X t 1J2m − J2mX1 2xX t 3 − J2mX2 2xX3 + X t 2J2m 2x2 + 2π0    = 0. 5.2.2.4 Condition LM4 Recall L = 1 2OF0. With respect to the basis (5.2.1), the induced L ⊗OF0 R-valued symmetric pairing on Λm ⊗OF0 R is represented by the matrix          S1 = 0 0 0 J2m 2 −J2m 0 0 0          . 0 0 0 2π0 (5.2.5) 0 0 0 The Condition LM4 translates to   t     t     X In   S1   X In   = 0 and half of the diagonal of Equivalently, we obtain   X In    2X t 3X3 + X t 1J2m − J2mX1 2xX t 3 − J2mX2 2xX3 + X t 2J2m 2x2 + 2π0   S1   X In   equals zero.    = 0, x2 + π0 = 0, half of the diagonal of 2X t 3X3 + X t 1J2m − J2mX1 equals zero. 5.2.2.5 Condition LM5 Since π ⊗ 1 acts as right multiplication by X on F, the wedge condition LM5 on F translates to ∧2(X + πIn) = 0. 90 5.2.2.6 Condition LM6 As in §4.1.2.6, the strengthened spin condition LM6 in this case implies that X1 = J2mX t 1J2m, 2πX t 3 = J2mX2. 5.2.2.7 A simplification of equations As in the case I = {0}, we can simplify the above equations and obtain the following. Proposition 5.2.2. The scheme U{m} is a closed subscheme of U′ {m} := Spec OF [X]/I, where I is the ideal generated by: tr(X + πIn) − 2π, ∧2(X + πIn), X t 1J2m + J2mX1, 2πX t 3 − J2mX2, half of the diagonal of 2X t 3X3 + X t 1J2m − J2mX1. Set (cid:101)X1 := X1 + πI2m, (cid:101)X :=    .    (cid:101)X1 X3 As X2 and x are determined by X1 and X3 by relations in I, we obtain the following proposition. Proposition 5.2.3. The scheme U′ {m} is isomorphic to Spec OF [ (cid:101)X]/(cid:101)I, where (cid:101)I is the ideal generated by: ∧2 ( (cid:101)X), J2m (cid:101)X1 + (cid:101)X t 1J2m, half of the diagonal of 2X t 3X3 + (cid:101)X t 1J2m − J2m (cid:101)X1. Definition 5.2.4. Denote by Ufl {m} the closed subscheme of U′ {m} = Spec OF [ (cid:101)X]/(cid:101)I defined by the ideal I fl ⊂ OF [ (cid:101)X] generated by: ∧2 ( (cid:101)X), J2m (cid:101)X1 + (cid:101)X t 1J2m, X t 3X3 + (cid:101)X t 1J2m. Note that (cid:101)I ⊂ (cid:101)I fl. 91 5.2.2.8 Global results We first give results for the schemes U{m} and Ufl {m}. Proposition 5.2.5. (1) Ufl {m} is smooth over OF of relative dimension n − 1 with geomet- rically integral special fiber. (2) U{m} and Ufl {m} have the same underlying topological space. Proof. The proof of (2) is similar as that of Lemma 4.1.19. Now we prove the smoothness of Ufl {m}. It is clear from the expression of (cid:101)I fl that (cid:101)X1 is determined by X3, and hence, OF [ (cid:101)X]/(cid:101)I fl ≃ Spec OF [X3] ≃ An−1 OF , which is smooth over OF of relative dimension n − 1. The special fiber of Ufl {m} is isomorphic to An−1 k , which is geometrically integral. As Ufl {m} is flat over OF , we may view Ufl {m} as an open subscheme of Mloc {m} containing the worst point. Then as in Lemma 5.1.6, we can show that the special fiber M{m} ⊗OF k has only two orbits under the action of H{m} ⊗OF0 deduce Theorem 1.2.2 and 1.2.3 in the case I = {m} and (R-P). k. Together with Proposition 5.2.5, we 92 CHAPTER 6 NORMAL FORMS OF HERMITIAN QUADRATIC MODULES Let us keep the notations as in §3.3. In this chapter, we will show that, under certain conditions, hermitian quadratic modules ´etale locally have a normal form up to similitude. This is a variant of [RZ96, Theorem 3.16] in our setting. This result will be important when we relate the local models to Shimura varieties. In the following, we let Nilp := NilpOF0 denote the category of noetherian1 OF0-algebras such that π0 is nilpotent. We set t := π + π. In particular, t = 0 if F/F0 is of (R-P) type. For an OF0-algebra R and a ∈ OF , we will simply use a to denote the element a ⊗ 1 in OF ⊗OF0 hermitian quadratic module (M, q, L ), we will use f to denote the associated symmetric R, if there is no confusion. For a pairing on M , as in Definition 3.2.1. 6.1 Hermitian quadratic modules of type Λm The results in this subsection are essentially contained in [Ans18, §9], with some modifi- cations to the proof. Lemma 6.1.1 (cf. [Ans18, Lemma 9.6]). Let R ∈ Nilp. Let (M, q, R) be an R-valued hermitian quadratic module over R. Assume there exist v, w ∈ M such that f (v, πw) = 1 in R. Then there exist v′, w′ in the R-submodule spanned by {v, w, πv, πw} such that q(v′) = q(w′) = f (v′, w′) = 0 and f (v′, πw′) = 1. Proof. For r ∈ R, we have q(v + rπw) = q(v) + rf (v, πw) + r2π0q(w) = (π0q(w))r2 + r + q(v), 1If R is noetherian, then a finitely generated R-module M is projective if and only if there exists a finite Zariski open cover {Spec Ri}i∈I of Spec R such that MRi is free. 93 which can be viewed as a quadratic function of r. As 4π0 is nilpotent on R by assumption, there exists a sufficiently large integer N such that the sum 1 − 2π0q(v)q(w) + 2π2 0q(v)2q(w)2 + · · · + (−1)N (cid:19) (cid:18)1/2 N 4N πN 0 q(v)N q(w)N in R is a square root of 1 − 4π0q(v)q(w). Note that (cid:0)1/2 N (cid:1)4N lies in R by a direct computation of the 2-adic valuation. In particular, r0 := −1 + (1 − 4π0q(v)q(w))1/2 2π0q(w) ∈ R, and it is a solution for the quadratic equation q(v + rπw) = 0. Replacing v by v + r0πw, we may assume q(v) = 0. Similarly, we may assume q(w) = 0 by replacing w by w + rπv for suitable r in R. Set r1 := (1 − f (x, y)f (v, π2w))−1 and r2 := −r1f (v, w). Note that f (v, π2w) = f (v, (tπ − π0)w) = tf (v, πw) − π0f (v, w) = t − π0f (v, w) is nilpotent in R, so r1 indeed exists in R. Set v′ := r1v + r2πv. Then the straightforward computation implies that f (v′, w) = r1f (v, w) + r2f (πv, w) = r1f (v, w) + r2f (v, πw) = r1f (v, w) + r2 = 0 and f (v′, πw) = r1f (v, πw) + r2f (πv, πw) = r1 + r2f (v, π2w) = 1. Lemma 6.1.2. Let R be an OF0-algebra and M be a finite free OF ⊗OF0 R-module of rank d ≥ 1. Suppose b : M × M → R is a perfect R-bilinear pairing. Then there exists v, w ∈ M such that b(v, πw) = 1. Proof. By assumption, we may choose an R-basis {v1, . . . , v2d} of M such that vd+i = πvi for 1 ≤ i ≤ d. This basis yields a dual basis {v∨ 1 , . . . , v∨ 2d} of M ∨ := HomR(M, R) such that 94 v∨ i (vj) = b(vi, vj) = δij. Since b is perfect, we can find elements {w1, . . . , w2d} in M such that b(wi, vj) = v∨ i (vj) = δij for 1 ≤ i, j ≤ 2d. Set v := wd+1 and w := v1. Then we have b(v, πw) = b(wd+1, vd+1) = v∨ d+1(vd) = 1. Lemma 6.1.3. Let R be an OF0-algebra and M be a finite free OF ⊗OF0 R-module of rank d ≥ 1. Suppose b : M × M → R is an R-bilinear pairing on M such that b(πm1, m2) = b(m1, πm2) (6.1.1) for any m1 and m2 in M . Let N be a free (OF ⊗OF0 R)-submodule of M such that b restricts to a perfect pairing on N . Denote by N ⊥ := {m ∈ M | b(m, n) = 0 for any n ∈ N } the (left) orthogonal complement of N with respect to b. Then N ⊥ is a projective (OF ⊗OF0 R)-module and M = N ⊕ N ⊥ as OF ⊗OF0 R-modules. Proof. By construction, we have an exact sequence of R-modules 0 → N ⊥ α−→ M β −→ HomR(N, R), (6.1.2) where α denotes the inclusion map and β denotes the map m (cid:55)→ (n (cid:55)→ b(m, n)) for m ∈ M and n ∈ N . By (6.1.1), the R-submodule N ⊥ is also an OF ⊗OF0 φ ∈ HomR(N, R), define πφ ∈ HomR(N, R) by setting (πφ)(n) := φ(πn) for n ∈ N . This R-submodule. For any endows HomR(N, R) with the structure of an OF ⊗OF0 R-module, and the exact sequence (6.1.2) becomes an exact sequence of OF ⊗OF0 β is surjective with a section HomR(N, R) → N ⊂ M . It follows that M = N ⊕ N ⊥ as R-modules. Since b is perfect on N , the map OF ⊗OF0 R-modules and N ⊥ is projective. 95 Lemma 6.1.4 (cf. [Ans18, Lemma 9.2]). Let R be an OF0-algebra and let M be a free OF ⊗OF0 R-module of rank d. Then the functor HQF (M ) : (Sch/R)op −→ Sets S (cid:55)→ {OS-valued hermitian quadratic forms on M ⊗R OS} is represented by the affine space Ad2 R of dimension d2 over R. Proof. Choose a basis e1, . . . , ed of M over OF ⊗OF0 R. This is also a basis of M ⊗R OS. By the properties of hermitian quadratic forms, we can see that any hermitian quadratic form q : M ⊗R OS → OS is determined by values q(ei) for 1 ≤ i ≤ d and f (ei, ej), f (ei, πej) for 1 ≤ i < j ≤ d. More precisely, for any element m = (cid:80)d i=1(aiei + biπei) ∈ M ⊗R OS for ai, bi ∈ OS, we have q(m) = q( d (cid:88) i=1 aiei) + f ( d (cid:88) aiei, d (cid:88) biπei) + q( biπei) d (cid:88) i=1 = d (cid:88) i=1 a2 i q(ei) + i=1 (cid:88) 1≤i j. Then we define a map q as in (6.1.3). We can check that q is an OS-valued hermitian quadratic form. The proof of Lemma 6.1.4 also implies that the scheme HQF (M ) is (non-canonically) isomorphic to Spec R[A, B]/I, where A, B are two d×d matrices, and I is the ideal generated 96 by Aij − Aji, Bkℓ + Bℓk − tAkℓ, Bii − tAii for 1 ≤ i, j ≤ d and 1 ≤ k < ℓ ≤ d. Definition 6.1.5. Let (M, q, L ) be an L -valued hermitian quadratic module of rank d over some OF0-algebra R. Then as an R-module, the rank of M is 2d. We define the discriminant as the morphism disc(q) : ∧2d R M → ∧2d R (M ∨ ⊗R L ) ≃ ∧2d R (M ∨) ⊗R L 2d induced by the morphism M → M ∨ ⊗R L , m (cid:55)→ f (m, −). Here M ∨ denotes the R-dual module HomR(M, R). Example 6.1.6. Assume d = 1. Let x ∈ M be a generator of M over OF ⊗OF0 with respect to the basis {x, πx}, the symmetric pairing f : M × M → L associated with q R. Then is given by the matrix     2q(x) tq(x) tq(x) 2π0q(x)   . Using the above basis, the discriminant map can be identified with the determinant of the previous matrix, as an element in L 2. Therefore, disc(q) = (4π0 − t2)q(x)2. We find that when d = 1, the discriminant is “divisible” by 4π0 − t2. More generally, we have the following lemma. Lemma 6.1.7 (cf. [Ans18, Lemma 9.4]). Assume d ≥ 1 is odd. Then there exists a functorial factorization ∧2d R M disc′(q) R M ∨ ⊗R L 2d ⊗OF0 ∧2d disc(q) R M ∨ ⊗R L 2d ∧2d j (4π0 − t2) Here the map j is induced by the natural inclusion of the ideal (4π0 − t2) in OF0. 97 (cid:47) (cid:47) (cid:15) (cid:15) (cid:52) (cid:52) Proof. It suffices to prove this in the universal case, i.e., R is the ring where I is the ideal generated by R = OF0[A, B]/I, Aij − Aji, Bkℓ + Bℓk − tAkℓ, Bii − tAii for 1 ≤ i, j ≤ d and 1 ≤ k < ℓ ≤ d, and M is equipped with the universal quadratic form q : M → R given by d (cid:88) (aiei + biπei)) := q( (cid:88) Aijaiaj + (cid:88) Bijaibj + π0 (cid:88) Aijbibj, i=1 1≤i,j≤d 1≤i,j≤d 1≤i,j≤d for some R-basis (ei, πei)1≤i≤d of M . Under the chosen basis, the associated symmetric bilinear form f is given by the matrix   C :=   (cid:101)A B Bt π0 (cid:101)A   ∈ M2d,2d(R), (6.1.4) where (cid:101)Aii := 2Aii for 1 ≤ i ≤ d, (cid:101)Aij := Aij for i ̸= j, and the transpose matrix Bt of B equals t (cid:101)A − B. We may identify disc(q) with the determinant of the above matrix C. To finish the proof, we need to show that the ideal (disc(q)) is contained in the ideal (4π0 − t2) in R. As (4π0 − t2) becomes the unit ideal in R[1/π0], it suffices to show that the ideal (disc(q)) is contained in (4π0 − t2) in the localization Rm, where m is the ideal (π0). Equivalently, we need to show that disc(q) is divisible by 4π0 − t2 in Rm/mk for all k ≥ 1. We will argue by induction on the rank d. If d = 1, this follows by the computation in Example 6.1.6. Note that in the ring Rm/mk, the element Bij = f (ei, πej) is a unit for i ̸= j and π0 is nilpotent. In particular, we may assume f (e1, πe2) = 1. Then by Lemma 6.1.1, we may assume f restricting to the submodule R⟨e1, e2, πe1, πe2⟩ is given by the matrix          0 0 0 0 1 0 −1 0 0 −1 1 0 0 0 0 0          . 98 The determinant of the above matrix is one. In particular, f is perfect on R⟨e1, e2, πe1, πe2⟩. Then we can write M = R⟨e1, e2, πe1, πe2⟩ ⊕ M ′, where M ′ is the orthogonal complement of R⟨e1, e2, πe1, πe2⟩ in M with respect to f . The rank of M ′ over OF ⊗OF0 is odd. By induction, disc(q|M ′) is divisible by 4π0 − t2. Hence, disc(q) = disc(q|M ′) is also R is d − 2, which divisible by 4π0 − t2. Definition 6.1.8. We call the morphism disc′(q) in Lemma 6.1.7 the divided discriminant of q. If disc′(q) is an isomorphism, then we say (M, q, L ) is a hermitian quadratic module of type Λm. Example 6.1.9 (cf. [Ans18, Definition 9.7]). Let R be an OF0-algebra. Define Mstd,2 := (OF ⊗OF0 R)⟨e1, e2⟩ with hermitian quadratic form qstd,2 : Mstd,2 → R determined by qstd,2(e1) = qstd,2(e2) = 0, fstd,2(e1, e2) = 0, fstd,2(e1, πe2) = 1. For an odd integer n = 2m + 1, we define Mstd,n := M ⊕m std,2 ⊕ (OF ⊗OF0 R)en as an orthogonal direct sum and qstd,n(en) := 1. Viewing disc′(qstd,n) as an element in R, then we have disc′(qstd,n) = 1. Hence, (Mstd,n, qstd,n, R) is a hermitian quadratic module over R of type Λm. Example 6.1.10. By direct computation of the determinants of matrices (5.1.6) and (5.2.5), the hermitian quadratic module (Λm, q, ε−1OF0) is of type Λm. Lemma 6.1.11. Let S be a scheme. Let G be a smooth group scheme over S. Let X be a scheme over S equipped with a G -action ρ : G ×S X → X. Assume ρ is simply transitive in the sense that for any S-scheme T , the set X(T ) is either empty or the action of G (T ) 99 on X(T ) is simply transitive. If the structure morphism X → S is surjective, then X is an ´etale G -torsor over S. Proof. As ρ is simply transitive, we have an isomorphism Φ : G ×S X ∼−→ X ×S X, (g, x) (cid:55)→ (ρ((g, x)), x) by [Sta24, 0499]. As G → S is a smooth cover of S and smoothness is an fpqc local property on the target, the isomorphism Φ implies that X → S is smooth. If X → S is surjective, then X → S is a smooth cover of S. Let s : X → G ×S X be the morphism induced by the identity section of G . Then the composite Φ ◦ s gives a section of X ×S X → X. By [Sta24, 055V], we can find an ´etale cover {Ui}i∈I of S such that X ×S Ui → Ui has a section for each i ∈ I. Hence, we deduce that X is an ´etale G -torsor over S. Theorem 6.1.12 (cf. [Ans18, Theorem 9.10]). Let (M, q, L ) be a hermitian quadratic module of type Λm of rank n = 2m + 1 over R. Then (M, q, L ) is ´etale locally isomorphic to (Mstd,n, qstd,n, R) up to similitude. In particular, (M, q, L ) is ´etale locally isomorphic to (Λm, q, ε−1OF0) ⊗OF0 R up to similitude. Proof. Denote Gm := Sim(Mstd,n). It suffices to show that the sheaf F := Sim((Mstd,n, qstd,n, R), (M, q, L )) of similitudes is an ´etale Gm-torsor over R. Clearly, F is represented by an affine scheme of finite type over R. We next prove that F is smooth over R. Over R[1/π0], the quadratic form is determined by the associated symmetric pairing, and both Mstd and M are self-dual with respect to the symmetric pairing. Then by the arguments in [RZ96, Appendix to Chapter 3], we see that F is smooth and surjective over R[1/π0]. Hence, to show the smoothness of F over R, it suffices to prove that the morphism F → Spec OF is (formally) smooth at points over Spec R/π0R. For any surjection S → S in NilpR with nilpotent kernel J and a similitude (φ, γ) ∈ F(S), we need to show that there exists a lift of (φ, γ) to S. We argue by induction on the rank n. We denote 100 by e1, . . . , en the standard basis of Mstd,n. We reorder the basis such that q(em+1) = 1 and R)⟨ei, en+1−i⟩ ≃ Mstd,2. We claim that there exist elements v1, . . . , vn in M ⊗R S (OF ⊗OF0 and a generator u ∈ L ⊗R S such that vi = φ(ei) in M ⊗R S and q(vm+1) = u, q(vi) = f (vi, vj) = 0 and f (vi, πvj) = uδi,n+1−j for 1 ≤ i < j ≤ n and i, j ̸= m + 1. Then the maps φ : ei (cid:55)→ vi and γ : 1 (cid:55)→ u define a lift of (φ, γ). Thus, it suffices to prove the claim. Suppose n = 1. Set v1 := φ(e1) ∈ M ⊗RS. Then v1 is a generator of M ⊗RS. Pick any lift v1 ∈ M of v1. As disc′(q) is an isomorphism, q(v1) is a generator of L . Let u = q(v1). This proves the claim for n = 1. For n ≥ 3, pick lifts v1, . . . , vn in M ⊗R S such that vi = φ(ei). Let f be the associated symmetric pairing of M . Then f (v1, πvn) is a generator in L ⊗R S, as its reduction in L ⊗R S is a generator. Set u = f (v1, πvn). Using the generator u, we may identify L ⊗R S with S, and we may assume that f (v1, πv2) = 1 in L ⊗R S ≃ S. Note that as elements q(v1), q(v2) and f (v1, v2) reduce to zero in S by properties of v1 and v2, they lie in the kernel J. Then the linear transformation in Lemma 6.1.1 does not change the reduction of v1 and v2, and hence, we may assume that q(v1) = q(vn) = f (v1, vn) = 0 and f (v1, πvn) = 1. Then f is perfect on the S-submodule N generated by v1, vn, πv1, πvn. Let N ⊥ be the orthogonal complement of N in M ⊗R S. Then N ⊥ ⊗R S is the OF ⊗OF0 M ⊗R S generated by v2, . . . , vn−1. For 2 ≤ i ≤ n − 1, we can write vi = w′ + w, where S-submodule in w′ ∈ N ⊥ and w ∈ N . As vi is orthogonal to N , we have w is orthogonal to N . Since f is perfect on N , we obtain w = 0. In particular, we may choose vi in N ⊥ as a lift of vi for 2 ≤ i ≤ n − 1. Now the claim follows by induction on the rank of M , and we deduce the (formal) smoothness of F over R. Note that the same proof implies that the group scheme Gm is smooth over R. As the Gm-action on F is simply transitive by construction, by Lemma 6.1.11, it remains to show that F is a surjective scheme over R. Since we have already shown that F is surjective over 101 R[1/π0], it suffices to prove the surjectivity of F over R/π0R. Then we may assume R = k is the algebraic closure of the residue field k of OF0 and L = k. We need to show that there exists a similitude isomorphism (φ, γ) between (Mstd,n, qstd,n, k) and (M, q, k). For the case n = 1, we can construct a similitude as in the previous paragraph. For n ≥ 3 odd, we first claim that there exist v and w in M such that f (v, πw) = 1. Otherwise, under a basis of the form (v1, . . . , vn, πv1, . . . , πvn), the pairing f corresponds to the 2n × 2n matrix       (cid:101)A 0 0 0 for some n × n matrix (cid:101)A, where (cid:101)Aii = 2q(vi) = 0 for 1 ≤ i ≤ n and (cid:101)Aij = f (vi, vj) for i ̸= j. Suppose for some indices i0 ̸= j0, we have f (vi0, vj0) ̸= 0. We may assume f (v1, v2) ̸= 0. Then by a suitable linear transformation of the basis v1, . . . , vn, we may assume that (cid:101)A is of the form       0 1 1 0 0       0 (cid:101)A1 In particular, M1 := (OF ⊗OF0 k)⟨v1, v2⟩ and M2 := (OF ⊗OF0 k)⟨v3, . . . , vn⟩ are orthogonal complement of each other. Then disc′(q) = disc(q|M1)disc′(q|M2). However, disc(q|M1) = det                   0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0. This contradicts the assumption that disc′(q) is a unit. Then we see f (vi, vj) = 0 for any i ̸= j, i.e., (cid:101)A is a diagonal matrix. Hence, M is an orthogonal direct sum of rank one 102 hermitian quadratic modules. This also contradicts disc′(q) ̸= 0. Then we conclude that there exist v and w in M such that f (v, πw) = 1. Then as in Lemma 6.1.1, we may assume that f restricting to (OF ⊗OF0 k)⟨v, w⟩ corresponds to the matrix  0   0       1 0 0 0 −1 0 0 −1 0 0 0  1         0 0 . Hence, (OF ⊗OF0 k)⟨v, w⟩ is isomorphic to Mstd,2. Its orthogonal complement is a hermitian quadratic module of type Λm of rank n − 2. Now we can finish the proof by induction on the rank of M . Theorem 6.1.13 (cf. [Ans18, Proposition 9.9]). The group functor Sim(Λm) is representable by an affine smooth group scheme over OF0 whose generic fiber is GU(V, h). Proof. By the proof of Theorem 6.1.12, the functor Sim(Λm) is representable by an affine smooth group scheme of finite type over OF0. It remains to prove the assertion for the generic fiber. Following the notations in §3.2, we denote by s the symmetric pairing on Λm. For any F0-algebra R, we have Sim(Λm)(R) = = = =             (φ, γ) φ is an automorphism of the OF ⊗OF0 R ∼−→ L ⊗OF0 γ : L ⊗OF0 q(φ(x)) = γ(q(x)) for x ∈ Λm ⊗OF0 R R = V ⊗F0 R R-module Λm ⊗OF0 R    φ ∈ GLF ⊗F0 R(V ⊗F0 R) γ : R ∼−→ R s(φ(x), φ(y)) = γ (s(x, y)) for x, y ∈ V ⊗F0 R    φ ∈ GLF ⊗F0 R(V ⊗F0 F ) s(φ(x), φ(y)) = c(φ)s(x, y) for x, y ∈ V ⊗F0 R and some c(φ) ∈ R× φ ∈ GLF ⊗F0 R(V ⊗F0 F ) h(φ(x), φ(y)) = c(φ)h(x, y) for x, y ∈ V ⊗F0 R and some c(φ) ∈ R×       103 = GU(V, h)(R). Therefore, the generic fiber of Sim(Λm) is GU(V, h). Corollary 6.1.14. The scheme Sim(Λm) is isomorphic to the parahoric group scheme at- tached to Λm. Proof. Let ˘F0 denote the completion of the maximal unramified extension of F0. By con- ) is the stabilizer of Λm in GU(V, h)( ˘F0), which is a struction, we know that Sim(Λ)(O ˘F0 parahoric subgroup by Proposition 2.4.1. As Sim(Λ) is smooth over OF0 by Theorem 6.1.13, the corollary follows by [BT84a, 1.7.6]. 6.2 Hermitian quadratic modules of type Λ0 Let R be an OF0-algebra. Recall that in Definition 3.2.3, we have defined the category CR of hermitian quadratic modules with ϕ. By a similar proof as in Lemma 6.1.4, we can show that for a fixed free OF ⊗OF0 R-module M of rank d, the moduli functor of all bilinear forms ϕ and quadratic forms q on M satisfying (3.2.4) in Definition 3.2.3 is representable by the affine space of dimension d2 over R. Let (M, q, L , ϕ) ∈ CR. Choose a basis (e1, . . . , ed, πe1, . . . , πed) of M . The pairing ϕ is then given by the matrix     (cid:101)A (cid:101)B t (cid:101)A − (cid:101)B π0 (cid:101)A   , where (cid:101)Aii = (t/π0)q(ei) and (cid:101)Bii = 2q(ei) for 1 ≤ i ≤ d, (cid:101)Aij = ϕ(ei, ej) and (cid:101)Bij = ϕ(ei, πej) for 1 ≤ i, j ≤ d and i ̸= j, and they satisfy (cid:101)A = − (cid:101)At + (t/π0) (cid:101)B and (cid:101)B = (cid:101)Bt. Definition 6.2.1. Let (M, q, L , ϕ) ∈ CR and the rank of M over R is 2d. We define the discriminant as the morphism disc(ϕ) : ∧2d R M → ∧2d R (M ∨ ⊗R L ) ≃ ∧2d R (M ∨) ⊗R L 2d induced by the morphism M → M ∨ ⊗R L , m (cid:55)→ ϕ(m, −). 104 Example 6.2.2. Assume d = 1. Let x ∈ M be a generator of M over OF ⊗OF0 (M, q, L ) is a hermitian quadratic module. Then we can define a bilinear form ϕ : M ×M → R. Suppose L given by the matrix    t/π0q(x) 2q(x) (t2 − 2π0)/π0q(x) tq(x)    with respect to the basis {x, πx}. Equipped with such ϕ, we have (M, q, L , ϕ) ∈ CR. Using the basis {x, πx}, we may view the discriminant map disc(ϕ) as the determinant of the above matrix. We have disc(ϕ) = 4π0 − t2 π0 q(x)2. Arguing similarly as in Lemma 6.1.7, we can show the following result. Lemma 6.2.3. Assume d ≥ 1 is odd. Then there exists a functorial factorization disc(ϕ) R M ∨ ⊗R L 2d ∧2d ∧2d R M disc′(ϕ) R M ∨ ⊗R L 2d ⊗OF0 ∧2d ( 4π0−t2 π0 j ) Here the map j is induced by the natural inclusion of the ideal ( 4π0−t2 π0 ) in OF0. Proof. As in the proof of Lemma 6.1.7, we can reduce to show that the determinant, which equals disc(ϕ), of a matrix of the form     (cid:101)A (cid:101)B t (cid:101)A − (cid:101)B π0 (cid:101)A   ∈ M2d,2d(R), is divisible by (4π0 − t2)/π0 in R, where (cid:101)Aii = (t/π0)q(ei) and (cid:101)Bii = 2q(ei) for 1 ≤ i ≤ d, (cid:101)Aij = ϕ(ei, ej) and (cid:101)Bij = ϕ(ei, πej) for 1 ≤ i, j ≤ d and i ̸= j, and they satisfy (cid:101)A = − (cid:101)At + (t/π0) (cid:101)B and (cid:101)B = (cid:101)Bt. If d = 1, then the lemma follows by Example 6.2.2. Suppose d ≥ 3. We may assume π0 is nilpotent in R and B12 = ϕ(e1, πe2) = 1 as in the proof of Lemma 6.1.7. As in Lemma 105 (cid:47) (cid:47) (cid:15) (cid:15) (cid:52) (cid:52) 6.1.1, replacing e1 by r1e1 + r2πe1 for suitable r1 and r2 in R, we may assume further that ϕ(e1, e2) = 0. Then restricting to the submodule ⟨e1, e2, πe1, πe2⟩, the pairing ϕ is given by the matrix          t π0 q(e1) t π0 t2−2π0 π0 q(e1) 0 2q(e1) 1 t π0 q(e2) 1 2q(e2) −1 tq(e1) 0 t2−π0 π0 t2−2π0 π0 q(e2) t tq(e2)          . By direct computation, the above is an invertible matrix, and hence the pairing ϕ is perfect on the module ⟨e1, e2, πe2, πe2⟩. Therefore, the orthogonal complement M ′ of ⟨e1, e2, πe2, πe2⟩ in M has rank n − 2 over OF ⊗OF0 R, and M ′ ∈ CR. Then we finish the proof by induction on the rank of M . Definition 6.2.4. Let R be an OF0-algebra. We say that a hermitian quadratic module (M, q, L , ϕ) ∈ CR over R is of type Λ0 if disc′(ϕ) is an isomorphism. Example 6.2.5. Let R be an OF0-algebra. (1) Suppose (M, q, R) is a hermitian quadratic module of rank one. Let x ∈ M be a generator and assume q(x) = 1. We can define a bilinear form ϕstd,1 : M × M → R as in Example 6.2.2. Then (M, q, L , ϕstd,1) ∈ CR. Viewing disc′(ϕstd,1) as an element in R, we have disc′(ϕstd,1) = 1. (2) Define Nstd,2 := (OF ⊗OF0 R)⟨e1, e2⟩ with hermitian quadratic form qstd,2 : Nstd,2 → R determined by qstd,2(e1) = qstd,2(e2) = 0, ϕstd,2(e1, e2) = 0, ϕstd,2(e1, πe2) = 1. For an odd integer n = 2m + 1, we define Nstd,n := N ⊕m std,2 ⊕ (OF ⊗OF0 R)en. 106 Here (OF ⊗OF0 direct sum is an orthogonal direct sum with respect to ϕstd,n := ϕ⊕m R)en is a hermitian quadratic module of rank one as in (1), and the std,2 ⊕ ϕstd,1. Viewing disc′(ϕstd,n) as an element in R, we have disc′(ϕstd,n) = 1. Hence, (Nstd,n, qstd,n, R, ϕstd,n) is a hermitian quadratic module over R of type Λ0. Example 6.2.6. Equipped with the following bilinear form ϕ(−, −) : Λ0 × Λ0 −→ L = ε−1OF0, (x, y) (cid:55)→ s(x, π−1y) = ε−1 TrF/F0 h(x, π−1y), the hermitian quadratic module (Λ0, q, ε−1OF0, ϕ) is of type Λ0. Theorem 6.2.7. Let (M, q, L , ϕ) be a hermitian quadratic module of type Λ0 of rank n = 2m + 1 over R. Then (M, q, L , ϕ) is ´etale locally isomorphic to (Nstd,n, qstd,n, R, ϕstd,n) up to similitude. In particular, (M, q, L , ϕ) is ´etale locally isomorphic to (Λ0, q, ε−1OF0, ϕ) ⊗OF0 up to similitude. R Proof. As in the proof of Theorem 6.1.12, it suffices to show that the representable sheaf F := Sim((Nstd,n, qstd,n, R, ϕstd,n), (M, q, L , ϕ)) of similitudes is surjective over R and smooth at points over Spec R/π0R. We first check that for any surjection S → S in NilpR with nilpotent kernel J and a similitude (φ, γ) ∈ F(S), there exists a lift of (φ, γ) to S. We denote by e1, . . . , en the standard basis of Nstd,n. We reorder the basis such that q(em+1) = 1 and (OF ⊗OF0 R)⟨ei, en+1−i⟩ ≃ Nstd,2. We claim that there exist lifts vi ∈ M ⊗R S of vi := φ(ei) for 1 ≤ i ≤ n and a generator u ∈ L ⊗R S such that q(vm+1) = u, q(vi) = ϕ(vi, vj) = 0 and ϕ(vi, πvj) = uδi,n+1−j for 1 ≤ i < j ≤ n and i, j ̸= m + 1. The the maps φ : ei (cid:55)→ vi and γ : 1 (cid:55)→ u defines a lift of (φ, γ) and (φ, γ) preserves ϕ. Thus it suffices to prove the claim. 107 Suppose n = 1. Pick any lift v1 of v1. As disc′(ϕ) is an isomorphism, q(v1) is a generator of L ⊗R S. Set u = q(v1). This proves the claim for n = 1. For n ≥ 3, pick any lifts v1, . . . , vn in M ⊗R S of v1, . . . , vn. As in the proof of Theorem 6.1.12, we may assume that L ⊗R S ≃ S and ϕ(v1, πvn) = 1 in S. Let r0 ∈ R be a solution of the quadratic equation q(vn)r2 + r + q(v1) = 0, which exists by arguments in Lemma 6.1.1. Since q(v1) and q(vn) lie in J, we have r0 ∈ J. Then v′ 1 := v1 + r0vn and v′ 1 = v1. So we may find a lift v′ n such that ϕ(v′ 1, v′ n) = 1. Set v′′ n := v′ n − q(v′ n)v′ 1. Then q(v′′ n) = 0 and v′′ n = vn. Set r1 := (1 − ϕ(v′ 1, v′′ n)ϕ(v′ 1, π2v′′ n))−1 and r2 := −r1ϕ(v′ 1, v′′ n). Since (φ, γ) preserves ϕ, we have ϕ(v′ 1, v′′ n) = γ(ϕstd,n(e1, en)) = 0. Thus, ϕ(v′ 1, v′′ n) and r2 are in J. Set v′′ 1 := r1v′ 1 + r2πv′ 1. Then v′′ 1 = v. As in Lemma 6.1.1, we have ϕ(v′′ 1 , πv′′ n) = 1 and ϕ(v′′ 1 , v′′ n) = 0. By replacing v1 by v′′ 1 and vn by v′′ n, we may assume that q(v1) = q(vn) = ϕ(v1, vn) = 0 and ϕ(v1, πvn) = 1. Then ϕ is perfect on the S-submodule N generated by v1, v2, πv1, πv2. Let N ⊥ be the orthogonal complement (with respect to ϕ) of N in M ⊗R S. As in the proof of Theorem 6.1.12, we may assume that lifts vi for 2 ≤ i ≤ n − 1 lie in N ⊥. The claim follows by induction on the rank of M , and hence, we deduce the smoothness of F over R. Next we prove the surjectivity of F over R. It suffices to prove that F has non-empty fibers over R/π0R. Then we may assume R = k is the algebraic closure of the residue field of OF0 and L = k. We need to show that there exists a similitude isomorphism (φ, γ) preserving ϕ between (Nstd,n, qstd,n, k, ϕstd,n) and (M, q, k, ϕ). Suppose n = 1. Then M ⊗R S = (OF ⊗OF0 S)v for some v. Define φ : Nstd ⊗R S −→ M ⊗R S = (OF ⊗OF0 S)v, γ :S −→ L ⊗R S e1 (cid:55)→ v, 1 (cid:55)→ q(v). As disc′(ϕ) is an isomorphism, q(v) is a generator. Since ϕ is determined by q in this case by computation in Example 6.2.2, the similitude (φ, γ) preserves ϕ. For n ≥ 3 odd, we claim 108 that there exist v and w in M ⊗R S such that ϕ(v, πw) = 1. This can be done using proof by contradiction as in Theorem 6.1.12. Set v′ := v + r0w, where r0 ∈ k is a solution for the quadratic equation q(v′) = q(w)r2 + r + q(v). Then ϕ(v′, πw) = ϕ(v, πw) + r0ϕ(w, πw) = 1 + 2r0q(w) = 1. The last equality holds since char k = 2. Set w′ := w − q(w)v′. Then q(w′) = 0. As in the previous paragraph, we may find suitable r1 and r2 such that v′′ := r1v′ + r2πv′ satisfies ϕ(v′′, πw) = 1 and ϕ(v′′, w′) = 0. Replacing v by v′′ and w by w′, we see that ϕ restricting to (OF ⊗OF0 k)⟨v, w⟩ acts the same as ϕstd,2. In particular, the subspace (OF ⊗OF0 k)⟨v, w⟩ is isomorphic to Nstd,2. Its orthogonal complement is a hermitian quadratic module of type Λ0 of rank n − 2. Now we can finish the proof by induction on the rank of M . Theorem 6.2.8. The group functor Sim((Λ0, ϕ)) of similitudes preserving ϕ is representable by an affine smooth group scheme over OF0 whose generic fiber is GU(V, h). Proof. By the proof of Theorem 6.2.7, the functor Sim((Λ0, ϕ)) is representable by an affine smooth group scheme over OF0. It remains to show the assertion for the generic fiber. Let R be an F -algebra. For any similitude (φ, γ) ∈ Sim(Λ0) and x, y ∈ Λ0 ⊗OF0 R = V ⊗F0 R, we have ϕ(φ(x), φ(y)) = ϕ(φ(x), π(π−1φ(y))) = q(φ(x) + φ(π−1y)) − q(φ(x)) − q(φ(π−1y)) = γ(q(x + π−1y) − q(x) − q(π−1y)) = γ(ϕ(x, y)). Hence, over the generic fiber, any similitude of Λ0 preserves ϕ. Then as in the proof of Theorem 6.1.13, we see that the generic fiber of Sim((Λ0, ϕ)) is GU(V, h). The same argument as in the proof of Corollary 6.1.14 implies the following. Corollary 6.2.9. The scheme Sim((Λ0, ϕ)) is isomorphic to the parahoric group scheme attached to Λ0. 109 CHAPTER 7 2-ADIC INTEGRAL MODELS OF SHIMURA VARIETIES In this chapter, we will constuct 2-adic integral models of Shimura varieties of abelian type with parahoric level structure. Our goal is to prove Theorem 1.2.7 in the Introduction. 7.1 p-divisible groups and Lau’s classification In this section, we review Lau’s work [Lau14] on the classification of 2-divisible groups in terms of Dieudonn´e displays. We generalize the construction of the natural “connection isomorphisms” for Dieudonn´e pairs in [KPZ24] to the case p = 2. We also compare Lau’s classification of p-divisible groups with Breuil-Kisin’s classfication. 7.1.1 Zink rings, frames and windows Let (R, mR, k) be an artinian local ring (or more generally an admissible ring in the sense of [Lau14, §1]) with residue field k. Denote by W (R) its associated Witt ring equipped with Frobenius φ and Verschiebung V . By [Lau14, §1B], the exact sequence 0 → W (mR) → W (R) → W (k) → 0 has a unique ring homomorphism section s : W (k) → W (R), which is φ-equivariant. Definition 7.1.1 ([Zin01]). The Zink ring of R is W(R) = sW (k)⊕(cid:99)W (mR), where (cid:99)W (mR) ⊂ W (mR) consists of elements (x0, x1, . . .) ∈ W (mR) such that xi = 0 for almost all i. The Zink ring W(R) is a φ-stable subring of W (R). If p = 2, W(R) is in general not stable under the Verschiebung V . We need to modify V as follows. The element p−[p] ∈ W (Zp) lies in the image of V because it maps to zero in Zp. Moreover, the element V −1(p−[p]) ∈ W (Zp) is a unit, since it maps to 1 in W (Fp). Define u0 :=    V −1(2 − [2]) if p = 2, 1 if p ≥ 3. (7.1.1) The image of u0 ∈ W (Zp)× in W (R)× is also denoted by u0. For x ∈ W (R), set V(x) := V (u0x). 110 Lemma 7.1.2 ([Lau14, Lemma 1.7]). The map V : W (R) → W (R) satisfies V(W(R)) ⊂ W(R). Moreover, there is an exact sequence 0 → W(R) V −→ W(R) w0−→ R → 0. Remark 7.1.3. We will call the map V : W(R) → W(R) the modified Verschiebung for W(R). Many statements about W(R) in the case p = 2 are proven by adapting the corresponding proofs for p > 2, with adjustments for the modified Verschiebung map. Now we recall the logarithm coordinates of the Witt ring, see [Lau14, §1C]. Let (S → R, δ) be a divided power extension of rings with kernel a ⊂ S. Denote by aN the additive group (cid:81) i∈N a, equipped with a W (S)-module structure x[a0, a1, . . .] := [w0(x)a0, w1(x)a1, . . .] for x ∈ W (S) and [a0, a1, . . .] ∈ (cid:81) i∈N a. Then the δ-divided Witt polynomials w′ n define an isomorphism of W (S)-modules Log : W (a) ∼−→ aN a = (a0, a1, . . .) (cid:55)→ [w′ 0(a), w′ 1(a), . . .] where w′ n(X0, . . . , Xn) = (pn − 1)!δpn(X0) + (pn−1 − 1)!δpn−1(X1) + · · · + Xn. For x ∈ W (a), we call Log(x) the logarithmic coordinate of x. In terms of logarithmic coordinates, the Frobenius and Verschiebung of W (a) act on aN as φ([a0, a1, . . .]) = [pa1, pa2, . . .], V ([a0, a1, . . .]) = [0, a0, a1, . . .]. (7.1.2) Moreover, Log induces an injective map Log : (cid:99)W (a) (cid:44)→ a(N), 111 which is bijective when the divided powers δ are nilpotent. Here, the group (cid:99)W (a) denotes the set of elements (a0, a1, . . .) ∈ W (a) such that ai = 0 for almost all i, and a(N) ⊂ aN denotes (cid:76) i∈N a. The ideal a ⊂ W (S) is by definition the set of elements whose logarithmic coordinates are of the form [a, 0, 0, . . .], a ∈ a. Definition 7.1.4. For a (Noetherian) complete local ring R with residue field k, we set W(R) := lim ←− n W(R/mn R). For a complete local ring R, we can define the modified Verschibung V on W(R) by passing to the limit. Then W(R) is a subring of W (R) := lim ←−n W (R/mn R), which is stable under φ and V. We also have W(R)/V(W(R)) ≃ R, see [Lau14, §1E]. Note that W(R) is p-adically complete by [Lau14, Proposition 1.14]. Here, we introduce notions of frames and windows following [Lau10, §2] and [Lau14, §2]. Definition 7.1.5. (1) A frame is a quintuple F = (S, I, R, σ, σ1), where S and R = S/I are rings, σ : S → S is a ring endomorphism with σ(a) ≡ ap mod pS, σ1 : I → S is a σ-linear map of S-modules whose image generates S as an S-module, and I + pS lies in the Jacobson radical of S. A frame is called a lifting frame if all projective R-modules of finite type can be lifted to projective S-modules. (2) A homomorphism of frames α : F −→ F ′ = (S′, I ′, R′, σ′, σ′ 1) is a ring homomorphism α : S → S′ with α(I) ⊂ I ′ such that σ′α = ασ and σ′ 1α = u · ασ1 for a unit u ∈ S′, which is then determined by α. We say that α is a frame u-homomorphism. If u = 1, then α is called strict. (3) Let F be a frame. A window over F (or F-window) is a quadruple P = (M, M1, F, F1), 112 where M is a projective S-module of finite type with a submodule M1 such that there exists a decomposition of S-modules M = L ⊕ T with M1 = L ⊕ IT , called a normal decomposition, and where F : M → M and F1 : M1 → M are σ-linear maps of S-modules with F1(ax) = σ1(a)F (x) for a ∈ I and x ∈ M , and F1(M1) generates M as an S-module. Remark 7.1.6. If F is a lifting frame, then the existence of a normal decomposition in (3) of the above definition is equivalent to that M/M1 is a projective R-module. A frame is a lifting frame if S is local or I-adic. A u-homomorphism α : F → F ′ induces a base change functor α∗ : (windows over F) −→ (windows over F ′) (7.1.3) from the category of windows over F to the category of windows over F ′. In terms of normal representations, the functor α∗ is given by (L, T, Ψ) (cid:55)→ (S′ ⊗S L, S′ ⊗S T, Ψ′) with Ψ′(s′ ⊗ l) = uσ′(s′) ⊗ Ψ(l) and Ψ′(s′ ⊗ t) = σ′(s′) ⊗ Ψ(t). Definition 7.1.7. A frame homomorphism α : F → F ′ is called crystalline if the functor α∗ is an equivalence of categories. Note that for a frame F = (S, I, R, σ, σ1), there is a unique element θ ∈ S such that σ(a) = θσ1(a) for all a ∈ I. For an S-module M , we write M (σ) = S ⊗σ,S M . Then for a window P = (M, M1, F, F1) over F, by [Lau14, Lemma 2.3], there exists a unique S-linear map V ♯ : M −→ M (σ) (7.1.4) such that V ♯(F1(x)) = 1 ⊗ x for x ∈ M1. It satisfies F #V ♯ = θ and V ♯F # = θ, where F # : M (σ) → M is the linearization of F . 113 Example 7.1.8. For a complete local ring R with perfect residue field, we will be interested in the following (lifting) frames: (1) the Dieudonn´e frame DR := (W(R), IR, R, φ, φ1), where IR = ker(w0 : W(R) → R) and φ1 : IR → W(R) is the inverse of V; (2) assume R = OK for some finite extension K of Qp with residue field k, choose a presentation R = S/ES, where S = W (k)[[u]] and E ∈ S is an Eisenstein polynomial with constant term p. Define the Breuil-Kisin frame B := (S, ES, R, φ, φ1), where φ : S → S acts on W (k) as usual Frobenius and sends u to up, and φ1(Ex) := φ(x) for x ∈ S. 7.1.2 Dieudonn´e displays and Dieudonn´e pairs Let R be a complete local ring with perfect residue field of characteristic p. By Re- mark 7.1.6, a window over DR (also called a Dieudonn´e display over R later) is a tuple (M, M1, F, F1), where (i) M is a finite free W(R)-module, (ii) M1 ⊂ M is a W(R)-submodule such that IRM ⊂ M1 ⊂ M and M/M1 is a projective R-module, (iii) F : M → M is a φ-linear map, (iv) F1 : M1 → M is a φ-linear map, whose image generates M as a W(R)-module, and which satisfies F1(V(w)m) = wF (m) (7.1.5) for any w ∈ W(R) and m ∈ M1. 114 Remark 7.1.9. For p > 2, windows over DR are the same as the Dieudonn´e displays over R used in [KP18, 3.1.3], and the ring W(R) here is denoted by (cid:99)W (R) in loc. cit.. For a Dieudonn´e display (M, M1, F, F1), by taking w = 1 and m ∈ M1 in the equation (7.1.5), we get F (m) = F1(V(1)m) = φV(1)F1(m) = pu0F1(m). Recall that u0 ∈ W (R)× is defined by (7.1.1). In particular, we can consider the condition (iv′) F1 : M1 → M is a φ-linear map, whose image generates M as a W(R)-module, and which satisfies F1(V(w)m) = wpu0F1(m) for any w ∈ W(R) and m ∈ M1. Let (cid:102)M1 be the image of the homomorphism φ∗(i) : φ∗M1 = W(R) ⊗φ,W(R) M1 → φ∗M = W(R) ⊗φ,W(R) M induced by the inclusion i : M1 (cid:44)→ M . Note that (cid:102)M1 and the notion of a normal decompo- sition depend only on M and M1, not on F and F1. Lemma 7.1.10. Suppose W(R) is p-torsion free (e.g. if R is p-torsion free, or pR = 0 and R is reduced). (1) Giving a Dieudonn´e display (M, M1, F, F1) over R is the same as giving (M, M1, F1) satisfying (i), (ii) and (iv′). In this case, we also refer to the tuple (M, M1, F1) as a Dieudonn´e display over R. (2) For a Dieudonn´e display (M, M1, F1) over R, the linearization F # 1 of F1 factors as φ∗M1 → (cid:102)M1 Ψ−→ M with Ψ a W(R)-module isomorphism. 115 (3) Given an isomorphism Ψ : (cid:102)M1 → M of W(R)-modules, there exists a unique Dieudonn´e display (M, M1, F1) over R, which produces the given (M, M1, Ψ) via the construction in (2). Proof. The proof closely follows [KP18, §3.1.3, Lemma 3.1.5], with adjustments for the modified Verschiebung V. We take this lemma as an example to illustrate how we modify the arguments concerning Dieudonn´e displays in [KP18] to deal with the case p = 2. (1) Given the tuple (M, M1, F1), set F (m) := F1(V(1)m) for m ∈ M . Clearly F : M → M is φ-linear. Then for w ∈ W(R) and m ∈ M , we have pu0F1(V(w)m) = F1(V(1)V(w)m) = F (V(w)m) = φV(w)F (m) = pu0wF (m). Since u0 ∈ W (R)× and W(R) is p-torsion free, we obtain that W(R) is (pu0)-torsion free, and hence F1(V(w)m) = wF (m). In particular, (M, M1, F, F1) is a Dieudonn´e display. (2) Let M = L ⊕ T be a normal decomposition for M . Since φ(IR) = pu0W(R) and W(R) is pu0-torsion free, we have (cid:102)M1 = φ∗(L) ⊕ pu0φ∗(T ) ≃ W(R)d, where d = rkW(R) M . Firstly, we show that F # 1 factors through (cid:102)M1. Let K denote the kernel of φ∗(i) : φ∗M1 → φ∗M . Note that F |M1 = pu0F1, and so pu0F # In particular, pu0F # 1 = F # ◦ φ∗(i). 1 vanishes on K. Since W(R) is pu0-torsion free, we conclude that F # 1 vanishes on K, and hence F # 1 factors through (cid:102)M1. Since F # obtain a surjective map Ψ : (cid:102)M1 → M between free W(R)-modules of the same rank. Hence, is surjective by definition, we 1 Ψ is an isomorphism. (3) Define F1 : M1 → M by F1(m1) := Ψ(1 ⊗ m1), 116 where 1 ⊗ m1 denotes the image of 1 ⊗ m1 ∈ W(R) ⊗φ,W(R) M1 = φ∗M1 in φ∗M . Then F1 is clearly φ-linear and its linearization F # 1 is surjective. Thus, we obtain a Dieudonn´e display (M, M1, F1). Definition 7.1.11 ([Hof23, §1.1]). Let R be a complete local ring. (1) A Dieudonn´e pair of type (n, d) over R is a pair (M, M1) of W(R)-modules such that M is a finite free W(R)-module of rank n, M1 is a W(R)-submodule of M and M/M1 is a finite free R-module of rank d. Sometimes, we simply say that (M, M1) is a Dieudonn´e pair. (2) A morphism between two Dieudonn´e pairs (M, M1) and (M ′, M ′ 1) is a homomorphism of W(R)-modules f : M → M ′ such that f (M1) ⊂ M ′ 1. Lemma 7.1.12. There exists a functor F : (M, M1) (cid:55)→ (cid:102)M1, from the category of Dieudonn´e pairs over R of type (n, d) to the category of finite free W(R)-modules of rank n, such that F is compatible with base change in R and there is a natural isomorphism (cid:102)M1[1/p] = (φ∗M )[1/p]. If W(R) is p-torsion free, then (cid:102)M1 is given by the construction in Lemma 7.1.10. Proof. (cf. [KPZ24, §5.1.1].) Let (M, M1) be a Dieudonn´e pair of type (n, d). Choose a normal decomposition M = L ⊕ T and a basis B = (e1, . . . , en) of M such that (e1, . . . , ed) is a basis of L and (ed+1, . . . , en) is a basis of T . Such a basis B is said to be adapted to the normal decomposition M = L ⊕ T . Set F((M, M1)) = (cid:102)M1 = (φ∗L) ⊕ (φ∗T ), which is a free W(R)-module of rank n. We denote by (cid:101)B = (φ∗e1, . . . , φ∗en) the basis of (cid:102)M1. Let (M ′, M ′ 1) be a second Dieudonn´e pair with a normal decomposition M ′ = L′ ⊕ T ′ and an adapted basis B′ = (e′ 1, . . . , e′ n). Let f be a morphism between (M, M1) and (M ′, M ′ 1). 117 Using the normal decompositions, we may express f as a block matrix    A B C D   ∈ Mn(W(R))  with respect to the bases B and B′, where the entries of C are in IR. Then we define F(f ) to be the morphism (cid:101)f : (cid:102)M1 → (cid:102)M ′ 1 given by the block matrix    φ(A) pu0φ(B) V−1(C) φ(D)    in terms of the bases (cid:101)B and (cid:101)B′. Using pu0V−1 = φ, it is straightforward to check that F is a well-defined functor. By construction, F is compatible with base change in R. There is a natural isomorphism (cid:102)M1[1/p] = (φ∗L)[1/p] ⊕ (φ∗T )[1/p] ∼−→ (φ∗M )[1/p] = (φ∗L)[1/p] ⊕ (φ∗T )[1/p] l + t (cid:55)→ l + pu0t. When W(R) is p-torsion free, the above isomorphism restricts to an injective map (cid:102)M1 (cid:44)→ φ∗M, and we recover the construction of (cid:102)M1 in Lemma 7.1.10. Lemma 7.1.13 (cf. [KPZ24, Lemma 5.1.3]). Let R be a complete local ring with residue field k. Suppose that W(R) is p-torsion free. Let (M, M1) be a Dieudonn´e pair over R with reduction (M0, M0,1) over k. Set aR := m2 R + pR. Then there exists a natural isomorphism c : (cid:102)M0,1 ⊗W (k) W(R/aR) ∼−→ (cid:102)M1 ⊗W(R) W(R/aR), which is called the “connection isomorphism”, fitting into a canonical commutative diagram (cid:102)M1 ⊗W(R) W(R/aR) (cid:47) φ∗(MR/aR) c ≃ (cid:102)M0,1 ⊗W (k) W(R/aR) φ∗(M0) ⊗W (k) W(R/aR), 118 (cid:47) (cid:47) (cid:47) (cid:79) (cid:79) where MR/aR := M ⊗W(R) W(R/aR) and horizontal maps are induced by taking the base change of the natural maps (cid:102)M0,1 → φ∗(M0) and (cid:102)M1 → φ∗(M ). Proof. Using Lemma 7.1.12, the construction of c and the proof of [KPZ24, Lemma 5.1.3] (replacing V −1 by V−1) also work for p = 2. 7.1.3 Lau’s classification of p-divisible groups One of the main results in [Lau14] is the following. Theorem 7.1.14. Let R be a complete local ring with perfect residue field of characteristic p. (1) There is an anti-equivalence of exact categories ΘR : (p-divisible groups over R) ∼−→ (Dieudonn´e displays over R) , which is compatible with base change in R. (2) For any p-divisible group G over R, there is a natural isomorphism ΘR(G )/IRΘR(G ) ≃ D(G )(R), where D(G ) denotes the contravariant Dieudonn´e crystal of G . (3) Let G be a p-divisible group over R. Write ΘR(G ) = (M, M1, F, F1). The Hodge filtration of ΘR(G ) is defined as M1/IRM ⊂ M/IRM. Then the isomorphism in (2) respects the Hodge filtrations on both sides. Remark 7.1.15. For p > 2, the functor ΘR recovers the anti-equivalence used in [KP18, 3.1.7] by sending a p-divisible group G over R to D(G )(W(R)). Note that when p > 2, W(R) → R has divided powers on IR by [Lau14, Lemma 1.16]. For p = 2, ΘR is not as explicit as in the case p > 2, but see the case when R is a ring of p-adic integers in §7.1.4. 119 Proof. (1) For any p-divisible group G over R, set ΘR(G ) := ΦR(G ∗), where G ∗ denotes the Cartier dual of G and ΦR denotes the equivalence in [Lau14, Corollary 5.4]. Then we see that ΘR is an anti-equivalence of exact categories. It commutes with base change in R by [Lau14, Theorem 3.9, 4.9]. (2) and (3) follow from [Lau14, Corollary 3.22, 4.10]. Note that we use contravariant Dieudonn´e crystals following [KP18], while Lau uses covariant Dieudonn´e crystals in [Lau14]. One can switch between contravariant and covariant Dieudonn´e crystals by taking Cartier duals. 7.1.4 Comparison with Breuil-Kisin’s classification Here the notation is as in Example 7.1.8 (2). In particular, we denote by OK the ring of integers for some finite extension K of Qp with residue field k. Let π be a uniformizer of OK satisfying E(π) = 0. Then there is a Frobenius-equivariant ring homomorphism κ : S = W (k)[[u]] → W (OK) sending u to [π], lifting the quotient map S → OK. Here [·] denotes the Teichm¨uller map OK → W (OK). Moreover, the image of κ lies in W(OK), see [Lau14, Remark 6.3]. Recall that B denotes the Breuil-Kisin frame in Example 7.1.8 (2). By [Lau14, Theorem 6.6], κ induces a crystalline homomorphism κ : B → DOK . That is, the induced functor κ∗ as in (7.1.3) gives an equivalence κ∗ : (windows over B) ∼−→ (windows over DOK ) = (Dieudonn´e displays over OK). Using the anti-equivalence ΘOK in Theorem 7.1.14, we obtain the anti-equivalence B(−) := κ−1 ∗ ◦ ΘOK : (p-divisible groups over OK) ∼−→ (windows over B). (7.1.6) 120 On the other hand, we have, by [Kis10, Theorem 1.4.2], a fully faithful contravariant functor M(−) : (p-divisible groups over OK) −→ BTφ S, where BTφ S denotes the category of Breuil-Kisin modules (M, φM) of E-height one, i.e., M is a finite free S-module and φM : φ∗M → M is an S-module homomorphism whose cokernel is killed by E. Proposition 7.1.16. There is an equivalence F : BTφ S −→ (windows over B) such that F ◦ M(−) is the equivalence B(−) in (7.1.6). In particular, M(−) is an anti- equivalence. Proof. The proposition is implicitly contained in [Lau10, §6, 7] (see also [KM16, §2]). To a Breuil-Kisin module (M, φM) in BTφ S, we can associate a triple (M, M1, F1), where M := φ∗M; M1 := M, viewed as a submodule of M via the unique map VM : M → φ∗M whose composition with φM is the multiplication by E(u); and F1 : M1 → M is given by x ∈ M (cid:55)→ 1 ⊗ x ∈ φ∗M. Then we see E(u)M ⊂ M1 ⊂ M. (7.1.7) Define F : M → M by sending m ∈ M to F1(E(u)m). Then (M, M1, F, F1) defines a window over B. Hence, we obtain a functor F : BTφ S −→ (windows over B). The functor F is an equivalence (cf. [Lau10, Lemma 8.2, 8.6]). Its inverse can be described as follows. Let (M, M1, F, F1) be a window over B. The S-module M1 is necessarily free, and hence the surjection F # 1 : φ∗M1 → M is an isomorphism. Let ϕ : M1 (cid:44)→ φ∗M1 denote the composition of the inclusion M1 (cid:44)→ M with the inverse of F # 1 . There is a unique S-linear map ψ : φ∗M1 → M1 such that ψϕ = E(u). Then (M1, ψ) defines an object in BTφ S and the 121 functor (M, M1, F, F1) (cid:55)→ (M1, ψ) is the inverse of F. Going through the proof of [KM16, Theorem 2.12], we have F ◦ M(−) = B(−). In particular, M(−) is also an equivalence. Definition 7.1.17. For (M, φM) ∈ BTφ S, the Hodge filtration of φ∗M is defined as M/E(u)φ∗M ⊂ φ∗M/E(u)φ∗M, where the inclusion is induced by (7.1.7). Corollary 7.1.18. Let G be a p-divisible group over OK. (1) There exists a natural isomorphism ΘOK (G ) ≃ φ∗M(G ) ⊗S,κ W(OK) as Dieudonn´e displays over OK. (2) There exists a natural isomorphism D(G )(OK) ≃ φ∗M(G ) ⊗S OK = φ∗M/E(u)φ∗M, which respects the Hodge filtrations on both sides. Proof. (1) It follows from the equality F ◦ M(−) = B(−) in Proposition 7.1.16 and the definition of base change of Dieudonn´e displays. (2) Denote by ψ the isomorphism in (1). By base change of ψ along the natural surjection W(OK) → OK, we obtain an isomorphism ΘOK (G )/IOK ΘOK (G ) ≃ φ∗M(G )/E(u)φ∗M. Since ψ is an isomorphism of Dieudonn´e displays, the above isomorphism respects the Hodge filtrations. By Theorem 7.1.14 (2) and (3), we obtain an isomorphism D(G )(OK) ≃ φ∗M(G ) ⊗S OK = φ∗M/E(u)φ∗M respecting the Hodge filtrations. 122 7.2 Deformation theory In this section, we extend the deformation theory of p-divisible groups in [KP18, §3] to the case p = 2. We also generalize the notion of very good Hodge embeddings for p = 2, allowing us to construct versal deformation of p-divisible groups with crystalline tensors (see Proposition 7.2.16). In Proposition 7.2.18, we establish a criterion for determining when a deformation is (GW , µy)-adapted in the sense of Definition 7.2.17. 7.2.1 Versal deformations of p-divisible groups The notations are as in §7.1. In this subsection, we aim to extend the construction of the versal deformation space of p-divisible groups in [KP18, §3.1] to the case p = 2. Firstly we generalize [Zin01, Theorem 3, 4], which deals with the case when R has residue characteristic p > 2 or 2R = 0. Theorem 7.2.1. Let k be a perfect field of characteristic p. Let (S → R, δ) be a nilpotent divided power extension of artinian local rings of residue field k, i.e., the kernel a of the surjection S → R is equipped with nilpotent divided powers δ. (1) Let P = (M, M1, F, F1) be a Dieudonn´e display over S and P = (M , M 1, F, F1) be the reduction of P over R. Denote by (cid:99)M1 the inverse image of M 1 under the homomor- phism M → M = W(R) ⊗W(S) M. Then F1 : M1 → M extends uniquely to a W(S)-module homomorphism (cid:98)F1 : (cid:99)M1 → M such that (cid:98)F1(aM ) = 0. Therefore, (cid:98)F1 restricted to (cid:99)W (a)M is given by (cid:98)F1([a0, a1, . . .]x) = [w0(u−1 0 )a1, w1(u−1 0 )a2, . . .]F (x) in logarithmic coordinates. 123 (2) Let P = (M, M1, F, F1) (resp. P ′ = (M ′, M ′ 1, F ′, F ′ Let P (resp. P ′ ) be the reduction over R. Assume that u : P → P is a morphism of 1)) be a Dieudonn´e display over S. ′ Dieudonn´e displays over R. Then there exists a unique morphism of quadruples u : (M, (cid:99)M1, F, (cid:98)F1) → (M ′, (cid:99)M ′ 1, F ′, (cid:98)F ′ 1) lifting u. Hence, we can associate a crystal to a Dieudonn´e display as follows: Let P = (M, M1, F, F1) be a Dieudonn´e display over R, (T → R, δ) be a divided power extension, then define the Dieudonn´e crystal D(P) evaluated at (T → R, δ) as D(P)(T ) := T ⊗w0,W(T ) (cid:102)M , where (cid:101)P = ( (cid:102)M , (cid:102)M1, (cid:101)F , (cid:101)F1) is any lifting of P over T . (3) Let C be the category of all pairs (P, F il), where P is a Dieudonn´e display over R and F il ⊂ D(P)(S) is a direct summand lifting the Hodge filtration M1/IRM (cid:44)→ M/IRM of D(P)(R). Then the category C is canonically isomorphic to the category of Dieudonn´e displays over S. Remark 7.2.2. The above theorem has a reformulation in terms of relative Dieudonn´e displays as in [Lau14, §2D, 2F]: the quadruple (M, (cid:99)M1, F, (cid:98)F1) defines a window over the relative Dieudonn´e frame DS/R. Proof. The proof adapts arguments in [Zin01, Theorem 3, 4] and [Zin02, Lemma 38, 42], with adjustments for V. (1) Choose a normal decomposition M = L ⊕ T . Then (cid:99)M1 = (cid:99)W (a)M + M1 = aT ⊕ L ⊕ IST. Using this decomposition, we can extend F1 by setting (cid:98)F1(aT ) = 0. We claim that (cid:98)F1(aL) = 0. Note that by formula (7.1.2), we have φ(a) = 0. Since F1 is φ-linear, we have (cid:98)F1(aL) = φ(a) (cid:98)F1(L) = 0. Thus, the extension (cid:98)F1 satisfies (cid:98)F1(aM ) = 0. It is unique since (cid:99)M1 = 124 (cid:99)W (a)M + M1 = aM + M1. For any [a0, a1, . . .] ∈ (cid:99)W (a) and x ∈ M , we have (cid:98)F1([a0, a1, . . .]x) = (cid:98)F1([a0, 0, 0, . . .]x) + (cid:98)F1(V [a1, a2, . . .]x) = 0 + F1(V(u−1 0 [a1, . . .])x) = F1(V([w0(u−1 0 )a1, w1(u−1 0 )a2, . . .])x) = [w0(u−1 0 )a1, w1(u−1 0 )a2, . . .]F (x). (2) For the uniqueness of u, it is enough to consider the case u = 0. Recall that for a Dieudonn´e display (M, M1, F, F1) over S, we have defined the map V ♯ : M → W(S)⊗φ,W(S)M in (7.1.4). For any integer N ≥ 1, we define (V N )♯ : M → M ⊗φN ,W(S) M as the composite M V ♯ −→ W(S) ⊗φ,W(S) M 1⊗V ♯ −−−→ W(S) ⊗φ2,W(S) M → · · · → W(S) ⊗φN ,W(S) M. Similarly, we can define maps (F N 1 )# and ( (cid:98)F N 1 )#. As in the proof of [Zin01, Theorem 3], we have a commutative diagram u M (V N )♯ (cid:99)W (a)M ′ ( (cid:98)F ′N 1 )# W(S) ⊗φN ,W(S) M 1⊗u (cid:47) (cid:47) W(S) ⊗φN ,W(S) (cid:99)W (a)M ′ By (1), for [a0, a1, . . .] ∈ (cid:99)W (a) and x ∈ M ′, we have (cid:98)F ′N 1 ([a0, . . .]x) = [ N −1 (cid:89) wi(u−1 0 )aN , i=0 N (cid:89) i=1 wi(u−1 0 )aN +1, . . .]F ′N (x). Since ai = 0 for almost all i, (cid:99)W (a)M ′ is annihilated by (cid:98)F ′N 1 for sufficiently large N . This shows u = 0 as desired. For the existence of u, we can repeat the proof of [Zin01, Theorem 3]. (3) Clearly we can get a lifting of the Hodge filtration of D(P)(R) from a Dieudonn´e display over S. On the other hand, given (P, F il) ∈ C, any lifting of P to S gives a unique quadruple (M, (cid:99)M1, F, (cid:98)F1) by (2). Let M1 ⊂ (cid:99)M1 be the inverse image of F il ⊂ M/ISM under the projection M → M/ISM , then we obtain a Dieudonn´e display (M, M1, F, (cid:98)F1|M1) over S. By (2), these two constructions are mutually inverse. 125 (cid:15) (cid:15) (cid:47) (cid:47) (cid:79) (cid:79) Now we fix a p-divisible group G0 over k, and let (D, D1, F, F1) be the corresponding Dieudonn´e display. Note that D is given by D(G0)(W ), see [Lau14, Corollary 2.34]. By Lemma 7.1.10, the Dieudonn´e display (D, D1, F, F1) corresponds to a triple (D, D1, Ψ0) for ∼−→ D. Next we will construct a versal deformation space of G0, an isomorphism Ψ0 : (cid:101)D1 equivalently a versal deformation space of the Dieudonn´e display (D, D1, Ψ0). Recall there is a canonical Hodge filtration on D ⊗W k = D(G0)(k): 0 → Homk(Lie G0, k) → D ⊗W k → Lie G ∗ 0 → 0. We think of D ⊗W k as a filtered k-module by setting Fil0(D ⊗W k) = D ⊗W k, Fil1(D ⊗W k) = Homk(Lie G0, k). This filtration corresponds to a parabolic subgroup P0 ⊂ GL(D ⊗W k). Fix a lifting of P0 to a parabolic subgroup P ⊂ GL(D). Write M loc = GL(D)/P and (cid:99)M loc = Spf R, (7.2.1) where (cid:99)M loc is the completion of GL(D)/P along the image of the identity in GL(D ⊗W k). Then R is a power series ring over W . Set M = D ⊗W W(R), and let M 1 ⊂ M/IRM be the direct summand corresponding to the parabolic subgroup gP g−1 ⊂ GL(D) over (cid:99)M loc, where g ∈ (GL(D)/P )(R) is the universal ∼→ M be a W(R)-module point. Let M1 ⊂ M be the preimage of M 1 in M and Ψ : (cid:102)M1 isomorphism reducing to Ψ0 modulo mR, where (cid:102)M1 is defined as in Lemma 7.1.10. Then the triple (M, M1, Ψ) gives a Dieudonn´e display over R reducing to (D, D1, Ψ0). By Theorem 7.1.14, the Dieudonn´e display (M, M1, Ψ) corresponds to a p-divisible group GR over R, which is a deformation of G0. Set aR := m2 R + pR. By Lemma 7.1.13, there exists a natural connection isomorphism c : (cid:101)D1 ⊗W W(R/aR) ∼−→ (cid:102)M1 ⊗W(R) W(R/aR). 126 Definition 7.2.3. The map Ψ is said to be constant modulo aR if the composite map (cid:101)D1 ⊗W W(R/aR) c−→ (cid:102)M1 ⊗W(R) W(R/aR) Ψ⊗1−−→ MR/aR ≃ D ⊗W W(R/aR) is equal to Ψ0 ⊗ 1. Lemma 7.2.4. If Ψ is constant modulo aR, then the deformation GR of G0 is versal. Proof. Recall that there exists a versal deformation ring Runiv for G0, which is a power series ring over W of the same dimension as R. The deformation GR is induced by a map Runiv → R. We want to show this is an isomorphism. It suffices to prove that the induced map on tangent spaces is an isomorphism. We have two Dieudonn´e displays over R/aR. One is obtained from (M, M1, FR, FR,1) (the Dieudonn´e display corresponding to (M, M1, Ψ)) by the base change along R → R/aR, the other is obtained from (D, D1, F, F1) by the base change along k → R/aR. If Ψ is constant modulo aR, then as in the proof of [KP18, Lemma 3.1.12], we know (cid:98)FR,1 = (cid:98)F1 on (cid:99)MR/aR,1, see the notation in Theorem 7.2.1. Hence, these two Dieudonn´e displays give rise to the same quadruple (MR/aR, (cid:99)MR/aR,1, FR/aR, (cid:98)FR/aR,1). Let G be a deformation over the ring k[ϵ] of dual numbers. Since k[ϵ] → k has trivial divided powers, it is a nilpotent divided power extension, then by Theorem 7.2.1 (1) and (2), the base change of (D, D1, F, F1) along the natural map k → k[ϵ] gives rise to a quadru- ple (Mk[ϵ], (cid:99)Mk[ϵ],1, Fk[ϵ], (cid:98)Fk[ϵ],1). By the proof of Theorem 7.2.1 (3), the Dieudonn´e display corresponding to G is of the form (Mk[ϵ], (cid:102)Fil, Fk[ϵ], (cid:98)Fk[ϵ],1), where (cid:102)Fil ⊂ (cid:99)Mk[ϵ],1 is the preimage of certain lifting Fil ⊂ (D ⊗W k) ⊗k k[ϵ] of the Hodge filtration of D. From the versality of the filtration M 1 ⊂ D⊗W R, there is a map α : R → k[ϵ] (necessarily factors through R/aR) such that the induced map D ⊗W R → D ⊗W k[ϵ] sends 127 M 1 to Fil. Then by the discussion in the previous paragraph, (Mk[ϵ], (cid:102)Fil, Fk[ϵ], (cid:98)Fk[ϵ],1) is the base change of (M, M1, FR, FR,1) along α. Thus, G is the base change of GR along α. In particular, Runiv → R induces an isomorphism of tangent spaces. Hence, we proved that GR is versal. Remark 7.2.5. Note that the functor F := Isom( (cid:102)M1, M ) of isomorphisms of finite free W(R)- modules between (cid:102)M1 and M is a GL(M )-torsor over W(R). Hence, the surjection W(R) ↠ W(R/aR) induces a surjection F(W(R)) ↠ F(W(R/aR)). This implies that an isomorphism Ψ, which is constant modulo aR, always exists. 7.2.2 Local models and local Hodge embeddings Before discussing the deformation of p-divisible groups with crystalline tensors, we will make a digression into local models and local Hodge embeddings in this subsection. Definition 7.2.6 ([KPZ24, Definition 3.1.2]). Let F/Qp be a complete discrete valued field. Let (G, {µ} , G) be a local model triple over F (see §3.4). (1) A pair (G, µ) is of (local) Hodge type if there is a closed immersion ρ : G (cid:44)→ GL(V ), where V is an F -vector space of dimension h, such that (i) ρ is a minuscule representation in the sense of [KP18, §1.2.9]. (ii) ρ ◦ µ is conjugate to the standard minuscule cocharacter µd of GL(VF ), where µd(t) := diag(t(d), 1(h−d)), t ∈ F . (iii) ρ(G) contains the scalars. Such a ρ will be said to give a (local) Hodge embedding ρ : (G, µ) (cid:44)→ (GL(V ), µd). (2) An integral Hodge embedding for (G, µ) is a closed immersion ρ : G (cid:44)→ GL(Λ) over OF , where Λ is a finite free OF -module, such that the base change ρ ⊗OF F is a Hodge embedding for (G, µ). 128 Lemma 7.2.7. Let (G, {µ} , G) be a local model triple over F . Suppose ρ : (G, µ) (cid:44)→ (GL(Λ), µd) is an integral Hodge embedding. Then ρ induces a closed immersion XG,µ = G/Pµ (cid:44)→ XGL(V ),µd ⊗F E = Gr(d, V )E, where Gr(d, V ) denotes the Grassmannian classifying subspaces of V of rank d. Let X G,µ be the (reduced) Zariski closure of XG,µ ⊂ Gr(d, V )E in Gr(d, Λ)OE . If X G,µ is normal, then X G,µ is isomorphic to Mloc Gr(d, Λ)OE is identified with the natural morphism Mloc G,µ, and the closed immersion X G,µ (cid:44)→ G,µ → Mloc ⊗OF OE induced by ρ. GL(Λ),µd Proof. See [KPZ24, Lemma 3.4.1]. Note that by [GL24], the condition in loc. cit. requiring the special fiber of X G,µ to be reduced is in fact implied by the remaining conditions. Definition 7.2.8 ([KPZ24, Definition 3.4.4]). Let ρ : (G, µ) (cid:44)→ (GL(Λ), µd) be an integral Hodge embedding over OF . We say that ρ is a good Hodge embedding, if the morphism Mloc G,µ −→ Mloc GL(Λ),µd ⊗OF OE induced by ρ is a closed immersion. By Lemma 7.2.7, ρ is good if the Zariski closure of XG,µ in Gr(d, Λ)OE is normal. From now on, we suppose that F/Qp is unramified and ρ : (G, µ) (cid:44)→ (GL(Λ), µd) is a good integral Hodge embedding over OF . In particular, we have a closed immersion Mloc G,µ (cid:44)→ Gr(d, Λ)OE . For any x ∈ Mloc G,µ(k), where k = Fp, we let RG = RG,x (resp. RE) denote the completion of Mloc G,µ (resp. Gr(d, Λ)OE ) at x. By our assumptions, RE is isomorphic to a power series ring over OEW (k) and RG is a (normal) quotient ring of RE. Then W(RE) and W(RG) are p-torsion free rings. Set M := Λ ⊗OF W(RE). Let M 1 ⊂ M/IRE M = Λ ⊗OF RE be the direct summand corresponding to the universal RE-valued point of Gr(d, Λ). Set M1 := the preimage of M 1 in M . 129 Then (M, M1) is a Dieudonn´e pair over RE. By the base change along RE ↠ RG, we obtain a Dieudonn´e pair (MRG, MRG,1) over RG. By Lemma 7.1.12, we can associate a free W(RG)-module (cid:102)MRG,1 with (cid:102)MRG,1[1/p] = (φ∗MRG)[1/p]. Definition 7.2.9. For any ring A and a finite free A-module N , we denote by N ⊗ the direct sum of all A-modules which can be formed from N by using the operations of taking tensor products, duals, symmetric and exterior powers. If N is equipped with a filtration, then N ⊗ is equipped with a filtration accordingly. If (sα) ⊂ N ⊗ and G ⊂ GL(N ) is the pointwise stabilizer of sα, we say G is the group scheme cut out by the tensors sα. Lemma 7.2.10 ([Kis10, Proposition 1.3.2]). Suppose that A is a discrete valuation ring of mixed characteristic and N is a finite free A-module. If G ⊂ GL(N ) is a closed A-flat subgroup whose generic fiber is reductive, then G is cut by a finite collection of tensors in N ⊗. Remark 7.2.11. By an argument of Deligne, the tensors in Lemma 7.2.10 can be taken in the submodule ⊕m,n≥0N ⊗m ⊗A (N ∨)⊗n. Here, N ∨ denotes the A-dual module HomA(N, A). Let ρ : G (cid:44)→ GL(Λ) be a Hodge embedding. Then G ⊂ GL(Λ) (via ρ) is cut out by a set of tensors (sα) ⊂ Λ⊗ by Lemma 7.2.10. Set (cid:101)sα := sα ⊗ 1 = φ∗(sα ⊗ 1) ∈ Λ⊗ ⊗OF W(RG) = φ∗M ⊗ RG . We may view ((cid:101)sα) as tensors in (φ∗MRG)⊗[1/p] = (cid:102)M ⊗ Corollary 3.2.11]), we have the following proposition. RG,1[1/p]. By [KPZ24, §5.2] (and [KP18, Proposition 7.2.12. Suppose that F/Qp is unramified and ρ : (G, µ) (cid:44)→ (GL(Λ), µd) is a good integral Hodge embedding over OF . Then (cid:101)sα ∈ (cid:102)M ⊗ RG,1. 130 Denote by (cid:101)sα,0 the reduction of (cid:101)sα in (cid:102)M ⊗ 0,1, where (cid:102)M0,1 = (cid:102)MRG,1 ⊗W(RG) W (k). By Lemma 7.1.13, we have a connection isomorphism cG : (cid:102)M0,1 ⊗W (k) W(RG/aRG) ∼−→ (cid:102)MRG,1 ⊗W(RG) W(RG/aRG). Definition 7.2.13. Under the assumptions in Proposition 7.2.12, we say that ρ is very good at x ∈ Mloc G,µ(k), if cG((cid:101)sα,0 ⊗ 1) = (cid:101)sα ⊗ 1. In this case, we say that the tensors ((cid:101)sα) are horizontal at x. We say ρ is a very good (integral) Hodge embedding if ρ is very good at every x ∈ Mloc G,µ(k). Definition 7.2.14 ([KPZ24, Definition 4.1.4]). For a scheme X over k and x ∈ X(k), we say that the tangent space TxX of X at x is spanned by smooth formal curves if the images of the tangent spaces by k-morphisms Spec k[[t]] → X with the closed point mapping to x generate the k-vector space TxX. We will use the following lemma in §7.3.3.3. Lemma 7.2.15 ([KPZ24, Proposition 5.3.10]). Assume ρ : (G, µ) (cid:44)→ (GL(Λ), µd) is a good integral Hodge embedding over Zp. Let x ∈ Mloc G,µ(k) be a closed point. If the tangent space of the special fiber Mloc G,µ ⊗OE k at x is spanned by smooth formal curves, then ρ is very good at x. We refer to [KPZ24, §5.3] for more properties of very good Hodge embeddings. 7.2.3 Deformations with crystalline tensors We continue to use the notation in §7.2.1, and as in [KP18, §3.2, 3.3], we may assume k is algebraically closed for simplicity. Let G0 be a p-divisible group over k. Denote D = D(G0)(W ). Let (sα,0) ⊂ D⊗ be a collection of φ-invariant tensors whose images in D(G0)(k)⊗ lie in Fil0 D(G0)(k)⊗. In this subsection, we assume the following conditions: (A1) there is an isomorphism Λ ⊗Zp W ≃ D for some free Zp-module Λ such that sα,0 ∈ Λ⊗; 131 (A2) the stabilizer group scheme G ⊂ GL(Λ) cut out by (sα,0) ⊂ Λ⊗ has reductive generic fiber G and G◦ is a parahoric group scheme over Zp. Note that the base change GW := G ⊗Zp W ⊂ GL(D) is cut out by (sα,0) ⊂ D⊗. In (7.2.1) of §7.2.1, we have defined M loc and (cid:99)M loc = Spf R. Let K ′/K0 be a finite extension and y : R → K ′ be a map such that sα,0 ∈ Fil0(D ⊗W K ′)⊗ for the filtration induced by y on D ⊗W K ′. By [Kis10, Lemma 1.4.5], the filtration is induced by a G-valued cocoharacter µy. We further impose the following assumption: (A3) there is a very good Hodge embedding (G, µ−1 y ) (cid:44)→ (GL(Λ), µd) for d = dimk Lie G0. Denote by E ⊂ K ′ the local reflex field of the G-conjugacy class of cocharacters {µy}. Write M loc G,y for the closure of the G-orbit G.y ⊂ M loc ⊗W E in M loc ⊗W OE. By assumption (A3) and Lemma 7.2.7, the scheme M loc G,y is isomorphic to the local model Mloc attached G,µ−1 y to the local model triple (G, {µ−1 y }, G), and hence M loc G,y is normal and only depends on the G-conjugacy class {µy} (not on y). We denote by (cid:99)M loc G,y = Spf RG the completion of M loc G,y along (the image of) the identity in GL(D ⊗W k). Then RG is a normal quotient ring of R ⊗W OE. Recall in §7.2.1, we constructed a versal deformation GR over R corresponding to a Dieudonn´e display (M, M1, Ψ), where Ψ is constant modulo aR. Set MRE := M ⊗W(R) W(RE), MRG := MRE ⊗W(RE ) W(RG). The tensors sα,0 ∈ D⊗ induce tensors in M ⊗ RG φ∗MRG and (sα,0) are φ-invariant. By [KP18, Corollary 3.2.11], we have (sα,0) ⊂ (cid:102)MRG,1. , still denoted as sα,0. Notice that (cid:102)MRG,1 ⊂ (Here we uses [Ans22, Proposition 10.3] to remove the condition (3.2.3) in [KP18].) Recall that the p-divisible group G0 over k corresponds to a Dieudonn´e display (D, D1, Ψ0 : (cid:101)D1 Since (cid:101)D1 = φ∗(D) and (sα,0) are φ-invariant, we have (sα,0) ⊂ (cid:101)D⊗ 1 . Set ∼→ D). aRE := m2 RE + πERE, 132 where πE ∈ OE is a uniformizer. In particular RE/aRE ≃ R/aR. Set aRG := m2 RG + πERG, Proposition 7.2.16 (cf. [KP18, §3.2.12]). Assume (A1) to (A3). (1) The scheme T := Isom(sα,0)( (cid:102)MRG,1, MRG) consisting of isomorphisms respecting tensors sα,0 is a trivial G-torsor over W(RG). (2) There exists an isomorphism ΨRG : (cid:102)MRG,1 ∼−→ MRG respecting sα,0 which lifts to an isomorphism ΨRE : (cid:102)MRE ,1 → MRE that is constant modulo aRE . Moreover, the p- divisible group GRE over RE corresponding to the Dieudonn´e display (MRE , MRE ,1, ΨRE ) is a versal deformation of G0. Proof. (1) This follows from [KP18, Corollary 3.2.11] and [Ans22, Proposition 10.3]. (2) By assumption (A3), the isomorphism ΨRG/aRG (cid:102)MRG,1 ⊗W(RG) W(RG/aRG) c−1 G−−→ (cid:101)D1 ⊗W W(RG/aRG) Ψ0⊗1−−−→ D ⊗W W(RG/aRG) = MRG/aRG preserves the tensors sα,0, and hence defines a point in T (W(RG/aRG)). Since T is a G- torsor, we can lift the point to a point in T (W(RG)), which corresponds to an isomorphism ΨRG : (cid:102)MRG,1 ∼−→ MRG respecting sα,0. By construction, ΨRG/aRG is the reduction of the isomorphism ΨRE /aRE (cid:102)MRE ,1 ⊗W(RE ) W(RE/aRE ) c−1 −−→ (cid:101)D1 ⊗W W(RE/aRE ) Ψ0⊗1−−−→ D ⊗W W(RE/aRE ) = MRE /aRE . Denote by F the GL(MRE )-torsor Isom( (cid:102)MRE ,1, MRE ) over W(RE). Then ΨRG and ΨRE /aRE define a point of F valued in W(RG) ×W(RG/aRG ) W(RE/aRE ) = W(RG ×RG/aRG RE/aRE ). We can lift this point to an W(RE)-valued point of F, which corresponds to an isomorphism ∼−→ MRE . Hence, ΨRE is constant modulo aRE . By Lemma 7.2.4 and the ΨRE : (cid:102)MRE ,1 discussion in [KP18, §3.2.12], the Dieudonn´e display (MRE , MRE ,1, ΨRE ) is versal. 133 Following [Zho20, §4], we make the following definition. Definition 7.2.17. Let G be a p-divisible group over OK deforming G0. We say that G is (GW , µy)-adapted if the tensors sα,0 lift to Frobenius invariant tensors (cid:101)sα ∈ ΘOK (G )⊗ such that the following two conditions hold: (1) There is an isomorphism ΘOK (G ) ≃ D ⊗W W(OK) sending (cid:101)sα to sα,0 ⊗ 1. (2) Under the canonical isomorphism D(G )(OK) ⊗OK K ≃ D ⊗W K, the filtration on D ⊗W K is induced by a G-valued cocharacter G-conjugate to µy. Proposition 7.2.18. Assume (A1) to (A3). View Spf RE as the versal deformation space of G0 by the construction in Proposition 7.2.16 (2). Then for any finite extension K/E, a map ξ : RE → OK factors through RG if and only if the p-divisible group Gξ = ξ∗GRE is (GW , µy)-adapted. Proof. (⇒) See [Zho20, Proposition 4.7] and [KZ24, Proposition 3.2.7]. (⇐) The proof goes as in [KP18, Proposition 3.2.17]. For completeness, we recall the arguments here. Suppose Gξ is (GW , µy)-adapted. Denote by sα ∈ D(G )(OK)⊗ the image of (cid:101)sα modulo IOK . Then the isomorphism in (1) of Definition 7.2.17 gives an isomorphism ∼→ D(G )(OK) taking sα,0 to sα. Hence, by (2) in Definition 7.2.17, DOK := D ⊗W OK this isomorphism induces a filtration on DOK corresponding to a map y′ : RG → OK and sα,0 ∈ Fil0 D⊗ OK . As RG depends only on the reduction of y and the conjugacy class of µy, we may assume y = y′ (and K ′ = K). The map y : RG → OK induces a Dieudonn´e display (MOK , MOK ,1, Ψ), and by the con- ∼→ MOK takes sα,0 to sα,0. Since y = y′, the struction of ΨRG, the isomorphism Ψ : (cid:102)MOK ,1 p-divisible group Gξ corresponds to a Dieudonn´e display (MOK , MOK ,1, Ψ′). As (cid:101)sα is Frobe- nius invariant and Ψ′ differs from the Frobenius a scalar (contained in G by assumption), then Ψ′ takes sα,0 to sα,0, and reduces to Ψ0 : (cid:101)D1 ∼→ D. Now we construct a Dieudonn´e display over S := OK[[T ]]. First consider the Dieudonn´e display (MS, MS,1, Ψ), the base change of (MOK , MOK ,1, Ψ) to S. The map S → OK ×k OK 134 given by T (cid:55)→ (0, π) is surjective, and hence so is W(S) → W(OK) ×W W(OK). Note that by Proposition 7.2.16, T is a (trivial) G-torsor. Since G is smooth, we have a surjection T (W(S)) ↠ T (W(OK) ×W W(OK)). That is, there exists an isomorphism ΨS : (cid:102)MS,1 ∼−→ MS which takes sα,0 to sα,0, and specializes to (Ψ, Ψ′) under T (cid:55)→ (0, π). We take MS to be the Dieudonn´e display associated to (MS, MS,1, ΨS). By versality, we may lift the map (y, ξ) : RE → OK ×k OK to a map (cid:101)ξ : RE → S which induces the Dieudonn´e display MS and MS is the base change of MRE by (cid:101)ξ. Now the rest of the proof is similar as in [KP18, Proposition 3.2.17]. Then we conclude that (cid:101)ξ factors though RG, and hence ξ does as well. In §7.3, we will construct (GW , µy)-adapted deformations of p-divisible groups associated to closed points in integral models of Shimura varieties, and apply Proposition 7.2.18 to describe the local structure of integral models of Shimura varieties. 7.3 Integral models of Shimura varieties of abelian type In this section, we will prove Theorem 1.2.7 in the Introduction. Following the strategy of [KP18; KPZ24], we first consider Shimura varieties ShK(G, X) of Hodge type. We construct their integral models SK(G, X) by using the Hodge embeddings into Siegel modular varieties, as in loc. cit.. Under certain assumptions (see Theorem 7.3.4), we apply the deformation theory developed in §7.2 to identify the formal neighborhood of SK(G, X) with that of the local model. Then we extend this construction of integral models to the case of Shimura varieties of abelian type by choosing suitable Hodge type lifts under certain conditions (see Theorem 7.3.9). We complete the proof of Theorem 1.2.7 by showing that these conditions are satisfied in Case (A) or (B). 135 7.3.1 Shimura varieties of Hodge type Let (G, X) be a Shimura datum, that is, G is a reductive group over Q and X is a G(R)-conjugacy class of h : S := ResC/RGm → GR satisfying axioms 2.1.1.1-2.1.1.3 in [Del79, §2.1]. Denote by µh : GmC → GC the associated Hodge cocharacter, defined by µh(z) = hC(z, 1). Set wh := µ−1 h µc−1 h (the weight homomor- phism), where c denotes the complex conjugation. Fix a Q-vector space V of dimension 2g with a perfect alternating pairing ψ : V ×V → Q. Let GSp = GSp(V, ψ) be the corresponding symplectic similitude group over Q, and let S± = S±(V, ψ) be the Siegel double space consisting of maps h : S → GSpR such that (1) The map S h−→ GSpR (cid:44)→ GL(VR) gives rise to a Hodge structure of type (−1, 0), (0, −1) on VR, i.e., VC = V −1,0 ⊕ V 0,−1. (2) The pairing (x, y) (cid:55)→ ψ(x, h(i)y) is (positive or negative) definite on VR. Then (GSp, S±) is a Shimura datum, which is called a Siegel Shimura datum. For the rest of the subsection, we assume (G, X) is of Hodge type, i.e., there exists an embedding of Shimura data ι : (G, X) (cid:44)→ (GSp(V, ψ), S±). Sometimes we will write G for GQp for simplicity. Let E = E(G, X) be the reflex field with ring of integers OE. Let p be a prime number. Let Af denote the ring of finite ad`eles over Q, and Ap f denote the ring of prime-to-p finite ad`eles, which we consider as the subgroup of Af with trivial component at p. Fix a place v|p of E, and let E denote the completion of E at v. Denote by OE,(v) (resp. OE) the localization (resp. completion) of OE at v. We write G for the base change GQp. Let G be the Bruhat-Tits group scheme over Zp associated with some x ∈ B(G, Qp), whose neutral component G◦ is parahoric. Set Kp = G(Zp) or G◦(Zp) and K = KpKp with Kp ⊂ G(Ap f ) 136 sufficiently small open compact subgroup. By general theory of Shimura varieties, these data yield a quasi-projective smooth algebraic variety ShK(G, X) canonically defined over E, whose C-points are given by ShK(G, X)(C) = G(Q)\X × G(Af )/K. We can also consider the projective limit of E-schemes Sh(G, X) = lim ←− K ShK(G, X), resp. ShKp(G, X) = lim ←− Kp ShKpKp(G, X), which carries a natural action of G(Af ) (resp. G(Ap f )). The projective limit exists since the transition maps are finite, hence affine. 7.3.1.1 Integral models for level G(Zp): construction Assume that (i) Kp = G(Zp); (ii) ιQp extends to a very good integral Hodge embedding (cid:101)ι : (G, µh) (cid:44)→ (GL(VZp), µg), where VZp ⊂ VQp is a self-dual Zp-lattice with respect to ψ. We let GSP denote the parahoric group scheme associated to the self-dual lattice VZp. Set VZ(p) := V ∩ VZp. Denote by GZ(p) the Zariski closure of G in GL(VZ(p)), then G is isomorphic p := GSP(Zp). Let K′p be a small enough open compact subgroup Zp. Set K′ f ) containing Kp, which leaves V (cid:98)Zp stable. Here (cid:98)Zp := (cid:81) Zℓ. Set K′ = K′ pK′p. ℓ̸=p to GZ(p) ⊗Z(p) of GSp(Ap Then the embedding ι induces a closed immersion ShK(G, X) (cid:44)→ ShK′(GSp, S±) ⊗Q E over E. The choice of VZ(p) gives rise to an interpretation of ShK′(GSp, S±) as a moduli space of polarized abelian varieties, and hence to a natural integral model SK′(GSp, S±) over Z(p) (cf. [Zho20, §6.3]). 137 Definition 7.3.1. The integral model SK(G, X) over OE,(v) of ShK(G, X) is the normaliza- tion of the (reduced) Zariski closure S − K (G, X) of ShK(G, X) in SK′(GSp, S±)OE,(v). We set The G(Ap f )-action on ShKp(G, X) extends to SKp(G, X). SKp(G, X) := lim ←− Kp SKpKp(G, X). 7.3.1.2 Hodge tensors and deformation theory Since GZ(p) has reductive generic fiber, by Lemma 7.2.10, we can find a finite collection of tensors (sα) ⊂ V ⊗ Z(p) = (V ∨ Z(p) )⊗ whose scheme-theoretic stabilizer in GL(VZ(p)) is GZ(p). Let h : A → SK(G, X) denote the pullback of the universal abelian scheme over SK′(GSp, S±). Denote by V = R1h∗Ω• the (relative) algebraic de Rham cohomology of A. Then the tensors (sα), by the de Rham isomorphism, give rise to a collection of (absolute) Hodge cycles sα,dR ∈ V ⊗ C , where VC is the complex analytic vector bundle attached to V, and sα,dR descends to V ⊗ by [KP18, Proposition 4.2.6] (i.e., sα,dR can be defined over OE,(v)). Recall that ˘E denotes the completion of the maximal unramified extension of E in Q p with residue field k. Let L/ ˘E be a finite extension. For a point x ∈ ShK(G, X)(L) specializing to x ∈ S − K (G, X)(k), we write Ax for the pullback of A to x and write Gx for the p- dR(Ax)⊗. We can divisible group associated with Ax. Then sα,dR pullbacks to sα,dR,x ∈ H 1 also obtain corresponding tensors sα,´et,x in TpG ∨⊗ x by the Betti-´etale comparison theorem. Here TpG ∨ x := HomZp(TpGx, Zp). The tensors sα,´et,x are Galois invariant and their scheme- theoretic stabilizer is isomorphic to G. Write Gx for the p-divisible group corresponding to x and Dx for D(Gx)(W ). Set V := TpG ∨ x ⊗Zp Qp. Then V is a crystalline representation of ΓL := Gal(L/L). The p-adic comparison isomorphism Bcris ⊗Zp TpG ∨ x ≃ Bcris ⊗K0 Dcris(V ), Dcris(V ) := (Bcris ⊗Qp V )ΓL, takes the Galois invariant tensors sα,´et,x to the φ-invariant tensors sα,0 ∈ Dcris(V )⊗. 138 Proposition 7.3.2. We have sα,0 ∈ D⊗ x , where we view D⊗ x as a W -submodule of the K0- vector space Dcris(V )⊗. Moreover, we have the following properties. (1) The tensors sα,0 ∈ D⊗ x lift to φ-invariant tensors (cid:101)sα,x ∈ ΘOL(Gx)⊗, which map into Fil0 D(Gx)(OL)⊗ along the natural projection ΘOL(Gx) → D(Gx)(OL) given by Theorem 7.1.14 (2). Denote by sα,x the image of (cid:101)sα,x. (2) There exists an isomorphism ΘOL(Gx) ≃ W(OL) ⊗Zp TpG ∨ x taking (cid:101)sα,x to sα,´et,x. In particular, there exists an isomorphism Dx ≃ W ⊗Zp TpG ∨ x taking sα,0 to sα,´et,x, and an isomorphism D(Gx)(OL) ≃ D(Gx)(W ) ⊗W OL taking sα,x to sα,0. Therefore, we can identify the group scheme GW ⊂ GL(Dx) defined by sα,0 with G ⊗Zp W , and there exists a GK0(= GW ⊗W K0)-valued cocharacter µy such that a) The filtration on Dx ⊗W L induced by the canonical isomorphism Dx ⊗W L ≃ D(Gx)(OL) ⊗OL L is given by a GK0-valued cocharacter GK0-conjugate to µy. b) µy induces a filtration on Dx which lifts the Hodge filtration on Dx ⊗W k = D(Gx)(k). Proof. As in [KP18, Proposition 3.3.8], the tensors (sα,´et,x) ⊂ TpG ∨⊗ x give rise to φ-invariant tensors sM α,x ⊂ M(Gx)⊗. The tensors sM α,x map to tensors (cid:101)sα,x in ΘOL(Gx)⊗ via the isomor- phism ΘOK (Gx) ≃ φ∗M(Gx) ⊗S,κ W(OK) 139 in Corollary 7.1.18 (1). Since the above isomorphism respects the Hodge filtrations by Corollary 7.1.18 (2), the tensors (cid:101)sα,x map into Fil0 D(Gx)(OL)⊗. The rest of the proof proceeds as in [KP18, Proposition 3.3.8, Corollary 3.3.10]. The above proposition implies that Gx is a (GW , µy)-adapted deformation of Gx in the sense of Definition 7.2.17. 7.3.1.3 Integral models for level G(Zp): properties Fix a parabolic subgroup P ⊂ GL(Dx) lifting P0 corresponding to the Hodge filtration of D(Gx)(k) = Dx ⊗W k. Let y = y(x) ∈ (GL(Dx)/P )) (L) correspond to the cocharacter µy as in Proposition 7.3.2 (2). Then as in §7.2.3, we obtain from y a closed subscheme M loc G,y ⊂ (GL(Dx)/P )O ˘E and formal local models (cid:99)M loc = Spf R, (cid:99)M loc G,y = Spf RG. comparison theorem, the scheme Isom(sα,sα,0)(V ∨ Note that RG is a quotient of RE = R ⊗W O ˘E. By Proposition 7.3.2 (2) and the Betti-´etale Zp ⊗Zp W, Dx) of tensor-preserving isomor- Zp ⊗Zp W ≃ Dx preserving phisms is a trivial G-torsor. Then we may choose an isomorphism V ∨ tensors such that the very good Hodge embedding (by our assumption on (cid:101)ι) (G ⊗Zp W, µh) (cid:101)ι (cid:44)→ (GL(VZp ⊗Zp W ), µg) ≃ (GL(V ∨ Zp ⊗Zp W ), µg) ≃ (GL(Dx), µg) induces a closed immersion Mloc G,µh ⊗OE O ˘E (cid:44)→ (GL(Dx)/P )O ˘E ∼→ Gr(g, Dx)O ˘E . Note that the Hodge filtration on Dx ⊗W L is induced by a G-valued cocharacter conjugate to µ−1 h . Hence, we can identify M loc G,y with Mloc G,µh ⊗OE ⊗O ˘E by Lemma 7.2.7, and so RG is normal. Proposition 7.3.3. Suppose that conditions (i) and (ii) in the beginning of §7.3.1.1 are satisfied. Let (cid:98)Ux be the completion of S − K (G, X)O ˘E at x. Then the irreducible component of (cid:98)Ux containing x is isomorphic to (cid:99)M loc G,y = Spf RG as formal schemes over O ˘E. Proof. We follow the arguments of [KP18, Proposition 4.2.2]. Note that GK0 ⊂ GL(Dx ⊗Zp Qp) contains scalars, since G ⊂ GL(VQ) contains the image of the weight homomorphism wh. As ιQp extend to a very good Hodge embedding, the 140 constructions and results in §7.2 can apply. In particular, by Proposition 7.2.16 (2), we can view Spf RE as a versal deformation space of Gx. Then the p-divisible group over (cid:98)Ux arising from the universal abelian scheme A gives rise to a natural map Φ : (cid:98)Ux → Spf RE, which is a closed embedding by Serre-Tate theorem. Let Z ⊂ (cid:98)Ux be the irreducible component containing x. Let x′ ∈ Z(L′) for some finite field extension L′ of ˘E. Then we can argue as in [KP18, Proposition 4.2.2] to show: sα,´et,x′ corresponds to sα,0 under the p-adic comparison isomorphism for the p-divisible group Gx′. Since the filtration on Dx ⊗W K ′ corresponding to Gx′ is given by a G-valued cocharacter which is conjugate to µy, by Proposition 7.3.2, Gx′ is (GW , µy)-adapted. By our assumption on the integral Hodge embedding (cid:101)ι and Proposition 7.3.2, the assumptions in Proposition G,y by Proposition 7.2.18. Since x′ 7.2.18 are satisfied. Hence, x′ is induced by a point of (cid:99)M loc is arbitrary, it follows that Φ(Z) ⊂ (cid:99)M loc G,y. They are equal, as Z and (cid:99)M loc G,y are of the same dimension. Theorem 7.3.4. Assume the following conditions: (i) Kp = G(Zp); (ii) ιQp extends to a very good integral Hodge embedding (cid:101)ι : (G, µh) (cid:44)→ (GL(VZp), µg), where VZp ⊂ VQp is a self-dual Zp-lattice with respect to ψ. Then the OE,(v)-schemes SK(G, X) and SKp(G, X) constructed in Definition 7.3.1 satisfy the following properties. (1) SKp(G, X) is an OE,(v)-flat, G(Ap f )-equivariant extension of ShKp(G, X). The integral model SK(G, X) is canonical in the sense of [PR24]. (2) For any discrete valuation ring R of mixed characteristic 0 and p, the natural map SKp(G, X)(R) → SKp(G, X)(R[1/p]) is a bijection. 141 (3) SK(G, X) fits into a local model diagram (cid:102)SK(G, X)OE π q SK(G, X)OE Mloc G,µh of OE-schemes, in which π is a G-torsor and q is G-equivariant and smooth of relative dimension dim G. (4) If in addition, we have G = G◦, then for each x ∈ SK(G, X)(k′) with k′/kE finite, there is a point y ∈ Mloc G,µh (k′) such that we have an isomorphism of henselizations Oh SK(G,X),x ≃ Oh Mloc G,µh ,y. Proof. Note that under the assumptions of the above theorem, we have Proposition 7.3.3, which extends [KP18, Proposition 4.2.2] to the case p = 2. Then the proofs of [KP18, Proposition 4.2.2, 4.2.7] and [KPZ24, Theorem 7.1.3] go through, and we obtain the theorem. We note that the assumption (B) in [KPZ24, Theorem 7.1.3] is not used in the proof. The integral model SK(G, X) is canonical by the construction in [PR24]. 7.3.1.4 Integral models for parahoric level G◦(Zp) Now we use previous results to study integral models with parahoric level structure. That is, the level at p is given by G◦(Zp). Write K◦ p = G◦(Zp) and K◦ = K◦ pKp. Note that there is a natural finite morphism of Shimura varieties ShK◦(G, X) → ShK(G, X). Definition 7.3.5. The integral model SK◦(G, X) for parahoric level K◦ is the normalization of SK(G, X) in ShK◦(G, X). We also set SK◦ p(G, X) := lim ←− Kp SK◦ pKp(G, X). Let Gsc denote the simply connected cover of Gder and set C = ker(Gsc → Gder). For a finite prime ℓ and c ∈ H 1(Q, C), we write cℓ for the image of c in H 1(Qℓ, C). We introduce 142 (cid:118) (cid:118) (cid:38) (cid:38) the following assumption: If c ∈ H 1(Q, C) satisfies cℓ = 0 for all ℓ ̸= p, then cp = 0. (7.3.1) Proposition 7.3.6. Assume that conditions (i) and (ii) in Theorem 7.3.4 and condition (7.3.1) are satisfied. (1) Assume Kp is sufficiently small. Then the covering SK◦(G, X) → SK(G, X) is ´etale, and splits over an unramified extension of OE. (2) The geometrically connected components of SK◦ p(G, X) are defined over the maximal extension of E that is unramified at primes above p. Proof. The proof follows the same argument as in [KP18, Proposition 4.3.7, 4.3.9]. 7.3.2 Shimura varieties of abelian type Let (G, X) be a Shimura datum of Hodge type with a Hodge embedding ι : (G, X) (cid:44)→ (GSp(V, ψ), S±). Denote by G the base change GQp. Let G◦ be the parahoric group scheme associated to some point x ∈ B(G, Qp). Assume (i) Kp = G(Zp); (ii) ιQp extends to a very good integral Hodge embedding (cid:101)ι : (G, µh) (cid:44)→ (GL(VZp), µg), where VZp ⊂ VQp is a self-dual Zp-lattice with respect to ψ; (iii) G satisfies condition (7.3.1); (iv) The center ZG of G is an R-smooth torus (see [KZ24, §2.4]). Assume (G2, X2) is a Shimura datum of abelian type such that there is a central isogeny inducing an isomorphism of Shimura data (Gad, X ad) ∼−→ (Gad Gder → Gder 2 , X ad 2 ). Here, 2 X ad denotes the Gad(R)-conjugacy class of had : S h−→ GR → Gad R for some h ∈ X; X ad 2 is similar. 143 As usual, we denote K◦ p := G◦(Zp) ⊂ G(Qp) and G2 := G2,Qp. Let x2 ∈ B(G2, Qp) be a lift of xad 2 = xad in the identification B(Gad scheme associated to x2. Write K◦ 2,p = G◦ 2 , Qp) = B(Gad, Qp). Let G◦ 2 be the parahoric group 2 (Zp). Denote by E2 the reflex field of (G2, X2) and set E′ := E · E2, recall E denotes the reflex field of (G, X). We fix a place v′ of E′ above v. Denote by E′ the completion of E′ at v′. Fix a connected component X + ⊂ X. Denote by ShK◦ p(G, X)+ the geometrically con- nected component containing the image of X + × 1 in G(Q)\X × G(Af )/K◦ pKp. lim ←− Kp By Proposition 7.3.6 (2), ShK◦ p(G, X)+ is defined over the maximal extension Ep of E that is unramified at primes above p. We denote by SK◦ p(G, X)+ the component of SK◦ p(G, X) extending ShK◦ p(G, X)+, which is defined over OEp,(v). 7.3.2.1 Integral models of Shimura varieties of abelian type We recall the notation of [Del79]. Let H be a group equipped with an action of a group ∆, and let Γ ⊂ H be a ∆-stable subgroup. Suppose we are given a ∆-equivariant map φ : Γ → ∆ where ∆ acts on itself by inner automorphisms, and suppose that for γ ∈ Γ, φ(γ) acts on H as conjugation by γ. Then the elements of the form (γ, φ(γ)−1) form a normal subgroup of the semi-direct product H ⋊ ∆. We denote by H ∗Γ ∆ the quotient of H ⋊ ∆ by this normal subgroup. For a subgroup H ⊂ G(R), denote by H+ the preimage in H of the connected component Gad(R)+ of the identity in Gad(R). We write Gad(Q)+ = Gad(Q) ∩ Gad(R)+. Lemma 7.3.7. Suppose S is an affine Q-scheme, and let SZp be a flat affine Zp-scheme with generic fiber S ⊗Q Qp. Then there exists a Z(p)-scheme SZ(p), which is unique up to isomorphism, with generic fiber S and SZ(p) ⊗Z(p) Zp = SZp. 144 have A ⊗Zp Proof. Let A (resp. B) be the affine coordinate ring of SZp (resp. S). By assumption, we Qp = B ⊗Q Qp. Then we can take SZ(p) to be Spec A ∩ B, where the intersection Zp = SZp Qp = B ⊗Q Qp. Any Z(p)-scheme T with generic fiber S and T ⊗Z(p) happens in A⊗Zp is necessarily isomorphic to Spec A ∩ B. By applying the above lemma to the group schemes G and G◦ over Zp, we obtain Z(p)- smooth affine group schemes GZ(p) and G◦ := G◦ model of the parahoric group scheme associated to xad ∈ B(Gad, Qp). Let Gad = GZ(p)/Z, where Z denotes the Zariski closure in GZ(p) of the center Z of the Q-group G. As we assume . Similarly, let Gad◦ = Gad◦ Z(p) be the Z(p)- Z(p) that the center ZG of G is an R-smooth torus, we have Gad◦ is the neutral component of Gad, see [KP18, Lemma 4.6.2] and [KZ24, Proposition 2.4.14]. Following [KP18, §4.6.3], we set A (GZ(p)) := G(Ap f )/Z(Z(p))− ∗G◦(Z(p))+/Z◦(Z(p)) Gad◦(Z(p))+, A (G) := G(Af )/Z(Q)− ∗G(Q)+/Z(Q) Gad(Q)+, and A (GZ(p))◦ := G◦(Z(p))− +/Z ◦(Z(p))− ∗G◦(Z(p))+/Z◦(Z(p)) Gad◦(Z(p))+, A (G)◦ := G(Q)− +/Z(Q)− ∗G(Q)+/Z(Q) Gad(Q)+. Here, G◦(Z(p))− + is the closure of G◦(Z(p))+ in G(Ap f ), and Z ◦ is the Zariski closure of Z in G◦. Similarly, we have A (G2,Z(p)) and A (G2,Z(p)). Since Gad◦ is the neutral component of (we assume ZG is an R-smooth torus), the action of A (GZ(p)) on ShK◦ p(G, X) extends p(G, X). There is an injection by [KP18, Lemma 4.6.10], Gad Z(p) to SK◦ A (GZ(p))◦\A (G2,Z(p)) (cid:44)→ A (G)◦\A (G2)/K◦ 2,p. Let J ⊂ G2(Qp) be a set of coset representatives for the image of the above injection. Definition 7.3.8. The integral model SK◦ 2,p (G2, X2) for ShK◦ 2,p (G2, X2) is [[SK◦ p(G, X)+ × A (G2,Z(p))]/A (GZ(p))◦]|J|. 145 The scheme SK◦ 2,p (G2, X2) is priori defined over OE′p,(v), but it descends to an OE′,(v′)- scheme with a G2(Ap f )-action, see [KP18, Corollary 4.6.15]. Theorem 7.3.9. Assume that conditions (i) to (iv) in the beginning of §7.3.2 are satisfied. (1) The E-scheme ShK◦ 2,p (G2, X2) admits a G2(Ap f )-equivariant extension to a flat normal OE′,(v′)-scheme SK◦ 2,p (G2, X2). Any sufficiently small Kp 2 ⊂ G2(Ap f ) acts freely on SK◦ 2,p (G2, X2), and the quotient SK◦ 2 (G2, X2) := SK2,p(G2, X2)/Kp 2 is a flat normal OE′,(v′)-scheme extending ShK◦ 2 (G2, X2). (2) For any discrete valuation ring R of mixed characteristic 0 and p, the map SK◦ 2,p (G2, X2)(R) → SK◦ 2,p (G2, X2)(R[1/p]) is a bijection. (3) There is a diagram of OE′-schemes (cid:102)S ad K◦ 2,p π q SK◦ 2,p (G2, X2)OE′ Mloc G◦ 2 ,µh2 ⊗OE2 OE′, where π is a G2(Ap f )-equivariant Gad 2,Zp-equivariant, and for any suf- ficiently small Kp 2 ⊂ G2(Ap f ), the map (cid:102)S ad K◦ ⊗OE2 OE′ induced by q is 2,Zp-torsor, q is Gad /Kp 2 → Mloc G◦ 2 ,µh2 2,p smooth of relative dimension dim Gad 2 . If in addition, we have G = G◦, then π reduces to a Gad◦ 2,Zp-torsor. Proof. Under the assumptions of the above theorem, we can construct the integral model SK◦ 2,p (G2, X2) as in Definition 7.3.8. The properties of SK◦ 2,p (G2, X2) are deduced from Theorem 7.3.4 by following the arguments in [KPZ24, Proposition 7.1.14] (cf. [KP18, §4.4- 4.6]). Note that arguments in [KP18, §4.4-4.6] also work for p = 2. 146 (cid:120) (cid:120) (cid:39) (cid:39) Remark 7.3.10. For a Shimura datum (G2, X2) of abelian type as in Theorem 7.3.9, we expect that the integral model SK◦ 2 (G2, X2) is canonical in the sense of [PR24], which would imply that SK◦ 2 (G2, X2) is independent of the choice of a Shimura datum (G, X), as well as the choice of a symplectic embedding (G, X) (cid:44)→ (GSp, S±). 7.3.2.2 Proof of Theorem 1.2.7 in Case (A) Now we start with a Shimura datum (G2, X2) of abelian type with reflex field E2, and denote by K◦ 2,p ⊂ G2(Qp) the parahoric subgroup associated to some x2 ∈ B(G2, Qp). Lemma 7.3.11. Suppose that (Gad 2 , X ad 2 ) has no factor of type DH, G2 is unramified over Qp, and K◦ 2,p is contained in some hyperspecial subgroup. Then there exists a Shimura datum (G, X) of Hodge type, together with a central isogeny Gder → Gder 2 inducing an isomorphism (Gad, X ad) ≃ (Gad 2 , X ad 2 ), such that the following conditions hold. (1) π1(Gder) is trivial. (2) Any prime v2|p of E2 splits completely in E′ = E · E2. (3) X∗(Gab)IQp is torsion free, where Gab denotes the quotient G/Gder and IQp denotes the inertia subgroup of Gal(Q p/Qp). (4) Conditions (i) to (iv) in the beginning of §7.3.2 are satisfied. Proof. As discussed in [KZ24, 2.4.5], the proof of [Edi92, Theorem 4.2] implies that a tamely ramified torus is R-smooth. As we assume G2 is unramified (in particular, G2 is tamely ramified), by [KP18, Lemma 4.6.22], it remains to show that there exists a Hodge embedding ι : (G, X) (cid:44)→ (GSp(V, ψ), S±) satisfying condition (ii) in the beginning of §7.3.2. Since π1(Gder) is trivial by our choice of (G, X), we may assume that, by Zarhin’s trick and [KP18, Corollary 2.3.16], there exists a good integral Hodge embedding (cid:101)ι : (G, µh) (cid:44)→ (GL(Λ), µg) extending ιQp, where Λ ⊂ VQp is a self-dual Zp-lattice with respect to ψQp. Denote GSp := GSp(V, ψ)Qp. By our assumptions and Theorem 7.4.1, there is a tame Galois extension 147 F/Qp with Galois group Γ such that in the diagram B(G, Qp) ι B(GSp, Qp) B(GF , F ) (cid:47) B(GSpF , F ) of Bruhat-Tits buildings, we have • the image of x ∈ B(G, Qp) in B(GF , F ) is hyperspecial, and determines a reductive group H over OF satisfying G ≃ (ResOF /ZpH)Γ; • the point ι(x) is hyperspecial corresponding to the self-dual lattice Λ, and its image in B(GSpF , F ) is hyperspecial corresponding to the lattice (cid:101)Λ := Λ ⊗Zp OF , which is self-dual with respect to the pairing ψF . By [DD11, Lemma 3.1], there exist a totally real number field F/Q and a place w above p such that Fw ≃ F . Let (cid:101)V denote the Q-vector space V ⊗Q F. We pick an element a ∈ F such that its image in F generates the different ideal δF/Qp. Then (cid:101)V is equipped with a perfect alternating pairing given by (cid:101)ψ(x, y) := TrF/Q(a−1ψF (x, y)) for x, y ∈ (cid:101)V . Then (cid:101)Λ is self-dual with respect to (cid:101)ψ, and the closed immersion (cid:101)ι : G (cid:44)→ ResOF /ZpH (cid:44)→ GL((cid:101)Λ) extends the Hodge embedding G (cid:44)→ GSp (cid:44)→ GSp( (cid:101)V , (cid:101)ψ)Qp ⊂ GL( (cid:101)VQp). As π1(Gder) is trivial and G is unramified over Qp, the Pappas-Zhu local model for (G, µh) is isomorphic to Mloc G,µh 2 , X ad (Gad , and (cid:101)ι is a good integral Hodge embedding by [KP18, Proposition 2.3.7]. As 2 ) has no factor of type DH, the closed immersion ResOF /ZpH (cid:44)→ GL((cid:101)Λ) gives a very good integral Hodge embedding by [KPZ24, Proposition 5.3.10, Theorem 1.2.3]. Since G = (ResOF /ZpH)Γ, we obtain that (cid:101)ι is also very good by [KPZ24, Corollary 5.3.4]. We then obtain a desired Hodge embedding by replacing ι by the Hodge embedding (G, X) (cid:44)→ (GSp( (cid:101)V , (cid:101)ψ), S±( (cid:101)V , (cid:101)ψ)). 148 (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) Corollary 7.3.12. Under the same assumptions as in Lemma 7.3.11, the integral model SK◦ 2,p (G2, X2) constructed in Definition 7.3.8 is defined over OE2,(v2) for some fixed prime v2|p of E2. Moreover, we have G = G◦, and the conclusions of Theorem 7.3.9 hold. In particular, if κ is a finite extension of κ(v2) and y ∈ SK◦ 2,p (G2, X2)(κ), then there exists z ∈ Mloc G◦ 2 ,µh2 (κ) such that we have an isomorphism of henselizations Oh S K◦ 2,p (G2,X2),y ≃ Oh Mloc G◦ 2 ,µh2 ,z. Proof. By Theorem 7.3.9 and Lemma 7.3.11 (4), the integer model SK◦ 2,p (G2, X2) is con- structed using the Shimura datum (G, X) chosen in Lemma 7.3.11. By Lemma 7.3.11 (2), there exists a prime v2|p of E2 extending to the prime v′ of E′, and we have OE2,(v2) ≃ OE′,(v′). Hence, the scheme SK◦ 2,p (G2, X2) is defined over OE2,(v2). Since π1(Gder) is trivial by Lemma 7.3.11 (1), we have π1(G) = X∗(Gab), and π1(G)IQp particular, we have G = G◦. is torsion-free by Lemma 7.3.11 (3). In By Theorem 7.3.9 and Corollary 7.3.12, we obtain Theorem 1.2.7 in Case (A). Note that the group G in Theorem 1.2.7 is denoted by G2 here. 7.3.3 Integral models of unitary Shimura varieties In this subsection, we consider Shimura varieties in Case (B) of §1.2.2. We show that, in this case, the assumptions in Theorem 7.3.9 are satisfied, allowing us to construct integral models of Shimura varieties for which the conclusions of Theorem 7.3.9 hold. 7.3.3.1 Let n = 2m + 1 ≥ 3 be an odd integer. Let F/Q be an imaginary quadratic extension such that 2 is ramified in F. Then F := F ⊗Q Q2 is a ramified quadratic extension of Q2 with residue field F2. Let (V, h) be an n-dimensional non-degenerate F/Q-hermitian space of signature (n − 1, 1). Denote by G := GU(V, h) 149 the unitary similitude group over Q attached to (V, h). Suppose that K2 ⊂ G(Q2) is a special parahoric subgroup in the sense of Bruhat-Tits theory. For an open compact subgroup of the form K = K2K2 ⊂ G(Af ), where K2 ⊂ G(A2 f ) is open compact and sufficiently small, we can associate a Shimura variety ShK(G, X) of level K as in [PR09, §1.1]. Then ShK(G, X) is a quasi-projective smooth variety of dimension n − 1 over the reflex field F. Denote by ShK(G, X)F the base change of ShK(G, X) to F . 7.3.3.2 Unitary local models Note that the vector space V := V⊗FF equipped with the F/Q2-hermitian form h := hQ2 defines a unitary similitude group G = GQ2 over Q2. Lemma 7.3.13. For any non-degenerate hermitian form h′ on V , we have G ≃ GU(V, h′). Proof. By the classification of hermitian spaces over local fields (see, for example, [Jac62, Theorem 3.1]), there are two isomorphism classes of n-dimensional non-degenerate hermitian spaces over Q2, classified by discriminants in Q× 2 /NF/Q2(F ×). Let a ∈ Q× 2 be an element not in NF/Q2(F ×). Define a hermitian form ha on V by setting ha(x, y) := ah(x, y) for x, y ∈ V . Since disc(ha) = andisc(h) and n is odd, the hermitian spaces (V, h) and (V, ha) represent the two isomorphism classes of n-dimensional non-degenerate hermitian spaces over Q2. Moreover, multiplication by a induces an isomorphism between GU(V, h) and GU(V, ha). Hence, the lemma follows. By Lemma 7.3.13, we may assume that the hermitian form h is split, that is, there exists an F -basis e1, . . . , en of V such that h(ei, ej) = δi,n+1−j. Then we are in the situation of the first part of the thesis. Up to conjugation, we may assume that the special parahoric subgroup K2 ⊂ G(Q2) corresponds to I = {0} or {m} by Theorem 1.2.1. Let GI denote the special parahoric group scheme corresponding to I = {0} or {m}. By [PR09, 1.2.3], GI is a 150 Bruhat-Tits stabilizer group scheme. Let µ denote the geometric cocharacter G m,F → GF ≃ GLn,F × G m,F given by z (cid:55)→ (diag(z, 1(n−1)), z). Let Mloc GI ,µ be the local model attached to (GI, µ) by Theorem 3.4.4. By Proposition 3.4.5, this is isomorphic to the unitary local model Mloc I in Theorem 1.2.2. Lemma 7.3.14. Let ΛI ⊂ V be the lattice as in Theorem 1.2.1 corresponding to the spe- cial parahoric subgroup K2 ⊂ G(Q2). Then there exists a good integral Hodge embedding (GI, µ) (cid:44)→ (GL(ΛI), µn). Proof. By the concrete description of the parahoric group scheme GI in Chapter 6, there is a closed immersion ι : GI (cid:44)→ GL(ΛI). The base change ιQp is the standard Hodge embedding G = GU(V, h) (cid:44)→ GL(V ), which sends the conjugacy class {µ} to {µn}. As G contains the scalars, ι is an integral Hodge embedding. Moreover, ι is good, since it induces a closed immersion Mloc GI ,µ ≃ Mloc I (cid:44)→ Gr(n, ΛI)OF by our construction of Mloc I . The following theorem is a key ingredient in the construction of very good Hodge em- beddings for (GI, µ). Theorem 7.3.15. For any closed point x ∈ Mloc I (k), the tangent space of the special fiber Mloc I ⊗OF k at x is spanned by smooth formal curves (see Definition 7.2.14). The proof of Theorem 7.3.15 is divided into the following two cases. The case I = {m} By Theorem 1.2.2 (2), the local model Mloc {m} is smooth over OF . Clearly Theorem 7.3.15 holds in this case by the infinitesimal lifting property of smooth morphisms. The case I = {0} By Theorem 1.2.2 (1), Mloc {0} is OF -smooth on the complement of a single closed point, which we will call the worst point. To prove Theorem 7.3.15 in this case, it suffices to prove the tangent space of Mloc {0} ⊗OF k at the worst point is spanned by smooth formal curves. 151 Definition 7.3.16. Let X be an affine scheme of finite type over k. Let x ∈ X(k) be a k-point. We may express X as a closed subscheme of Ad = Spec k[T1, . . . , Td] defined by an ideal a ⊂ k[T1, . . . , Td] such that x is the origin of Ad. (1) For a polynomial f ∈ k[T1, . . . , Td], write f = (cid:80)N i=r fi as a decomposition into homo- geneous polynomials with fr ̸= 0. Denote by f ∗ (resp. f (1)) the lowest degree term fr (resp. f1). If r ≥ 2, set f1 = 0. (2) Denote by a∗ (resp. a(1)) the ideal in k[T1, . . . , Td] generated by f ∗ (resp. f (1)), for all f ∈ a. The tangent cone T CxX (resp. schematic tangent space T sch x X) of X at x is the scheme Spec k[T1, . . . , Td]/a∗ (resp. Spec k[T1, . . . , Td]/a(1)). Note that the definition of T CxX (resp. T sch x X) is independent of the embeddings of X in affine spaces. See [Mum99, Chapter III, §3, 4]. Clearly T sch x X is a linear subspace of Ad and there is a closed immersion T CxX (cid:44)→ T sch x X. Note that there is a natural bijection x X(k) and the tangent space TxX, see [Mum99, §4]. Concretely, for between the k-points T sch any z ∈ T sch x X(k) corresponding to a k-algebra homomorphism z : k[T1, . . . , Td]/a(1) → k, we can associate a k-algebra homomorphism tz : k[T1, . . . , Td]/a → k[t]/(t2) via Ti (cid:55)→ z(Ti)t. The morphism tz defines a tangent vector of X at x. Lemma 7.3.17. Let X be a reduced affine scheme of finite type over k. Let x ∈ X(k). Assume that there exists a closed immersion i : X (cid:44)→ Ad such that X is defined by a homogeneous ideal a and i(x) is the origin O of Ad. Then the set T CxX(k) spans the k- vector space TxX. Proof. Without loss of generality, we may assume that i does not factor through any (proper) linear subspace of Ad. As X is reduced, the image i(X) is not contained in any (proper) linear subspace of Ad. Since a is homogeneous, X is isomorphic to the tangent cone T CxX and i is identified with the embedding T CxX (cid:44)→ T sch x X (cid:44)→ T sch O Ad. Let W denote the subspace in TxX spanned by T CxX(k). Then we have a linear subspace W sch ⊂ Ad such 152 that W sch(k) = W . We obtain a factorization i : X (cid:44)→ W sch (cid:44)→ T sch x X (cid:44)→ Ad. Since i : X (cid:44)→ Ad does not factor through any proper linear subspace of Ad, it forces that W sch = T sch x X = Ad, and hence, W = TxX. Corollary 7.3.18. Under the same assumptions as in Lemma 7.3.17, the tangent space TxX is spanned by smooth formal curves. Proof. Denote X = Spec R = Spec k[T1, . . . , Td]/a. By assumption, the tangent cone T CxX is isomorphic to X. Recall that for a k-point z ∈ T CxX(k) corresponding to z : R = k[T1, . . . , Td]/a → k, the associated tangent vector tz ∈ X(k[t]/(t2)) is given by the k-algebra homomorphism R → k[t]/(t2) sending Ti (cid:55)→ z(Ti)t. Define a k-algebra homomorphism (cid:101)tz : k[T1, . . . , Td] → k[[t]] via Ti (cid:55)→ z(Ti)t. For any homogeneous polynomial f ∈ a, we have (cid:101)tz(f ) = f (z(T1)t, . . . , z(Td)t) = tdeg f f (z(T1), . . . , z(Td)) = 0. Hence, the map (cid:101)tz factors through R/a. In other words, the tangent vector tz lifts to the smooth formal curve (cid:101)tz ∈ X(k[[t]]). Now the corollary follows from Lemma 7.3.17 immediately. Recall that, by Theorem 1.2.3 (1), there is an open affine neighborhood Uloc {0} of Mloc {0} containing the worst point such that Uloc {0} ⊗OF k is defined by a homogeneous ideal under the obvious closed embedding Uloc {0} ⊗OF k (cid:44)→ Spec k[A|B], which sends the worst point to the origin. By Corollary 7.3.18, we obtain the following. Corollary 7.3.19. The tangent space of Mloc {0} ⊗OF k at the worst point is spanned by smooth formal curves. This proves Theorem 7.3.15 in the case I = {0}. 153 7.3.3.3 Proof of Theorem 1.2.7 in Case (B) Let us keep the notation as in §7.3.3.1. Let a ∈ F× be an element such that a = −a. Then the hermitian form h on V induces a perfect alternating Q-bilinear form ψ on V by setting ψ(x, y) := TrF/Q(a−1h(x, y)), for x, y ∈ V. Denote by GSp(V, ψ) the symplectic similitude group over Q associated with the above pair- ing. Then we obtain an embedding ι1 : G (cid:44)→ GSp(V, ψ), which also induces an embedding of Shimura data ι1 : (G, X) (cid:44)→ (GSp(V, ψ), S±(V, ψ)). By Lemma 7.3.14, there exists a good integral Hodge embedding (cid:101)ι1 : (GI, µ) (cid:44)→ (GL(ΛI), µn) extending ι1,Q2. By Theorem 7.3.15 and Lemma 7.2.15, (cid:101)ι1 is very good. Denote by Λ# the dual lattice of ΛI with respect to ψ. Set Λ := (ΛI)4 ⊕ (Λ# I )4 ⊂ V 8. Using Zarhin’s trick I ⊂ V as in the proof of [KPZ24, Proposition 7.2.10 (3)], there exists a non-degenerate alternating pairing ψ′ on V8 such that Λ is self-dual with respect to ψ′ Q2 , and an embedding of Shimura data ι : (G, X) (cid:44)→ (GSp(V8, ψ′), S±(V8, ψ′)) such that ι extends to a very good integral Hodge embedding (GI, µ) (cid:44)→ (GL(Λ), µ8n). Denote (GSp, S±) := (GSp(V8, ψ′), S±(V8, ψ′)). Then we obtain an embedding of Shimura data ι : (G, X) (cid:44)→ (GSp, S±). Moreover, the embedding ιQ2 extends to a very good integral Hodge embedding by previous discussion. Note that for odd unitary similitude groups, the parahoric group scheme corre- sponding to K2 is connected by [PR09, 1.2.3]. In particular, the assumptions in Theorem 7.3.4 are satisfied and we obtain the following theorem. 154 Theorem 7.3.20. There exists a normal flat OF -scheme SK(G, X) extending ShK(G, X) such that the conclusions of Theorem 7.3.4 hold for SK(G, X). This finishes the proof of Theorem 1.2.7 in Case (B). 7.4 Bruhat-Tits group schemes and tame Galois fixed points In this section, we show that, for an unramified group G over a 2-adic field F , if a stabilizer group scheme G satisfies G(OF ) ⊂ H for some hyperspecial subgroup H of G(F ), then G can be written as the tame Galois fixed points of the Weil restriction of scalars of a reductive group scheme. This result is used in the proof of Lemma 7.3.11 to construct very good Hodge embeddings in Case (A). Let F be a complete discrete valued field with residue characteristic p = 2. Let G be a connected reductive group over F . Denote by B(G, F ) (resp. B(G, F )) the extended (resp. “classical”) Bruhat-Tits building. Recall that for a finite tame Galois extension K/F with Galois group Γ, the inclusion B(G, F ) (cid:44)→ B(G, K) of buildings identifies the image with the fixed point set B(G, K)Γ. For x ∈ B(G, F ), we use GK x to denote the Bruhat-Tits group scheme over OK attached to the image of x in B(G, K). Theorem 7.4.1. Assume G is unramified. Let G = Gf be the Bruhat-Tits group scheme attached to some facet f in B(G, F ) whose closure contains a hyperspecial point. Then there exist a point x ∈ B(G, F ) and a finite tame Galois extension K/F with Galois group Γ such that G ⊗F K is split, G = Gx, and (the image of ) x is hyperspecial in B(G, K). Moreover, we have an isomorphism of (smooth) OF -group schemes G ≃ (ResOK /OF GK x )Γ extending the isomorphism G ≃ (ResK/F GK)Γ. 155 The proof of Theorem 7.4.1 We first consider the case when G is split, absolutely simple, and simply connected. Fix a maximal torus T and a Borel subgroup B containing T . Let ∆ = {α1, . . . , αn} be the subset of simple roots with respect to (T, B) in the root system Φ = Φ(T, B). Denote by Φ+ = Φ ∩ Z≥0∆ the set of positive roots. Note that there is a perfect pairing ⟨−, −⟩ : X∗(T ) × X ∗(T ) → Z between the cocharacter group X∗(T ) and the character group X ∗(T ) of T . There is an isomorphism between the apartment A of B(G, F ) corresponding to T and V := X∗(T )R such that the origin in V corresponds to a special vertex, which is also hyperspecial, in A. Moreover, a chamber C of A is given by C = (cid:8)x ∈ V | 0 < ⟨x, α⟩ < 1 for all α ∈ Φ+(cid:9) . For 1 ≤ i ≤ n, denote by ωi ∈ V the fundamental coroot corresponding to αi ∈ ∆. Then the chamber C has n + 1 vertices v0, . . . , vn, where v0 = 0 and vi = ωi/ci for 1 ≤ i ≤ n, where ci is a positive integer such that (cid:80)n i=1 ciαi is the highest root in Φ. Since G(F ) acts transitively on the set of chambers in A (see, for example, [Tit79, §1.8]), we may assume that f is contained in the closure of C. By assumption, the closure f of f contains a hyperspecial vertex vf . Note that vf is some vertex vi for which ci = 1. If f consists of only a single point, there is nothing to prove. Hence, we may assume that f strictly contains vf . Let y ∈ V be the barycenter of the (sub)facet determined by the vertices in f except vf . Then y is of the form y = 1 m2d y1, where m is an odd integer, d ≥ 0 is an integer, and y1 ∈ Z∆. Set x := 1 m2d+1 + 1 vf + m2d+1 m2d+1 + 1 y = 1 m2d+1 + 1 vf + 2 m2d+1 + 1 y1 Then x lies in the line segment between vf and y, and hence in f . Since G is simply connected, we have Gx = Gf . 156 Let F1 be a finite extension of F with ramification index m2d+1 + 1. Denote by ρ : B(G, F ) (cid:44)→ B(G, F1), the natural inclusion of buildings. Then we see that ρ(x) = vf + 2y1 ∈ vf + X∗(T ). Thus, ρ(x) is a hyperspecial point in B(G, F1). As p = 2, the extension F1/F is tame. Let K be the Galois closure of F1/F . Then K is a tame Galois extension of F . Note that the image of ρ(x) in B(G, K) is also hyperspecial. The pair (K, x) satisfies the conclusion of Theorem 7.4.1. Next we consider the case when G is unramified, absolutely simple and simply connected. Let F1/F be an unramified Galois extension over which G is split. Denote by Γ1 the Galois group of F1/F . Then the facets in B(G, F ) correspond to Γ1-invariant facets in B(G, F1). Let f1 be the Γ1-invariant facet in B(G, F1) corresponding to f . The closure of the facet f1 contains a hyperspecial point, which is the image of vf in B(G, F1). Let y1 be the barycenter of f1. Then y1 is a fixed point of Γ1 and we have Note that y1 is of the form G = (ResOF1 /OF GF1 y1 )Γ1. y1 = 1 m2d (vf + y2), where m is odd and y2 ∈ X∗(T ) for a maximal torus T in the split group GF1. Since y1 and vf are fixed by Γ1, so is any point in the line segment of y1 and vf . Set x := 1 m2d+1 + 1 vf + m2d+1 m2d+1 + 1 y1 = 3 m2d+1 + 1 vf + 2 m2d+1 + 1 y2. Then x lies in the line segment between y1 and vf , and hence is fixed by Γ1. We obtain that x corresponds to a point in B(G, F ) and Gx = Gf . Let F2 be a finite (tame) extension of F1 with ramification index m2d+1 + 1. Then the image of x in B(G, F2) is of the form 157 3vf + 2y2 ∈ 3vf + X∗(T ). Since 3vf is hyperspecial, x is hyperspecial in B(G, F2). Let K be the Galois closure of F2/F . Then K is a tame Galois extension of F and the pair (K, x) satisfies the conclusion of Theorem 7.4.1. In particular, Theorem 7.4.1 holds when G is unramified, absolutely simple and simply connected. 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