TY - JOUR
AU - Cho, Sungmun
AB - Abstract In this article, we explain a simple and uniform construction of a smooth integral model associated to a quadratic, (anti)-hermitian, and (anti)-quaternionic hermitian lattice defined over an arbitrary local field. As one major application, we explain a conjectural recipe for computing local densities case by case, which is an essential factor in the classification of forms as above over the ring of integers of a number field, by introducing one conjecture about the number of rational points of the special fiber of a smooth integral model. 1 Introduction A long standing central problem in the arithmetic theory of hermitian (or quadratic) forms is the classification of hermitian (or quadratic) forms over the ring of integers of a number field. If we let $$\mathfrak{O}$$ be the ring of integers in a number field $$k$$, then for a totally definite hermitian (or quadratic) $$\mathfrak{O}$$-lattice $$(L, H)$$, the genus of $$(L, H)$$, denoted by $$\mathrm{gen}(L, H)$$, is defined as the set of (equivalence classes of) hermitian (or quadratic) lattices that are locally equivalent to $$(L, H)$$. Since the local-global principle does not hold for a hermitian (or quadratic) lattice $$(L, H)$$, the set $$\mathrm{gen}(L, H)$$ is not trivial in general. However, it is well-known that $$\mathrm{gen}(L, H)$$ is a finite set. The computation of the total mass of $$(L, H)$$ ($$=\sum_{(L^{\prime}, H^{\prime}) \in \mathrm{gen}(L, H)} \frac{1}{\#\mathrm{Aut}_{\mathfrak{O}}(L^{\prime}, H^{\prime})}$$) is an essential ingredient for enumerating all elements of the set $$\mathrm{gen}(L, H)$$ explicitly. The total mass of $$(L, H)$$ can be expressed as a product of local factors, the so-called local densities, by the celebrated Smith–Minkowski–Siegel mass formula. The local density is defined as follows. To simplify notation, we consider a quadratic $$A$$-lattice $$(L, h)$$ at this moment. Here, $$A$$ is the ring of integers of a local field $$F$$ with a uniformizer $$\pi$$ and the residue field $$\kappa$$. The local density was originally defined as the limit of a certain sequence [10], which is described below: βL=12⋅limN→∞q−NdimG#G_′(A/πNA). Here $$\underline{G}^{\prime}$$ is a naive integral model of the orthogonal group $$\mathrm{O}(V, h)$$, where $$V=L\otimes_AF$$, such that $$\underline{G}^{\prime}(R)=\mathrm{Aut}_{R}(L\otimes_AR, h\otimes_AR)$$ for any commutative $$A$$-algebra $$R$$, and $$q$$ is the cardinality of $$\kappa$$. This limit stabilizes after finite steps, and the formula is found in [5] and [8] for $$\mathbb{Z}_p$$. However, for a given quadratic lattice defined over a finite (especially ramified) extension of $$\mathbb{Z}_2$$, it is a nontrivial task to compute the above limit. Later, W. T. Gan and J.-K. Yu found another approach for computing local densities in [6]. (We explain their approach here briefly and for a detailed exposition including a general lattice such as a hermitian lattice, see Section 3 of [6].) The local density of a quadratic $$A$$-lattice $$(L, h)$$ can also be defined as an integral of a certain volume form $$\omega^{\mathrm{ld}}$$ associated to $$\underline{G}^{\prime}$$, which is described below: βL=12∫G_′(A)|ωld|. On the other hand, the following integral is known. ∫G_′(A)|ωcan|=q−dimG⋅#G_(κ). Here $$\underline{G}$$ is the unique smooth affine group scheme (called a smooth integral model) of $$\mathrm{O}(V, h)$$ such that $$\underline{G}(R)=\underline{G}^{\prime}(R)$$ for any étale $$A$$-algebra $$R$$ (Proposition 3.7 in [6]) and $$\omega^{\mathrm{can}}$$ is a volume form associated to $$\underline{G}$$. Therefore, in order to obtain an explicit formula for the local density, it suffices to determine $$\underline{G}$$ and the number of rational points of the special fiber of $$\underline{G}$$ (i.e., $$\# \underline{G}(\kappa)$$); compare two volume forms $$\omega^{\mathrm{ld}}$$ and $$\omega^{\mathrm{can}}$$. The difference between the two volume forms can be calculated directly from the construction of the smooth integral model $$\underline{G}$$. Therefore, constructing the smooth integral model $$\underline{G}$$ and investigating the special fiber of $$\underline{G}$$ will lead us to an explicit formula. The local density formula together with a smooth integral model $$\underline{G}$$ and its special fiber has been fully studied in [6] $$(p\neq 2)$$, [2] (quadratic lattice with $$A/\mathbb{Z}_2$$ unramified), and [3] and [4] (ramified hermitian lattice with $$A/\mathbb{Z}_2$$ unramified). Indeed, the constructions of $$\underline{G}$$ in [2–4], when $$A/\mathbb{Z}_2$$ is unramified, are much more complicated than that of [6] when $$p\neq 2$$. More precisely, if $$p\neq 2$$, then the construction of $$\underline{G}$$ is based on the dual lattice $$L^{\#}$$ of $$L$$. In the case that $$A/\mathbb{Z}_2$$ is unramified, one needs to consider a series of sublattices of $$L$$ in order to construct $$\underline{G}$$. Therefore, if $$A/\mathbb{Z}_2$$ is ramified, then it is natural to expect that one needs much more involved sublattices of $$L$$ beyond a series of sublattices of $$L$$ used in the unramified case in order to construct $$\underline{G}$$. One can also expect that a construction of $$\underline{G}$$ in a quadratic lattice is different from that of a hermitian lattice, as in [2–4]. There are two main parts in this article. In the first main part (Section 3), we explain a uniform construction of a smooth integral model $$\underline{G}$$ associated to a quadratic or a hermitian lattice over any non-Archimedean local field without taking account of any sublattice of $$L$$. Our construction of $$\underline{G}$$ is simple and canonical (independent of the choice of a basis of $$L$$). As one major application, we obtain the following theorem about the local density for a given lattice $$(L, h)$$: Theorem 1.1. (Theorem 3.25) Let $$\tilde{G}$$ (respectively $$G$$) be the special fiber (respectively generic fiber) of $$\underline{G}$$ and let $$q$$ be the cardinality of $$\kappa$$. Then the local density of ($$L,h$$) is βL=1[G:Go]qN⋅q−dim G⋅#G~(κ), where $$G^o$$ is the identity component of $$G$$ and $$N$$ is a suitable integer. □ Therefore, in order to compute the local density after constructing $$\underline{G}$$, we only have to know two ingredients in the above theorem; $$q^N$$ and $$\#\tilde{G}(\kappa)$$. Here $$q^N$$ quantifies the difference between the two volume forms $$\omega^{\mathrm{ld}}$$ and $$\omega^{\mathrm{can}}$$ and this can be computed easily based on the construction of $$\underline{G}$$, as explained in Theorem 3.25. Thus, once we construct $$\underline{G}$$, the remaining challenging ingredient is the computation of $$\#\tilde{G}(\kappa)$$. The second main part of this article (Section 4) is devoted to explaining a conjectural recipe for computing $$\#\tilde{G}(\kappa)$$ case by case. For a given quadratic lattice $$(L, h)$$, let $$L=\bigoplus_{i\geq 0}L_i$$ be a Jordan splitting of $$(L, h)$$. Then there exists a morphism as algebraic groups $$\varphi_{i} : \tilde{G} \rightarrow \mathrm{O}(\bar{V_i}, \bar{h_i})$$ which is naturally induced from the structure of $$(L, h)$$ (Proposition 4.9). Here, $$\bar{V_i}$$ and $$\bar{h_i}$$ are defined in Proposition 4.9. Let φ=∏iφi:G~⟶∏iO(Vi¯,hi¯). The formation of $$\varphi$$ is compatible with finite unramified extension of $$A$$. To compute $$\#\tilde{G}(\kappa),$$ we propose the following conjecture. Conjecture 1.2. (Conjecture 4.10) (1) The morphism $$\varphi$$ is surjective. (2) The smooth (possibly disconnected) algebraic group $$U := (\mathrm{ker~}\varphi)_{\rm{red}}$$ is unipotent with constant component group. □ Applying this general conjecture after any unramified extension of $$A$$ implies the surjectivity in (1) on $$\kappa'$$-points for all finite extensions $$\kappa'$$ of $$\kappa$$. Unipotence implies the finite étale component group $$\pi_0(U) = U/U^0$$ has order $$2^{\beta}$$ for some $$\beta$$, and the identity component $$U^0$$ is isomorphic to an affine space $$\mathbb{A}^{\ell}$$ for some $$\ell$$ (as for any smooth unipotent group over a perfect field of characteristic 2). Here, $$\ell={\mathrm{dim}}$$$$\tilde{G}$$ - $$\sum_i \mathrm{dim}\,\,\, \mathrm{O}(\bar{V_i}, \bar{h_i})$$. The dimension of $$\tilde{G}$$ is the same as the dimension of $$G$$ which is $$n(n-1)/2$$, where $$G$$ is the generic fiber of $$\underline{G}$$ and $$n$$ is the rank of $$L$$, since $$\underline{G}$$ is flat over $$A$$. Once one verifies the conjecture, we obtain that #G~(κ)=∏i#(O(Vi¯,hi¯)(κ))⋅qℓ⋅2β. The conjecture tells us more than $$\#\tilde{G}(\kappa)$$. Since we conjecture that $$\varphi_{\kappa'}$$ is surjective for every finite extension $$\kappa'$$ of $$\kappa$$, it follows that the local density is given by a rational function of $$q$$ as we extend the base ring $$A$$ to finite unramified extensions $$A'$$ (cf. Remark 4.11). This had not been expected before this article. The conjecture appears to be very difficult in the general case. For the special cases when $$A$$ is a finite unramified extension of $$\mathbb{Z}_2$$ in [2–4], we proved this by writing down formal matrix forms of an element of $$\tilde{G}(\kappa)$$ and the group homomorphism $$\varphi_{\kappa}$$ explicitly based on a matrix form of an element of $$\underline{G}(A)$$. In the general case, it seems unlikely that such formal matrix forms can be written because the classification of hermitian (or quadratic) lattices over a ramified extension of $$\mathbb{Z}_2$$ is a highly involved problem [9, 12, 13]. However, the construction of $$\underline{G}$$ studied in this article gives an explicit matrix form of an element of $$\underline{G}(A)$$ which yields explicit formal matrix forms of $$\varphi_{\kappa}$$ as well as an element of $$\tilde{G}(\kappa)$$, case by case. In Remark 4.12, we explain a possible framework to prove the conjecture case by case, based on the proof of the conjecture in a specific case as given in Theorem 4.4 (unimodular quadratic lattices with odd rank). In order to show how the framework in Remark 4.12 works case by case, we provide Examples 4.13–4.16 (non-unimodular quadratic lattices) and prove the conjecture in those cases. Especially, Examples 4.14–16 and Remark 4.17 serve as evidence for constancy of the component group of $$(\mathrm{ker~}\varphi)_{\rm{red}}$$. Example 4.14 serves as evidence that the special fiber (and the local density) depends on lower ramification groups of the Galois group as well as the ramification index, which had not been known before this article. In Remark 4.17, we explain one expected phenomenon about the number of component groups (i.e., the value of $$\beta$$) in terms of “distance” between Jordan components. This article does not render the papers [2–4, 6] obsolete since we do not offer any new method to prove the conjecture. One of the main contributions in those articles is to prove special cases of the conjecture (for an unramified extension of $$\mathbb{Z}_2$$). Consequently, by assuming that the above conjecture is true, this article gives a new formula in terms of geometrically meaningful invariants for computing the local density of a given quadratic (or hermitian) lattice case by case, especially, defined over an arbitrary ramified extension of $$\mathbb{Z}_2$$. One strength of this conjectural recipe is that all computations for local densities of given lattices are reduced to computations on finitely generated $$A$$-modules (conditionally depending on Conjecture 4.10 and Remark 4.12), limiting the language of schemes to as little as possible. This article is organized as follows. After fixing the notations in Section 2, we explain a uniform construction of a smooth integral model $$\underline{G}$$ in Section 3. The constructions of $$\underline{G}$$ in [2–4, 6] are special and simple cases (at most $$\alpha=2$$) of the construction studied in this article, which is explained in Remark 3.23. Then we give the local density formula in Theorem 3.25. In the first part of Section 4 (4.1–4.7), we work out completely the mass of unimodular quadratic lattices in odd dimension. In the second part of Section 4 (4.8-4.17), we explain a conjectural recipe for computing $$\#\tilde{G}(\kappa)$$ in the general case by introducing Conjecture 4.10 and Remark 4.12 about the number of rational points of the special fiber of $$\underline{G}$$. We wish to point out that our method also works for symplectic lattices, hermitian lattices, anti-hermitian lattices, quaternionic hermitian lattices, and anti-quaternionic hermitian lattices over any non-Archimedean local field. 2 Notations This section is taken from [6]. 2.0.1 Let $$F$$ be a non-Archimedean local field with $$A$$ its ring of integers and $$\kappa$$ its residue field. Let $$p$$ be the residue characteristic of $$F$$. Choose a uniformizing element $$\pi$$ of $$A$$. 2.0.2 Let $$(K, \sigma)$$ be one of the following $$F$$-algebras with involution: $$K=F$$ with $$\textit{char } F\neq 2$$, $$\sigma$$ identity; $$K=E$$ a separable quadratic extension, $$\sigma$$ the unique nontrivial automorphism of $$E/F$$; $$K=F\oplus F$$, $$\sigma(x,y)=(y,x)$$; $$K=D$$ is the unique nonsplit quaternion algebra over $$F$$, $$\sigma$$ the standard involution. 2.0.3 Let $$B$$ be a maximal $$A$$-order in $$K$$. Then $$B$$ is uniquely determined. If $$K=E$$ is a ramified quadratic extension of $$F$$, or if $$K=D$$, we let $$\pi_K$$ be a uniformizer of $$K$$; in all other cases, we let $$\pi_K=\pi$$. Note that $$\pi$$ in $$K=F\oplus F$$ means $$(\pi, \pi)$$. 2.0.4 Let $$\epsilon$$ be either $$1$$ or $$-1$$. The triple $$(K, \sigma, \epsilon)$$ will be fixed throughout this article, and by a hermitian form we always mean a $$(\sigma, \epsilon)$$-hermitian form. If $$(K, \epsilon, p)\neq (F, 1, 2)$$, we consider a $$B$$-lattice $$L$$ (i.e., a free right$$B$$-module of finite rank) of rank $$n$$ with a hermitian form $$h : L \times L \rightarrow B.$$ Our convention is h(v⋅a,w⋅b)=σ(a)h(v,w)b, h(w,v)=ϵσ(h(v,w)). If $$(K, \epsilon, p)= (F, 1, 2)$$, we consider a $$B (=A)$$-lattice $$L$$ with a quadratic form $$h : L \rightarrow B$$, and we write $$h(v, w)=1/2\cdot (h(v+w)-h(v)-h(w))$$. When $$K=F$$, the characteristic of $$F$$ is not 2 as explained in Section 2.0.2. We assume that $$V=L\otimes_AF$$ is nondegenerate with respect to $$h$$. The right $$B$$-module $$L$$ is also regarded as a left $$B$$-module by the rule $$a\cdot v=v\cdot \sigma(a)$$. Definition 2.1. We define the dual lattice of $$L$$, denoted by $$L^{\#}$$, as L#={x∈L⊗AF:h(x,L)⊂B}. □ The pair ($$L, h$$) is fixed throughout this article. Lemma 2.2. For a given pair ($$L, h$$), we have the following identity: (L#)#=L. □ This lemma can be deduced from the fact that $$L^{\#}$$ is isomorphic to $$\mathrm{Hom}_B(L, B)$$ with its right $$B$$-module structure via $$\sigma$$, from which the double duality follows automatically, see for example, II.2.2 of [11]. 2.2.1 Let $$G$$ be the reductive algebraic group over $$F$$ such that G(F)=AutK⊗FF(V⊗FF,h⊗FF) for any commutative $$F$$-algebra $$\mathfrak{F}$$. Then $$G$$ is a classical group, not necessarily connected. Indeed, it is essential that $$G$$ is smooth in our construction of the smooth integral model $$\underline{G}$$ due to Theorem 3.1 below. Thus we exclude the case in Subsection 2.0.2 that $$G$$ is an orthogonal group over a field of characteristic 2, that is, the case $$K=F$$ with $$\textit{char } F= 2$$ and $$\sigma$$ is the identity. We denote by $$\underline{\mathrm{GL}_K(V)}_{/F}$$ the $$F$$-group scheme whose group of $$\mathfrak{F}$$-valued points is $$\mathrm{GL}_{K\otimes_F\mathfrak{F}}(V\otimes_F\mathfrak{F})$$ for any commutative $$F$$-algebra $$\mathfrak{F}$$. If $$K$$ is commutative, this is just the Weil restriction $$\mathrm{Res}_{K/F}\underline{\mathrm{GL}_K(V)}$$. 3 Smooth integral model We start with the following theorem which is crucially used in this article. Theorem 3.1. (Proposition 3.7 in [6]) Assume that $$A$$ is a discrete valuation ring. Let $$\underline{G}^{\prime}$$ be an affine group scheme over $$A$$ of finite type with smooth generic fiber $$G$$. Then there exists a unique smooth affine group scheme $$\underline{G}$$ over $$A$$ with generic fiber $$G$$ such that G_(R)=G_′(R)for any étale A-algebra R. □ Let $$\underline{G}^{\prime}$$ be the naive integral model of $$G$$ such that for any commutative $$A$$-algebra $$R$$, $$\underline{G}^{\prime}(R)=\mathrm{Aut}_{B\otimes_AR}(L\otimes_AR, h\otimes_AR)$$. Then, the above theorem guarantees the existence of a unique smooth integral model $$\underline{G}$$ of $$G$$ such that $$\underline{G}(R)=\underline{G}^{\prime}(R)$$ for any étale $$A$$-algebra $$R$$. In this section, we give an explicit construction of the smooth integral model $$\underline{G}$$. We briefly explain the strategy of our construction. The naive integral model $$\underline{G}^{\prime}$$ is defined as the closed subgroup scheme of the group of invertible elements in $$\mathrm{End}_{B}(L)$$ stabilizing the hermitian form $$h$$. We will first construct a certain congruence subgroup of $$\mathrm{End}_{B}(L)$$ (denoted by $$\widetilde{T}$$) from a sequence of functors $$T^m$$ from the category of flat $$A$$-algebras to the category of abelian groups. Then we will show that the closed subgroup scheme of the group of invertible elements in $$1+\widetilde{T}$$ (denoted by $$\underline{M}$$) stabilizing the hermitian form $$h$$ is the desired smooth integral model $$\underline{G}$$. We will show that the functors $$T^m$$ are represented by a polynomial ring in some number of variables over $$A$$, in the following sense. Firstly, $$T^m$$ is represented as a sheaf on the small fppf site over $$A$$. Secondly, a polynomial ring in some number of variables over $$A$$ means a ring of the form $$A[t_1, \cdots, t_l]$$, where $$t_i$$’s are algebraically independent over $$A$$. Finally, by saying that a functor $$T^m$$ is represented by a polynomial ring $$A[t_1, \cdots, t_l]$$, we mean that $$T^m$$ is the functor of points of an affine space $$\mathbb{A}^l$$. 3.2 Constructions of $$T^0$$ and $$H^0$$ based on [6] We first define a functor $$T^0$$ from the category of commutative flat $$A$$-algebras to the category of rings. For any commutative flat $$A$$-algebra $$R$$, set T0(R)={X∈EndB⊗AR(L⊗AR):X(L#⊗AR)⊂L#⊗AR}. The functor $$T^0$$ is represented by a unique flat $$A$$-algebra which is a polynomial ring over $$A$$ in $$n^2\cdot[K:F]$$ variables (cf. Section 5.2 of [6]). Moreover, it is easy to see that $$T^0$$ has the structure of a ring scheme since $$T^0(R)$$ is closed under addition and multiplication. For future use, we state the following proposition. Proposition 3.3. For a flat $$A$$-algebra $$R$$, choose $$X \in T^0(R)$$. We define the adjoint $$X^{ad}$$ of $$X$$ characterized as $$h(X(v), w)=h(v, X^{ad}(w))$$, where $$v, w \in L \otimes_A R$$. Then we have that Xad∈T0(R) and (Xad)ad=X. □ Proof. By Lemma 2.2, it is easy to see that $$X^{ad}$$ stabilizes $$L^{\#}$$ as well as $$L$$. This completes the proof. ■ The proposition yields that $$X^{ad}\cdot Y\in T^0(R)$$, where $$X, Y \in T^0(R)$$. This will be used later on. We define another functor $$H^0$$ from the category of commutative flat $$A$$-algebras to the category of abelian groups. For any commutative flat $$A$$-algebra $$R$$, set H0(R)={f : f is an hermitian form on L⊗AR such that f(L⊗AR,L#⊗AR)⊂B⊗AR}. The functor $$H^0$$ is also represented by a flat $$A$$-algebra which is a polynomial ring over $$A$$ in $$n^2\cdot[K:F]-\mathrm{dim~}G$$ variables (cf. Section 5.4 of [6]). Moreover, it is easy to see that $$H^0$$ has the structure of an abelian group scheme since $$H^0(R)$$ is closed under addition. The fixed hermitian form $$h$$ is an element of $$H^0(A)$$. 3.4 Constructions of three functors $$\boldsymbol{\varphi_{0}}$$, $$\boldsymbol{\psi_{0}}$$, $$\boldsymbol{\bar{\varphi}_{0}}$$ 3.4.1 We define a morphism of functors $$\varphi_0 : T^0 \rightarrow H^0$$ defined on a flat $$A$$-algebra $$R$$ by φ0,R:T0(R)⟶H0(R), X↦h∘X. Here, $$h\circ X$$ is a hermitian form on $$L\otimes_AR$$ such that $$h\circ X(v, w)=h(X(v), X(w))$$ for $$v, w \in L\otimes_AR$$. Then $$\varphi_0$$ is represented by a morphism of $$A$$-schemes. 3.4.2 Let $$R$$ be a commutative $$A$$-algebra. Since $$T^0$$ is represented by a polynomial ring in some number of variables, the fiber of the identity in the morphism $$T^0(R[\varepsilon]/\varepsilon^2) \rightarrow T^0(R)$$ is naturally identified with $$T^0(R)$$. Similarly, the fiber of the fixed hermitian form $$h$$ in the morphism $$H^0(R[\varepsilon]/\varepsilon^2) \rightarrow H^0(R)$$ is identified with $$H^0(R)$$. Then the morphism $$\varphi_0$$ induces a morphism of functors $$\psi_0 : T^0 \rightarrow H^0$$ defined on a flat $$A$$-algebra $$R$$ by ψ0,R:T0(R)⟶H0(R), ψ0,R(X)(v,w)=h(v,X(w))+h(X(v),w)). Both $$T^0(R)$$ and $$H^0(R)$$ have $$R$$-module structures and that $$\psi_{0, R}$$ is an $$R$$-module homomorphism. Like $$\varphi_0$$, the map $$\psi_{0}$$ is represented by a morphism of $$A$$-schemes. Indeed, $$\psi_0$$ is the differential of $$\varphi_0$$ and is $$A$$-linear. 3.4.3 We define a morphism of functors $$\bar{\varphi}_0 : T^0 \rightarrow \mathrm{coker}(\psi_0)$$ defined on a flat $$A$$-algebra $$R$$ by φ¯0,R:T0(R)⟶H0(R)⟶coker(ψ0)(R). Here, the first map is $$\varphi_{0, R}$$ and the second map is the quotient map. Note that $$\mathrm{coker}(\psi_0)$$ is a representable sheaf on the small fppf site over $$A$$ such that $$\mathrm{coker}(\psi_0)(R)=H^0(R)/\mathrm{Im~}\psi_{0, R}$$ since $$\psi_{0, R}$$ is a morphism of $$R$$-modules (cf. Lemma 6.3.3 in [6]). Then $$\mathrm{coker}(\psi_0)(R)$$ is an abelian group since it is a quotient as $$R$$-modules. The map $$\bar{\varphi}_{0, R}$$ is then a group homomorphism since $$\varphi_{0, R}(X+Y)=\varphi_{0, R}(X)+\varphi_{0, R}(Y)+\psi_{0, R}(X^{ad}\cdot Y)$$ for $$X, Y \in T^0(R)$$. Here, $$X^{ad}\cdot Y \in T^0(R)$$ by Proposition 3.3 and thus $$\psi_{0, R}(X^{ad}\cdot Y) \in \mathrm{Im~}\psi_{0, R}$$. We point out that $$\bar{\varphi}_{0, R}$$ is not a morphism of $$R$$-modules even though it is a morphism of abelian groups. 3.5 Construction of $$T^1$$ We define the functor $$T^1$$ from the category of commutative flat $$A$$-algebras to the category of abelian groups as follows: T1(R)={ker(φ¯0,A)∩ker(φ¯0,A)adif R=A;R⊗AT1(A)if R is a flat A-algebra.  Note that $$\mathrm{ker}(\bar{\varphi}_{0, A})\cap \mathrm{ker}(\bar{\varphi}_{0, A})^{ad}$$ can also be characterized as the set $$\{X\in \mathrm{ker~}\bar{\varphi}_{0, A} : X^{ad} \in \mathrm{ker~}\bar{\varphi}_{0, A}\}$$ by using Proposition 3.3. To show that the functor $$T^1$$ is well-defined, it is enough to show that the set $$\mathrm{ker}(\bar{\varphi}_{0, A})\cap \mathrm{ker}(\bar{\varphi}_{0, A})^{ad}$$ is an $$A$$-module, which is obvious. We can also see that $$T^1(R)$$ is an $$R$$-submodule of $$T^0(R)$$ for a flat $$A$$-algebra $$R$$ since $$T^0(R)=R\otimes_AT^0(A)$$. The abelian group $$T^1(R)$$ has the following description for an étale $$A$$-algebra $$R$$. Theorem 3.6. Let $$R$$ be a flat $$A$$-algebra. Then $$R\otimes_AT^1(A) \subseteq \mathrm{ker}(\bar{\varphi}_{0, R})\cap \mathrm{ker}(\bar{\varphi}_{0, R})^{ad}$$ as $$R$$-submodules of $$T^0(R)$$. The equality holds assuming that $$R$$ is étale over $$A$$. □ Proof. The first claim is obvious. For the second claim, let $$R$$ be an étale local ring over $$A$$. Such $$R$$ is finite over $$A$$ since any étale local ring $$R$$ over a henselian local ring is finite by Proposition 4 of Section 2.3 in [1] and $$A$$ is henselian by the assumption made in Subsection 2.0.1. Then the morphism $$\textit{Spec R}\rightarrow \textit{Spec A}$$ is a Galois covering (cf. Section 6.2, Example B in [1]). We remark that $$T^0(R)=R\otimes_AT^0(A)$$ and $$H^0(R)=R\otimes_AH^0(A)$$. It is, then, easy to show that the set $$\mathrm{ker}(\bar{\varphi}_{0, A})\cap \mathrm{ker}(\bar{\varphi}_{0, A})^{ad}$$ is stabilized by any element of $$\mathrm{Aut}_A(R)$$. Therefore, the above set descends to an $$A$$-submodule $$\mathcal{M}$$ of $$T^0(A)$$ by Galois Descent. Choose $$X\in \mathcal{M}$$. Since $$\varphi_{0, R}(X), \varphi_{0, R}(X^{ad})\in \mathrm{Im~}\psi_{0, R} \cap H^0(A)$$, we can see that $$\varphi_{0, A}(X), \varphi_{0, A}(X^{ad})\in \mathrm{Im~}(\psi_{0, A})$$. We conclude that $$X\in T^1(A)$$. Thus $$\mathcal{M} \subseteq T^1(A)$$, which concludes the proof. ■ In the above theorem, the equality does not hold for a general flat $$A$$-algebra $$R$$. For example, assume that $$(L, h)$$ is a quadratic $$\mathbb{Z}_2$$-lattice of rank 1 and let $$R$$ be a ramified quadratic extension of $$\mathbb{Z}_2$$. Then $$R\otimes_AT^1(A)$$ is the ideal of $$R$$ generated by 2, whereas $$\mathrm{ker}(\bar{\varphi}_{0, R})\cap \mathrm{ker}(\bar{\varphi}_{0, R})^{ad}$$ is the maximal ideal of $$R$$ generated by a uniformizer. Thus if we define $$T^1(R)$$ to be $$\mathrm{ker}(\bar{\varphi}_{0, R})\cap \mathrm{ker}(\bar{\varphi}_{0, R})^{ad}$$ for any flat $$A$$-algebra $$R$$, then the functor $$T^1$$ may not be represented by a polynomial ring. We now claim that the functor $$T^1$$ is represented by a unique polynomial ring in some number of variables. More precisely, Lemma 3.7. The functor $$T^1$$ is represented by a flat $$A$$-algebra which is a polynomial ring over $$A$$ in $$n^2\cdot[K:F]$$ variables. □ Proof. Let $$R$$ be a flat $$A$$-algebra. Since $$T^1(R)=R\otimes_AT^1(A)$$ is a finitely generated free $$R$$-module, $$T^1$$ is represented by a polynomial ring in some number of variables over $$A$$. Thus the relative dimension of $$T^1$$, considered as an affine space, over $$\textit{Spec A}$$ is the same as the dimension of the generic fiber of $$T^1$$ over $$\textit{Spec F}$$, which is the same as the dimension of $$T^1(F)$$ as an $$F$$-vector space. We claim that $$T^1(F)~~(=F\otimes_AT^1(A))~~=T^0(F)$$. It is easy to show that $$F\otimes_AT^1(A)=\{X\in \mathrm{ker~}\bar{\varphi}_{0, F} : X^{ad} \in \mathrm{ker~}\bar{\varphi}_{0, F}\}$$. Since $$\psi_{0, F}$$ is surjective, The latter is the same as $$T^0(F)$$, whose dimension as an $$F$$-vector space is $$n^2\cdot [K:F]$$. The surjectivity of $$\psi_{0, F}$$ follows from smooothness of the generic fiber $$G$$. thus $$\psi_{0, F}$$ is not surjective if $$(L, q)$$ is a quadratic $$A$$-lattice and the characteristic of $$F$$ is $$2$$. However, this case is excluded along the article as mentioned in Sections 2.0.2, 2.0.4, and 2.2.1. ■ There is a natural morphism from $$T^1$$ to $$T^0$$ mapping $$X$$ to $$X$$ itself, where $$X\in T^1(R)$$ for a flat $$A$$-algebra $$R$$. This morphism is represented by a morphism of schemes and we denote it by $$\iota_0$$. We note that $$\iota_0$$ is an injection on flat $$A$$-algebras, but not an immersion of schemes. For example, if $$R$$ is a torsion $$A$$-algebra, then $$\iota_{0, R}$$ is no longer injective. Let $$\varphi_1$$ (resp. $$\psi_1$$) be the morphism from $$T^1$$ to $$H^0$$ induced by $$\varphi_0$$ (respecticvely $$\psi_0$$) composed with $$\iota_0$$, illustrated in the following commutative diagram of morphisms of schemes: $$ $$ 3.8 Construction of the functor $$\boldsymbol{T^{m+1}}$$ on the category of étale $$A$$-algebras In this section and the next, our aim is to recursively define functors $$T^{m+1}$$ from the category of flat $$A$$-algebras to the category of abelian groups and recursively define morphisms of functors $$\iota_m : T^{m+1} \rightarrow T^m$$, $$\varphi_m, \psi_m : T^m \rightarrow H^0$$, and $$\bar{\varphi}_m : T^m \rightarrow \mathrm{coker}(\psi_m)$$ for all $$m\geq 0$$. First we define them on the category of étale $$A$$-algebras, and then in Section 3.11, we extend them to the category of all flat $$A$$-algebras. We define the functor $$T^{m+1}$$, for all $$m \geq 0$$, from the category of étale $$A$$-algebras to the category of abelian groups as follows: Tm+1(R)=ker(φ¯m,R)∩ker(φ¯m,R)ad with the following morphisms: {ιm−1,R:Tm(R)⟶Tm−1(R), X↦X;φm,R,ψm,R:Tm(R)⟶H0(R), φm,R=φm−1,R∘ιm−1,R, ψm,R=ψm−1,R∘ιm−1,R;φ¯m,R:Tm(R)⟶H0(R)⟶coker(ψm).  The set $$\mathrm{ker}(\bar{\varphi}_{m, R})\cap \mathrm{ker}(\bar{\varphi}_{m, R})^{ad}$$ can be characterized as the set $$ \{X\in \mathrm{ker~}\overline{\varphi}_{m, R} : X^{ad} \in \mathrm{ker~}\overline{\varphi}_{m, R}\}$$ by using Proposition 3.3. For convenience, let $$T^{-m}=T^0$$ and let $$\varphi_{-m, R}=\varphi_{0, R}$$ and $$\psi_{-m, R}=\psi_{0, R}$$ for all $$m\geq 0$$. We will show that the above functors are well-defined in the following theorem. Theorem 3.9. Let $$R$$ be an étale $$A$$-algebra. Then for any integer $$m\geq 0$$, we have the followings: (1) The morphisms $$\iota_{m-1, R}, \varphi_{m, R},$$ and $$\psi_{m, R}$$ are well-defined and $$\psi_{m, R}$$ is a group homomorphism under addition; (2) $$X^{ad}\cdot Y\in T^m(R)$$ with $$X, Y\in T^m(R)$$; (3) $$\bar{\varphi}_{m, R}$$ is a group homomorphism under addition; (4) $$T^{m+1}(R)$$ is a group under addition. □ Proof. The statement (4) is a direct consequence of the statements (1)–(3). We prove this by induction. When $$m=0$$, the theorem follows from Proposition 3.3 and Section 3.4.3. Suppose that the theorem is true for all $$n\leq m-1$$, where $$m \geq 1$$, so that Tm(R)=ker(φ¯m−1,R)∩ker(φ¯m−1,R)ad is well-defined and a group under addition. Then the statement (1) is obvious. To prove the statement (2), let $$X, Y\in T^{m}(R)$$ and choose $$Z\in T^{m-1}(R)$$ such that $$\varphi_{m-1, R}(X^{ad})=\psi_{m-1, R}(Z)$$. To show that $$X^{ad}\cdot Y\in T^m(R)$$, we consider the following identity: φm−1,R(Xad⋅Y)=ψm−1,R(Yad⋅Z⋅Y). Since $$Z\cdot Y$$ and $$Y^{ad}\cdot Z\cdot Y \in T^{m-1}(R)$$ by hypothesis of induction (statement (2)), we conclude that $$\overline{\varphi}_{m-1, R}(X^{ad}\cdot Y)=0$$. Similarly, $$\overline{\varphi}_{m-1, R}(Y^{ad}\cdot X)=0$$ so $$X^{ad}\cdot Y\in T^m(R)$$. To show that $$\overline{\varphi}_{m, R}$$ is a group homomorphism, choose $$X, Y \in T^m(R)$$. Let us consider the following identity: φm,R(X+Y)=φm,R(X)+φm,R(Y)+ψm,R(Xad⋅Y). Since $$X^{ad}\cdot Y\in T^m(R)$$, $$\overline{\varphi}_{m, R}$$ is a group homomorphism. This completes the proof. ■ Corollary 3.10. Let $$R$$ be an étale $$A$$-algebra. Let $$X\in T^{m+1}(R)$$ and let $$Y\in T^{m}(R)$$. Then $$\iota_{m, R}(X)\cdot Y\in \mathrm{ker~}\overline{\varphi}_{m, R}$$. □ Proof. Let $$\varphi_{m, R}\left( \iota_{m, R}(X)\right)=\psi_{m, R}(Z)$$ for some $$Z\in T^m(R)$$. Then $$\varphi_{m, R}(\iota_{m, R}(X)\cdot Y)=\psi_{m, R}(Y^{ad}\cdot Z\cdot Y)$$ and $$Y^{ad}\cdot Z\cdot Y\in T^m(R)$$ by the above theorem. This completes the proof. ■ 3.11 Constructions of the functors $$\boldsymbol{T^{m+1}}$$ and $$\boldsymbol{\widetilde{T}}$$ on the category of flat $$\boldsymbol{A}$$-algebras We extend the functor $$T^{m+1}$$ to a functor from the category of commutative flat $$A$$-algebras to the category of abelian groups as follows: Tm+1(R)={ker(φ¯m,A)∩ker(φ¯m,A)adif R=A;R⊗ATm+1(A)if R is a flat A-algebra.  By similar arguments used in Subsection 3.5 and Theorem 3.6 and Lemma 3.7, combined with induction on $$m$$, we can show that (1) the set $$\mathrm{ker}(\bar{\varphi}_{m, A})\cap \mathrm{ker}(\bar{\varphi}_{m, A})^{ad}$$ is an $$A$$-module so that $$T^{m+1}$$ is well-defined; (2) $$\iota_{m, R} : T^{m+1}(R) \rightarrow T^{m}(R)$$ is an injective $$R$$-module homomorphism for a flat $$A$$-algebra $$R$$; (3) for an étale $$A$$-algebra $$R$$, R⊗ATm+1(A)(=Tm+1(R))=ker(φ¯m,R)∩ker(φ¯m,R)ad, as $$R$$-submodules of $$T^{m}(R)$$; (4) $$T^{m+1}$$ is represented by a flat $$A$$-algebra which is a polynomial ring over $$A$$ in $$n^2\cdot[K:F]$$ variables; (5) the morphisms $$\iota_{m-1, R}, \psi_{m, R}, \bar{\varphi}_{m, R}$$, and $$\varphi_{m, R}$$, for a flat $$A$$-algebra $$R$$, are represented by morphisms of $$A$$-schemes. First three are morphisms of group schemes under addition, whereas the last is not. Note that $$T^m(F)=T^0(F)$$ for all $$m$$. Let us consider the following sequence induced by $$\iota_m$$: T0(R)⊇T1(R)⊇⋯⊇Tm(R)⊇⋯ for a flat $$A$$-algebra $$R$$. We emphasize that all ranks of $$T^i(R)$$, as finitely generated free $$R$$-modules, are the same, which is $$n^2\cdot[K:F]$$. Indeed, there is a suitable integer $$\eta$$ which makes the above sequence stabilize and this is proved in the following theorem. Theorem 3.12. There exists an integer $$\eta ~(\geq 0)$$ such that $$T^m=T^\eta$$ for all $$m \geq \eta$$. □ Proof. Let $$M=T^0(A)/T^1(A)$$. It is then clear that $$M$$ is a torsion $$A$$-module since the rank of $$T^0(A)\otimes_AF$$ is the same as that of $$T^1(A)\otimes_AF$$. Let $$l$$ be the smallest non-negative integer such that $$\pi^l\cdot M=0$$. Let T′(R)=π2lT0(R) for a flat $$A$$-algebra $$R$$. Then $$T'$$ is represented by a flat $$A$$-algebra which is a polynomial ring in some number of variables over $$A$$. Let $$\varphi'$$ and $$\psi'$$ be morphisms from $$T'$$ to $$H^0$$ induced from $$\varphi_0$$ and $$\psi_0$$, respectively. We choose an element $$\pi^{2l}\cdot X\in T'(R)$$ for an étale $$A$$-algebra $$R$$. Since $$\pi^l\cdot X\in T^1(R)$$, we have that $$\varphi_{0, R}(\pi^l\cdot X)=\psi_{0, R}(Y)$$ for a certain $$Y\in T^0(R)$$. Then, φR′(π2l⋅X)=π2l⋅φR′(πl⋅X)=π2l⋅ψ0,R(Y)=ψR′(π2l⋅Y) with $$\pi^{2l}\cdot Y\in T'(R)$$. Here, the first equality follows from the fact that $$\varphi'_R(aX)=a^2\varphi'_R(X)$$ for $$a\in A$$ and the last equality follows from the fact that $$\psi'_R$$ is $$A$$-linear. Therefore, $$T'(R)\subset T^m(R)$$ for every integer $$m$$, where $$R$$ is an étale $$A$$-algebra. On the other hand, if $$T^l=T^{l+1}$$ for certain non-negative integer $$l$$, then it is clear that $$T^l=T^{l'}$$, for all $$l'\geq l$$. This fact yields that a sequence of the form $$T^l(R)=T^{l+1}(R)\nsupseteq T^{l+2}(R)$$, for a flat $$A$$-algebra $$R$$, cannot happen. Thus we conclude the existence of an integer $$\eta ~(\geq 0)$$ stabilizing the sequence explained just before this theorem. ■ Let $$\alpha$$ be the smallest non-negative integer satisfying that $$T^m=T^{\alpha}$$ for all $$m \geq \alpha$$. We finally define the functor T~:=Tα, equipped with two morphisms $$\varphi_{\alpha}$$ and $$\psi_{\alpha}$$ mapping to $$H^0$$. We note that Theorem 3.12 and its proof only assert stability of a sequence of functors $$T^m$$. In order to find $$\alpha$$, one can construct $$T^{m}(A)$$ explicitly starting from $$m=0$$. Then $$\alpha$$ is the first integer $$m$$ such that $$T^{m}(A)=T^{m+1}(A)$$ (equivalently, the map $$\overline{\varphi}_{m, A} : T^m(A) \rightarrow H^0(A)/\mathrm{Im~}\psi_{m, A}$$ is zero). Indeed, we expect that $$\alpha$$ is at most $$e'+1$$, where the ramification index $$e=2e'$$ or $$e=2e'-1$$, as this is true in the unramified case (cf. Remark 3.23) and for the examples covered in Section 4.1 and Examples 4.13–4.16. 3.13 Construction of $$\boldsymbol{\widetilde{H}}$$ Recall that $$\psi_{\alpha, R} : \widetilde{T}(R) \longrightarrow H^0(R)$$ is $$R$$-linear for a flat $$A$$-algebra $$R$$. Define the functor $$\widetilde{H}$$ from the category of commutative flat $$A$$-algebras to the category of abelian groups as follows: H~(R)=Im ψα,R. Theorem 3.14. The functor $$\widetilde{H}$$ is represented by a flat $$A$$-algebra which is a polynomial ring in some number of variables over $$A$$ of $$n^2\cdot[K:F]-\mathrm{dim~}G$$ variables and the map $$\psi_{\alpha, R} : \widetilde{T}(R) \rightarrow \widetilde{H}(R)$$ is represented by a morphism of schemes, denoted by $$\widetilde{\psi}$$. In addition, the map $$\widetilde{\psi_{R_{\kappa}}} : \widetilde{T}(R_{\kappa}) \rightarrow \widetilde{H}(R_{\kappa})$$ is surjective for any $$\kappa$$-algebra $$R_{\kappa}$$. □ Proof. Since $$\widetilde{T}(R)=R\otimes_A\widetilde{T}(A)$$ and $$\psi_{\alpha, R}$$ is $$R$$-linear for a flat $$A$$-algebra $$R$$, we have that $$\widetilde{H}(R)=R\otimes_A\widetilde{H}(A)$$. Therefore, $$\widetilde{H}$$ is represented by a polynomial ring in some number of variables over $$A$$. Since $$\widetilde{H}(F)=H^0(F)$$, the dimension of $$\widetilde{H}(F)$$ as an $$F$$-vector space is $$n^2\cdot [K:F]-\mathrm{dim~}G$$, which is the same as the relative dimension of $$\widetilde{H}$$ over $$\textit{Spec A}$$. The representability of $$\psi_{\alpha, R}$$ is obvious. To show that $$\widetilde{\psi_{R_{\kappa}}} : \widetilde{T}(R_{\kappa}) \rightarrow \widetilde{H}(R_{\kappa})$$ is surjective for a $$\kappa$$-algebra $$R_{\kappa}$$, we choose a flat $$A$$-algebra $$R$$ such that $$R\otimes_A\kappa=R_{\kappa}$$. Then we have the following commutative diagram: $$ $$ Since $$\widetilde{T}$$ and $$\widetilde{H}$$ are represented by affine spaces, two vertical maps are surjective. One can also show surjectivity of two vertical maps by using Hensel’s lemma since $$\widetilde{T}$$ and $$\widetilde{H}$$ are smooth. In addition, $$\widetilde{\psi_{R}}$$ is surjective by the definition of $$\widetilde{H}$$. Therefore, $$\widetilde{\psi_{R_{\kappa}}}$$ is surjective as well. ■ Theorem 3.15. Let $$R$$ be a flat $$A$$-algebra. Then the image of $$\widetilde{T}(R)$$, under the map $$\varphi_{\alpha, R}$$, is contained in $$\widetilde{H}(R)$$. Therefore, the morphism $$\varphi_{\alpha} : \widetilde{T} \rightarrow H^0$$ factors through $$\widetilde{H}$$. □ Proof. Let $$R$$ be an étale $$A$$-algebra. Suppose that there is $$X\in T^{\alpha}(R)=\widetilde{T}(R)$$ such that $$\varphi_{\alpha, R}(X) \notin \widetilde{H}(R)$$. Then $$\overline{\varphi}_{\alpha, R}(X)\neq 0$$ and so $$X$$ is not contained in $$T^{\alpha+1}(R)$$. This contradicts the definition of $$\alpha$$. Thus $$\varphi_{\alpha, R}(X) \in \widetilde{H}(R)$$ for any $$X\in \widetilde{T}(R)$$. Let $$R$$ be a flat $$A$$-algebra. Choose $$X\in T^{\alpha}(R)$$. We can write $$X=\sum_ir_i\cdot X_i$$ with $$r_i\in R$$ and $$X_i\in T^{\alpha}(A)=\widetilde{T}(A)$$. Then $$\varphi_{\alpha, R}(X)=\sum_ir_i^2\cdot \varphi_{\alpha, A}(X_i)+\sum_{i<j}r_ir_j\cdot \psi_{\alpha, A}(X_i^{ad}\cdot X_j)$$. Since $$\varphi_{\alpha, A}(X_i)$$, $$\psi_{\alpha, A}(X_i^{ad}\cdot X_j) \in \widetilde{H}(A)$$ and $$\widetilde{H}(R)=R\otimes_A\widetilde{H}(A)$$, $$\varphi_{\alpha, R}(X)$$ is contained in $$\widetilde{H}(R)$$. ■ Let $$\widetilde{\varphi}$$ be the morphism from $$\widetilde{T}$$ to $$\widetilde{H}$$ induced from $$\varphi_{\alpha}$$. We have finally constructed two morphisms $$\widetilde{\varphi}$$ and $$\widetilde{\psi}$$ from $$\widetilde{T}$$ to $$\widetilde{H}$$. 3.16 Construction of $$\underline{G}$$ We define two functors from the category of commutative flat $$A$$-algebras to the category of sets as follows: M_(R)={1+X:X∈T~(R)},H_(R)={h+f:f∈H~(R)}. Here, we consider $$\underline{M}(R)$$ (respectively $$\underline{H}(R)$$) as a subset of $$\mathrm{End}_{B\otimes_AR}(L \otimes_A R)$$ (resp. $$H^0(R)$$) and $$1$$ in the description of $$\underline{M}(R)$$ is the identity of multiplication in $$\mathrm{End}_{B\otimes_AR}(L \otimes_A R)$$. The set $$\mathrm{End}_{B\otimes_AR}(L \otimes_A R)$$ (respectively $$H^0(R)$$) has an obvious additive structure and the addition in the description of $$\underline{M}(R)$$ (respectively $$\underline{H}(R)$$) comes from this. Note that $$1$$ in the description of $$\underline{M}(R)$$ might not be contained in $$\widetilde{T}(R)$$ if $$A$$ has residue characteristic $$2$$, and similarly for $$h$$ in the description of $$\underline{H}(R)$$. However, in odd residue characteristic, $$1$$ is contained in $$\widetilde{T}(R)$$ and $$h$$ is also contained in $$\widetilde{H}(R)$$. Indeed, (denoting, for the moment, the identity map by $$I$$), since $$1/2 \cdot I\in T^0(A)$$ and $$\psi_{0, A}(1/2\cdot I)=h$$ and $$\varphi_{0, A}(I)=h$$, we find that $$I\in T^1(A)$$ and also $$1/2\cdot I\in T^1(A)$$. We can go onto $$T^2$$, etc. Then these two functors are represented by flat $$A$$-algebras which are polynomial rings in some number of variables over $$A$$. Note that $$h$$ is the fixed hermitian form throughout this article. Lemma 3.17. The functor $$\underline{M}$$ is a functor from the category of commutative flat $$A$$-algebras to the category of monoids under multiplication. □ Proof. Choose $$1+X, 1+X' \in \underline{M}(R)$$ for a flat $$A$$-algebra $$R$$. Then it suffices to show that $$(1+X)(1+X')-1\in \widetilde{T}(R)$$. Note that $$\widetilde{T}(R)=\widetilde{T}(A)\otimes_AR$$. Thus by Theorem 3.9 applied to $$\widetilde{T}(A)$$, $$\widetilde{T}(R)$$ is closed under multiplication. Since $$\widetilde{T}(R)$$ is also an additive group, we have that $$X+X'+X\cdot X'\in \widetilde{T}(R)$$. ■ For any commutative $$A$$-algebra $$R$$, consider the group of units M_∗(R)={m∈M_(R):there exists m−1∈M_(R) such that m⋅m−1=m−1⋅m=1}. The definition of $$\underline{M}^{\ast}$$ given here is a specific case of the general construction of the group of units in a monoid (sheafified). Then $$\underline{M}^{\ast}$$ is represented by a group scheme and is an open subscheme of $$\underline{M}$$, with generic fiber $$M^{\ast}=\underline{\mathrm{GL}_K(V)}_{/F}$$. Here the notation $$\underline{\mathrm{GL}_K(V)}_{/F}$$ is explained in Section 2.2.1. Thus $$\underline{M}^{\ast}$$ is smooth over $$A$$ since $$\underline{M}$$ is smooth over $$A$$. The proofs of these are similar to Section 3.2 of [2] or Remark 3.2 of [3], by using the following Lemma so we skip them. Lemma 3.18. For a flat $$A$$-algebra $$R$$, let $$m\in \underline{M}(R)$$ such that $$m^{-1}\in \mathrm{Aut}_{B\otimes_AR}(L\otimes_AR)$$. Then such $$m^{-1}$$ is an element of $$\underline{M}(R)$$. □ Proof. By Lemma 3.2 of [2], $$m^{-1}\in T^0(R)$$. Let $$m=1+\sum_ir_i\cdot X_i$$ and $$m^{-1}=1+\sum_is_j\cdot Y_j$$ with $$X_i\in \widetilde{T}(A)$$, $$Y_j\in T^0(A)$$ and $$r_i, s_j\in R$$. Then $$\sum_is_j\cdot Y_j=-\sum_ir_i\cdot X_i-(\sum_ir_i\cdot X_i)\cdot (\sum_is_j\cdot Y_j)$$ and $$\sum_is_j\cdot Y_j^{ad}=-\sum_ir_i\cdot X_i^{ad}-(\sum_ir_i\cdot X_i^{ad})\cdot (\sum_is_j\cdot Y_j^{ad})$$. Thus, by Corollary 3.10 with an induction on $$n$$, we have that $$\sum_is_j\cdot Y_j\in T^{n}(R)$$ for all $$n\geq 0$$. ■ Theorem 3.19. The group $$\underline{M}^{\ast}(R)$$ acts on the right on $$\underline{H}(R)$$ by $$(h+f)\cdot m=(h+f)\circ m $$, for any flat $$A$$-algebra $$R$$. Then this action is represented by an action morphism H_×M_∗⟶H_. □ Proof. Let $$R$$ be a flat $$A$$-algebra. It suffices to show that $$(h+f)\circ m-h\in \widetilde{H}(R)$$. Let $$m=1+ X$$, where $$X\in \widetilde{T}(R)$$. Then $$h\circ m-h=\widetilde{\varphi_R}(X)+\widetilde{\psi_R}(X)$$. Let $$f=\widetilde{\psi_R}(Y)$$ for some $$Y\in \widetilde{T}(R)$$. Notice that we can always find such $$Y$$ since $$\widetilde{\psi_R} : \widetilde{T}(R)\rightarrow \widetilde{H}(R)$$ is surjective. Then $$f\circ m =\widetilde{\psi_R}(Y)+\widetilde{\psi_R}(X^{ad}\cdot Y)+\widetilde{\psi_R}(Y\cdot X)+\widetilde{\psi_R}(X^{ad}\cdot Y\cdot X)$$. By Theorem 3.9 together with the fact that $$\widetilde{T}(R)=R\otimes_A\widetilde{T}(A)$$, $$X^{ad}\cdot Y$$, $$Y\cdot X,$$ and $$ X^{ad}\cdot Y\cdot X \in \widetilde{T}(R)$$. This completes our claim. The proof of the representability of the action morphism is similar to the argument explained in Theorem 3.4 of [3] and so we skip it. ■ Theorem 3.20. Let $$\rho$$ be the morphism $$\underline{M}^{\ast} \rightarrow \underline{H}$$ defined by $$\rho(m)=h \circ m$$. Then $$\rho$$ is smooth of relative dimension dim $$G$$. □ Proof. The theorem follows from Theorem 5.5 in [6] and the following lemma. ■ Lemma 3.21. The morphism $$\rho \otimes \kappa : \underline{M}^{\ast}\otimes \kappa \rightarrow \underline{H}\otimes \kappa$$ is smooth of relative dimension dim $$G$$. □ Proof. The proof is based on Lemma 5.5.2 in [6]. It is enough to check the statement over the algebraic closure $$\bar{\kappa}$$ of $$\kappa$$. By [7], III.10.4, it suffices to show that, for any $$m \in \underline{M}^{\ast}(\bar{\kappa})$$, the induced map on the Zariski tangent space $$\rho_{\ast, m}:T_m \rightarrow T_{\rho(m)}$$ is surjective. We identify $$T_m$$ with $$\widetilde{T}(\bar{\kappa})$$ and $$T_{\rho(m)}$$ with $$\widetilde{H}(\bar{\kappa})$$. Let $$m=1+Y$$ for some $$Y\in \widetilde{T}(\bar{\kappa})$$. The map $$\rho_{\ast, m}:T_m \rightarrow T_{\rho(m)}$$ is then X↦ψκ¯~(X+Yad⋅X)=ψκ¯~(mad⋅X). Since $$\widetilde{\psi_{\bar{\kappa}}} : \widetilde{T}(\bar{\kappa}) \longrightarrow \widetilde{H}(\bar{\kappa})$$ is surjective by Theorem 3.14, it suffices to show that the following map T~(κ¯)⟶T~(κ¯), X↦mad⋅X is bijective. To prove the above, it is enough to show that $$\widetilde{T}(\bar{\kappa})\longrightarrow \widetilde{T}(\bar{\kappa}), ~~~~~X\mapsto m^{ad}\cdot X$$ is well-defined so that the inverse map $$X \mapsto (m^{-1})^{ad} \cdot X$$ is well-defined as well. We consider a map $$\underline{M}(R)\times \widetilde{T}(R)\longrightarrow \widetilde{T}(R), ~~~~~ (m, X)\mapsto m^{ad}\cdot X$$, for a flat $$A$$-algebra $$R$$. It is easy to show that this map is well-defined by using Theorem 3.9 together with the fact that $$\widetilde{T}(R)=R\otimes_A\widetilde{T}(A)$$ and is represented by a morphism of schemes. Therefore, a map $$\underline{M}(\bar{\kappa})\times \widetilde{T}(\bar{\kappa})\longrightarrow \widetilde{T}(\bar{\kappa}), ~~~~~ (m, X)\mapsto m^{ad}\cdot X$$ is well-defined as well. This completes the proof. ■ Let $$\underline{G}$$ be the stabilizer of $$h$$ in $$\underline{M}^{\ast}$$. That is, $$\underline{G}$$ is the fiber of $$h$$ along the smooth morphism $$\rho : \underline{M}^{\ast} \rightarrow \underline{H}, \rho(m)=h \circ m$$. Thus $$\underline{G}$$ is smooth, being the fiber of a smooth morphism. Therefore we have the following theorem. Theorem 3.22. The group scheme $$\underline{G}$$ with generic fiber $$G$$ is smooth, and $$\underline{G}(R)=\mathrm{Aut}_{B\otimes_AR}(L\otimes_A R,h\otimes_A R)$$ for any étale $$A$$-algebra $$R$$. In conclusion, $$\underline{G}$$ is the desired smooth integral model of $$G$$. □ Proof. We choose an element $$1+X\in \mathrm{Aut}_{B\otimes_AR}(L\otimes_A R,h\otimes_A R)$$ for any étale $$A$$-algebra $$R$$. Then we only need to show that $$1+X \in \underline{M}^{\ast}(R)$$, in other words, $$ X\in \widetilde{T}(R)$$. Firstly, it is clear that $$1+X\in T^0(R)$$ so $$X\in T^0(R)$$. Secondly, since $$h\circ (1+X)=h$$, we have that $$\varphi_{0, R}(X)=\psi_{0, R}(-X).$$ On the other hand, $$\varphi_{0, R}(X^{ad})=\psi_{0, R}(-X^{ad}).$$ Therefore, by using induction, we conclude that $$ X\in T^m(R)$$ for all $$m\geq 0$$ so $$ X\in \widetilde{T}(R)$$. ■ Remark 3.23. We explain how constructions of smooth integral models in [2–4, 6] can be recovered from the construction studied in this article, by finding $$\alpha$$. Recall that $$\alpha$$ is defined in the paragraph following Theorem 3.12. We use terminology of [2–4] in the following $$(2)$$ and $$(3)$$, respectively. (1) If $$p\neq 2$$, then $$\alpha=0$$. Indeed, $$\alpha=0$$ for all cases covered in [6]. (2) Let $$(K, \epsilon, p)=(F, 1, 2)$$ and we assume that $$F/\mathbb{Q}_2$$ is unramified. This is the case covered in [2]. Then $$\alpha$$ is at most $$2$$. More precisely, if all Jordan components of $$(L, h)$$ are of type II, then $$\alpha=0$$; if at least one Jordan component is free of type I and if there is no Jordan component which is bound of type I, then $$\alpha=1$$; if at least one Jordan component is bound of type I, then $$\alpha=2$$. Here, we say that a Jordan component is of type I if its norm equals its scale, otherwise it is of type II. We also say that a Jordan component is $$free$$ if both adjacent Jordan components are of type II, otherwise it is bound. (3) Let $$(K, \epsilon, p)=(E, 1, 2)$$. We assume that $$F/\mathbb{Q}_2$$ is unramified and that $$E/F$$ is ramified. This is the case covered in [3] and [4]. Then $$\alpha$$ is at most $$2$$. More precisely, when $$E/F$$ satisfies Case 1, $$\alpha=1$$ if at least one Jordan component is of type I, and $$\alpha=0$$ otherwise. When $$E/F$$ satisfies Case 2, $$\alpha=0$$ if all Jordan components are of type II, $$\alpha=2$$ if at least one Jordan component $$L_i$$ with $$i$$ even is of type I, and $$\alpha=1$$ otherwise. Here, the type of each Jordan component $$L_i$$ is as explained in the above case (2), when $$i$$ is even. When $$i$$ is odd, we refer to Remark 2.6 of [3] for the definition of each type. In all cases, $$\underline{M}^{\ast}$$ and $$\underline{H}$$ (or $$\underline{Q}$$ in [2]) defined in [2–4, 6] are exactly the same as $$\underline{M}^{\ast}$$ and $$\underline{H}$$ defined in this article. □ 3.24 The local density formula One major application of smooth integral models is a computation of local densities. For a detailed explanation of a relation between local densities and smooth integral models, see the introduction of this paper and Section 3 of [6]. As we have seen in Section 3.16, the construction of $$\underline{G}$$ simply follows from those of $$\widetilde{T}$$ and $$\widetilde{H}$$. Furthermore, from the fact that $$\widetilde{T}(R)=R\otimes_A\widetilde{T}(A)$$ for a flat $$A$$-algebra $$R$$ (cf. Section 3.11), the only issue in the construction of $$\underline{G}$$ is the construction of $$\widetilde{T}(A)$$. This implies that $$\underline{G}$$ is constructed from some computation with $$R=A$$. We briefly explain where $$\widetilde{T}(A)$$ and $$\widetilde{H}(A)$$ can be found in this paper. Recall that we have the sequence T0(A)⊇T1(A)⊇⋯⊇Tm(A)⊇⋯ at the paragraph before Theorem 3.12. Here, $$T^0(A)$$ is defined at the beginning of Section 3.2, $$T^{1}(A)$$ is defined at the beginning of Section 3.5, and $$T^{m}(A)$$ is defined at the beginning of Section 3.11. This sequence stabilizes for a nonnegative integer $$\alpha$$ by Theorem 3.12 and such $$\alpha$$ can be computed as explained at the paragraph following Theorem 3.12. Finally $$\widetilde{T}(A)=T^{\alpha}(A)$$ at the paragraph following Theorem 3.12 and $$\widetilde{H}(A)$$ is defined at the beginning of Section 3.13. Theorem 3.25. Let $$\tilde{G}$$ be the special fiber of $$\underline{G}$$ and let $$q$$ be the cardinality of $$\kappa$$. Let $$M'=\mathrm{End}_{B}(L)$$ and $$H'= \{\textit{$f$ : $f$ is hermitian form on $L$}\}$$. Notice that $$M'/\widetilde{T}(A)$$ and $$H'/\widetilde{H}(A)$$ are finitely generated torsion $$A$$-modules. Let qN=#(H′/H~(A))#(M′/T~(A)). Here, $$\#$$ stands for the cardinality and $$N$$ is an integer. Then the local density of ($$L,h$$) is βL=1[G:Go]qN⋅q−dim G⋅#G~(κ), where $$G^o$$ is the identity component of $$G$$. □ By Theorem 3.25, once one constructs the smooth integral model $$\underline{G}$$ for a given lattice $$(L, h)$$, the only real challenge to compute the local density is $$\#\tilde{G}(\kappa)$$. In the next section, we will explain a conjectural recipe for computing $$\#\tilde{G}(\kappa)$$, by introducing one conjecture (Conjecture 4.10) concerning properties of a specific map of algebraic groups. 4 Calculation of local densities In the first part of this section (4.1–4.7), we work out completely the mass of unimodular quadratic lattices in odd dimension. In the second part (4.8–4.17), we explain a conjectural recipe for computing $$\#\tilde{G}(\kappa)$$ in the general case. We note that our conjectural recipe is conditional on Conjecture 4.10. Let $$A$$ be a finite extension of $$\mathbb{Z}_2$$ and $$\pi$$ be a uniformizing element of $$A$$. Let $$e=2e'$$ or $$e=2e'-1>1$$ (we assume this to exclude known cases) be the ramification index. Let $$(L, h)$$ be a quadratic lattice (i.e., $$K=F$$ and $$\epsilon=1$$). We use the symbol $$A(a, b)$$ to denote the $$A$$-lattice $$A\cdot e_1+A\cdot e_2$$ with the symmetric bilinear form having Gram matrix $$\begin{pmatrix} a&1\\ 1&b \end{pmatrix}$$ . We denote by $$(t)$$ the $$A$$-lattice of rank 1 equipped with the symmetric bilinear form having Gram matrix $$(t)$$. Suppose that L=⨁iAi(0,0)⊕A(πs,rπ2e−s)⊕(t) of rank $$2m+3$$ with $$r\in A$$ and $$t\equiv 1 \ \ \mathrm{mod}\ \ \pi$$. Here, $$s\leq e$$, $$s$$ is odd if $$s<e$$, and $$A_i(0,0)=A(0,0)$$. A quadratic lattice $$L$$ having the above description exhausts all unimodular quadratic lattices with odd rank $$>1$$ by the following facts extracted from Examples 93:9 and 93:18 of [13]: (1) (cf. 93:18.(v) of [13]) If the rank of $$L$$ is at least $$5$$, then L≅A(0,0)⊕⋯. (2) (cf. 93:18.(ii), (iv) of [13]) If the rank of $$L$$ is $$3$$, then L≅A(0,0)⊕(t) orL≅A(b,4rb−1)⊕(t). Here, $$b, r\in A$$ such that the exponential order of $$b$$ is odd and at most $$e$$, and $$t\equiv 1 \ \ \mathrm{mod}\ \ \pi$$. (3) (cf. 93:9 of [13]) A(2α,0)≅A(2α+2β,0) for any $$\alpha, \beta\in A$$. If we choose $$\beta=-\alpha$$, then this implies that $$A(2\alpha,0)\cong A(0,0)$$. 4.1 Constructions of $$\widetilde{T}$$ and $$\widetilde{H}$$ We first construct the smooth integral model $$\underline{G}$$ associated to $$L$$. As mentioned in Section 3.24, a key ingredient of the construction of $$\underline{G}$$ is the explicit description of $$\widetilde{T}(A)$$. Basically, the procedure to find $$\widetilde{T}(A)$$ gives certain congruence conditions on the entries of an element of $$T^0(A)$$. A matrix form of an element of $$\widetilde{T}(A)$$ is described below. For $$X\in \widetilde{T}(A)$$, X={(x0y0π[(e−s)/2]z0πe′w0π[(e−s)/2]x1π[(e−s)/2]y1πe−sz1π[e−(s−1)/2]w1x2y2π[(e−s)/2]z2πe′w2πe′x3πe′y3π[e−(s−1)/2]z32w3)if s<e odd;(xπe′yπe′z2w)if s=e.  Here, $$x_0$$ is a $$(2m \times 2m)$$-matrix, etc., if $$s<e$$ odd, and $$x$$ is a $$(2m+2) \times (2m+2)$$-matrix, etc., if $$s=e$$. The notion $$[(e-s)/2]$$ means the smallest integer greater than or equal to $$(e-s)/2$$. Then a matrix form of an element of $$\underline{M}(A)$$ is $$1+X$$. A matrix form of an element of $$\widetilde{H}(A)$$ is described below. For $$f\in \widetilde{H}(A)$$, f={(a0b0π[(e−s)/2]c0πe′d0tb02b1π[(e−s)/2]c1πe′d1t(π[(e−s)/2]c0)t(π[(e−s)/2]c1)π2e−sc2π[e−(s−1)/2]d2t(πe′d0)t(πe′d1)t(π[e−(s−1)/2]d2)4d3)if s<e odd;(aπe′bt(πe′b)4c)if s=e.  Here, the diagonal entries of $$a_0$$ are divisible by $$2$$, where $$a_0$$ is a $$(2m \times 2m)$$-matrix, etc., if $$s<e$$ odd, and the diagonal entries of $$a$$ are divisible by $$2$$, where $$a$$ is a $$(2m+2) \times (2m+2)$$-matrix, etc., if $$s=e$$. The matrix transpose of $$b_0$$ is denoted by $${}^tb_0$$. The above matrix descriptions yield the value of $$q^N$$ by Theorem 3.25. Lemma 4.2. Let $$q^{N_1}=\#(M'/\widetilde{T}(A))$$ and $$q^{N_2}=\#(H'/\widetilde{H}(A))$$. Then the factor $$q^N$$ in Theorem 3.25 is given as follows: (1) If $$s<e$$ is odd, then {N1=2(2m+1)(e′+[(e−s)/2])+2[e−(s−1)/2]+2e−s;N2=(2m+1)(e′+[(e−s)/2])+[e−(s−1)/2]+(2m+5)e−s;N=N2−N1=(2m+1)(e−e′−[(e−s)/2])−[e−(s−1)/2]+2e.  (2) If $$s=e$$, then {N1=2(2m+2)e′+e;N2=(2m+2)(e+e′)+2e;N=N2−N1=(2m+2)(e−e′)+e.  □ 4.3 The special fiber $$\boldsymbol{\tilde{G}}$$ We define two sublattices $$B(L), Z(L)$$ of $$L$$ as follows: (1) $$B(L)$$, the sublattice of $$L$$ such that $$B(L)/2L$$ is the kernel of the additive polynomial $$h$$ mod 2 on $$L/2L$$. (2) $$Z(L)$$, the sublattice of $$B(L)$$ such that $$Z(L)/\pi B(L)$$ is the kernel of the quadratic form $$\frac{1}{2}h$$ mod $$\pi$$ on $$B(L)/\pi B(L)$$. The idea behind the definition of the above two sublattices of $$L$$ is to extract a quadratic form over the residue field $$\kappa$$ involving the hyperbolic component (i.e., the component $$\oplus_i A(0,0)$$) of $$L$$. Let $$\bar{V}=B(L)/Z(L)$$ be a $$\kappa$$-vector space and $$\bar{h}$$ denote the nonsingular quadratic form $$\frac{1}{2}h$$ mod $$\pi$$ on $$\bar{V}$$. Since the action of an element of $$\underline{G}(A)$$ on $$L$$ preserves $$B(L)$$ and $$Z(L)$$, each element of $$\underline{G}(A)$$ determines an element of $$\mathrm{GL}(B(L)/Z(L))$$. Then the element of $$\mathrm{GL}(B(L)/Z(L))$$ determined by an element of $$\underline{G}(A)$$ fixes $$\bar{h}$$ on $$B(L)/Z(L) (=\bar{V})$$. Since $$\underline{G}$$ is smooth, the map $$\underline{G}(A)\rightarrow \tilde{G}(\kappa)$$ is surjective by Hensel’s lemma. Now, we choose an element $$g\in \tilde{G}(\kappa)$$ and a lift $$\tilde{g} \in \underline{G}(A)$$. Since $$\tilde{g}$$ induces an element of $$\mathrm{O}(\bar{V}, \bar{h})(\kappa)$$ as explained above, we have a group homomorphism $$\varphi_{\kappa}$$ from $$\tilde{G}(\kappa)$$ to $$\mathrm{O}(\bar{V}, \bar{h})(\kappa)$$. It is easy to see that this map is well-defined, that is, independent of a lift $$\tilde{g}$$ of $$g$$, by observing a formal matrix description of $$\varphi_{\kappa}$$ which will be given in the proof of the following theorem. Theorem 4.4. Recall that $$L$$ is a unimodular quadratic lattice of rank $$2m+3$$ defined over a (ramified) finite extension $$A$$ of $$\mathbb{Z}_2$$. Firstly, the group homomorphism φκ:G~(κ) ⟶ O(V¯,h¯)(κ) is surjective. Secondly, $$\mathrm{ker~}\varphi_{\kappa}$$ is isomorphic to $$(\mathbb{A}^{\ell}\times (\mathbb{Z}/2\mathbb{Z})^{\beta})(\kappa)$$ as sets, where $$\mathbb{A}^{\ell}$$ is an affine space of dimension $$\ell$$. Here, $$\ell=\frac{(2m+3)(2m+2)}{2}-\mathrm{dim~}\mathrm{O}(\bar{V}, \bar{h})$$, and $$\beta=2$$ (respectively $$\beta=1$$) if $$s<e$$ is odd (respectively if $$s=e$$). Therefore, #G~(κ)=#(O(V¯,h¯)(κ))⋅qℓ⋅2β, where $$q$$ is the cardinality of $$\kappa$$. □ Proof. We represent the given quadratic form $$h$$ by a symmetric matrix $$\delta=\begin{pmatrix} \delta^{\prime}&0\\0&\delta^{\prime\prime} \end{pmatrix}$$ , where $$\delta^{\prime}$$ is $$(2m \times 2m)$$-matrix (respectively $$(2m+2) \times (2m+2)$$-matrix) if $$s<e$$ (respectively $$s=e$$). We explain our strategy of the proof. For the first part, we will write an element of $$\tilde{G}(\kappa)$$ as a formal matrix and then describe the image of this formal matrix under the morphism $$\varphi_{\kappa}$$. We will then choose a suitable subgroup $$H$$ of $$\tilde{G}(\kappa)$$ such that the restriction map $$\varphi_{\kappa}|_{H}$$ is surjective. For the second part, we will enumerate all equations defining $$\mathrm{ker~}\varphi_{\kappa}$$. It follows by considering a formal matrix equation $${}^tg\delta g=\delta$$, where $$g$$ is a $$\kappa$$-point of the special fiber of $$\underline{M}^{\ast}$$. Let $$\tilde{M}$$ be the special fiber of $$\underline{M}^{\ast}$$ and $$g$$ be an element of $$\tilde{G}(\kappa)$$. Recall that a formal matrix form of $$g$$, as an element of $$\tilde{M}(\kappa)$$, is g={(x0y0π[(e−s)/2]z0πe′w0π[(e−s)/2]x11+π[(e−s)/2]y1πe−sz1π[e−(s−1)/2]w1x2y21+π[(e−s)/2]z2πe′w2πe′x3πe′y3π[e−(s−1)/2]z31+2w3)if s<e odd;(xπe′yπe′z1+2w)if s=e.  We note that the above matrix description is valid for elements of $$\underline{G}(A)$$ inside $$\underline{M}(A)$$, where $$\underline{M}(A)$$ is defined in Section 3.16. However, as mentioned in Section 5.3 of [6], we formally use this matrix description for $$g\in \tilde{G}(\kappa)$$. Then under the morphism $$\varphi_{\kappa}$$, $$g$$ maps to {(x00x1 (resp. x3)1)if s<e odd and e=2e′−1 (resp. e=2e′);(x) (resp. (x0z1))if s=e and e=2e′−1 (resp. e=2e′).  We define the subgroup $$H$$ of $$\tilde{G}(\kappa)$$ by setting as follows: {y0=w0=0,x2=x3=0,y1=w1=y2=z2=w2=y3=z3=w3=0if e=2e′−1, s<e odd;y=0,z=0,w=0if e=2e′−1, s=e;y0=z0=0,x1=x2=0,y1=z1=w1=y2=z2=w2=y3=z3=0if e=2e′, s<e odd;no contributionif e=2e′, s=e.  Then by observing the equations defining $$H$$, we can easily show that the restriction map $$\varphi_{\kappa}|_{H}$$ is surjective, which implies the surjectivity of $$\varphi_{\kappa}$$. Note that the equations defining $$H$$, as a subgroup of $$\tilde{M}(\kappa)$$, can be obtained by considering each block of the formal matrix equation $${}^t g \delta g=\delta$$, where a formal matrix form of $$g \left(\in \tilde{M}(\kappa)\right)$$ is as described above. Here, the formal matrix equation $${}^t g \delta g=\delta$$ is interpreted as follows (cf. Remark 3.5 of [3]). We compute the formal matrix product $${}^t g \delta g$$. By Theorem 3.19, the formal matrix $${}^t g \delta g$$ should be of the form $$\delta+\tilde{f}$$ such that a formal matrix form of $$\tilde{f}$$ is as follows: f={(a~0b~0π[(e−s)/2]c~0πe′d~0tb~02b~1π[(e−s)/2]c~1πe′d~1t(π[(e−s)/2]c~0)t(π[(e−s)/2]c~1)π2e−sc~2π[e−(s−1)/2]d~2t(πe′d~0)t(πe′d~1)t(π[e−(s−1)/2]d~2)4d~3)if s<e odd;(a~πe′b~t(πe′b~)4c~)if s=e.  Here, each $$i$$-th diagonal entry of $$\tilde{a}_0$$ has $$2$$ as a factor, denoted by $$2\tilde{\gamma}_0^i$$, where $$\tilde{a}_0$$ is a $$(2m \times 2m)$$-matrix, etc., if $$s<e$$ odd, and each $$i$$-th diagonal entry of $$\tilde{a}$$ has $$2$$ as a factor, denoted by $$2\tilde{\gamma}^i$$, where $$\tilde{a}$$ is a $$(2m+2) \times (2m+2)$$-matrix, etc., if $$s=e$$. We now let $$\pi$$ be zero in each nondiagonal entry of $$(\tilde{a}_0, \tilde{a})$$ and in each entry of formal matrices $$(\tilde{\gamma}_0^i, \tilde{b}_0, \tilde{c}_0, \tilde{d}_0, \tilde{b}_1, \tilde{c}_1, \tilde{d}_1, \tilde{c}_2, \tilde{d}_2, \tilde{d}_3, \tilde{\gamma}^i, \tilde{b}, \tilde{c})$$, denoted by $$(\tilde{a}_0, \cdots, \tilde{c})$$. Then these entries are elements in $$\kappa$$. Let $$(a_0, \gamma_0^i, b_0, c_0, d_0, b_1, c_1, d_1, c_2, d_2, d_3, a, \gamma^i, b, c)$$, denoted by $$(a_0, \cdots, c)$$, be the reduction of $$(\tilde{a}_0, \cdots, \tilde{c})$$ as explained above, that is, by letting $$\pi$$ be zero in the entries of formal matrices as described above. Then $$(a_0, \cdots, c)$$ is an element of $$\tilde{H}(\kappa)$$. Thus the formal matrix equation $${}^t g \delta g=\delta$$ equals that each entry of $$(a_0, \cdots, c)$$ is zero. Next, $$\mathrm{ker~}\varphi_{\kappa}$$ is obtained, as a subgroup of $$\tilde{G}(\kappa)$$, by setting: {x0=id,x1=0if e=2e′−1 and s<e odd;x=idif e=2e′−1 and s=e;x0=id,x3=0if e=2e′ and s<e odd;x=id,z=0if e=2e′ and s=e.  We now state the equations defining $$\mathrm{ker~}\varphi_{\kappa}$$, as a subgroup of $$\tilde{M}(\kappa)$$, by observing each block of the formal matrix equation $${}^t g \delta g=\delta$$ as follows. Here, the formal matrix equation is to be interpreted as explained above. If $$e=2e'-1$$ and $$s<e$$ is odd, δ′y0+tx2=0,z0=0,δ′w0+tx3=0,ty0δ′y0/2+y12+y2=0,z2+ty1=0,ty0δ′w0+w2+ty3=0,z12+z1=0,w1+z3=0,w32+w3=0. If $$e=2e'-1$$ and $$s=e$$, δ′y+tz=0,w2+w=0. If $$e=2e'$$ and $$s<e$$ is odd, δ′y0+tx2=0,δ′z0+tx1=0,w0=0,ty0δ′y0/2+y2+πe/2y32=0,ty0δ′z0+z2+ty1=0,w2+ty3=0,z12+z1=0,w1+z3=0,w32+w3=0. If $$e=2e'$$ and $$s=e$$, y=0,w2+w=0. The claim regarding $$\mathrm{ker~}\varphi_{\kappa} $$ directly follows by observing the above equations. For example, if $$e=2e'-1$$ and $$s=e$$, then $$\mathrm{ker~}\varphi_{\kappa} $$ is identified with $$\mathrm{Spec~}\kappa[x, y, z, w]/(x-id, \delta'y+{}^tz, w^2+w)(\kappa)$$. Thus it is isomorphic to $$(\mathbb{A}^{2m+2}\times (\mathbb{Z}/2\mathbb{Z})^{1})(\kappa)$$. The other cases can be proved by using a similar argument. ■ The proof of this theorem can easily be generalized in the language of algebraic groups by working with $$\kappa$$-algebra points rather than $$\kappa$$ points without any serious obstruction. Thus we can see that the group homomorphism $$\varphi_{\kappa}$$ is a surjective morphism of algebraic groups $$\varphi : \tilde{G} ~ \longrightarrow ~ \mathrm{O}(\bar{V}, \bar{h})$$ with a kernel $$\mathbb{A}^{\ell}\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ as algebraic varieties. We explain this point of view in Proposition 4.9, Conjecture 4.10, and Remark 4.12 below with more details. Remark 4.5. We describe Im $$\varphi_{\kappa}$$ as follows. $$ $$ □ Theorem 4.6. By combining Lemma 4.2 and Theorem 4.4 with Theorem 3.25, the local density of ($$L,h$$) is βL=1/2⋅qN⋅q−dim G⋅#G~(κ)=2β−1⋅qN−dim O(V¯,h¯)⋅#(O(V¯,h¯)(κ)), where N={(2m+1)(e−e′−[(e−s)/2])−[e−(s−1)/2]+2eif s<e odd;(2m+2)(e−e′)+eif s=e.  Here, $$\beta$$ is defined in Theorem 4.4 and $$\mathrm{O}(\bar{V}, \bar{h})(\kappa)$$ is as explained in Remark 4.5. As always, $$q$$ is the cardinality of $$\kappa$$. □ Remark 4.7. (1) As promised in the introduction, we use less scheme language. All computations can be done over finitely generated $$A$$-modules. (2) As mentioned in the paragraph just before Remark 4.5, Theorem 4.4 can be generalized to give an algebraic group structure of $$\tilde{G}$$ by working with $$\kappa$$-algebra points. Indeed, $$\tilde{G}$$ satisfies the short exact sequence 1⟶ker φ⟶G~⟶φO(V¯,h¯)red⟶1 where $$\mathrm{ker~}\varphi \cong \mathbb{A}^{\ell}\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ as varieties over $$\kappa$$ and $$\mathrm{O}(\bar{V}, \bar{h})_{\rm{red}}$$ is the reduced subgroup scheme of $$\mathrm{O}(\bar{V}, \bar{h})$$. For a detailed or scheme-theoretic explanation of the construction of the morphism $$\varphi$$, see Proposition 4.9. One can also construct a morphism from $$\tilde{G}$$ to $$(\mathbb{Z}/2\mathbb{Z})^{\beta}$$ as algebraic groups and the construction of this morphism is similar to that of Sections 4.2 and 4.3 of [2]. Thus, we obtain the following short sequence 1⟶RuG~⟶G~⟶O(V¯,h¯)red×(Z/2Z)β⟶1, where the unipotent radical $$R_u\tilde{G} $$ is isomorphic to $$\mathbb{A}^{\ell}$$ as varieties over $$\kappa$$. However, we only need $$\# \tilde{G}(\kappa)$$ in order to compute the local density. Thus, for our purpose it is enough to work with $$\kappa$$-points rather than an algebraic group structure. □ 4.8 A conjectural recipe for computing $$\boldsymbol{\#\tilde{G}(\kappa)}$$ in the general case In this subsection, we work out the general idea of Theorem 3.25 for arbitrary quadratic lattices, admitting an arbitrary ramification index $$e$$. Let $$L=\bigoplus_{i\geq 0}L_i$$ be a Jordan splitting of a quadratic lattice $$(L, h)$$. We define three sublattices $$A_i, B_i, Z_i$$ of $$L$$ for each $$i$$ as follows: (1) $$A_i$$, $$A_i=\{x\in L:h(x, L)\in \pi^iA\};$$ (2) $$B_i$$, the sublattice of $$L$$ such that $$B_i/2 A_i$$ is the kernel of the additive polynomial $$1/\pi^i \cdot h$$ mod 2 on $$A_i/2 A_i$$; (3) $$Z_i$$, the sublattice of $$B_i$$ such that $$Z_i/\pi B_i$$ is the kernel of the quadratic form $$1/2\cdot 1/\pi^i \cdot h$$ mod $$\pi$$ on $$B_i/\pi B_i$$. As in Section 4.3, the idea behind the definition of the above three sublattices of $$L$$ is to extract a quadratic form over the residue field $$\kappa$$ involving the hyperbolic component (i.e., the component $$\oplus_i A(0,0)$$) of each $$L_i$$. Proposition 4.9. Let $$\bar{V_i}=B_i/Z_i$$ be a $$\kappa$$-vector space and $$\bar{h_i}$$ denote the nonsingular quadratic form $$1/2\cdot 1/\pi^i \cdot h$$ mod $$\pi$$ on $$\bar{V_i}$$. Then there exists a unique morphism of algebraic groups φi:G~⟶O(Vi¯,hi¯) defined over $$\kappa$$ such that for all étale local $$A$$-algebras $$R$$ with residue field $$\kappa_R$$ and every $$\tilde{m} \in \underline{G}(R)$$ with reduction $$m\in \tilde{G}(\kappa_R)$$, $$\varphi_i(m)\in \mathrm{GL}(B_i\otimes_AR/Z_i\otimes_AR)$$ is induced by the action of $$\tilde{m}$$ on $$L\otimes_AR$$ (which preserves $$B_i\otimes_AR$$ and $$Z_i\otimes_AR$$). The formation of $$\varphi_i$$ is compatible with finite unramified extension on $$A$$. □ Proof. We define three representable functors $$\mathrm{End}_A(B_i, Z_i)$$, $$\mathrm{Aut}_A(B_i, Z_i)$$, and $$\widetilde{\mathrm{Aut}_A(B_i, Z_i)}$$. (1) The first functor $$\mathrm{End}_A(B_i, Z_i)$$ is a functor from the category of commutative flat $$A$$-algebras to the category of rings described as follows: EndA(Bi,Zi)(R)={m∈EndR(Bi⊗AR)|m stabilizes Zi⊗AR} for any commutative flat $$A$$-algebra $$R$$. Since $$B_i/Z_i$$ is a $$\kappa$$-vector space, there is a suitable basis of $$B_i$$ such that $$B_i=B_i^1\oplus B_i^2$$ and $$Z_i=\pi B_i^1\oplus B_i^2$$. Based on this basis, each element $$m\in \mathrm{Aut}_A(B_i, Z_i)(R)$$, for a flat $$A$$-algebra $$R$$, can be written as a matrix $$\begin{pmatrix} x&\pi y\\ z& w \end{pmatrix}$$. Then by Lemma 3.1 of [2], the functor $$\mathrm{End}_A(B_i, Z_i)$$ is represented by a unique flat $$A$$-algebra which is a polynomial ring in some number of variables over $$A$$. (2) The second functor $$\mathrm{Aut}_A(B_i, Z_i)$$ is a functor from the category of commutative $$A$$-algebras to the category of groups described as follows: AutA(Bi,Zi)(R) ={m∈EndA(Bi,Zi)(R):thereexistsm−1∈EndA(Bi,Zi)(R)suchthatm⋅m−1 =m−1⋅m=1} for any commutative flat $$A$$-algebra $$R$$. Note that $$\mathrm{Aut}_A(B_i, Z_i)$$ is the group of units in the ring scheme $$\mathrm{End}_A(B_i, Z_i)$$. Then $$\mathrm{Aut}_A(B_i, Z_i)$$ is an open subscheme of $$\mathrm{End}_A(B_i, Z_i)$$ and $$\mathrm{Aut}_A(B_i, Z_i)$$ is a smooth group scheme over $$A$$. The proof of this is similar to Section 3.2 of [2] by using Lemma 3.2 in [2]. (3) The third functor $$\widetilde{\mathrm{Aut}_A(B_i, Z_i)}$$ is defined as the special fiber of $$\mathrm{Aut}_A(B_i, Z_i)$$. Then for a $$\kappa$$-algebra $$R$$, each element of $$\widetilde{\mathrm{Aut}_A(B_i, Z_i)}(R)$$ can also be written as a formal matrix $$\begin{pmatrix} x&\pi y\\ z& w \end{pmatrix}$$ . For the multiplication in $$\widetilde{\mathrm{Aut}_A(B_i, Z_i)}(R)$$, we refer to the description of Section 5.3 of [6]. Let $$R$$ be an étale local $$A$$-algebra with $$\kappa_R$$ as its residue field. Note that such $$R$$ becomes finite over $$A$$ since any étale local algebra $$R$$ over a henselian local ring is finite by Proposition 4 of Section 2.3 in [1] and $$A$$ is henselian. For such a finite field extension $$\kappa_R$$ of $$\kappa$$, $$R$$ is uniquely determined up to isomorphism. Since $$\mathrm{Aut}_A(B_i, Z_i)$$ is smooth over $$A$$, the map $$\mathrm{Aut}_A(B_i, Z_i)(R)\rightarrow \widetilde{\mathrm{Aut}_A(B_i, Z_i)}(\kappa_R)$$ is surjective by Hensel’s lemma. Now, we choose an element $$m\in \widetilde{\mathrm{Aut}_A(B_i, Z_i)}(\kappa_R)$$ and a lift $$\tilde{m} \in \mathrm{Aut}_A(B_i, Z_i)(R)$$. Since $$\tilde{m}$$ preserves $$B_i\otimes_AR$$ and $$Z_i\otimes_AR$$, it determines an element of $$\mathrm{GL}_R(B_i\otimes_AR/Z_i\otimes_AR)=\mathrm{GL}_{\kappa}(B_i/Z_i)(\kappa_R)$$. Thus we have a group homomorphism from $$\widetilde{\mathrm{Aut}_A(B_i, Z_i)}(\kappa_R)$$ to $$\mathrm{GL}_{\kappa}(B_i/Z_i)(\kappa_R)$$ with the following formal matrix description: AutA(Bi,Zi)~(κR)⟶GLκ(Bi/Zi)(κR), (xπyzw)↦(x). Indeed, the above formal matrix description shows that the map is well-defined, that is, independent of a lift $$\tilde{m}$$ of $$m$$. The formal matrix description also implies that this map is uniquely represented by a morphism of algebraic groups from $$\widetilde{\mathrm{Aut}_A(B_i, Z_i)}$$ to $$\mathrm{GL}_{\kappa}(B_i/Z_i)$$ over $$\kappa$$ and its proof is similar to the argument explained in two paragraphs written before Theorem 4.1 of [2]. On the other hand, for a flat $$A$$-algebra $$R$$, an element of $$\underline{G}(R)$$ stabilizes $$B_i\otimes_AR$$ and $$Z_i\otimes_AR$$. This fact tells us the existence of a morphism from $$\underline{G}$$ to $$\mathrm{Aut}_A(B_i, Z_i)$$ as group schemes over $$A$$. Thus we have a morphism of algebraic groups from $$\tilde{G}$$ to $$\widetilde{\mathrm{Aut}_A(B_i, Z_i)}$$ over $$\kappa$$. Furthermore, if $$R$$ is an étale local $$A$$-algebra $$R$$ with $$\kappa_R$$ as its residue field, then an element of $$\underline{G}(R)$$ determines an element of $$\mathrm{Aut}_A(B_i, Z_i)(R)$$ which also determines an element of $$\mathrm{GL}_{\kappa}(B_i/Z_i)(\kappa_R)$$. Then the element in $$\mathrm{GL}_{\kappa}(B_i/Z_i)(\kappa_R)$$ fixes the quadratic form $$\bar{h_i}\otimes_{\kappa}\kappa_R$$ defined over $$\bar{V_i}\otimes_{\kappa}\kappa_R$$. In conclusion, for an étale local $$A$$-algebra $$R$$, we have the following commutative diagram: $$ $$ such that the image of an element of $$\tilde{G}(\kappa_R)$$ in $$\mathrm{GL}_{\kappa}(B_i/Z_i)(\kappa_{R})$$ is contained in $$\mathrm{O}(\bar{V_i}, \bar{h_i})(\kappa_{R})$$. In this diagram, all horizontal arrows are uniquely represented by morphisms of group schemes over $$A$$ or over $$\kappa$$. By using the argument explained in two paragraphs written before Theorem 4.1 of [2] once again, we have a morphism of algebraic groups from $$\tilde{G}$$ to $$\mathrm{O}(\bar{V_i}, \bar{h_i})$$. ■ Let φ=∏iφi:G~⟶∏iO(Vi¯,hi¯). To compute $$\#\tilde{G}(\kappa)$$ we propose the following conjecture. Conjecture 4.10. (1) The morphism $$\varphi$$ is surjective. (2) The smooth $$($$possibly disconnected$$)$$ algebraic group $$U := (\mathrm{ker~}\varphi)_{\rm{red}}$$ is unipotent with constant component group. □ Applying this general conjecture after any unramified extension of $$A$$ implies the surjectivity in (1) on $$\kappa'$$-points for all finite extensions $$\kappa'$$ of $$\kappa$$. Unipotence implies the finite étale component group $$\pi_0(U) = U/U^0$$ has order $$2^{\beta}$$ for some $$\beta$$, and the identity component $$U^0$$ is isomorphic to an affine space $$\mathbb{A}^{\ell}$$ for some $$\ell$$ (as for any smooth unipotent group over a perfect field of characteristic 2). Here, $$\ell={\mathrm{dim}}$$$$\tilde{G}$$ - $$\sum_i {\it(}\mathrm{dim} \,\,\, \mathrm{O}(\bar{V_i}, \bar{h_i}){\it)}$$. The dimension of $$\tilde{G}$$ is the same as the dimension of $$G$$ which is $$n(n-1)/2$$, where $$G$$ is the generic fiber of $$\underline{G}$$ and $$n$$ is the rank of $$L$$, since $$\underline{G}$$ is flat over $$A$$. Once one verifies the conjecture, we obtain that #G~(κ)=∏i#(O(Vi¯,hi¯)(κ))⋅qℓ⋅2β. Here, $$q$$ is the cardinality of $$\kappa$$. This, combined with Theorem 3.25, yields the computation of the local density. Remark 4.11. The conjecture tells us more than $$\#\tilde{G}(\kappa)$$. Let $$A'$$ be a finite unramified extension of $$A$$ with the residue field $$\kappa'$$. We compare the local density associated to a quadratic $$A'$$-lattice $$(L', h')=(L\otimes_AA', h\otimes_AA')$$ with that of a quadratic $$A$$-lattice $$(L, h)$$. Recall from Section 3 that the construction of the smooth integral model is based on those of $$\widetilde{T}$$ and $$\widetilde{H}$$. Theorem 3.6 for $$T^1$$ and a similar statement for $$T^{m+1}$$ in Section 3.11 yield that $$\widetilde{T}$$ and $$\widetilde{H}$$ associated to $$(L', h')$$ can be obtained from $$\widetilde{T}$$ and $$\widetilde{H}$$ associated to $$(L, h)$$, by taking the base change to $$A'$$ respectively. Thus the smooth integral model associated to $$(L', h')$$ can be obtained from the smooth integral model associated to $$(L, h)$$, by taking the base change to $$A'$$. In addition, the integer $$N$$ in Theorem 3.25 is not changed when we vary $$A$$ to $$A'$$. We can also see that $$\bar{V_i}$$ associated to $$(L', h')$$ is the base change of $$\bar{V_i}$$ associated to $$(L, h)$$ to $$\kappa'$$. Thus the morphism $$\varphi$$ associated to $$(L', h')$$ is the base change of the morphism $$\varphi$$ associated to $$(L, h)$$ to $$\kappa'$$. In conclusion, under the conjecture, the local density is given by a rational function of $$q$$ as we extend the base ring $$A$$ to finite unramified extensions $$A'$$. □ Remark 4.12. The conjecture is proved when $$A/\mathbb{Z}_2$$ is unramified in [2] and when $$(L, h)$$ is unimodular of odd rank (except that the kernel is unipotent, but it is easy to prove by using a similar argument used in Lemma 4.11 and Theorem 4.12 of [2]) in Theorem 4.4. The proof is based on formal matrix interpretations of an element of $$\tilde{G}$$ and the morphism $$\varphi$$. In the general case, the classification of quadratic (or hermitian) lattices over a finite ramified extension of $$\mathbb{Z}_2$$ is a highly involved problem [9, 12, 13] and so it seems unlikely that formal matrix forms of an element of $$\tilde{G}$$ and the morphism $$\varphi$$ can be written. For this reason, it seems unlikely that the conjecture admits a uniform proof in the general case. It seems to us that the conjecture might be verified case by case. We explain a possible framework to prove it. For a given specific lattice, one can write an explicit matrix form of an element of $$\underline{G}(A)$$. This matrix form yields formal matrix forms of an element of $$\tilde{G}$$ and the morphism $$\varphi$$ explicitly, as in the proof of Theorem 4.4. Using formal matrix interpretations of these, one can then prove surjectivity by choosing a certain suitable subgroup $$H$$ of $$\tilde{G}$$ as in Theorem 4.4 and Example 4.13 such that the restriction of $$\varphi$$ to $$H$$ is surjective. The second part of the conjecture follows from enumerating all equations defining $$\mathrm{ker~}\varphi$$ as a subgroup of $$\tilde{M}$$. Here, $$\tilde{M}$$ is the special fiber of $$\underline{M}^{\ast}$$. We explain a strategy to compute these equations. If $$g$$ is an element of $$\tilde{M}$$ and $$\delta$$ is a symmetric matrix associated to a given quadratic form $$h$$, then equations defining $$\mathrm{ker~}\varphi$$, as a subgroup of $$\tilde{M}$$, are obtained from a formal matrix equation $${}^tg\delta g=\delta$$ such that $$\varphi(g)=id$$ as in Theorem 4.4. Here, the formal matrix equation $${}^tg\delta g=\delta$$ is interpreted by using a similar argument explained in the middle of the proof of Theorem 4.4. It is well known that every connected smooth unipotent group is isomorphic to an affine space as algebraic varieties. Thus $$(\mathrm{ker~}\varphi)_{\rm{red}}$$ consists of two parts: the identity component (isomorphic to an affine space) and the component group. Indeed, we expect that $$\mathrm{ker~}\varphi$$ is reduced, that is, $$\mathrm{ker~}\varphi=(\mathrm{ker~}\varphi)_{\rm{red}}$$. As in the last paragraph of the proof of Theorem 4.4, the equations defining the identity component of $$(\mathrm{ker~}\varphi)_{\rm{red}}$$ eliminate certain variables which appear in the coordinate ring of $$\tilde{M}$$. Indeed, we expect that all equations arising from non-diagonal entries (as well as non-diagonal blocks) of a formal matrix equation $${}^tg\delta g=\delta$$ are of this type. Thus we can ignore such equations with the associated variables so that less equation and variables remain to be analyzed. We expect that such remaining equations arise from diagonal entries (rather than diagonal blocks) of a formal matrix equation $${}^tg\delta g=\delta$$ as in Equations (A4) and (A5) in Appendix A of [2]. □ We provide an example to compute the local density of a non-unimodular quadratic lattice by using the argument explained in the above remark. Example 4.13. Assume that the ramification index $$e$$ is $$2$$. Let $$L=\left(\oplus_i A_i(0,0)\oplus (1)\right)\oplus \left(\oplus_j \pi A_j(0,0)\right) $$ of rank $$2m+1+2m'$$. Here, $$\pi A_j(0,0)$$ is a quadratic lattice with Gram matrix $$\begin{pmatrix} 0&\pi\\ \pi&0 \end{pmatrix}$$. Then a Jordan splitting of $$L$$ is $$L=L_0\oplus L_1$$. We represent the given quadratic form $$h$$ by a symmetric matrix $$\begin{pmatrix} \delta&0&0\\0&1&0\\0&0&\pi \delta' \end{pmatrix}$$, where $$\delta$$ is a $$(2m \times 2m)$$-matrix and $$\delta'$$ is a $$(2m' \times 2m')$$-matrix. Then we have the following descriptions of elements $$X\in \widetilde{T}(A)$$ and $$f\in \widetilde{H}(A)$$: X=(x0πy0πz0πx1π2y1π2z1x2πy2z2), f=(a0πb0πc0π⋅tb04b1π2c1π⋅tc0π2⋅tc1πc2). Here, $$x_0$$ and $$a_0$$ are $$(2m \times 2m)$$-matrices, and $$z_2$$ and $$c_2$$ are $$(2m' \times 2m')$$-matrices, etc. In addition, the diagonal entries of $$a_0$$ and $$c_2$$ are divisible by $$2$$. Then the exponent $$N$$ of $$q^N$$ (cf. Theorem 3.25) is computed as follows: N=(2(m′)2+4mm′+6m+9m′+4)−(4m+6m′+4mm′+2)=2(m′)2+2m+3m′+2. To compute $$\# \tilde{G}(\kappa)$$, we describe $$\varphi_{\kappa}=\varphi_{0, \kappa}\times \varphi_{1, \kappa} : \tilde{G}(\kappa) \longrightarrow \mathrm{O}(\bar{V_0}, \bar{h_0})(\kappa)\times \mathrm{O}(\bar{V_1}, \bar{h_1})(\kappa)$$ explicitly in terms of formal matrices. Recall that a formal matrix form of an element $$g\in \tilde{G}(\kappa)$$, as an element of $$\tilde{M}(\kappa)$$, is $$\begin{pmatrix}x_0&\pi y_0&\pi z_0\\\pi x_1&1+\pi^2 y_1&\pi^2 z_1\\x_2&\pi y_2&z_2\end{pmatrix}$$ . Then under the morphism $$\varphi_{\kappa}$$, $$g$$ maps to $$\begin{pmatrix}x_0&0\\x_1&1\end{pmatrix} \times \begin{pmatrix}z_2\end{pmatrix}$$. If we define the subgroup $$H$$ of $$\tilde{G}(\kappa)$$ by setting $$z_0=0, y_1=0, z_1=0, x_2=0, y_2=0$$, then we can easily show that $$\varphi_{\kappa}|_{H}$$ is surjective by observing the equations defining $$H$$, which implies the surjectivity of $$\varphi_{\kappa}$$. The equations defining $$H$$, as a subgroup of $$\tilde{M}(\kappa)$$, can be obtained by observing each block of the formal matrix equation $${}^t g \delta g=\delta$$, where $$g \left(\in \tilde{M}(\kappa)\right)$$ is as described above. Here, the formal matrix equation $${}^tg\delta g=\delta$$ is interpreted by using a similar argument explained in the middle of the proof of Theorem 4.4. In addition, the formal matrix equation $${}^t g \delta g=\delta$$ yields equations defining the kernel of $$\varphi_{\kappa}$$ as follows: δy0=0,δz0+tx2δ′=0,u¯y1+(u¯y1)2=0,z1+ty2δ′=0. Here, $$\bar{u}$$ is the reduction of $$u$$ in $$\kappa$$, where $$u$$ is the unit in $$A$$ such that $$\pi^2=2u$$. Thus, ker φκ=(Aℓ×(Z/2Z)1)(κ)=(A4mm′+2m′×Z/2Z)(κ), that is, ℓ=4mm′+2m′ and β=1. In the proof, we work with $$\kappa$$-points rather than arbitrary $$\kappa$$-algebra points. However, all arguments can be easily generalized to those with $$\kappa$$-algebra points. Therefore, we verified Conjecture 4.10 in this example. In conclusion, $$\# \tilde{G}(\kappa)=\#(\mathrm{O}(2m+1, \bar{h_0})(\kappa))\cdot \#(\mathrm{O}(2m', \bar{h_1})(\kappa))\cdot q^{4m m'+2m'}\cdot 2$$. Thus by Theorem 3.25, the local density of $$(L, h)$$ is βL=1/2⋅qN⋅q−dim G⋅#G~(κ)=qm+4m′+2−2m2⋅#(O(2m+1,h0¯)(κ))⋅#(O(2m′,h1¯)(κ)). □ In the following three examples, we work with arbitrary $$\kappa$$-algebra points rather than $$\kappa$$-points in order to affirm the component group as a constant group. Example 4.14. In this example, there are extra variables needed in the description of elements of $$\widetilde{T}(A)$$ and $$\widetilde{H}(A)$$ beyond those arising from entries of matrices. The example also serves as evidence that the special fiber (and the local density) depends on lower ramification groups of the Galois group as well as the ramification index. Let $$L=(1)\oplus (2)$$. Assume that the ramification index $$e$$ is $$2$$. By Hensel’s lemma, we may choose a unit $$u$$ in $$A$$ such that $$\pi^2=2u$$ and $$u=1+u'\pi$$. Since the ramification index is $$2$$, there are two cases for $$A/\mathbb{Z}_2$$, called Case 1 and Case 2, depending on lower ramification groups of the Galois group $$\mathrm{Gal}(F/\mathbb{Q}_2)$$ (cf. Section 2.1 of [3]). By loc. cit., $$u'$$ is a unit if $$A/\mathbb{Z}_2$$ satisfies Case 1. If $$A/\mathbb{Z}_2$$ satisfies Case 2, then $$u'$$ is not a unit. Then we have the following descriptions of elements $$X\in \widetilde{T}(R)$$ and $$f\in \widetilde{H}(R)$$ for a flat $$A$$-algebra $$R$$: X=(πx02y0z0πw0), f=(2πa02πb02πb04πc0) such that x0+z0=πx0′,y0+w0=πw0′,y0+z0=πy0′,a0+c0=πa0′. For the structure of $$\tilde{G}$$, it is easy to see that the morphism $$\varphi$$ is trivial. If we express an element $$g\in \tilde{G}(R_{\kappa})$$, for a $$\kappa$$-algebra $$R_{\kappa}$$, as a formal matrix $$\begin{pmatrix} 1+\pi x_0&2 y_0\\ z_0& 1+\pi w_0\end{pmatrix}$$ with extra three variables $$x_0'$$, $$w_0'$$, $$y_0'$$, then the kernel of $$\varphi_{R_{\kappa}}$$, as a subgroup of $$\tilde{M}(R_{\kappa})$$, is defined by the following equations: x0=y0=z0=w0,x0+u′x02=0,(x0′+w0′+y0′)+(x0′+w0′)2=0,y0′=0. Thus, as varieties over $$\kappa$$, G~≅{A1×(Z/2Z)2ifA/Z2satisfiesCase1;A1×(Z/2Z)1ifA/Z2satisfiesCase2.  Therefore, this is evidence that ramification index and congruence conditions on $$\widetilde{T}$$ and $$\widetilde{H}$$ do not determine the special fiber. The special fiber depends on lower ramification groups of the Galois group $$\mathrm{Gal}(F/\mathbb{Q}_2)$$ as well. □ In the following examples, we give more evidence for constancy of the component group of $$(\mathrm{ker~}\varphi)_{\rm{red}}$$. In Remark 4.17, we explain our expectation about noninterference of compotent groups, in terms of the ramification index and so-called “distance” between Jordan components. Example 4.15. This example is non-unimodular but rather easy. We compute the special fiber in the example as preparation for Remark 4.17. Let $$L=(1)\oplus (\pi)$$. Assume that $$\pi^2=2u$$ for a unit $$u$$ in $$A$$ (so $$e=2$$). Then we have the following descriptions of elements $$X\in \widetilde{T}(R)$$ and $$f\in \widetilde{H}(R)$$ for a flat $$A$$-algebra $$R$$: X=(2x02πy02z02w0), f=(4a02πb02πb04πc0). For the structure of $$\tilde{G}$$, it is easy to see that the morphism $$\varphi$$ is trivial. If we express an element $$g\in \tilde{G}(R_{\kappa})$$, for a $$\kappa$$-algebra $$R_{\kappa}$$, as a formal matrix $$\begin{pmatrix}1+2x_0&2\pi y_0\\ 2 z_0&1+2w_0\end{pmatrix}$$, then the kernel of $$\varphi_{R_{\kappa}}$$, as a subgroup of $$\tilde{M}(R_{\kappa})$$, is defined by the following equations: x0+x02=0,w0+w02=0,y0=z0. Thus, $$\tilde{G}$$ is isomorphic to $$\mathbb{A}^{1}\times (\mathbb{Z}/2\mathbb{Z})^{2}$$ as varieties over $$\kappa$$, that is, $$\beta=2$$. □ Example 4.16. Assume that $$\pi^2=2u$$ for a unit $$u$$ in $$A$$ (so $$e=2$$). Let $$L^a=(1)\oplus (4\pi^a)$$ for a nonnegative integer $$a$$. Then we have the following descriptions of elements $$X\in \widetilde{T}(R)$$ and $$f\in \widetilde{H}(R)$$ for a flat $$A$$-algebra $$R$$: X=(2x04πay0z02w0), f=(4a04πab04πab016πac0). Equations defining the kernel of $$\varphi_{R_{\kappa}}$$, as a subgroup of $$\tilde{M}(R_{\kappa})$$ for a $$\kappa$$-algebra $$R_{\kappa}$$, are x0+x02+πaz02=0,w0+w02+πay02=0,y0=z0. Thus, as varieties over $$\kappa$$, G~≅{A1×(Z/2Z)1if a=0;A1×(Z/2Z)2if a>0.  □ Remark 4.17. Let $$J$$ and $$J'$$ be Jordan components of a quadratic lattice $$L$$. Assume that $$J$$ (respectively $$J'$$) is $$\pi^n$$-modular (respectively $$\pi^{n+i}$$-modular). Then we say that the distance between $$J$$ and $$J'$$ is $$|i|$$. We keep using the notation $$L^a$$ from the above example. For the lattice $$L^0$$ (respectively $$L^a$$ with $$a>0$$), the component group is determined by two equations $$x_0+x_0^2+z_0^2=0, w_0+w_0^2+y_0^2=0$$ (respectively $$x_0+x_0^2=0, w_0+w_0^2=0$$) which are $$(1\times 1)$$ and $$(2\times 2)$$ blocks of a formal matrix equation. The difference between these two lattices $$L^0$$ and $$L^a$$ with $$a>0$$ is the distance between two Jordan components. If the distance between Jordan components is strictly greater than $$2e$$, then two terms $$z_0^2$$ and $$y_0^2$$ are ignored in the above equations. In other words, two Jordan components with distance greater than $$2e$$ do not interrupt each other in a sense of identifying the connected components. Indeed, the $$(1\times 1)$$ (respectively $$(2\times 2)$$) block of $$\underline{G}$$ associated to $$L^a$$ with $$a>0$$ is the same as $$\underline{G}$$ associated to the quadratic lattice $$(1)$$ (respectively $$(4\pi^a)$$) of rank $$1$$. The congruence conditions of the remained blocks of $$\underline{G}$$ associated to $$L^a$$ are the same as those of $$T^0$$. It seems that this phenomenon occurs in the general case with any ramification index. For example, consider the lattice $$M=\left((1)\oplus (\pi)\right)\oplus \left((8)\oplus (16)\right)\oplus \left((64\pi)\oplus (128)\right)$$ with $$e=2$$. Here the distance between $$(\pi)$$ and $$(8)$$, and between $$(16)$$ and $$(64\pi)$$ is $$5>2e$$. Then the block, corresponding to $$(1)\oplus (\pi)$$ (respectively $$(8)\oplus (16)$$, resp. $$(64\pi)\oplus (128)$$), of $$\underline{G}$$ associated to $$M$$ is the same as $$\underline{G}$$ associated to $$(1)\oplus (\pi)$$ (respectively $$(8)\oplus (16)$$, respectively $$(64\pi)\oplus (128)$$), and the congruence conditions of the remained blocks are the same as those of $$T^0$$ associated to $$M$$. We can then enumerate all equations defining $$\tilde{G}$$ (the morphism $$\varphi$$ is trivial) and show that G~≅A15×(Z/2Z)β as varieties over $$\kappa$$. The value of $$\beta$$ is described as follows: β={2+2+2=6ifA/Z2satisfiesCase1;2+1+2=5ifA/Z2satisfiesCase2.  Here, $$\mathbb{A}^{3}\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ is determined by three diagonal blocks corresponding to $$(1)\oplus (\pi)$$, $$(8)\oplus (16)$$, and $$(64\pi)\oplus (128)$$ (which can be read off from Examples 4.14–4.15), and $$\mathbb{A}^{12}$$ is determined by the remained non-diagonal blocks (which is easy to identify). Thus the value of $$\beta$$ associated to $$M$$ is the same as the sum of $$\beta$$’s associated to $$(1)\oplus (\pi)$$, $$(8)\oplus (16)$$, and $$(64\pi)\oplus (128)$$. The case when $$A/\mathbb{Z}_2$$ is unramified in [2] (and the case of hermitian lattices in [3] and [4]) also provides evidence of the expectation explained in this remark. □ For other lattices such as hermitian lattices, one can still construct a morphism of algebraic groups over $$\kappa$$ as in Proposition 4.9 φ=∏iφi:G~⟶∏i(classical group). One method to find classical groups in the above is the following: Each Jordan component of a Jordan splitting of a given lattice $$(L, h)$$ involves a hyperbolic part (possibly empty as Examples 4.14–4.16). Let $$g$$ be an element of $$\tilde{M}$$ (which is the special fiber of $$\underline{M}^{\ast}$$) and $$\delta$$ be a matrix associated to a given form $$h$$. Then we consider the formal matrix equation $$\sigma({}^tg)\delta g=\delta$$ as in Theorem 4.4 and Example 4.13. Now we observe the block of the above formal matrix equation corresponding to the hyperbolic part of each Jordan component. This block should not be difficult to figure out and give information about the form of classical groups. The articles [3] and [4] give an explicit construction of such morphism for ramified hermitian lattices defined over an unramified extension of $$\mathbb{Z}_2$$. In this case, such classical groups are orthogonal groups or symplectic groups. We conjecture that, under these assumptions, Conjecture 4.10 still holds. As in Proposition 4.9, we expect that the morphism $$\varphi$$ is constructed by choosing certain sublattices of $$L$$ and using this, one can show that the formation of $$\varphi$$ is compatible with finite unramified extension on $$A$$. The conjecture can also be proved by using the similar argument explained in Theorem 4.4 and Remark 4.12. In addition, one can see that the local density is given by a rational function with respect to the cardinality of the residue field, as one varies a base ring $$A$$ to a finite unramified extension $$A'$$ (cf. Remark 4.11). Funding This work was supported by JSPS KAKENHI Grant No. 16F16316. Acknowledgments The author had a motivation and an initial idea by hearing the discussion of Andrew Fiori and Gabriele Nebe, and by discussing with Brian Conrad during the AIM workshop “Algorithms for lattices and algebraic automorphic forms”, May 2013. The author would like to show deep appreciation to Wai Kiu Chan, Brian Conrad, Andrew Fiori, Abhinav Kumar, Gabriele Nebe, Rudolf Scharlau, Rainer Schulze-Pillot, and John Voight for valuable comments and discussions during the workshop. The author also thanks the organizers of the workshop and the AIM for inviting him and for their hospitality. Section 3, which is a main section of this article, was inspired by a discussion with Radhika Ganapathy. This article was improved by discussing with Wee Teck Gan during the author’s visit to 2013 Pan Asian Number Theory (PANT) conference, Hanoi, July 2013, Vietnam. He thanks the organizers of 2013 PANT conference and VIASM for inviting him and for their hospitality as well. The author would like to express his deep gratitude to Juan Marcos Cerviño, Wai Kiu Chan, Brian Conrad, Wee Teck Gan, Radhika Ganapathy, Benedict Gross, Manish Mishra, Gopal Prasad, Rudolf Scharlau, Rainer Schulze-Pillot, Olivier Taïbi, Sandeep Varma, Jiu-Kang Yu, and his wife Bogume Jang for their encouragement and support on various aspects of the article and the author’s mathematical life, and to the referee for many valuable comments and corrections. Present address: Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan Communicated by Prof. Wee Teck Gan References [1] Bosch S. , Lütkebohmert W. and Raynaud. M. “Néron Models.” Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 21 . Berlin : Springer , 1990 . Google Scholar CrossRef Search ADS [2] Cho S. “Group Schemes and Local Densities of Quadratic Lattices in Residue Characteristic 2.” Compositio Mathematica 151 , no. 5 ( 2015 ): 793 – 827 . 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TI - A Uniform Construction of Smooth Integral Models and a Conjectural Recipe for Computing Local Densities
JO - International Mathematics Research Notices
DO - 10.1093/imrn/rnw340
DA - 2017-02-26
UR - https://www.deepdyve.com/lp/oxford-university-press/a-uniform-construction-of-smooth-integral-models-and-a-conjectural-m3vEzrZrzg
SP - 1
EP - 3907
VL - Advance Article
IS - 12
DP - DeepDyve
ER -