Between the genus and the $\Gamma$-genus of an integral quadratic $\Gamma$-form

TL;DR

This paper investigates the classification of quadratic forms with group actions over global function fields, revealing differences between the genus and the mma-genus due to the failure of Witt cancellation.

Contribution

It introduces a cohomological approach to classify mma-forms and compares their genus and mma-genus, highlighting the non-injectivity caused by Witt cancellation failure.

Findings

mma-genus does not inject into the genus.

Witt cancellation failure affects classification.

Cohomological methods classify forms over mbda-forms.

Abstract

Let \Gamma be a finite group and (V,q) be a regular quadratic \Gamma-form defined over an integral domain $\mathcal{O}_S$ of a global function field (of odd characteristic). We use flat cohomology to classify the quadratic \Gamma-forms defined over $\mathcal{O}_S$ that are locally \Gamma-isomorphic for the flat topology to (V,q) and compare between the genus c(q) and the \Gamma-genus c_{\Gamma}(q) of q. We show that c_{\Gamma}(q) should not inject in c(q). The suggested obstruction arises from the failure of the Witt cancellation theorem for $\mathcal{O}_S$.

Equations

$${\mathcal{O}}_S:=\{x\in K:v_{\mathfrak{p}}(x)\ge 0\forall \mathfrak{p}\notin S\}.$$

$$\forall \gamma \in \Gamma ,A\in \text{GL}(V):{}^{\gamma }A=\gamma A{\gamma }^{-1}.$$

$$B_q(u,v)=q(u+v)-q(u)-q(v).$$

$$\phi :\mathfrak{Forms}(X_0)\to \mathfrak{Tors}({\text{Aut}}_{X_0}):X\mapsto {\text{Iso}}_{X_0,X}$$

$$E_{\Lambda }(R^{\prime })=\{\phi \in L^{\vee }\otimes R^{\prime }:{\phi }^{\otimes m}=h\}$$

$$E_{\Lambda }=\text{Iso}(R\oplus R,R\oplus L),$$

$${\underline{\text{O}}}_V(R)=\{A\in \text{GL}(V\otimes R):q\circ A=q\}.$$

$$\Gamma \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\gamma A{\gamma }^{-1}=A⟺D(\gamma )D(A)D(\gamma {)}^{-1}=D(A),$$

$${\text{Cl}}_S(\underline{G}):=\underline{G}({\mathbb{A}}_S)\backslash \underline{G}(\mathbb{A})/G(K)$$

$$1\to {\text{Cl}}_S(\underline{G})\to H_{\text{fl}}^1({\mathcal{O}}_S,\underline{G})\to H^1(K,G)\times \prod _{\mathfrak{p}\notin S}{H}_{\text{fl}}^1({\hat{\mathcal{O}}}_{\mathfrak{p}},{\underline{G}}_{\mathfrak{p}})$$

$$\forall \mathfrak{p}\notin S:H_{\text{fl}}^1({\hat{\mathcal{O}}}_{\mathfrak{p}},{\underline{G}}_{\mathfrak{p}})\hookrightarrow H_{\text{fl}}^1({\hat{K}}_{\mathfrak{p}},G_{\mathfrak{p}}),$$

$$1\to {\text{Cl}}_S(\underline{G})\to H_{\text{fl}}^1({\mathcal{O}}_S,\underline{G})\to H^1(K,G),$$

$$1\to \tilde{\underline{G}}\stackrel{i}{\to }\widetilde{{\underline{G}}^{\prime }}\to \tilde{\underline{Q}}\to 1$$

$${\text{Cl}}_S(\underline{G})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\displaystyle x^{\prime}\in\text{Cl}_{S}(\underline{G}^{\prime})\$$

$$\Rightarrow \exists y\in H_{\text{eˊt}}^1({\mathcal{O}}_S,\underline{G}):\psi (y)=i^{\prime }(x^{\prime })\Rightarrow m^{\prime }(\psi (y))=h(m(y))=0$$

$$\Rightarrow m(y)=0\Rightarrow \exists x\in {\text{Cl}}_S(\underline{G}):\psi (i(x)=y)=i^{\prime }(x^{\prime })\Rightarrow \psi (x)=x^{\prime }.$$

$$1\to {\underline{\mu }}_m\to {\underline{\mathbb{G}}}_m\to {\underline{\mathbb{G}}}_m\to 1$$

$$1\to {\mathcal{O}}_S^{\times }/({\mathcal{O}}_S^{\times }{)}^m\to H_{\text{eˊt}}^1({\mathcal{O}}_S,{\underline{\mu }}_m)\to {}_m\text{Pic }({\mathcal{O}}_S)\to 1$$

$$C^{\text{af}}=\{y^2=x^3+x\}\text{and:}{\mathcal{O}}_S={\mathbb{F}}_{11}[C^{\text{af}}].$$

$$\underline{\textbf{SO}}_{V}=\underline{\textbf{SO}}_{2}=\left\{\left(\begin{array}[]{cc}x&-y\\ y&x\end{array}\right):x^{2}+y^{2}=1\right\}$$

$$1\to \underline{N}\to \underline{R}:=R_{{\mathcal{O}}_S(i)/{\mathcal{O}}_S}({\underline{\mathbb{G}}}_m)\stackrel{\det }{\to }{\underline{\mathbb{G}}}_m\to 1.$$

$$1\to H_{\text{eˊt}}^1({\mathcal{O}}_S,\underline{N})\to \text{Pic }({\mathcal{O}}_S(i))\to \text{Pic }({\mathcal{O}}_S)$$

$$\underline{\textbf{SO}}_{V}^{\Gamma}=\left\{\left(\begin{array}[]{cc}x&0\\ 0&x\end{array}\right):x^{2}=1\right\}\cong\underline{\mu}_{2}.$$

$$L\otimes L=⟨x^2,xy,y^2⟩=⟨x^2,xy,x^3+x⟩\subseteq ⟨x⟩.$$

$$c_{\Gamma }^{+}(q)=c^{+}(q^{\prime })\stackrel{\text{eqref}genusaskernel}{\cong }\ker [H_{\text{eˊt}}^1({\mathcal{O}}_S,{\underline{\mathbb{G}}}_m)\to H^1(K,{\mathbb{G}}_m)].$$

$$c_{\Gamma }^{+}(q)\cong \text{Pic }({\mathcal{O}}_S)\to c(q)\cong \text{Pic }({\mathcal{O}}_S)/2$$

$$\Gamma\hookrightarrow\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&S_{n-2}\end{array}\right)$$

Full text

††2010 Mathematics Subject Classification: Primary 11Gxx; Secondary 11Rxx.††Key words and phrases: Hasse principle, integral quadratic forms, étale and flat cohomology, genus.

1 Introduction

Let $C$ be a geometrically connected and smooth projective curve defined over the finite field $\mathbb{F}_{q}$ ($q$ is odd). Let $K=\mathbb{F}_{q}(C)$ be its function field and let $\Omega$ denote the set of all closed points of $C$. For any point $\mathfrak{p}\in\Omega$ let $v_{\mathfrak{p}}$ be the induced discrete valuation on $K$, $\hat{\mathcal{O}}_{\mathfrak{p}}$ the complete discrete valuation ring with respect to $v_{\mathfrak{p}}$ and $\hat{K}_{\mathfrak{p}}$ its fraction field. Any Hasse set of $K$ namely, a non-empty finite set $S\subset\Omega$, gives rise to an integral domain of $K$ called a Hasse domain:

[TABLE]

Any $\mathcal{O}_{S}$-scheme is underlined, being omitted in the notation of its generic fiber.

Let $\Gamma$ be a finite group, faithfully represented by $\rho:\Gamma\hookrightarrow\textbf{GL}(V)$ where $V$ is a projective $\mathcal{O}_{S}$-module of rank $n\geq 1$. We briefly write $\gamma$ instead of $\rho(\gamma)$ when no confusion may occur and assume $|\Gamma|$ is prime to $\text{char}(K)$. Then $\Gamma$ acts on $\textbf{GL}(V)$ on the left by conjugation:

[TABLE]

Let $V$ be equipped with a degree two homogeneous $\mathcal{O}_{S}$-form $q:V\to\mathcal{O}_{S}$, turning $(V,q)$ into an integral quadratic $\mathcal{O}_{S}$-space, represented by a bilinear map $B_{q}:V\times V\to\mathcal{O}_{S}$ such that:

[TABLE]

We assume $(V,q)$ is $\mathcal{O}_{S}$-regular, i.e., that the induced homomorphism $V\to V^{\vee}:=\text{Hom}(V,\mathcal{O}_{S})$ is an isomorphism. We say it is rationally isotropic (or just isotropic), if there exists a non-zero $v\in V$ for which $q(v)=0$. It is considered a $\Gamma$-form if it satisfies $q\circ\gamma=q$ for all $\gamma\in\Gamma$. Two integral forms $(V,q)$ and $(V^{\prime},q^{\prime})$ are said to be $R$-isomorphic where $R$ is an extension of $\mathcal{O}_{S}$ if there exists an $R$-isomorphism $A:V^{\prime}\cong V$ such that $q\circ A=q^{\prime}$. $A$ is called an $R$-isometry. Two $\Gamma$-forms are said to be $\Gamma$-isomorphic over $R$ if there exists an $R$-isometry between them which is $\Gamma$-equivariant.

The classification of quadratic forms (without a group action) defined over an integral domain of a function field was initially studied by L. Gerstein ([Ger]) and J. S. Hsia ([Hsia]) in the late seventies. Later on, J. Morales proved in [Mor] that there are only finitely many $\Gamma$-isomorphism classes of $\Gamma$-forms defined over $\mathbb{Z}$ of a given discriminant. He also showed that the classical Hasse-Minkowski theorem, stating that two forms defined over a global field $K$ are $K$-isomorphic if and only if they are $\hat{K}_{v}$-isomorphic at any place $v$, does not hold for $\Gamma$-forms with $\Gamma$-isomorphisms when $K=\mathbb{Q}$. Recently, E. Bayer-Fluckiger, N. Bhaskhar and R. Parimala showed in [BBP] that this principle does hold, however, for $\Gamma$-forms when $K$ is a global function field.

The failure of this local-global principle in the case of integral forms (even without a group action) is measured by the genus of such a form. This term and its generalization to integral $\Gamma$-forms are defined as follows:

Definition 1.1**.**

The genus $c(q)$ of an $\mathcal{O}_{S}$-form $q$ is the set of isomorphism classes of $\mathcal{O}_{S}$-forms that are $K$ and $\hat{\mathcal{O}}_{\mathfrak{p}}$-isomorphic to $q$ for any prime $\mathfrak{p}\notin S$.

The $\Gamma$-genus $c_{\Gamma}(q)$ of a $\Gamma$-form $q$ defined over $\mathcal{O}_{S}$ is the set of $\Gamma$-isomorphism classes of $\Gamma$-forms defined over $\mathcal{O}_{S}$ that are $K$ and $\hat{\mathcal{O}}_{\mathfrak{p}}$-isomorphic to $q$ for any prime $\mathfrak{p}\notin S$, and these isomorphisms are $\Gamma$-ones.

We denote by $c^{+}(q)$ and $c_{\Gamma}^{+}(q)$ the genus and the $\Gamma$-genus of $q$ respectively, with respect to proper (i.e., of $\det=1$) isomorphisms only.

This paper was motivated by the following question, as was posed to me by B. Kunyavskiĭ:

Question 1.2**.**

Suppose two integral $\Gamma$-forms share the same $\Gamma$-genus and they are $\mathcal{O}_{S}$-isomorphic (the $\Gamma$-action is forgotten). Would they necessarily be also $\Gamma$-isomorphic ?

Any integral $\Gamma$-form representing a class in $c_{\Gamma}(q)$ clearly represents a class in $c(q)$ as well. So the map $\psi:c_{\Gamma}(q)\to c(q)$ is well-defined, and we may rephrase Question 1.2 as follows:

Is $\psi$ always an injection ? Because if the answer is no, and only then, this would mean that there exist two integral $\Gamma$-forms representing two distinct classes in $c_{\Gamma}(q)$, though being $\mathcal{O}_{S}$-isomorphic.

After showing in Section 2 that [$H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{SO}}_{V}^{\Gamma})$] $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V}^{\Gamma})$ [properly] classifies the integral $\Gamma$-forms that are locally $\Gamma$-isomorphic to $(V,q)$, where [$\underline{\textbf{SO}}_{V}^{\Gamma}$] $\underline{\textbf{O}}_{V}^{\Gamma}$ stands for the [special] orthogonal group of $(V,q)$, we compare in Section 3 between the genus and the $\Gamma$-genus of $(V,q)$, and give a necessary and sufficient condition for the positive answer to Question 1.2. In order to provide a counter-example, we refer in Section 4 more concretely to the case in which $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is the special orthogonal group of another isotropic form $(V^{\prime},q^{\prime})$. Based on a result established in [Bit1] and [Bit2], stating that if $q$ is isotropic of rank $\geq 3$, then $c(q)\cong\text{Pic~{}}(\mathcal{O}_{S})/2$, we show that if $q^{\prime}$ is isotropic of rank $2$ then it may possess twisted forms which are only stably isomorphic, namely, that become isomorphic over (any) regular non-trivial extension of $V^{\prime}$. So our suggested obstruction to Question 1.2 arises from the failure of the Witt cancellation theorem over $\mathcal{O}_{S}$.

Acknowledgements: I would like to thank my Post Doc. host in Camille Jordan Institute in the University Lyon1, P. Gille, for his useful advice and support, and B. Kunyavskiĭ for valuable discussions concerning the topics of the present article.

2 The classification via flat cohomology

The following general framework that appears in [CF, §2.2.4] allows us to derive some known facts about the classification of integral forms via flat cohomology, to integral $\Gamma$-forms.

Proposition 2.1**.**

Let $R$ be a scheme and $X_{0}$ be an $R$-form, namely, an object of a fibered category of schemes defined over $R$. Let $\textbf{Aut}_{X_{0}}$ be its $R$-group of automorphisms. Let $\mathfrak{Forms}(X_{0})$ be the category of $R$-forms that are locally isomorphic for some topology to $X_{0}$ and let $\mathfrak{Tors}(\text{Aut}_{X_{0}})$ be the category of $\text{Aut}_{X_{0}}$-torsors in that topology. The functor

[TABLE]

is an equivalence of fibered categories.

We first implement this Proposition on split torsion $R$-groups. For a non-negative integer $m$ we consider the $R$-group $\underline{\mu}_{m}:=\text{Spec}\,R[t]/(t^{m}-1)$. The pointed-set $H^{1}_{\text{fl}}(R,\underline{\mu}_{m})$ classifies $\underline{\mu}_{m}$-torsors, namely, $R$-groups that are locally isomorphic to $\underline{\mu}_{m}$ for the flat topology. We briefly introduce another description of these elements, as can be found for example in [AG, §5.1].

An $m$-degree $R$-Kummer pair is a couple $\Lambda=(L,h)$ consisting of a rank $1$ projective $R$-module $L$ and an isomorphism $h:R\cong L^{\otimes m}$. It gives rise to a $\underline{\mu}_{m}$-torsor $E_{\Lambda}$ assigning to any extension $R^{\prime}/R$ the group:

[TABLE]

where $L^{\vee}:=\text{Hom}(R,L)$.

In particular, for $m=2$, we set $X_{0}$ to be the quadratic $R$-algebra $R\oplus R$ with the standard involution $(r_{1},r_{2})\mapsto(r_{1},-r_{2})$, thus $\textbf{Aut}_{X_{0}}=\underline{\mu}_{2}$. Let $L$ be a fractional ideal of order $2$ in $\text{Pic~{}}(R)$. Then any $2$-degree Kummer pair $\Lambda=(L,h)$ gives rise to an $R$-algebra $X=R\oplus L$ with multiplication defined by $(0,l_{1})\cdot(0,l_{2})=(h^{-1}(l_{1}\otimes l_{2}),0)$, viewed as an $R$-form, whence according to Proposition 2.1, $\Lambda$ corresponds to the $\underline{\mu}_{2}$-torsor

[TABLE]

in which the isomorphism induced by $h$ is $\varphi:(r_{1},r_{2})\mapsto(r_{1},l)$ where $l$ is such that $l\otimes l=h(r_{2})$, i.e., $\varphi\otimes\varphi=h$.

Let $\underline{\textbf{O}}_{V}$ be the orthogonal group of $(V,q)$, namely, the $\mathcal{O}_{S}$-group assigning to any $\mathcal{O}_{S}$-algebra $R$ the group of self isometries of $q$ defined over $R$:

[TABLE]

The pointed-set $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V})$ classifies the integral quadratic forms of rank $n$ (cf. [Knu, IV.5.3.1]), being all locally isomorphic for the flat topology to $(V,q)$. We may generalize this to $\Gamma$-forms as follows: Suppose $(V,q)$ is a $\Gamma$-form. Then restricting $\varphi$ in Proposition 2.1 to the $\mathcal{O}_{S}$-group of $\Gamma$-automorphisms $\underline{\textbf{O}}_{V}^{\Gamma}$ for the flat topology, the corresponding forms are the integral $\Gamma$-forms that are locally $\Gamma$-isomorphic to $(V,q)$ in the flat topology. Modulo $\mathcal{O}_{S}$-isomorphisms we get $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V}^{\Gamma})$.

As $2\in\mathcal{O}_{S}^{\times}$ and $(V,q)$ is $\mathcal{O}_{S}$-regular, $\underline{\textbf{O}}_{V}$ is smooth and its connected component, the special orthogonal group of $(V,q)$, is $\underline{\textbf{SO}}_{V}=\ker[\underline{\textbf{O}}_{V}\stackrel{{\scriptstyle\det}}{{\rightarrow}}\underline{\mu}_{2}]$, containing only the proper isometries (cf. [Con, Thm. 1.7, Cor. 2.5]). The push-forward homomorphism $\det_{*}:H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V})\to H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\mu}_{2})$ induced by flat cohomology, assigns to any quadratic form $(V^{\prime},q^{\prime})$ (taken up to an $\mathcal{O}_{S}$-isomorphism) the class of its discriminant module $D(q^{\prime})=D(V^{\prime},q^{\prime})=(\wedge^{n}V^{\prime},\det(q^{\prime}))$ (see [Knu, IV, 4.6, 5.3.1]). Let $\underline{\textbf{SO}}_{V}^{\Gamma}$ be its $\Gamma$-invariant subgroup. Our assumption that $|\Gamma|\in\mathcal{O}_{S}^{\times}$ guarantees that $\underline{\textbf{SO}}_{V}^{\Gamma}$ remains smooth (cf. [CGP2, Proposition A.8.10(2)]). Any representative $(V^{\prime},q^{\prime})$ of a class in $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{SO}}_{V}^{\Gamma})$ represents a class in $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V}^{\Gamma})$, though $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{SO}}_{V}^{\Gamma})$ should not embed in $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V}^{\Gamma})$ (notice that these pointed-sets do not have to be groups). More precisely, any class in $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{SO}}_{V}^{\Gamma})$ is represented by a triple $(V^{\prime},q^{\prime},\theta^{\prime})$ where $\theta^{\prime}$ is the trivialization of $D(q^{\prime})$, namely, an isomorphism $\theta^{\prime}:D(q^{\prime})\otimes S\cong D(q)$ where $S$ is some $2$-degree flat extension of $\mathcal{O}_{S}$. Any proper $\Gamma$-isometry $A:V^{\prime}\cong V:q\circ A=q^{\prime}$ induces an isomorphism $D(A):D(q^{\prime})\otimes S\cong D(q)$. The question is whether these additional data $D(q^{\prime})$ and $D(A)$ (when $q^{\prime}$ is a $\Gamma$-form) are also $\Gamma$-equivariant.

In order to answer this question we may consider as mentioned before the $\mathcal{O}_{S}$-module $D(q^{\prime})$, being of rank $1$ and isomorphic to $\mathcal{O}_{S}$ over some at most $2$-degree étale extension, as a $2$-degree Kummer pair $(L=D(q^{\prime}),h)$, giving rise to the $\underline{\mu}_{2}$-torsor $E_{q^{\prime}}$ for which $E_{q^{\prime}}(\mathcal{O}_{S})=\{\varphi\in L^{\vee}:\varphi\otimes\varphi=h\}$. Since ${q^{\prime}}$ is a $\Gamma$-form, $\Gamma$ admits the following commutative diagram:

[TABLE]

from which we see that $\Gamma$ acts on $E_{q^{\prime}}(\mathcal{O}_{S})$ through its determinant in $\underline{\mu}_{2}(\mathcal{O}_{S})$. But $\underline{\mu}_{2}(\mathcal{O}_{S})$ is the automorphism group of $\mathcal{O}_{S}\oplus\mathcal{O}_{S}$ with respect to its standard involution $\tau=(\text{id},-\text{id})$ and $E_{q^{\prime}}(\mathcal{O}_{S})$ is stable (not fixed point-wise) under $\tau$, as $-l\otimes-l=l\otimes l$. Furthermore, the correspondence $A\leadsto D(A)$ is functorial, thus referring to any $\gamma\in\Gamma$ as to an isometry one has

[TABLE]

i.e. $D(A)$ is $\Gamma$-invariant as well. So Proposition 2.1 is also applied to the proper classification.

Corollary 2.2**.**

Given an integral $\Gamma$-form base-point $(V,q)$, the pointed set $H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V}^{\Gamma})$ $\mathrm{[}H^{1}_{\text{fl}}(\mathcal{O}_{S},\underline{\textbf{SO}}_{V}^{\Gamma})\mathrm{]}$ $\mathrm{[}$properly$\mathrm{]}$ classifies the integral $\Gamma$-forms that are locally $\Gamma$-isomorphic to $(V,q)$ for the flat topology.

3 The [proper] genus and [proper] $\Gamma$-genus

Consider the ring of $S$-integral adèles $\mathbb{A}_{S}:=\prod_{\mathfrak{p}\in S}\hat{K}_{\mathfrak{p}}\times\prod_{\mathfrak{p}\notin S}\hat{\mathcal{O}}_{\mathfrak{p}}$, being a subring of the adèles $\mathbb{A}$. The $S$-class set of an affine and flat $\mathcal{O}_{S}$-group $\underline{G}$ is the set of double cosets:

[TABLE]

(where for any prime $\mathfrak{p}$ the geometric fiber $\underline{G}_{\mathfrak{p}}$ of $\underline{G}$ is taken). According to Nisnevich ([Nis, Theorem I.3.5]) $\underline{G}$ admits the following exact sequence of pointed sets

[TABLE]

in which the left exactness reflects the fact that $\text{Cl}_{S}(\underline{G})$ is the genus of $\underline{G}$, namely the set of (classes of) $\underline{G}$-torsors that are generically and locally at $\mathfrak{p}\notin S$ isomorphic to $\underline{G}$. If $\underline{G}$ admits the property

[TABLE]

then due to Corollary 3.6 in [Nis] the Nisnevich’s sequence (3.1) simplifies to

[TABLE]

which indicates that any two $\underline{G}$-torsors belong to the same genus if and only if they are $K$-isomorphic.

Remark 3.1**.**

Since $\text{Spec}\,\mathcal{O}_{S}$ is normal, i.e., is integrally closed locally everywhere (due to the smoothness of $C$), any finite étale covering of $\mathcal{O}_{S}$ arises by its normalization in some separable unramified extension of $K$ (see [Len, Theorem 6.13]). Consequently, if $\underline{G}$ is a finite $\mathcal{O}_{S}$-group, then $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{G})$ is embedded in $H^{1}(K,\underline{G})$.

Lemma 3.2**.**

Let $\varphi:\underline{G}\to\underline{G}^{\prime}$ be a monomorphism of smooth affine $\mathcal{O}_{S}$-groups and let $\widetilde{\underline{Q}}$ be the sheafification of $\underline{Q}:=\text{coker}(\varphi)$. Then

• •

The map $\underline{G}^{\prime}(\mathcal{O}_{S})\to\widetilde{\underline{Q}}(\mathcal{O}_{S})$ is surjective iff $\ \ker[\text{Cl}_{S}(\underline{G})\stackrel{{\scriptstyle\psi}}{{\rightarrow}}\text{Cl}_{S}(\underline{G}^{\prime})]=~{}1$.

• •

If, moreover, $\underline{G}$ is locally of finite presentation, $\underline{G}$ and $\underline{G}^{\prime}$ admit property (3.2), $\underline{Q}$ is a finite $\mathcal{O}_{S}$-group and $G^{\prime}(K)\to Q(K)$ is surjective, then $\psi$ is surjective.

Proof.

As a pointed set, $\text{Cl}_{S}(\underline{G})$ is bijective to the first Nisnevich’s cohomology set $H^{1}_{\text{Nis}}(\mathcal{O}_{S},\underline{G})$ (cf. [Nis, I. Theorem 2.8]), classifying $\underline{G}$-torsors for the Nisnevich’s topology. But Nisnevich’s covers are étale, so $\text{Cl}_{S}(\underline{G})$ is a subset of $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{G})$. The monomorphism $\varphi$ does not have to be a closed immersion hence $\underline{Q}$ may not be representable, and so we may not be able to apply flat cohomology on the obtained short exact sequence of $\mathcal{O}_{S}$-schemes. Restricting, however, to the small site of flat extensions of $\mathcal{O}_{S}$, we have $\widetilde{\underline{G}}(R)\subseteq\widetilde{\underline{G}^{\prime}}(R)$ for any such extension $R/\mathcal{O}_{S}$, where $\widetilde{\underline{G}}$ and $\widetilde{\underline{G}^{\prime}}$ stand for the sheafifications of $\underline{G}$ and $\underline{G}^{\prime}$, respectively. Then flat cohomology applied to the exact sequence of flat sheaves

[TABLE]

yields a long exact sequence in which $H^{1}_{\text{fl}}(\mathcal{O}_{S},\widetilde{*})=H^{1}_{\text{fl}}(\mathcal{O}_{S},*)=H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},*)$ for both smooth groups $*=\underline{G}^{\prime}$ and $\underline{G}$, whence $\widetilde{\underline{G}^{\prime}}(\mathcal{O}_{S})=\underline{G}^{\prime}(\mathcal{O}_{S})\to\widetilde{\underline{Q}}(\mathcal{O}_{S})$ is surjective if and only if $\ker[H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{G})\xrightarrow{\psi}H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{G}^{\prime})]=1$, being equivalent by restriction to $\text{Cl}_{S}(\underline{G})$, to $\ker[\text{Cl}_{S}(\underline{G})\xrightarrow{\psi}\text{Cl}_{S}(\underline{G}^{\prime})]=1$, since any twisted form in $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{G})$ can be $\mathcal{O}_{S}$-isomorphic to $\underline{G}$ only if it belongs to $\text{Cl}_{S}(\underline{G})$.

Now suppose $\underline{G}$ is locally of finite presentation and $\underline{Q}$ is representable as a finite $\mathcal{O}_{S}$-group. Then $\underline{Q}$ is smooth as well (cf. [SGA3, VI${}_{\text{B}}$, Proposition 9.2 xii]) and so given furthermore that $\underline{G}$ and $\underline{G}^{\prime}$ admit property (3.2) and $G^{\prime}(K)\to Q(K)$ is surjective, étale cohomology applied to sequence (3.4) over $\mathcal{O}_{S}$ and over $K$ extends the exactness of sequence (3.3) to the commutative diagram

[TABLE]

in which $m^{\prime\prime}$ is injective as $\underline{Q}$ is finite, by Remark 3.1. We then get the surjectivity of $\psi$:

[TABLE]

∎

We return now to our integral $\Gamma$-form $(V,q)$ for which $\text{Cl}_{S}(\underline{\textbf{O}}_{V}^{\Gamma})=c_{\Gamma}(q)$ and $\text{Cl}_{S}((\underline{\textbf{O}}_{V}^{\Gamma})^{0})=c_{\Gamma}^{+}(q)$. Since $\underline{\textbf{O}}_{V}^{\Gamma}$ and $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ are smooth, their flat cohomology sets over $\text{Spec}\,\mathcal{O}_{S}$ coincide with the étale ones (cf. [SGA4, VIII Corollaire 2.3]). According to Lemma 3.2, if $\underline{\textbf{O}}_{V}(\mathcal{O}_{S})\to(\widetilde{\underline{\textbf{O}}_{V}/\underline{\textbf{O}}_{V}^{\Gamma}})(\mathcal{O}_{S})$ is surjective then $\ker[c_{\Gamma}(q)\xrightarrow{\psi}c(q)]=1$, which means that for any $[q^{\prime}]\in c_{\Gamma}(q)$, if $q^{\prime}$ is not $\Gamma$-isomorphic to $q$ then neither is it $\mathcal{O}_{S}$-isomorphic to it. This still does not imply that $c_{\Gamma}(q)$ injects into $c(q)$ since both are not necessarily groups, so there may be two classes of forms other than $[q]$, which are distinct in $c_{\Gamma}(q)$, yet are $\mathcal{O}_{S}$-isomorphic. We summarize by

Proposition 3.3**.**

Question 1.2 is answered in the affirmative if and only if $\underline{\textbf{O}}_{V^{\prime}}(\mathcal{O}_{S})\to(\widetilde{\underline{\textbf{O}}_{V^{\prime}}/\underline{\textbf{O}}_{V^{\prime}}^{\Gamma}})(\mathcal{O}_{S})$ is surjective for any $[(V^{\prime},q^{\prime})]\in c(q)$.

Lemma 3.4**.**

The group scheme $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is reductive.

Proof.

The group scheme $\underline{\textbf{SO}}_{V}$ is reductive as it is smooth, affine and all its fibers are reductive. As mentioned before, its affine subgroup $\underline{\textbf{SO}}_{V}^{\Gamma}$ is smooth as well, so we may refer to its neutral component $(\underline{\textbf{SO}}_{V}^{\Gamma})^{0}$ as defined in [SGA3, VIB, Théorème 3.10]. The reduction $(\overline{\textbf{SO}}_{V})_{\mathfrak{p}}$ defined over the residue field $k_{\mathfrak{p}}$ at any prime $\mathfrak{p}$ is reductive, hence according to [CGP2, Proposition A.8.12] its subgroup $((\overline{\textbf{SO}}_{V})_{\mathfrak{p}}^{\Gamma})^{0}=(\overline{\textbf{SO}}_{V}^{\Gamma})^{0}_{\mathfrak{p}}$ remains reductive, so that $(\underline{\textbf{SO}}_{V}^{\Gamma})^{0}$ is reductive. The latter being a smooth, open and connected subgroup of $\underline{\textbf{O}}_{V}^{\Gamma}$ coincides with $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$, thus it is reductive. ∎

Remark 3.5**.**

The group $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ admits property (3.2) by Lang’s Theorem (recall that all residue fields are finite). Being reductive over $\text{Spec}\,\mathcal{O}_{S}$ (Lemma 3.4), this property holds for $\underline{\textbf{O}}_{V}^{\Gamma}$ as well if $\underline{\textbf{O}}_{V}^{\Gamma}/(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is representable as a finite $\mathcal{O}_{S}$-group (see the proof of Proposition 3.14 in [CGP1]).

Following sequence (3.3) and Remark 3.5 we then get:

Corollary 3.6**.**

$c^{+}_{\Gamma}(q)\cong\ker[H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},(\underline{\textbf{O}}_{V}^{\Gamma})^{0})\to H^{1}(K,(\textbf{O}_{V}^{\Gamma})^{0})]$*.

If $\underline{\textbf{O}}_{V}^{\Gamma}/(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is a finite $\mathcal{O}_{S}$-group then $c_{\Gamma}(q)\cong\ker[H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V}^{\Gamma})\to H^{1}(K,\textbf{O}_{V}^{\Gamma})]$.*

Corollary 3.7**.**

If $(V,q)$ of rank $\geq 3$ is regular and isotropic then $c^{+}(q)=c(q)$.

Proof.

Any representative $(V^{\prime},q^{\prime})$ of a class in $c(q)$ is $\mathcal{O}_{S}$-regular, thus $\underline{\textbf{O}}_{V^{\prime}}/\underline{\textbf{SO}}_{V^{\prime}}\cong\underline{\mu}_{2}$ (cf. [Con, Thm. 1.7], notice that $\underline{\mathbb{Z}/2}:=\text{Spec}\,\mathbb{Z}[t]/(t(t-1))$ is isomorphic to $\underline{\mu}_{2}$ as $2\in\mathcal{O}_{S}^{\times}$). Moreover, being $K$-isomorphic to $q$ (due to Corollary 3.6), $q^{\prime}$ is isotropic as well, therefore as $\text{rank}(V)\geq 3$, the map $\underline{\textbf{O}}_{V^{\prime}}(\mathcal{O}_{S})\xrightarrow{\det}\underline{\mu}_{2}(\mathcal{O}_{S})=\mu_{2}(K)=\{\pm 1\}$ is surjective (cf. [Bit2, Lemma 4.3]). So setting $\underline{G}=\underline{\textbf{SO}}_{V^{\prime}}$ and $\underline{G}^{\prime}=\underline{\textbf{O}}_{V^{\prime}}$ in Lemma 3.2, we get that $\ker[c^{+}(q^{\prime})\xrightarrow{\psi}c(q^{\prime})=c(q)]=1$ and $\psi$ is surjective. This holds as mentioned before to any $[q^{\prime}]\in c(q)$, which amounts to $\psi$ being the identity. ∎

Remark 3.8**.**

When $|S|=1$, i.e., $S=\{\infty\}$ where $\infty$ is an arbitrary closed point of $K$, Lemma 3.7 is automatic for any regular $q$ of rank $\geq 3$, since in that case $(V,q)$ must be isotropic (see the proof of [Bit1, Prop. 4.4]).

Lemma 3.9**.**

*Suppose that $\underline{\textbf{O}}_{V}^{\Gamma}\cong(\underline{\textbf{O}}_{V}^{\Gamma})^{0}\rtimes\underline{Q}$ where $\underline{Q}$ is a finite $\mathcal{O}_{S}$-group.

Then $\ker[c_{\Gamma}^{+}(q)\xrightarrow{\psi}c_{\Gamma}(q)]=1$ and $\psi$ is surjective.*

Proof.

As $\underline{Q}$ embeds as a semi-direct factor in $\underline{\textbf{O}}_{V}^{\Gamma}$, $\underline{\textbf{O}}_{V}^{\Gamma}(\mathcal{O}_{S})$ surjects onto $\underline{Q}(\mathcal{O}_{S})$ and $\textbf{O}_{V}^{\Gamma}(K)$ onto $Q(K)$. Given furthermore that $\underline{Q}$ is a finite $\mathcal{O}_{S}$-group, both $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ and $\underline{\textbf{O}}_{V}^{\Gamma}$ admit property (3.2) (see Remark 3.5), so all conditions in Lemma 3.2 are satisfied and the assertion follows. ∎

Lemma 3.10**.**

If $\text{rank}(V)=2$ then $c_{\Gamma}(q)\subseteq c(q)$, i.e., Question 1.2 is then answered positively.

Proof.

If $\text{rank}(V)=2$ then $\underline{\textbf{SO}}_{V}$ is a one dimensional torus. We first claim that $\underline{\textbf{O}}_{V}^{\Gamma}$ cannot be $\underline{\textbf{SO}}_{V}$. Indeed, since $q$ is a $\Gamma$-form, $\Gamma$ embeds by definition in $\underline{\textbf{O}}_{V}(\mathcal{O}_{S})$. As $(\underline{\textbf{O}}_{V}/\underline{\textbf{SO}}_{V})(\mathcal{O}_{S})=\underline{\mu}_{2}(\mathcal{O}_{S})$ does not commute with $\underline{\textbf{O}}_{V}(\mathcal{O}_{S})$, $\Gamma$ cannot have a non-trivial image in it and still stabilizing $\underline{\textbf{SO}}_{V}$, i.e., it must embed in $\underline{\textbf{SO}}_{V}(\mathcal{O}_{S})$ only. But then being finite, the $\Gamma$-image must be the group $\{\pm I_{2}\}$, which does not kill $\underline{\mu}_{2}(\mathcal{O}_{S})$. So either $\underline{\textbf{O}}_{V}^{\Gamma}=\underline{\textbf{O}}_{V}$ for which the assertion is trivial, or $\underline{\textbf{O}}_{V}^{\Gamma}$ is a finite group, for which $c_{\Gamma}(q)\stackrel{{\scriptstyle\eqref{genus as kernel}}}{{\cong}}\ker[H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\textbf{O}}_{V}^{\Gamma})\to H^{1}(K,\textbf{O}_{V}^{\Gamma})]$ is trivial according to Remark 3.1. ∎

As mentioned in the proof of Lemma 3.10, the inequality $c_{\Gamma}(q)\subsetneq c(q)$ for $\text{rank}(V)=2$ may occur only when $\underline{\textbf{O}}_{V}^{\Gamma}$ is finite. For example, suppose $\underline{\textbf{O}}_{V}^{\Gamma}=\underline{\mu}_{m}$. Then étale cohomology applied to the related Kummer exact sequence of smooth $\mathcal{O}_{S}$-groups

[TABLE]

yields the exactness of

[TABLE]

where the right non-trivial term stands for the $m$-torsion part of $\text{Pic~{}}(\mathcal{O}_{S})$. According to Remark 3.1, since $\text{Spec}\,\mathcal{O}_{S}$ is normal, both $\mathcal{O}_{S}^{\times}/(\mathcal{O}_{S}^{\times})^{m}$ and the preimage of ${{}_{m}}\text{Pic~{}}(\mathcal{O}_{S})$ in $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mu}_{m})$ are embedded in $K^{\times}/(K^{\times})^{m}\cong H^{1}(K,\mu_{m})$. The following example demonstrates this embedding, which yields the above inequality.

Example 3.11**.**

Consider the elliptic curve $C=\{Y^{2}Z=X^{3}+XZ^{2}\}$ defined over $\mathbb{F}_{11}$. Removing the closed point $\infty=(0:1:0)$ results in an affine curve with coordinate ring:

[TABLE]

Let $V=\mathcal{O}_{S}^{2}$ be generated by the standard basis over $\mathcal{O}_{S}$ endowed with the form $q$ represented by $B_{q}=1_{2}$. Then

[TABLE]

is a one dimensional $\mathcal{O}_{S}$-torus and as $-1$ is not a square, it is isomorphic to the non-split norm torus $\underline{N}:=R^{(1)}_{\mathcal{O}_{S}(i)/\mathcal{O}_{S}}(\underline{\mathbb{G}}_{m})$, fitting into the exact sequence of $\mathcal{O}_{S}$-tori:

[TABLE]

Then since $\underline{R}(\mathcal{O}_{S})\xrightarrow{\det}\mathcal{O}_{S}^{\times}=\mathbb{F}_{11}^{\times}$ is surjective ($x^{2}+y^{2}$ gets any value in $\mathbb{F}_{11}^{\times}$), étale cohomology together with the Shapiro’s Lemma gives rise to the exact sequence:

[TABLE]

from which we see that $c^{+}(q)=\text{Cl}_{S}(\underline{\textbf{SO}}_{V})=H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\textbf{SO}}_{V}\cong\underline{N})$ (cf. [Bit1, Prop. 4.2]) is far from being trivial (say, by the Hasse-Weil bound: $|\text{Pic~{}}(\mathcal{O}_{S})|=|C^{\text{af}}(\mathbb{F}_{11})|<11+1+2\sqrt{11}<19$ while: $|\text{Pic~{}}(\mathcal{O}_{S}(i))|=|C^{\text{af}}(\mathbb{F}_{11}(i))|>121+1-2\sqrt{121}=100$, see in Example 4.2). As $\underline{\textbf{O}}_{V}/\underline{\textbf{SO}}_{V}\cong\underline{\mu}_{2}$ and $\underline{\textbf{O}}_{V}(\mathcal{O}_{S})\xrightarrow{\det}\underline{\mu}_{2}(\mathcal{O}_{S})$ is surjective by $\text{diag}(1,-1)\mapsto-1$, setting $\underline{G}=\underline{\textbf{SO}}_{V}$ and $\underline{G}^{\prime}=\underline{\textbf{O}}_{V}$ in Lemma 3.2 we get that $c(q)=c^{+}(q)$, thus is not trivial as well.

Now let $\Gamma=S_{3}=\langle\tau,\sigma\rangle$ be represented in $V$ by: \tau\mapsto\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right) and: \sigma\mapsto\left(\begin{array}[]{cc}5&-8\\ 8&5\end{array}\right). One can easily check that $q$ is a $\Gamma$-form and that

[TABLE]

For $L=\langle x,y\rangle\in\text{Pic~{}}(\mathcal{O}_{S})$ one has:

[TABLE]

But $x=y^{2}-x^{3}\in L\otimes L$, thus $L\otimes L=\langle x\rangle$. So the $2$-degree Kummer pair $(L,h)$ gives rise to the $\underline{\mu}_{2}$-torsor $\mathcal{O}_{S}\oplus L$ being isomorphic to $\mathcal{O}_{S}^{2}$ over $\mathcal{O}_{S}[1/\sqrt{x}]$ which is not contained in $K$. The same happens for the other $\underline{\mu}_{2}$-torsors, i.e., $c_{\Gamma}(q)\stackrel{{\scriptstyle\eqref{genus as kernel}}}{{\cong}}\ker[H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mu}_{2})\to H^{1}(K,\mu_{2})]=1\subsetneq c(q)$.

4 An explicit obstruction

The criterion exhibited in Proposition 3.3 for Question 1.2 to be answered in the affirmative, namely, $\underline{\textbf{O}}_{V^{\prime}}(\mathcal{O}_{S})\to(\widetilde{\underline{\textbf{O}}_{V^{\prime}}/\underline{\textbf{O}}_{V^{\prime}}^{\Gamma}})(\mathcal{O}_{S})$ is surjective for any $[(V^{\prime},q^{\prime})]\in c(q)$, is somewhat vague. We would like to refer to the case in which $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is the special orthogonal group of another isotropic $\Gamma$-form. It is shown in [Bit1, Proposition 4.4] for $|S|=1$ and more generally in [Bit2, Theorem 4.6] for any finite $S$, that if $\text{rank}(V)\geq 3$ ($q$ is isotropic), then $c(q)\cong\text{Pic~{}}(\mathcal{O}_{S})/2$. For $\text{rank}(V)=2$, however, this genus might be larger. This means that there may be two integral forms of rank $2$ that are only stably isomorphic, i.e., become isomorphic after being extended by any non-trivial regular common extension. This failure of the Witt Cancellation Theorem over $\mathcal{O}_{S}$, invokes a case in which Question 1.2 is answered negatively, namely, when $\text{rank}(V)\geq 3$ and $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is the special orthogonal group of another integral $\Gamma$-form $(V^{\prime},q^{\prime})$ of rank $2$, whose genus decreases over $V$.

Proposition 4.1**.**

Let $(V,q)$ be a regular $\Gamma$-form of rank $\geq 3$ such that $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is the special orthogonal group of an isotropic form of rank $2$, being a semi-direct factor in $\underline{\textbf{O}}_{V}^{\Gamma}$, while the quotient is a finite $\mathcal{O}_{S}$-group. If $-1\in(\mathbb{F}_{q}^{\times})^{2}$ and $\text{exp}(\text{Pic~{}}(\mathcal{O}_{S}))>2$ then $c_{\Gamma}(q)$ does not inject into $c(q)$, i.e., Question 1.2 is then answered negatively.

Proof.

Given that $\text{rank}(V^{\prime})=2$ and $-1\in(\mathbb{F}_{q}^{\times})^{2}$, $\underline{\textbf{O}}_{V^{\prime}}^{0}\cong\underline{\mathbb{G}}_{m}$, so one has:

[TABLE]

This kernel is isomorphic due to Shapiro’s Lemma and Hilbert’s 90 Theorem to $\text{Pic~{}}(\mathcal{O}_{S})$. Since $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}$ is a normal semi-direct factor in $\underline{\textbf{O}}_{V}^{\Gamma}$ and the quotient is a finite $\mathcal{O}_{S}$-group, by Lemma 3.9 $\ker[c_{\Gamma}^{+}(q)\xrightarrow{\psi}c_{\Gamma}(q)]=1$, so $\ker[c_{\Gamma}(q)\to c(q)]$ cannot vanish, because if it would, the composition, being a morphism of abelian groups

[TABLE]

would be injective, which is impossible whenever $\text{exp}(\text{Pic~{}}(\mathcal{O}_{S}))>2$. ∎

Example 4.2**.**

Let $C$ be an elliptic $\mathbb{F}_{q}$-curve such that $-1\in(\mathbb{F}_{q}^{\times})^{2}$ and $\text{exp}(C(\mathbb{F}_{q}))>2$. Let $\infty$ be an $\mathbb{F}_{q}$-rational point. Then $\text{Pic~{}}(\mathcal{O}_{S})\cong C(\mathbb{F}_{q})$ (cf. e.g., [Bit1, Example 4.8]). Let $(V,q)$ be the quadratic space generated by the standard basis and represented by $B_{q}=1_{n}$ for $n\geq 4$. Then as mentioned before, $c(q)\cong\text{Pic~{}}(\mathcal{O}_{S})/2\cong C(\mathbb{F}_{q})/2$. Let the permutations in $\Gamma=S_{n-2}$ be canonically represented by monomial matrices in the lower right $(n-2)\times(n-2)$ block of $\underline{\textbf{O}}_{V}\subset\underline{\textbf{GL}}(V)$:

[TABLE]

turning $q$ into a $\Gamma$-form. Now if $n$ is even, then $\underline{\textbf{SO}}_{V}^{\Gamma}\cong\underline{\mathbb{G}}_{m}\times\{\pm I_{n-2}\}$. Otherwise, if $n$ is odd, then $\underline{\textbf{SO}}_{V}^{\Gamma}$ is a semi-direct product of $\underline{\mathbb{G}}_{m}\times\{I_{n-2}\}\cong~{}\underline{\mathbb{G}}_{m}$ and $\text{diag}(1,-1)\times\{-I_{n-2}\}\cong\underline{\mu}_{2}$. In both cases, $(\underline{\textbf{O}}_{V}^{\Gamma})^{0}=(\underline{\textbf{SO}}_{V}^{\Gamma})^{0}\cong\underline{\mathbb{G}}_{m}$ is a normal semi-direct factor in $\underline{\textbf{O}}_{V}^{\Gamma}$ and the quotient is a finite $\mathcal{O}_{S}$-group, therefore according to Proposition 4.1 $c_{\Gamma}(q)$ cannot inject into $c(q)$.