Spinor groups with good reduction

Abstract

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$. We show that for a given $n\geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\mathsf{G}_{2}$.

1 Introduction

Let $K$ be a field. The purpose of this paper is to present new results in the framework of the following general problem:

1. () (When) can one equip $K$ with a natural set $V$ of discrete valuations such that for a given absolutely almost simple simply connected algebraic $K$ -group $G$ , the set of $K$ -isomorphism classes of (inner) $K$ -forms of $G$ having good reduction at all $v\in V$ (respectively, at all $v\in V\setminus S$ , where $S\subset V$ is an arbitrary finite subset) is finite?

(We refer to the subsection Notation and conventions below for the definition of good reduction.)

Although the analysis of Abelian varieties defined over a global field and having good reduction at a given set of places of the field has been one of the central topics in arithmetic geometry for a long time, particularly since the work of Faltings [Fal83], similar questions in various situations involving linear algebraic groups have received less attention so far. Thus, before formulating our results, we would like to include a brief account of the previous work in this direction and discuss connections between () and several other topics of active current research, which should provide the reader with some context for () and its variations.

Historically, the consideration of forms of linear algebraic groups with good reduction can be traced back to the work of Harder [Har67], Colliot-Thélène and Sansuc [CS79] and Gross [Gro96]. We note that in [Har67], the focus was primarily on a number field $K$ . In this case, combining well-known results about the properness of the global-to-local map for the Galois cohomology of algebraic groups [Ser97, ch. III, 4.6] with the fact that an absolutely almost simple algebraic group with good reduction over a $p$ -adic field is necessarily quasi-split, one can see that a set $V$ consisting of almost all nonarchimedean places of $K$ satisfies $(\ast )$ ; see [Gro96, Con15] for more precise results over $K=\mathbb{Q}$ , and also [JL15].

The case where $K=k(x)$ is the field of rational functions in one variable over a field $k$ and $V$ consists of the discrete valuations $v_{p(x)}$ corresponding to all irreducible polynomials $p(x)\in k[x]$ was considered by Raghunathan and Ramanathan [RR84] (see also [CGP12, Theorem 2.1]): their result implies that if $G_{0}$ is a (connected) semi-simple group over $k$ and $G$ is obtained from $G_{0}$ by the base change $K/k$ , then any $K$ -form $G$ that splits over $\bar{k}(x)$ (where $\bar{k}$ is a separable closure of $k$ ) and has good reduction at all $v\in V$ is obtained by base change from a certain $k$ -form of $G_{0}$ . In the same notation, a description of the $K$ -forms of $G$ that (split over $\bar{k}(x)$ and) have good reduction at all $v\in V\setminus \{v_{x}\}$ was obtained by Chernousov, Gille and Pianzola [CGP12], which played a crucial role in the proof of the conjugacy of the analogues of Cartan subalgebras in certain infinite-dimensional Lie algebras [CNPY16]. (We note that if $k$ has characteristic zero, then every semi-simple $K$ -group $G$ becomes quasi-split over $\bar{k}(x)$ , which implies that those $G$ that have good reduction at all $v\in V\setminus \{v_{x}\}$ automatically split over $\bar{k}(x)$ .)

In all these instances, $K$ was the fraction field of a certain Dedekind ring and $V$ was the set of discrete valuations associated with the nonzero prime ideals of this ring, making the situation ‘one-dimensional’. Although the finiteness questions have not been fully answered yet even in the one-dimensional case (cf. Theorem 1.4 below and subsequent remarks), the current work on the following problems necessitate the analysis of () in a more general (higher-dimensional) setting.

• The genus problem. Given an absolutely almost simple simply connected algebraic $K$ -group $G$ , its genus $\mathbf{gen}_{K}(G)$ is defined to be the set of $K$ -isomorphism classes of $K$ -forms $G^{\prime }$ of $G$ that have the same isomorphism classes of maximal $K$ -tori as $G$ (the latter means that every maximal $K$ -torus $T$ of $G$ is $K$ -isomorphic to some maximal $K$ -torus $T^{\prime }$ of $G^{\prime }$ , and vice versa). One expects $\mathbf{gen}_{K}(G)$ to be finite for any $G$ over an arbitrary finitely generated field $K$ . This has been proven for groups of all types when $K$ is a number field [PR09, Theorem 7.5], and for inner forms of type $\mathsf{A}_{\ell }$ over arbitrary finitely generated fields, cf. [CRR13, CRR16a, CRR1]. The connection between the genus and good reduction described in [CRR16b, Theorem 5] and [CRR2] implies that (under minor additional technical assumptions) the fact that $K$ possesses a set $V$ of discrete valuations such that () holds for a given $G$ implies that $\mathbf{gen}_{K}(G)$ is finite. Thus, () provides a natural approach to proving the finiteness of the genus.

• The Hasse principle. For a given algebraic $K$ -group $G$ and a given set $V$ of valuations of $K$ , one considers the global-to-local map in Galois cohomology:

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G),\end{eqnarray}$$

(where $K_{v}$ is the completion of $K$ with respect to $v$ ), and one says that the Hasse principle holds for $G$ with respect to $V$ if $\unicode[STIX]{x1D703}_{G,V}$ is injective (cf. [Ser97, ch. III, § 4.7]). Although the Hasse principle does hold in many important situations over a number field $K$ (in which case, one takes for $V$ the set of all valuations of $K$ , including archimedean ones; see [PR94, ch. VI]), it may fail for general semi-simple groups [Ser97, 4.7], but for any algebraic group $G$ over a number field $K$ , the map $\unicode[STIX]{x1D703}_{G,V}$ is proper (in other words, the deviation from the Hasse principle is always finite). It is important to point out that the affirmative answer to () would enable one to significantly extend this result, namely one shows that if () holds for a given $K$ , $V$ and $G$ , then for the corresponding adjoint group $\overline{G}$ the map $\unicode[STIX]{x1D703}_{\overline{G},V}$ is proper (cf. § 5.2). See Conjecture B and its discussion below for more concrete statements/results.

• Eigenvalue rigidity. Let $G_{1}$ and $G_{2}$ be two semi-simple algebraic groups over a field $F$ of characteristic zero and let $\unicode[STIX]{x1D6E4}_{i}\subset G_{i}(F)$ be a Zariski-dense subgroup for $i=1,2$ . In [PR09], the notion of weak commensurability of $\unicode[STIX]{x1D6E4}_{1}$ and $\unicode[STIX]{x1D6E4}_{2}$ , based on the consideration of eigenvalues of semi-simple elements of those subgroups, was introduced and developed. A detailed analysis of weakly commensurable arithmetic groups was used in [PR09] to derive geometric consequences. It is currently expected that certain results allowing one to (almost) recover some characteristics of an arithmetic group from the information about the eigenvalues of its elements should be valid for arbitrary finitely generated Zariski-dense subgroups; this phenomenon is called eigenvalue rigidity [Rap14]. Central to it is the following finiteness conjecture [PR15, Rap14]: let $G_{1}$ and $G_{2}$ be absolutely simple (adjoint) algebraic groups over a field $F$ of characteristic zero, and let $\unicode[STIX]{x1D6E4}_{1}\subset G_{1}(F)$ be a finitely generated Zariski-dense subgroup with trace field ¹   $K=K_{\unicode[STIX]{x1D6E4}_{1}}$ . Then there are only finitely many $F/K$ -forms $\mathscr{G}_{2}^{(1)},\ldots ,\mathscr{G}_{2}^{(r)}$ of $G_{2}$ such that any finitely generated Zariski-dense subgroup $\unicode[STIX]{x1D6E4}_{2}\subset G_{2}(F)$ that is weakly commensurable to $\unicode[STIX]{x1D6E4}_{1}$ is $G_{2}(F)$ -conjugate to a subgroup of $\mathscr{G}_{2}^{(j)}(K)\subset G_{2}(F)$ for some $j=1,\ldots ,r$ . Note that the field $K$ is finitely generated, hence admits a divisorial set of places $V$ (see below). Then if () holds for $K$ , $V$ and an absolutely almost simple simply connected group $G$ of the same type as $G_{2}$ , the above finiteness conjecture is valid, cf. [CRR2].

While various forms of the Hasse principle (including the cohomological form cited above) have been studied extensively for a long time, the genus problem and eigenvalue rigidity emerged relatively recently as an extension of the research carried out in [PR09] in connection with some geometric problems for isospectral and length-commensurable locally symmetric spaces. Thus, the fact that () has strong consequences for all three problems, combined with the previous work done in the Dedekind case, makes the analysis of () in the general case worthwhile.

To make () more tractable, one can specialize it to a suitable class of fields $K$ or restrict the type of an absolutely almost simple algebraic $K$ -group $G$ in its statement (or do both). It is most interesting to investigate () for finitely generated fields $K$ . In this case, $K$ possesses natural sets of discrete valuations called divisorial. More precisely, let $\mathfrak{X}$ be a normal irreducible scheme of finite type over $\mathbb{Z}$ (if $\operatorname{char}K=0$ ) or over a finite field (if $\operatorname{char}K>0$ ) such that $K$ is the field of rational functions on $\mathfrak{X}$ (we will refer to $\mathfrak{X}$ as a model of $K$ ). It is well-known that to every prime divisor $\mathfrak{Z}$ of $\mathfrak{X}$ there corresponds a discrete valuation $v_{\mathfrak{Z}}$ on $K$ (cf. [Cut18, 12.3] and [Har77, ch. II, § 6]). Then

$$\begin{eqnarray}V(\mathfrak{X})=\{v_{\mathfrak{Z}}\mid \mathfrak{Z}\text{ prime divisor of }\mathfrak{X}\}\end{eqnarray}$$

is called the divisorial set of places of $K$ corresponding to the model $\mathfrak{X}$ . Any set of places $V$ of $K$ of this form (for some model $\mathfrak{X}$ ) will be simply called divisorial. Note that any two divisorial sets $V_{1}$ and $V_{2}$ are commensurable, i.e.  $V_{i}\setminus (V_{1}\,\cap \,V_{2})$ is finite for $i=1,2$ , and for any finite subset $S$ of a divisorial set $V$ the set $V\setminus S$ contains a divisorial set. We then can formulate the following more precise version of ().

Conjecture A. Let $K$ be a finitely generated field and let $V$ be a divisorial set of places. Then for a given absolutely almost simple algebraic $K$ -group $G$ , the set of $K$ -isomorphism classes of (inner) $K$ -forms of $G$ that have good reduction at all $v\in V$ is finite.

Rapoport suggested that a finiteness statement of such nature is likely to be true for arbitrary reductive $K$ -groups, namely the number of $K$ -isomorphisms classes of reductive $K$ -groups of bounded dimension (or rank) that have good reduction at all $v\in V$ should be finite. At this point, Conjecture A is known over arbitrary finitely generated only for inner forms of type $\mathsf{A}_{\ell +1}$ with $(\ell +1)$ prime to $\operatorname{char}K$ (cf. [CRR16a, CRR16b]). In fact, before the current paper Conjecture A was not known for any other type over fields other than global.

One of the goals of the paper is to prove Conjecture A for the spinor groups of quadratic forms, and also some unitary groups and groups of type $\mathsf{G}_{2}$ , over two-dimensional global fields of characteristic $\neq \,2$ . Following Kato [Kat86], by a two-dimensional global field, we mean either the function field $K=k(C)$ of a smooth geometrically integral curve $C$ over a number field $k$ , or the function field $K=k(S)$ of a smooth geometrically integral surface $S$ over a finite field $k$ .

Theorem 1.1. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$ . Fix an integer $n\geqslant 5$ . Then the set of $K$ -isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate quadratic forms in $n$ variables over $K$ that have good reduction at all $v\in V$ is finite.

We will now indicate some consequences of this result for two of the three problems discussed above. First, we have the following finiteness result for the genus of spinor groups over two-dimensional global fields.

Theorem 1.2. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $G=\operatorname{Spin}_{n}(q)$ , where $q$ is an $n$ -dimensional nondegenerate quadratic form over $K$ . If either $n\geqslant 5$ is odd or $n\geqslant 10$ is even and $q$ is $K$ -isotropic, then $\mathbf{gen}_{K}(G)$ is finite.

(In fact, we prove that the number of $K$ -isomorphism classes of spinor groups $G^{\prime }=\operatorname{Spin}_{n}(q^{\prime })$ of nondegenerate $n$ -dimensional quadratic forms $q^{\prime }$ over $K$ that have the same isomorphism classes of maximal $K$ -tori as $G$ is finite for any $n\geqslant 5$ ; see Theorem 5.1.)

Another application deals with the global-to-local map in Galois cohomology. Namely, the techniques developed to prove Theorem 1.1 also yield the following.

Theorem 1.3. With notation as in Theorem 1.1, for $G=\operatorname{SO}_{n}(q)$ the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G)\end{eqnarray}$$

is proper, i.e. the pre-image of a finite set is finite.

Moreover, our techniques in fact yield similar results for some groups $G$ of the form $\operatorname{SL}_{1,D}$ (cf. Theorem 5.7), some unitary groups and groups of type $\mathsf{G}_{2}$ (cf. §§ 8–9), and a partial result for some spinor groups (cf. Proposition 5.8). Based on these results, we propose the following conjecture.

Conjecture B. Let $K$ be a two-dimensional global field and let $V$ be a divisorial set of places of $K$ . Then for any absolutely almost simple (or even semi-simple) algebraic group $G$ , the natural map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G)\end{eqnarray}$$

is proper. In particular,

$$\begin{eqnarray}\unicode[STIX]{x0428}(G,V):=\ker \unicode[STIX]{x1D703}_{G,V}\end{eqnarray}$$

is finite.

(We note that the use of twisting shows that proving the finiteness of $\unicode[STIX]{x0428}(G,V)$ is the most essential part of Conjecture B.)

Of course, it would be quite tempting to extend this conjecture to all finitely generated fields, but for this more evidence needs to be developed. At this point, the only result that goes beyond two-dimensional global fields is that for any finitely generated field $K$ , its divisorial set of places $V$ , and any $n$ prime to $\operatorname{char}K$ , the map $H^{1}(K,G)\rightarrow \prod _{v\in V}H^{1}(K_{v},G)$ is proper for $G=\operatorname{PGL}_{n}$ , which follows from the finiteness of the unramified Brauer group $_{n\!}\operatorname{Br}(K)_{V}$ (cf. [CRR16a, CRR16b]).

Applications to weakly commensurable Zariski-dense subgroups will be given in [CRR2].

Theorem 1.1 will be derived from a more general Theorem 2.1 (see § 3) that (potentially) enables one to answer () for spinor groups over arbitrary finitely generated fields. We will also use this result to prove a finiteness statement for spinor groups, as well as some unitary groups and groups of type $\mathsf{G}_{2}$ , with good reduction over a class of fields that are not finitely generated. This class includes the function fields of $p$ -adic curves that have received a great deal of attention in recent years (we refer the reader to [Bru10, CPS12, Par14, PS98] and references therein for various results involving division algebras, quadratic forms and algebraic groups over those fields), but is in fact much larger. We will formulate our results using a generalization of Serre’s condition $(\text{F})$ (see [Ser97, ch. III, § 4]) offered in [Rap18]. Let $K$ be a field and let $m\geqslant 1$ an integer prime to $\operatorname{char}K$ . We then introduce the following condition on $K$ :

$(\text{F}_{m}^{\prime })$

For every finite separable extension $L/K$ , the quotient $L^{\times }/{L^{\times }}^{m}$ is finite.

(Note that if $L^{\times }/{L^{\times }}^{m}$ is finite for every finite separable extension $L/K$ , then it is finite for any finite extension of $K$ ; see [Rap18, Lemma 2.8]). Combining Theorem 2.1 with the results on the finiteness of unramified cohomology with $\unicode[STIX]{x1D707}_{m}$ -coefficients over fields satisfying $(\text{F}_{m}^{\prime })$ (see [Rap18]), we obtain the following.

Theorem 1.4. Let $C$ be a smooth geometrically integral curve over a field $k$ of characteristic $\neq 2$ that satisfies condition $(\text{F}_{2}^{\prime })$ and let $K=k(C)$ be its function field. Denote by $V$ the set of discrete valuations of $K$ corresponding to the closed points of $C$ . Then the number of $K$ -isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate quadratic forms $q$ over $K$ in $n\geqslant 5$ variables that have good reduction at all $v\in V$ is finite.

Theorem 1.4 is likely to extend to absolutely almost simple simply connected groups of all types over the function fields of curves defined over a field that satisfies condition $(\text{F})$ ; see Conjecture 7.3. In this regard, we observe that in §§ 8–9 we extend the above results for spinor groups to the special unitary groups of Hermitian forms over quadratic extensions of the base field and to groups of type $\mathsf{G}_{2}$ (for the same fields $K$ and the same sets of valuations $V$ ).

The paper is organized as follows. In § 2, we develop some formalism involving the Picard group associated with a set of discrete valuations and then use it to formulate a general result (Theorem 2.1) that reduces the proof of the finiteness of the set of isomorphism classes of spinor groups having good reduction to the finiteness of certain unramified cohomology groups. We prove Theorem 2.1 in § 3, where we also give an application of our method to the properness of the global-to-local map in Galois cohomology (Theorem 3.4). In § 4, we combine Theorem 2.1 with the finiteness results for unramified cohomology of two-dimensional global fields to prove Theorem 1.1. It should be pointed out that these finiteness results are known in degrees ${\leqslant}2$ and easily follow from the theorems of Poitou–Tate in degrees ${\geqslant}4$ , but have not appeared in the literature in degree three. We derive the required fact in degree three (Corollary 6.2) from the more general Theorem 6.1. For several reasons (see the discussion in § 4.1), we give two proofs of the characteristic zero case of Theorem 6.1: that given in § 6 is based on the referee’s suggestions, whereas that in Appendix A is our original argument. The positive characteristic case of Theorem 6.1 is treated in § 7, along with the finiteness results involving fields of type $(\text{F}_{m}^{\prime })$ and the proof of Theorem 1.4. Theorems 1.2 and 1.3 are proved in § 5. Finally, in §§ 8–9, we present finiteness results for special unitary groups and groups of type $\mathsf{G}_{2}$ .

Notation and conventions. For a field $k$ , we will denote by $\bar{k}$ a fixed separable closure. Given a discrete valuation $v$ of $k$ , we let $k_{v}$ and $k^{(v)}$ denote the completion and the residue field of $k$ at $v$ , respectively. We recall that a $\operatorname{Gal}(\bar{k}/k)$ -module $M$ is said to be unramified at $v$ if for some (equivalently, any) extension $w$ of $v$ to $\bar{k}$ , the inertia subgroup of the decomposition group $\operatorname{Dec}_{w}\subset \operatorname{Gal}(\bar{k}/k)$ acts trivially on $M$ . In addition, if $G$ is an absolutely almost simple linear algebraic group defined over $k$ , we will say that $G$ has good reduction at $v$ if there exists a reductive group scheme $\mathscr{G}$ over the valuation ring ${\mathcal{O}}_{v}$ of $k_{v}$ whose generic fiber $\mathscr{G}\otimes _{{\mathcal{O}}_{v}}k_{v}$ is isomorphic to $G\otimes _{k}k_{v}$ (this definition, involving completions, is convenient for applications to the Hasse principle).

We will follow the conventions outlined in [GMS03, ch. II, § 7.8] regarding Tate twists of Galois modules. Namely, suppose $v$ is a discrete valuation of $k$ , $n$ an integer prime to $\operatorname{char}k^{(v)}$ and $M$ a finite $\operatorname{Gal}(\bar{k}/k)$ -module satisfying $nM=0$ . For an integer $d$ , one defines $M(d)$ to be $\unicode[STIX]{x1D707}_{n}^{\otimes d}\otimes M$ if $d\geqslant 0$ and $\operatorname{Hom}(\unicode[STIX]{x1D707}_{n}^{\otimes (-d)},M)$ if $d<0$ . In particular, $M(-1)=\operatorname{Hom}(\unicode[STIX]{x1D707}_{n},M)$ . In the case where $M$ is torsion (but not necessarily finite) without any elements of order equal to the residue characteristic, one writes $M=\underset{\longrightarrow }{\lim }M^{\prime }$ , where $M^{\prime }$ are the finite submodules of $M$ , and sets $M(d)=\underset{\longrightarrow }{\lim }M^{\prime }(d)$ . As usual, we will use $\unicode[STIX]{x1D707}_{n}^{\otimes d}$ to denote $\mathbb{Z}/n\mathbb{Z}(d)$ for all $d$ .

Finally, we recall that if $n$ is an integer prime to $\operatorname{char}k$ , then isomorphism $k^{\times }/{k^{\times }}^{n}\simeq H^{1}(k,\unicode[STIX]{x1D707}_{n})$ from Kummer theory is induced by sending an element $a\in k^{\times }$ to the cohomology class of the 1-cocycle

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{n,a}(\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D70E}(\sqrt[n]{a})/\sqrt[n]{a}\quad \text{for }\unicode[STIX]{x1D70E}\in \operatorname{Gal}(\bar{k}/k).\end{eqnarray}$$

When $n=2$ , we will denote this cocycle simply by $\unicode[STIX]{x1D712}_{a}.$

2 The Picard group associated with a set of discrete valuations

Suppose that a field $K$ is equipped with a set $V$ of discrete valuations that satisfies the following condition:

1. (A) For any $a\in K^{\times }$ , the set $V(a):=\{v\in V\mid v(a)\neq 0\}$ is finite.

(It is worth noting that $(\text{A})$ automatically holds for a divisorial set of valuations $V$ of a finitely generated field $K$ .) We now let $\operatorname{Div}(V)$ denote the free Abelian group on the set $V$ , the elements of which will be called ‘divisors’. The fact that $V$ satisfies $(\text{A})$ enables one to associate to any $a\in K^{\times }$ the corresponding ‘principal divisor’

$$\begin{eqnarray}(a)=\mathop{\sum }_{v\in V}v(a)\cdot v.\end{eqnarray}$$

Let $\text{P}(V)$ denote the subgroup of $\operatorname{Div}(V)$ formed by all principal divisors. We call the quotient $\operatorname{Div}(V)/\text{P}(V)$ the Picard group of $V$ and denote it by $\operatorname{Pic}(V)$ .

Next, we recall (cf. [GMS03, ch. II]) that for a discrete valuation $v$ of a field $K$ and a finite Galois module $M$ that is unramified at $v$ and order of which is prime to $\operatorname{char}K^{(v)}$ , one has a residue map

$$\begin{eqnarray}\unicode[STIX]{x2202}_{i,v}^{M}:H^{i}(K,M)\longrightarrow H^{i-1}(K^{(v)},M(-1))\quad (i\geqslant 1).\end{eqnarray}$$

In particular, for every $n$ prime to $\operatorname{char}K^{(v)}$ and every $d$ we have the residue map

$$\begin{eqnarray}\unicode[STIX]{x2202}_{i,v}^{n,d}:H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes d})\longrightarrow H^{i-1}(K^{(v)},\unicode[STIX]{x1D707}_{n}^{\otimes (d-1)}).\end{eqnarray}$$

An element of $H^{i}(K,M)$ (in particular, of $H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes d})$ ) is said to be unramified if it lies in the kernel of the relevant residue map.

In this section, we only consider cohomology with coefficients in $\unicode[STIX]{x1D707}_{2}=\{\pm 1\}$ . Then for a discrete valuation $v$ of $K$ , the corresponding residue map is defined whenever $\operatorname{char}K^{(v)}\neq 2$ and will be denoted by

$$\begin{eqnarray}\unicode[STIX]{x2202}_{v}^{i}:H^{i}(K,\unicode[STIX]{x1D707}_{2})\longrightarrow H^{i-1}(K^{(v)},\unicode[STIX]{x1D707}_{2}).\end{eqnarray}$$

We now make the following assumption:

1. (B) $\operatorname{char}K^{(v)}\neq 2$ for all $v\in V$ .

We then define the $i$ th unramified cohomology group of $K$ with respect to $V$ by

$$\begin{eqnarray}H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}=\mathop{\bigcap }_{v\in V}\operatorname{Ker}\unicode[STIX]{x2202}_{v}^{i}.\end{eqnarray}$$

With this notation, we have the following result.

Theorem 2.1. Let $K$ be a field equipped with a set $V$ of discrete valuations satisfying conditions $(\text{A})$ and $(\text{B})$ and let $n\geqslant 5$ be an integer. Assume that:

1. (1) the quotient $\operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V)$ is finite; and

2. (2) the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ are finite for all $i=1,\ldots ,\ell :=[\log _{2}n]+1$ .

Then the number of $K$ -isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate quadratic forms $q$ over $K$ in $n$ variables that have good reduction at all $v\in V$ is

$$\begin{eqnarray}{\leqslant}|\operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V)|\cdot \mathop{\prod }_{i=1}^{\ell }|H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}|\end{eqnarray}$$

(in particular, finite).

We postpone the proof of Theorem 2.1 until the next section, and recall now the connection between $\operatorname{Pic}(V)$ and the idèles, which is well-known in the classical setting (cf. [CF10, ch. II, §17]). Given a field $K$ endowed with a set $V$ of discrete valuations that satisfies (A), we define the group of idèles $\mathbb{I}(K,V)$ as the restricted direct product of the multiplicative groups $K_{v}^{\times }$ for $v\in V$ with respect to the groups of units $U_{v}={\mathcal{O}}_{v}^{\times }$ :

Furthermore, we let

$$\begin{eqnarray}\mathbb{I}_{0}(K,V)=\mathop{\prod }_{v\in V}U_{v}\end{eqnarray}$$

be the subgroup of integral idèles. As $V$ satisfies (A), one can consider the diagonal embedding $K^{\times }{\hookrightarrow}\mathbb{I}(K,V)$ , the image of which will be called the group of principal idèles and denoted also by $K^{\times }$ . Then

$$\begin{eqnarray}\unicode[STIX]{x1D708}:\mathbb{I}(K,V)\rightarrow \operatorname{Div}(V),\quad (x_{v})\mapsto \mathop{\sum }_{v\in V}v(x_{v})\cdot v\end{eqnarray}$$

is a surjective group homomorphism with kernel $\ker \unicode[STIX]{x1D708}=\mathbb{I}_{0}(K,V)$ . Thus, we obtain the following.

Lemma 2.2. The map $\unicode[STIX]{x1D708}$ induces a natural identification of the quotient $\mathbb{I}(K,V)/\mathbb{I}_{0}(K,V)K^{\times }$ with $\operatorname{Pic}(V)$ .

3 Proof of Theorem 2.1 and its variations

3.1 Two facts about the Witt ring

Let $F$ be a field of characteristic $\neq 2$ . We let $W(F)$ and $I(F)$ denote the Witt ring of $F$ and its fundamental ideal, respectively. For a nondegenerate quadratic form $q$ over $F$ , $[q]$ denotes the corresponding class in $W(F)$ . As usual, the quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}$ with $a_{i}\in F^{\times }$ will be denoted by $\langle a_{1},\ldots ,a_{n}\rangle$ , whereas $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ will be used to denote the $d$ -fold Pfister form $\langle 1,-a_{1}\rangle \otimes \cdots \otimes \langle 1,-a_{d}\rangle$ . Clearly, for any $d\geqslant 1$ , the $d$ th power $I(F)^{d}$ is additively generated by the classes of $d$ -fold Pfister forms.

Lemma 3.1. Let $q$ be a nondegenerate quadratic form over $F$ such that $[q]\in I(F)^{d}$ for some $d\geqslant 1$ . Then for any $\unicode[STIX]{x1D706}\in F^{\times }$ , we have $[\unicode[STIX]{x1D706}q]\in I(F)^{d}$ and $[\unicode[STIX]{x1D706}q]+I(F)^{d+1}=[q]+I(F)^{d+1}$ .

Proof. The fact that $[\unicode[STIX]{x1D706}q]\in I(F)^{d}$ is obvious. Furthermore, we have

$$\begin{eqnarray}[q]-[\unicode[STIX]{x1D706}q]=[q\bot (-\unicode[STIX]{x1D706}q)]=[\langle 1,-\unicode[STIX]{x1D706}\rangle \otimes q]\in I(F)^{d+1},\end{eqnarray}$$

as required. ◻

Now, let $F$ be a field complete with respect to a discrete valuation $v$ such that the characteristic of the residue field $F^{(v)}$ is $\neq 2$ . We let $U(F)$ denote the group of units in $F$ , and fix a uniformizer $\unicode[STIX]{x1D70B}$ . Then one can define the first and second residue maps

$$\begin{eqnarray}\unicode[STIX]{x2202}_{i}:W(F)\rightarrow W(F^{(v)}),\quad i=1,2,\end{eqnarray}$$

which are homomorphisms of additive groups uniquely characterized by the conditions

$$\begin{eqnarray}\unicode[STIX]{x2202}_{1}\langle u\rangle =\langle \bar{u}\rangle ,\quad \unicode[STIX]{x2202}_{1}\langle \unicode[STIX]{x1D70B}u\rangle =0\quad \text{and}\quad \unicode[STIX]{x2202}_{2}\langle u\rangle =0,\quad \unicode[STIX]{x2202}_{2}\langle \unicode[STIX]{x1D70B}u\rangle =\langle \bar{u}\rangle ,\end{eqnarray}$$

for any $u\in U(F)$ (where $\bar{u}$ denote the image of $u$ in $F^{(v)}$ ) (see [Lam05, ch. VI, § 1] or [Mil70, § 5] for the details). Let $W_{0}(F)$ be the subring of $W(F)$ generated by the classes of $\langle u\rangle$ for $u\in U(F)$ . Then $W_{0}(F)=\operatorname{Ker}\unicode[STIX]{x2202}_{2}$ , whereas $\unicode[STIX]{x2202}_{1}$ yields an isomorphism between $W_{0}(F)$ and $W(F^{(v)})$ . Thus, we have the following split exact sequence

$$\begin{eqnarray}0\rightarrow W_{0}(F)\longrightarrow W(F)\stackrel{\unicode[STIX]{x2202}_{2}}{\longrightarrow }W(F^{(v)})\rightarrow 0.\end{eqnarray}$$

Let $I_{0}(F)=W_{0}(F)\,\cap \,I(F)$ . Milnor [Mil70, § 5] shows that for any $d\geqslant 1$ , the restriction of $\unicode[STIX]{x2202}_{2}$ to $I^{d}(F)$ (the $d$ th power of $I(F)$ ) yields the exact sequence

$$\begin{eqnarray}0\rightarrow I_{0}^{d}(F)\longrightarrow I^{d}(F)\stackrel{\unicode[STIX]{x2202}_{2}}{\longrightarrow }I^{d-1}(F^{(v)})\rightarrow 0.\end{eqnarray}$$

This, in particular, gives the following result.

Lemma 3.2. For any $d\geqslant 1$ , we have $I(F)^{d}\,\cap \,W_{0}(F)=I_{0}(F)^{d}$ .

3.2 The Milnor isomorphism and unramified classes

Again let $F$ be a field of characteristic $\neq 2$ . It is a consequence of Voevodsky’s proof of the Milnor conjecture (see [OVV07, Voe03, Voe11]) that for $d\geqslant 1$ , there are natural isomorphisms of Abelian groups

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{F,d}:I(F)^{d}/I(F)^{d+1}\longrightarrow H^{d}(F,\unicode[STIX]{x1D707}_{2}).\end{eqnarray}$$

Explicitly, $\unicode[STIX]{x1D6FE}_{F,d}$ is defined by sending the class of the Pfister form $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ to the cup-product $\unicode[STIX]{x1D712}_{a_{1}}\,\cup \,\cdots \,\cup \,\unicode[STIX]{x1D712}_{a_{d}}$ , where for $a\in F^{\times }$ , we let $\unicode[STIX]{x1D712}_{a}$ be the corresponding 1-cocycle given by Kummer theory.

Lemma 3.3. Let $F$ be a field complete with respect to a discrete valuation $v$ such that $\operatorname{char}F^{(v)}\neq 2$ and let $d\geqslant 1$ . If $q$ is a nondegenerate quadratic form over $F$ such that $[\unicode[STIX]{x1D706}q]\in I(F)^{d}\,\cap \,W_{0}(F)$ for some $\unicode[STIX]{x1D706}\in F^{\times }$ (notation as in the previous subsection), then $[q]\in I(F)^{d}$ and the cohomology class $\unicode[STIX]{x1D6FE}_{F,d}([q])\in H^{d}(F,\unicode[STIX]{x1D707}_{2})$ is unramified at $v$ .

Proof. It immediately follows from Lemma 3.1 that $[q]\in I(F)^{d}$ and $\unicode[STIX]{x1D6FE}_{F,d}([q])=\unicode[STIX]{x1D6FE}_{F,d}([\unicode[STIX]{x1D706}q])$ . Thus, it is enough to show that if $[q]\in I(F)^{d}\,\cap \,W_{0}(F)$ , then $\unicode[STIX]{x1D6FE}_{F,d}([q])$ is unramified at $v$ . According to Lemma 3.2, we have $I(F)^{d}\,\cap \,W_{0}(F)=I_{0}(F)^{d}$ , which is additively generated by the classes of Pfister forms $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ with $a_{i}\in U(F)$ . However, for any such form, the corresponding class

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{F,d}([\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle ])=\unicode[STIX]{x1D712}_{a_{1}}\cup \cdots \cup \unicode[STIX]{x1D712}_{a_{d}}\end{eqnarray}$$

is clearly unramified, and our claim follows. ◻

3.3 Proof of Theorem 2.1

Let $\{q_{i}\}\text{}_{i\in I}$ be a family of $n$ -dimensional nondegenerate quadratic forms over $K$ such that:

• ∙ for each $i\in I$ , the spinor group $G_{i}=\operatorname{Spin}_{n}(q_{i})$ has good reduction at all $v\in V$ ; and

• ∙ for $i,j\in I$ , $i\neq j$ , the forms $q_{i}$ and $q_{j}$ are not similar (i.e.  $q_{i}$ is not equivalent to any nonzero scalar multiple of $q_{j}$ ).

We wish to show that $I$ is finite and

(1) $$\begin{eqnarray}|I|\leqslant d_{0}d_{1}\cdots d_{\ell },\end{eqnarray}$$

where

$$\begin{eqnarray}d_{0}=|\text{Pic}(V)/2\cdot \operatorname{Pic}(V)|\quad \text{and}\quad d_{i}=|H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}|\quad \text{for }i=1,\ldots ,\ell =[\log _{2}n]+1.\end{eqnarray}$$

We begin by observing that the first of the above conditions implies that for each $i\in I$ and any $v\in V$ , there exists $\unicode[STIX]{x1D706}_{v}^{(i)}\in K_{v}^{\times }$ such that the form $\unicode[STIX]{x1D706}_{v}^{(i)}q_{i}\in W_{0}(K_{v})$ in the above notation. In addition, we may assume without loss of generality that for each $i\in I$ , we have $\unicode[STIX]{x1D706}_{v}^{(i)}=1$ for almost all $v$ ; then $\unicode[STIX]{x1D706}^{(i)}:=(\unicode[STIX]{x1D706}_{v}^{(i)})_{v\in V}\in \mathbb{I}(K,V)$ . Using Lemma 2.2, we see that

$$\begin{eqnarray}\mathbb{I}(K,V)/\mathbb{I}(K,V)^{2}\mathbb{I}_{0}(K,V)K^{\times }\simeq \operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V).\end{eqnarray}$$

Thus, there exists a subset $J_{0}\subset I$ of size ${\geqslant}|I|/d_{0}$ (if $I$ is infinite then this simply means that $J_{0}$ is also infinite) such that all $\unicode[STIX]{x1D706}^{(i)}$ for $i\in J_{0}$ have the same image in $\mathbb{I}(K,V)/\mathbb{I}(K,V)^{2}\mathbb{I}_{0}(K,V)K^{\times }$ . Now, fix $j_{0}\in J_{0}$ . For any $j\in J_{0}$ , we can write

$$\begin{eqnarray}\unicode[STIX]{x1D706}^{(j)}=\unicode[STIX]{x1D706}^{(j_{0})}(\unicode[STIX]{x1D6FC}^{(j)})^{2}\unicode[STIX]{x1D6FD}^{(j)}\unicode[STIX]{x1D6FF}^{(j)}\quad \text{with }\unicode[STIX]{x1D6FC}^{(j)}\in \mathbb{I}(K,V),~\unicode[STIX]{x1D6FD}^{(j)}\in \mathbb{I}_{0}(K,V),~\unicode[STIX]{x1D6FF}^{(j)}\in K^{\times },\end{eqnarray}$$

(assuming that the elements $\unicode[STIX]{x1D6FC}^{(j_{0})}$ , $\unicode[STIX]{x1D6FD}^{(j_{0})}$ and $\unicode[STIX]{x1D6FF}^{(j_{0})}$ are all trivial), and then set

$$\begin{eqnarray}\tilde{q}_{j}=\unicode[STIX]{x1D6FF}^{(j)}q_{j}\quad \text{and}\quad \tilde{\unicode[STIX]{x1D706}}^{(j)}=\unicode[STIX]{x1D706}^{(j)}(\unicode[STIX]{x1D6FF}^{(j)})^{-1}.\end{eqnarray}$$

We note that the forms $\tilde{q}_{j}$ for $j\in J_{0}$ remain pairwise nonsimilar, in particular, inequivalent. Furthermore, for any $v\in V$ we have $\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j}=\unicode[STIX]{x1D706}_{v}^{(j)}q_{j}$ , hence $[\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j}]\in W_{0}(K_{v})$ , and

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D706}}^{(j)}=\tilde{\unicode[STIX]{x1D706}}^{(j_{0})}(\unicode[STIX]{x1D6FC}^{(j)})^{2}\unicode[STIX]{x1D6FD}^{(j)}.\end{eqnarray}$$

Then for any $v\in V$ , the form

$$\begin{eqnarray}q(j,v):=\tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{0})}(\tilde{q}_{j}\bot (-\tilde{q}_{j_{0}}))=(\unicode[STIX]{x1D6FC}_{v}^{(j)})^{-2}(\unicode[STIX]{x1D6FD}^{(j)})^{-1}(\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j})\bot \tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{0})}(-\tilde{q}_{j_{0}})\end{eqnarray}$$

is equivalent to

$$\begin{eqnarray}(\unicode[STIX]{x1D6FD}_{v}^{(j)})^{-1}(\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j})\bot \tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{0})}(-\tilde{q}_{j_{0}}).\end{eqnarray}$$

As $\unicode[STIX]{x1D6FD}_{v}^{(j)}\in U(K_{v})$ , we see that $[q(j,v)]\in W_{0}(K_{v})\,\cap \,I(K_{v})$ . Now, invoking Lemma 3.3 with $d=1$ , we obtain that $\unicode[STIX]{x1D6FE}_{K_{v},1}([\tilde{q}_{j}]-[\tilde{q}_{j_{0}}])\in H^{1}(K_{v},\unicode[STIX]{x1D707}_{2})$ is unramified at $v$ . This being true for all $v\in V$ , we conclude that

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,1}([\tilde{q}_{j}]-[\tilde{q}_{j_{0}}])\in H^{1}(K,\unicode[STIX]{x1D707}_{2})_{V}.\end{eqnarray}$$

Then one can find a subset $J_{1}\subset J_{0}$ of size

$$\begin{eqnarray}{\geqslant}|J_{0}|/d_{1}\geqslant |I|/(d_{0}d_{1})\end{eqnarray}$$

(again, if $I$ is infinite this simply means that $J_{1}$ is also infinite) such that the elements $[\tilde{q}_{j}]-[\tilde{q}_{j_{0}}]\in I(K)$ for $j\in J_{1}$ have the same image under $\unicode[STIX]{x1D6FE}_{K,1}$ . Fix $j_{1}\in J_{1}$ . Then for any $j\in J_{1}$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,1}([\tilde{q}_{j}]-[\tilde{q}_{j_{1}}])=\unicode[STIX]{x1D6FE}_{K,1}(([\tilde{q}_{j}]-[\tilde{q}_{j_{0}}])-([\tilde{q}_{j_{1}}]-[\tilde{q}_{j_{0}}]))=0,\end{eqnarray}$$

implying that $[\tilde{q}_{j}]-[\tilde{q}_{j_{1}}]\in I(K)^{2}$ . Furthermore, we observe that

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D706}}^{(j)}=\tilde{\unicode[STIX]{x1D706}}^{(j_{1})}(\bar{\unicode[STIX]{x1D6FC}}^{(j)})^{2}\bar{\unicode[STIX]{x1D6FD}}^{(j)}\quad \text{with }\bar{\unicode[STIX]{x1D6FC}}^{(j)}\in \mathbb{I}(K,V),~\bar{\unicode[STIX]{x1D6FD}}^{(j)}\in \mathbb{I}_{0}(K,V),\end{eqnarray}$$

using which one shows that for each $v\in V$ the class of the form $\tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{1})}(\tilde{q}_{j}\bot (-\tilde{q}_{j_{1}}))$ lies in $W_{0}(K_{v})\,\cap \,I(K_{v})^{2}$ . Now, Lemma 3.3 with $d=2$ yields that

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,2}([\tilde{q}_{j}]-[\tilde{q}_{j_{1}}])\in H^{2}(K,\unicode[STIX]{x1D707}_{2})_{V}.\end{eqnarray}$$

Then there exists a subset $J_{2}\subset J_{1}$ of size ${\geqslant}|I|/(d_{0}d_{1}d_{2})$ such that the elements $[\tilde{q}_{j}]-[\tilde{q}_{j_{1}}]$ for $j\in J_{2}$ have the same image under $\unicode[STIX]{x1D6FE}_{K,2}$ . Consequently, fixing $j_{2}\in J_{2}$ , we will have $[\tilde{q}_{j}]-[\tilde{q}_{j_{2}}]\in I(K)^{3}$ for all $j\in J_{2}$ . Proceeding inductively, we construct a nested sequence of subsets

$$\begin{eqnarray}I\supset J_{0}\supset J_{1}\supset J_{2}\supset \cdots \supset J_{\ell }\end{eqnarray}$$

such that:

1. (a) $|J_{m}|\geqslant |I|/(d_{0}d_{1}\cdots d_{m})$ ; and

2. (b) fixing $j_{m}\in J_{m}$ , we will have $[\tilde{q}_{j}]-[\tilde{q}_{j_{m}}]\in I(K)^{m+1}$ for all $j\in J_{m}$ ,

for any $m=1,\ldots ,\ell$ . However, according to [Lam05, ch. X, Hauptsatz 5.1], the dimension of any positive-dimensional anisotropic form in $I(K)^{\ell +1}$ is ${\geqslant}2^{\ell +1}>2^{\log _{2}n+1}=2n$ . Thus, the fact that

$$\begin{eqnarray}[\tilde{q}_{j}]-[\tilde{q}_{j_{\ell }}]=[\tilde{q}_{j}\bot (-\tilde{q}_{j_{\ell }})]\in I(K)^{\ell +1}\end{eqnarray}$$

implies that the form $\tilde{q}_{j}\bot (-\tilde{q}_{j_{\ell }})$ is hyperbolic, and consequently $\tilde{q}_{j}$ and $\tilde{q}_{j_{\ell }}$ are equivalent. As $j\in J_{\ell }$ was arbitrary and the forms $\tilde{q}_{j}$ for $j\in J_{\ell }$ are pairwise inequivalent, we see that $J_{\ell }$ actually reduces to a single element. Then the inequality in part (a) yields the required estimation (1).◻

3.4 Another application of the method

The method developed to prove Theorem 2.1 can be used in various situations. Here we would like to indicate one application to the analysis of the global-to-local map in Galois cohomology.

Theorem 3.4. Let $K$ be a field equipped with a set $V$ of discrete valuations satisfying conditions $(\text{A})$ and $(\text{B})$ and let $n\geqslant 5$ be an integer. Assume that:

$(\bullet )$

for each $i=1,\ldots ,[\log _{2}n]+1=\ell$ , the kernel $\unicode[STIX]{x1D6FA}_{i}$ of the diagonal map

$$\begin{eqnarray}H^{i}(K,\unicode[STIX]{x1D707}_{2})\longrightarrow \mathop{\prod }_{v\in V}H^{i}(K_{v},\unicode[STIX]{x1D707}_{2})\end{eqnarray}$$

is finite of order $\unicode[STIX]{x1D714}_{i}$ .

Then for a nondegenerate $n$ -dimensional quadratic form $q$ over $K$ and the diagonal map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:H^{1}(K,\operatorname{SO}_{n}(q))\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},\operatorname{SO}_{n}(q)),\end{eqnarray}$$

the set $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D70B}(h))$ is finite of size ${\leqslant}\unicode[STIX]{x1D714}_{1}\cdots \unicode[STIX]{x1D714}_{\ell }$ for any $h\in H^{1}(K,\operatorname{SO}_{n}(q))$ . In particular, $\unicode[STIX]{x1D70B}$ is a proper map.

Proof. Let $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D70B}(h))=\{h_{i}\}\text{}_{i\in I}$ and let $q_{i}$ be the quadratic form obtained from $q$ by twisting using $h_{i}$ . It is well-known that the $K$ -equivalence classes of nondegenerate $n$ -dimensional quadratic forms over $K$ are in a natural bijective correspondence with the elements of $H^{1}(K,\text{O}_{n}(q))$ . As the map $H^{1}(K,\operatorname{SO}_{n}(q))\rightarrow H^{1}(K,\text{O}_{n}(q))$ is injective (cf. [PR94, § 6.6]), the forms $q_{i}$ for $i\in I$ are pairwise inequivalent. On the other hand, by our construction, the form $q_{i}$ for any $i\in I$ is equivalent over $K_{v}$ to the form $q_{0}$ obtained from $q$ by twisting using $h$ , for any $v\in V$ . Thus, for any $i,j\in I$ , the class $[q_{i}]-[q_{j}]$ is trivial in $W(K_{v})$ for any $v\in V$ . Thus, using the compatibility of the map $\unicode[STIX]{x1D6FE}_{F,d}$ with base change, we conclude that if $[q_{i}]-[q_{j}]\in I(K)^{d}$ , then $\unicode[STIX]{x1D6FE}_{K,d}([q_{i}]-[q_{j}])\in \unicode[STIX]{x1D6FA}_{d}$ .

We now proceed as in the proof of Theorem 2.1. Fix $j_{0}\in I$ . Then there exists a subset $J_{1}\subset I$ of size ${\geqslant}|I|/\unicode[STIX]{x1D714}_{1}$ such that all elements $[q_{j}]-[q_{j_{0}}]\in I(K)$ for $j\in J_{1}$ have the same image under $\unicode[STIX]{x1D6FE}_{K,1}$ . Fix $j_{1}\in J_{1}$ . Then for any $j\in J_{1}$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,1}([q_{j}]-[q_{j_{1}}])=\unicode[STIX]{x1D6FE}_{K,1}(([q_{j}]-[q_{j_{0}}])-([q_{j_{1}}]-[q_{j_{0}}]))=0,\end{eqnarray}$$

which means that $[q_{j}]-[q_{j_{1}}]\in I(K)^{2}$ . Inductively, we construct a sequence of subsets $I\supset J_{1}\supset J_{2}\supset \cdots \supset J_{\ell }$ such that:

$(\text{a}^{\prime })$

$|J_{m}|\geqslant |I|/\unicode[STIX]{x1D714}_{1}\cdots \unicode[STIX]{x1D714}_{m}$ ;

$(\text{b}^{\prime })$

fixing $j_{m}\in J_{m}$ , we will have $[q_{j}]-[q_{j_{m}}]\in I(K)^{m+1}$ for all $j\in J_{m}$ ,

for any $m=1,\ldots ,\ell$ . As in the proof of Theorem 2.1, we see that the fact that $[q_{j}]\,-\,[q_{j_{\ell }}]\in I(K)^{\ell +1}$ implies that $q_{j}$ and $q_{j_{\ell }}$ are actually equivalent. This means that $J_{\ell }$ reduces to a single element, and then the inequality in $(\text{a}^{\prime })$ yields the required estimation.◻

We observe that for any $i$ , we have the inclusion $\unicode[STIX]{x1D6FA}_{i}\subset H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ in the above notation. In particular, the finiteness of $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ implies that of $\unicode[STIX]{x1D6FA}_{i}$ , which enables us to apply Theorem 3.4.

4 Proof of Theorem 1.1

4.1 The proof

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be any divisorial set of places of $K$ associated with a (normal) model $\mathfrak{X}$ of finite type on which $2$ is invertible (thus, $\operatorname{char}K^{(v)}\neq 2$ for all $v\in V$ ). According to Theorem 2.1, it is enough to establish the finiteness of the quotient $\operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V)$ and of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for all $i\geqslant 1$ . However, it easily follows from the definitions that the group $\operatorname{Pic}(V)$ coincides with the usual Picard group $\operatorname{Pic}(\mathfrak{X})$ of the scheme $\mathfrak{X}$ . Thus, the finiteness of $\operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V)$ is a consequence of the following well-known statement.

Proposition 4.1. Let $\mathfrak{X}$ be an irreducible normal scheme of finite type over $\mathbb{Z}$ or a finite field and let $n\geqslant 2$ be an integer that is invertible on $\mathfrak{X}$ . Then the quotient $\operatorname{Pic}(\mathfrak{X})/n\cdot \operatorname{Pic}(\mathfrak{X})$ and the $n$ -torsion $_{n\!}\operatorname{Pic}(\mathfrak{X})$ are finite groups.

Sketch of proof.

As $n$ is invertible on $\mathfrak{X}$ , we have the following Kummer sequence of étale sheaves on $\mathfrak{X}$

$$\begin{eqnarray}1\rightarrow \unicode[STIX]{x1D707}_{n,\mathfrak{X}}\longrightarrow \mathbb{G}_{m,\mathfrak{X}}\stackrel{[n]}{\longrightarrow }\mathbb{G}_{m,\mathfrak{X}}\rightarrow 1,\end{eqnarray}$$

which gives rise to the long exact sequence

$$\begin{eqnarray}H^{1}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})\longrightarrow H^{1}(\mathfrak{X},\mathbb{G}_{m,\mathfrak{X}})\stackrel{[n]}{\longrightarrow }H^{1}(\mathfrak{X},\mathbb{G}_{m,\mathfrak{X}})\longrightarrow H^{2}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})\end{eqnarray}$$

in étale cohomology. As $\operatorname{Pic}(\mathfrak{X})=H^{1}(\mathfrak{X},\mathbb{G}_{m,\mathfrak{X}})$ (cf. [Mil80, ch. III, Proposition 4.9]), we have a surjection $H^{1}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}}){\twoheadrightarrow}_{n\!}\operatorname{Pic}(\mathfrak{X})$ and an injection $\operatorname{Pic}(\mathfrak{X})/n\cdot \operatorname{Pic}(\mathfrak{X}){\hookrightarrow}H^{2}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})$ . Thus, it is enough to prove the finiteness of $H^{i}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})$ for $i=1,2$ . In fact, these groups are finite for all $i$ . For $\mathfrak{X}$ of finite type over a finite field, this is a consequence of [SGA4½, Théorème 1.1 in ‘Théorèmes de finitude’] and the Leray spectral sequence [Mil80, ch. III, Theorem 1.18] in general and is [Mil80, ch. VI, Corollary 5.5] for $\mathfrak{X}$ smooth. For $\mathfrak{X}$ of finite type over $\mathbb{Z}$ , this is [CRR16a, Theorem 10.2].◻

It should be noted that Kahn [Kah06] in fact showed that the group $\operatorname{Pic}(\mathfrak{X})$ is finitely generated.

Combining the finiteness of $_{2\!}\operatorname{Pic}(V)$ with the fact that the group of $V$ -units

$$\begin{eqnarray}U(V)=\{a\in K^{\times }\mid v(a)=0\text{ for all }v\in V\}\end{eqnarray}$$

is finitely generated (cf. [Sam66]) and applying [CRR16a, Proposition 5.1(a)], we obtain that $H^{1}(K,\unicode[STIX]{x1D707}_{2})_{V}$ is finite. The finiteness of $H^{2}(K,\unicode[STIX]{x1D707}_{2})_{V}=_{2\!}\operatorname{Br}(K)_{V}$ was established in [CRR16a, § 10]. The finiteness of $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}$ is a new result (see Corollary 6.2 of the more general Theorem 6.1) whose proof requires separate arguments in the characteristic zero and positive characteristic cases. The proof for the characteristic zero case that is given in § 6 was suggested by one of the referees; it uses several powerful results, first and foremost, those of Kato [Kat86] on cohomological Hasse principles. We have included our original proof of Theorem 6.1 in Appendix A. It uses considerably less input; in particular, it does not rely on Kato’s local–global principle but is based instead on a modification of Jannsen’s [Jan89–90] proof of the latter. The reason we decided to keep this argument in the paper is that it appears to be more amenable to generalizations in the spirit of Jannsen’s proof [Jan16] of Kato’s local–global principle for higher-dimensional varieties that generalized his original argument [Jan89–90]. The finiteness of $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for two-dimensional global fields of positive characteristic will be established in § 7 (we note that this result is implicitly contained in [Rap18, Proposition 4.4 and Remark 4.5]).

As the cohomological dimension of a two-dimensional global field of positive characteristic is ${\leqslant}3$ (cf. [Ser97, ch. II, 4.2]), the groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})$ vanish for all $i\geqslant 4$ . We will now show that in characteristic zero, the finiteness of $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 4$ easily follows from the results of Poitou–Tate (cf. [Ser97, ch. II, 6.3]), which will complete the proof of Theorem 1.1.

Proposition 4.2. Let $C$ be a smooth (but not necessarily projective) geometrically integral curve over a number field $k$ , and $K=k(C)$ . Then for $i\geqslant 4$ , the groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V_{0}}$ , where $V_{0}$ is the set of geometric places of $K$ associated with the closed points of $C$ , are finite.

To apply the proposition in our situation, we observe that given a divisorial set of places $V$ of a two-dimensional global field $K$ of characteristic zero, there exists a smooth geometrically connected curve $C$ defined over a number field $k$ such that $K=k(C)$ and for the set of geometric places $V_{0}$ of $K$ associated with the closed points of $C$ , we have the inclusion

$$\begin{eqnarray}H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}\subset H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V_{0}}.\end{eqnarray}$$

For the proof of the proposition, we first need to review the exact sequence for the étale cohomology of a curve, which will also be used in Appendix A.

4.2 The fundamental exact sequence

Let $C$ be a geometrically integral smooth affine curve over an arbitrary field $k$ and let $n\geqslant 2$ be an integer prime to $\operatorname{char}k$ . Then for any integer $d$ , we consider the Hochschild–Serre spectral sequence in étale cohomology

$$\begin{eqnarray}E_{2}^{p,q}=H^{p}(k,H^{q}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d}))\Rightarrow H^{p+q}(C,\unicode[STIX]{x1D707}_{n}^{\otimes d}).\end{eqnarray}$$

As $C$ is affine, we have $H^{p}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d})=0$ for $p\geqslant 2$ (cf. [Stacks19, Lemma 65.3], [Mil80, ch. V, § 2]). Thus, for each $\ell \geqslant 1$ , the spectral sequence yields the following short exact sequence:

(2) $$\begin{eqnarray}H^{\ell }(k,\unicode[STIX]{x1D707}_{n}^{\otimes d})\stackrel{\unicode[STIX]{x1D704}_{k,n}^{d,\ell }}{\longrightarrow }H^{\ell }(C,\unicode[STIX]{x1D707}_{n}^{\otimes d})\stackrel{\unicode[STIX]{x1D714}_{k,n}^{d,\ell }}{\longrightarrow }H^{\ell -1}(k,H^{1}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d})).\end{eqnarray}$$

Furthermore, let $V_{0}$ denote the set of places of $K=k(C)$ associated with the closed points of $C$ . Then, because $n$ is prime to $\operatorname{char}k$ , it follows from the Bloch–Ogus spectral sequence (see [BO74, CHK97, JSS14]) that for each $\ell \geqslant 1$ , the natural map $H^{\ell }(C,\unicode[STIX]{x1D707}_{n}^{\otimes d})\rightarrow H^{\ell }(K,\unicode[STIX]{x1D707}_{n}^{\otimes d})_{V_{0}}$ is surjective. (This is a consequence of the fact that the $E_{2}$ -terms in the Bloch–Ogus spectral sequence satisfy $E_{2}^{p,q}=0$ for all $p>\dim C=1$ and all $q$ , whereas $E_{2}^{0,q}=H^{q}(K,\unicode[STIX]{x1D707}_{n}^{\otimes d})_{V_{0}}$ .)

Let

$$\begin{eqnarray}M(n,d)=H^{1}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d}).\end{eqnarray}$$

Equivalently,

(3) $$\begin{eqnarray}M(n,d)=\operatorname{Hom}(\unicode[STIX]{x1D70B}_{1}(C\otimes _{k}\bar{k},\bar{\unicode[STIX]{x1D702}}),\unicode[STIX]{x1D707}_{n}^{\otimes d})\quad \text{where }\bar{\unicode[STIX]{x1D702}}=\operatorname{Spec}\bar{K}.\end{eqnarray}$$

It follows that we have the following equality for the Tate twist

$$\begin{eqnarray}M(n,d)(\ell )=M(n,d+\ell ).\end{eqnarray}$$

4.3 Proof of Proposition 4.2

We may assume that $C$ is affine, and it is enough to show that $H^{i}(C,\unicode[STIX]{x1D707}_{n}^{\otimes d})$ is finite for all $i\geqslant 4$ . According to Poitou–Tate, for a number field $k$ and any finite Galois module $M$ , the natural homomorphism

$$\begin{eqnarray}H^{i}(k,M)\longrightarrow \mathop{\prod }_{v\in V_{\infty }^{k}}H^{i}(k_{v},M),\end{eqnarray}$$

where $V_{\infty }^{k}$ is the set of all real places of $k$ , is an isomorphism for any $i\geqslant 3$ (see [Ser97, ch. II, 6.3, Theorem B]); in particular, the group $H^{i}(k,M)$ is finite. As $H^{1}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d})$ is finite by [SGA4, Exposé XVI, Théorème 5.2], it follows that the left-most and the right-most terms in the exact sequence (2) are finite. Thus, the middle term is also finite for all $i\geqslant 4$ , as required.◻

5 Proof of Theorems 1.2 and 1.3.

5.1 The finiteness of genus

Theorem 1.2 will be derived from the following result.

Theorem 5.1. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $G=\operatorname{Spin}_{n}(q)$ , where $q$ is a nondegenerate quadratic form over $K$ of dimension $n\geqslant 5$ . Then the number of $K$ -isomorphism classes of the spinor groups $G^{\prime }=\operatorname{Spin}_{n}(q^{\prime })$ of nondegenerate $n$ -dimensional quadratic forms $q^{\prime }$ over $K$ that have the same isomorphism classes of maximal $K$ -tori as $G$ , is finite.

To prove Theorem 5.1, we need the following general statement.

Theorem 5.2 [CRR16b, Theorem 5].

Let $G$ be an absolutely almost simple simply connected algebraic group over a field $K$ and let $v$ be a discrete valuation of $K$ . Assume that the residue field $K^{(v)}$ is finitely generated and that $G$ has good reduction at $v$ . Then any $G^{\prime }\in \mathbf{gen}_{K}(G)$ also has good reduction at $v$ . Moreover, the reduction ${\text{}\underline{G}^{\prime }}^{(v)}$ lies in the genus $\mathbf{gen}_{K^{(v)}}(\text{}\underline{G}^{(v)})$ of the reduction $\text{}\underline{G}^{(v)}$ .

Let $V$ be a divisorial set of places of our two-dimensional global field $K$ of characteristic $\neq \,2$ . Using the fact that $V$ satisfies condition (A), it is not difficult to see that the set $v\in V$ where $G=\operatorname{Spin}_{n}(q)$ does not have good reduction, is finite. Picking a divisorial set of places in the complement of this finite set, we may assume without loss of generality that $G$ has good reduction at all $v\in V$ . Then according to Theorem 5.2, every $G^{\prime }\in \mathbf{gen}_{K}(G)$ has good reduction at all $v\in V$ . Thus, Theorem 1.1 yields the finiteness of the number of $K$ -isomorphism classes of the spinor groups $G^{\prime }=\operatorname{Spin}_{n}(q^{\prime })$ lying in $\mathbf{gen}_{K}(G)$ , which is precisely the assertion of Theorem 5.1.

Now, Theorem 1.2 for quadratic forms of odd dimension immediately follows from Theorem 5.1. To consider the case of isotropic quadratic forms $q$ of even dimension, we need to combine the latter with part (a) of the following proposition, which implies that every group in $\mathbf{gen}_{K}(G)$ is of the form $\operatorname{Spin}_{n}(q^{\prime })$ .

Proposition 5.3. Let $K$ be an infinite finitely generated field of characteristic $\neq 2$ , let $G=\operatorname{Spin}_{n}(q)$ where $q$ is a nondegenerate quadratic form over $K$ of even dimension $n\geqslant 10$ and let $H=\widetilde{\operatorname{SU}}_{m}(D,h)$ (universal cover) where $D$ is a central division algebra over $K$ of degree $d\geqslant 1$ with an orthogonal involution $\unicode[STIX]{x1D70F}$ and $h$ is a nondegenerate $m$ -dimensional $\unicode[STIX]{x1D70F}$ -Hermitian form.

1. (a) If $d>1$ and $q$ is $K$ -isotropic, then $H\notin \mathbf{gen}_{K}(G)$ .

2. (b) If $d>2$ , then $H\notin \mathbf{gen}_{K}(G)$ for any $q$ .

(Note that $d$ is a power of $2$ .)

We will first establish one general fact that involves the notion of generic tori, which we now recall for the reader’s convenience. Let $\mathscr{G}$ be a semi-simple algebraic group over a field $\mathscr{K}$ . Fix a maximal $\mathscr{K}$ -torus $\mathscr{T}$ of $\mathscr{G}$ , and let $\unicode[STIX]{x1D6F7}(\mathscr{G},\mathscr{T})$ and $W(\mathscr{G},\mathscr{T})$ denote the corresponding root system and the Weyl group. The natural action of the absolute Galois group $\operatorname{Gal}(\mathscr{K}^{\text{sep}}/\mathscr{K})$ , where $\mathscr{K}^{\text{sep}}$ is a fixed separable closure of $\mathscr{K}$ , on the character group $X(\mathscr{T})$ of $\mathscr{T}$ gives rise to a group homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\mathscr{T}}:\operatorname{Gal}(\mathscr{K}^{\text{sep}}/\mathscr{K})\rightarrow \operatorname{Aut}(\unicode[STIX]{x1D6F7}(\mathscr{G},\mathscr{T}))\end{eqnarray}$$

that factors through the Galois group $\operatorname{Gal}(\mathscr{K}_{\mathscr{T}}/\mathscr{K})$ of the minimal splitting field of $\mathscr{T}$ in $\mathscr{K}^{\text{sep}}$ inducing an injective homomorphism $\bar{\unicode[STIX]{x1D703}}_{\mathscr{T}}:\operatorname{Gal}(\mathscr{K}_{\mathscr{T}}/\mathscr{K})\rightarrow \operatorname{Aut}(\unicode[STIX]{x1D6F7}(\mathscr{G},\mathscr{T}))$ . We say that $\mathscr{T}$ is generic over $\mathscr{K}$ if $\unicode[STIX]{x1D703}_{\mathscr{T}}(\operatorname{Gal}(\mathscr{K}^{\text{sep}}/\mathscr{K}))\supset W(\mathscr{T},\mathscr{G})$ .

Proposition 5.4. Let $\mathscr{D}$ be a central division algebra of degree $d>1$ over a field $\mathscr{K}$ of characteristic $\neq 2$ with an orthogonal involution $\unicode[STIX]{x1D70F}$ , and let $h$ be a nondegenerate $m$ -dimensional $\unicode[STIX]{x1D70F}$ -Hermitian form. Then $\mathscr{H}=\operatorname{SU}_{m}(\mathscr{D},h)$ does not contain a maximal $\mathscr{K}$ -torus of the form $\mathscr{T}=\mathscr{T}_{1}\mathscr{T}_{2}$ (almost direct product over $\mathscr{K}$ ) where $\mathscr{T}_{1}=\mathbb{G}_{m}$ and $\mathscr{T}_{2}$ is isomorphic to a generic torus in a $\mathscr{K}$ -group of type $\mathsf{D}_{n/2-1}$ with $n=dm$ .

Proof. We begin with the following lemma.

Lemma 5.5. Notations as in Proposition 5.4, let $r$ denote the Witt index of $h$ . Then for any nontrivial $\mathscr{K}$ -split torus $\mathscr{S}$ of $\mathscr{H}$ , the centralizer $C_{\mathscr{H}}(\mathscr{S})$ is an almost direct product $\mathscr{S}_{0}\mathscr{H}_{1}\cdots \mathscr{H}_{s}$ over $\mathscr{K}$ , where $\mathscr{S}_{0}$ is the central torus of $C_{\mathscr{H}}(\mathscr{S})$ and each $\mathscr{H}_{i}$ is a semi-simple $\mathscr{K}$ -group either of inner type $\mathsf{A}_{\ell -1}$ with $\ell \leqslant dr$ or of type $\mathsf{D}_{\ell }$ with $\ell \leqslant d(m-2)/2$ (and not a triality form when $\ell =4$ ).

Proof. Let $\unicode[STIX]{x1D712}_{0}=1,\unicode[STIX]{x1D712}_{1},-\unicode[STIX]{x1D712}_{1},\ldots ,\unicode[STIX]{x1D712}_{t},-\unicode[STIX]{x1D712}_{t}$ be all the weights for the action of $\mathscr{S}(\mathscr{K})$ on $W=\mathscr{D}^{m}$ , and denote by $W_{\unicode[STIX]{x1D712}}$ the weight subspace corresponding to the character $\unicode[STIX]{x1D712}$ of $\mathscr{S}$ . Set

$$\begin{eqnarray}W_{0}=W_{\unicode[STIX]{x1D712}_{0}}\quad \text{and}\quad W_{i}=W_{\unicode[STIX]{x1D712}_{i}}\,\oplus \,W_{-\unicode[STIX]{x1D712}_{i}}\quad \text{for }i=1,\ldots ,t.\end{eqnarray}$$

Then $W$ has the following orthogonal decomposition

$$\begin{eqnarray}W=W_{0}\bot W_{1}\bot \cdots \bot W_{t}.\end{eqnarray}$$

Let

$$\begin{eqnarray}m_{0}=\dim _{\mathscr{D}}W_{0}\quad \text{and}\quad m_{i}=\dim _{\mathscr{D}}W_{\unicode[STIX]{x1D712}_{i}}=\dim _{\mathscr{D}}W_{-\unicode[STIX]{x1D712}_{i}}\quad \text{for }i=1,\ldots ,t,\end{eqnarray}$$

so that $m_{i}\leqslant r\leqslant m/2$ and $\dim _{\mathscr{D}}W_{i}=2m_{i}$ . Then

$$\begin{eqnarray}C_{\mathscr{H}}(\mathscr{S})=\mathscr{F}_{0}\times \mathscr{F}_{1}\times \cdots \times \mathscr{F}_{t},\end{eqnarray}$$

where $\mathscr{F}_{0}=\operatorname{SU}_{m_{0}}(\mathscr{D},h|W_{0})$ and $\mathscr{F}_{i}$ for $i>0$ is the stabilizer of the subspaces $W_{\unicode[STIX]{x1D712}_{i}}$ and $W_{-\unicode[STIX]{x1D712}_{i}}$ in $\operatorname{SU}_{2m_{i}}(\mathscr{D},h|W_{i})$ . As $m_{i}\geqslant 1$ and $t>0$ as $\mathscr{S}$ is nontrivial, we see that $m_{0}\leqslant m-2$ , and therefore $\mathscr{F}_{0}$ is of type $\mathsf{D}_{\ell }$ with

$$\begin{eqnarray}\ell =dm_{0}/2\leqslant d(m-2)/2\end{eqnarray}$$

(obviously, not a triality form). Furthermore, for $i>0$ , the group $\mathscr{F}_{i}$ is isomorphic to $\operatorname{GL}(W_{\unicode[STIX]{x1D712}_{i}})$ (or $\operatorname{GL}(W_{-\unicode[STIX]{x1D712}_{i}})$ ). Thus, its semi-simple part is $\operatorname{SL}(W_{\unicode[STIX]{x1D712}_{i}})$ , which is an inner form of type $\mathsf{A}_{\ell -1}$ with

$$\begin{eqnarray}\ell =dm_{i}\leqslant dr.\Box\end{eqnarray}$$

We will now complete the proof of Proposition 5.4. Assume the contrary, i.e.  $\mathscr{H}$ contains a maximal $\mathscr{K}$ -torus $\mathscr{T}=\mathscr{T}_{1}\mathscr{T}_{2}$ as in the proposition. Applying the lemma to $\mathscr{S}=\mathscr{T}_{1}$ , we obtain that

$$\begin{eqnarray}\mathscr{T}=\mathscr{T}_{1}\mathscr{T}_{2}\subset \mathscr{S}_{0}\mathscr{H}_{1}\cdots \mathscr{H}_{s},\end{eqnarray}$$

where $\mathscr{S}_{0}$ is the central torus of $C_{\mathscr{H}}(\mathscr{T}_{1})$ and the $\mathscr{H}_{i}$ are semisimple $\mathscr{K}$ -groups of the types specified in the lemma. Clearly, $\mathscr{T}\not \subset \mathscr{S}_{0}$ as otherwise $\mathscr{H}$ would be quasi-split over $\mathscr{K}$ , which it is not because $d>1$ . As $\mathscr{T}_{1}\subset \mathscr{S}_{0}$ , we conclude that $\mathscr{T}_{2}\not \subset \mathscr{S}_{0}$ . It follows that there exists $i\in \{1,\ldots ,s\}$ such that

$$\begin{eqnarray}\mathscr{T}_{2}\not \subset \hat{\mathscr{H}}_{i}:=\mathscr{S}_{0}\mathscr{H}_{1}\cdots \mathscr{H}_{i-1}\mathscr{H}_{i+1}\cdots \mathscr{H}_{s}.\end{eqnarray}$$

This means that for the quotient map $\unicode[STIX]{x1D70B}_{i}:C_{\mathscr{H}}(\mathscr{T}_{1})\rightarrow C_{\mathscr{H}}(\mathscr{T}_{1})/\hat{\mathscr{H}}_{i}=:\bar{\mathscr{H}}_{i}$ , we have $\unicode[STIX]{x1D70B}_{i}(\mathscr{T}_{2})\neq \{e\}$ . Being generic over $\mathscr{K}$ , the torus $\mathscr{T}_{2}$ is $\mathscr{K}$ -irreducible, i.e. contains no proper $\mathscr{K}$ -defined subtori. Thus, the restriction of $\unicode[STIX]{x1D70B}_{i}$ gives an isogeny of $\mathscr{T}_{2}$ onto a $\mathscr{K}$ -subtorus $\bar{\mathscr{T}}_{i}$ of $\bar{\mathscr{H}}_{i}$ . Moreover, because $\dim \mathscr{T}_{2}=\operatorname{rk}\mathscr{H}-1$ and $\operatorname{rk}\bar{\mathscr{H}}_{i}\leqslant \operatorname{rk}\mathscr{H}-1$ , we see that $\bar{\mathscr{T}}_{i}$ is actually a maximal $\mathscr{K}$ -torus of $\bar{\mathscr{H}}_{i}$ . Let $\mathscr{L}$ be the common minimal splitting field of $\mathscr{T}_{2}$ and $\bar{\mathscr{T}}_{i}$ . As $\mathscr{T}_{2}$ is $\mathscr{K}$ -generic in a group of type $\mathsf{D}_{n/2-1}$ where $n=dm$ , we have

(4) $$\begin{eqnarray}|\text{Gal}(\mathscr{L}/\mathscr{K})|\geqslant 2^{n/2-2}\cdot (n/2-1)!.\end{eqnarray}$$

Now, assume that $\mathscr{H}_{i}$ is an inner form of type $\mathsf{A}_{\ell -1}$ with $\ell \leqslant dr$ . As $r\leqslant m/2$ , we obtain

$$\begin{eqnarray}|\text{Gal}(\mathscr{L}/\mathscr{K})|\leqslant |W(\bar{\mathscr{H}}_{i},\bar{\mathscr{T}}_{i})|=\ell !\leqslant (n/2)!.\end{eqnarray}$$

Comparing with (4), we obtain the inequality

$$\begin{eqnarray}2^{n/2-2}\leqslant n/2,\end{eqnarray}$$

which fails for all $n\geqslant 10$ .

Next, let $\mathscr{H}_{i}$ be of type $\mathsf{D}_{\ell }$ with $\ell \leqslant d(m-2)/2\leqslant n/2-2$ as $d\geqslant 2$ . As it is not a triality form when $\ell =4$ , we have

$$\begin{eqnarray}|\text{Gal}(\mathscr{L}/\mathscr{K})|\leqslant 2\cdot |W(\bar{\mathscr{H}}_{i},\bar{\mathscr{T}}_{i})|=2^{\ell }\cdot \ell !\leqslant 2^{n/2-2}\cdot (n/2-2)!.\end{eqnarray}$$

Thus, comparing with (4), we obtain the inequality

$$\begin{eqnarray}2^{n/2-2}\cdot (n/2-1)!\leqslant 2^{n/2-2}\cdot (n/2-2)!\end{eqnarray}$$

which never holds for $n\geqslant 6$ .

Thus, the assumption that $\mathscr{H}$ contains a maximal $\mathscr{K}$ -torus $\mathscr{T}=\mathscr{T}_{1}\mathscr{T}_{2}$ leads to a contradiction in both cases, proving our claim.◻

Proof of Proposition 5.3.

(a) Write $q=q_{1}\bot q_{2}$ , where $q_{1}$ is a two-dimensional hyperbolic form. Then $T_{1}=\operatorname{Spin}_{2}(q_{1})$ is a one-dimensional split torus $\mathbb{G}_{m}$ . As $K$ is infinite and finitely generated, we can find a maximal $K$ -torus $T_{2}$ of $\operatorname{Spin}_{n-2}(q_{2})$ that is generic over $K$ (cf. [PR03, PR17]). Consider the maximal $K$ -torus $T=T_{1}T_{2}$ of $G$ . Then it follows from Proposition 5.4 that $T$ is not isomorphic to a maximal $K$ -torus of $H$ , hence $H\notin \mathbf{gen}_{K}(G)$ .

(b) Again, write $q=q_{1}\bot q_{2}$ where $q_{1}=\langle a_{1},a_{2}\rangle$ is a binary form. Set $L=K(\sqrt{-a_{1}a_{2}})$ . If $L=K$ , then our claim follows from part (a). Thus, we may assume that $L$ is a quadratic extension of $K$ . Then $T_{1}=\operatorname{Spin}_{2}(q_{1})$ is the one-dimensional norm torus $\text{R}_{L/K}^{(1)}(\mathbb{G}_{m})$ . Furthermore, let $T_{2}$ be a maximal $K$ -torus of $\operatorname{Spin}_{n-2}(q_{2})$ that is generic over $L$ (cf. [PR03, PR17]). Then the maximal $K$ -torus $T=T_{1}T_{2}$ is not isomorphic to a maximal $K$ -torus of $H$ . Indeed, assume the contrary, i.e.  $T\subset H$ . Note that over $L$ , the torus $T$ is an almost direct product $T_{1}^{\prime }T_{2}^{\prime }$ where $T_{1}^{\prime }$ is a one-dimensional split torus and $T_{2}^{\prime }$ is isomorphic to a generic torus in a group of type $\mathsf{D}_{n/2-1}$ . At the same time, since $d>2$ , the group $H$ is $L$ -isomorphic to $\widetilde{\text{SU}}_{m^{\prime }}(\mathscr{D},h^{\prime })$ where $\mathscr{D}$ is a central division algebra over $\mathscr{K}=L$ of degree $d^{\prime }>1$ and $h^{\prime }$ is an $m^{\prime }$ -dimensional Hermitian form (note that $d^{\prime }m^{\prime }=dm$ ). However, this contradicts Proposition 5.4. Thus, again $H\notin \mathbf{gen}_{K}(G)$ .◻

Remark 5.6. It follows from Proposition 5.3(b) that to complete the proof of Theorem 1.2 for all even-dimensional form, it would be sufficient to demonstrate that $\mathbf{gen}_{K}(G)$ cannot contain a group of the form $H=\widetilde{\operatorname{SU}}_{m}(D,h)$ where $D$ is a central quaternion division algebra over $K$ . The proof of Proposition 5.3(b) given above shows that this boils down to proving that for any quaternion division algebra $D$ over $K$ , the given quadratic form $q$ contains a binary subform $q_{1}=\langle a_{1},a_{2}\rangle$ such that the field $K(\sqrt{-a_{1}a_{2}})$ does not split $D$ , which seems very likely at least when the dimension of $q$ is sufficiently large (note that those $d\in K^{\times }\setminus {K^{\times }}^{2}$ for which $K(\sqrt{d}){\hookrightarrow}D$ must be represented by the ternary quadratic form associated with $D$ ).

5.2 Properness of the global-to-local map

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places. As we already mentioned, for each $i\geqslant 1$ , the kernel

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{i}=\ker \biggl(H^{i}(K,\unicode[STIX]{x1D707}_{2})\longrightarrow \mathop{\prod }_{v\in V}H^{i}(K_{v},\unicode[STIX]{x1D707}_{2})\biggr)\end{eqnarray}$$

is contained in $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ , hence is finite ² (cf. § 4.1). Applying now Theorem 3.4, we conclude that for $q$ a nondegenerate quadratic form in $n\geqslant 5$ variables and $G=\operatorname{SO}_{n}(q)$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G)\end{eqnarray}$$

is proper, which is precisely the assertion of Theorem 1.3.

To put this result in a more general context, we recall the following general observation [CRR15, Rap14]: Let $G$ be an absolutely almost simple simply connected algebraic group over a field $K$ , and let $V$ be a set of discrete valuations of $K$ that satisfies condition (A). Assume that for any finite subset $S\subset V$ , the set of $K$ -isomorphism classes of inner $K$ -forms of $G$ having good reduction at all $v\in V\setminus S$ is finite. Then for the corresponding adjoint group $\overline{G}$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\overline{G},V}:H^{1}(K,\overline{G})\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},\overline{G})\end{eqnarray}$$

is proper. This means, in particular, that for $n$ odd, Theorem 1.3 directly follows from Theorem 1.1. On the other hand, for $n$ even, the latter implies neither Theorem 1.3 nor the corresponding fact for the simply connected and adjoint groups $\widetilde{G}=\operatorname{Spin}_{n}(q)$ and $\overline{G}=\operatorname{PSO}_{n}(q)$ (thus, in this case Theorem 1.3 is an independent result). Later, we will see that the analogues of Theorem 1.3 are valid for special unitary groups of Hermitian forms over a quadratic extension of the base (two-dimensional global) field and for the group of type $\mathsf{G}_{2}$ over such a field (note that all these groups are simply connected); see Theorems 8.4 and 9.1(iii). We close this subsection with some additional results on the properness of the global-to-local map for simply connected groups.

Theorem 5.7. Let $K$ be a two-dimensional global field, let $V$ be its divisorial set of places and let $m$ be a square-free integer prime to $\operatorname{char}K$ . Then for a central simple $K$ -algebra $A$ of degree $m$ and $G=\operatorname{SL}_{1,A}$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G)\end{eqnarray}$$

is proper.

Proof. We may assume without loss of generality that $\operatorname{char}K^{(v)}$ is prime to $m$ for all $v\in V$ . It is well-known (cf. [KMRT98, 29.4]) that for any extension $F/K$ there is a bijection

$$\begin{eqnarray}\unicode[STIX]{x1D708}_{F}:H^{1}(F,G)\rightarrow F^{\times }/\operatorname{Nrd}_{A_{F}/F}(A_{F}^{\times })\quad \text{where }A_{F}=A\,\otimes _{K}\,F,\end{eqnarray}$$

and this bijection is functorial in $F$ . Thus, letting $A_{v}=A_{K_{v}}$ for $v\in V$ , we need to show that the map

$$\begin{eqnarray}\unicode[STIX]{x1D704}:K^{\times }/\operatorname{Nrd}_{A/K}(A^{\times })\rightarrow \mathop{\prod }_{v\in V}K_{v}^{\times }/\operatorname{Nrd}_{A_{v}/K_{v}}(A_{v}^{\times })\end{eqnarray}$$

has finite kernel. For any field extension $F/K$ , we have a homomorphism

$$\begin{eqnarray}F^{\times }\rightarrow H^{3}(F,\unicode[STIX]{x1D707}_{m}^{\otimes 2}),\quad a\mapsto [A_{F}]\cup \unicode[STIX]{x1D712}_{m,a},\end{eqnarray}$$

where $[A_{F}]$ is the class of $A_{F}$ in $_{m}\operatorname{Br}(F)=H^{2}(F,\unicode[STIX]{x1D707}_{m})$ . As $m$ is square-free, according to [MS82, Theorem 12.2], the kernel of this map coincides with $\operatorname{Nrd}_{A_{F}/F}(A_{F}^{\times })$ , so we have an injective homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{F}:F^{\times }/\operatorname{Nrd}_{A_{F}/F}(A_{F}^{\times })\rightarrow H^{3}(F,\unicode[STIX]{x1D707}_{m}^{\otimes 2}).\end{eqnarray}$$

Then we have the following commutative diagram

where $\unicode[STIX]{x1D6E5}_{V}=\prod _{v\in V}\unicode[STIX]{x1D6FF}_{K_{v}}$ . The fact that the unramified cohomology group $H^{3}(K,\unicode[STIX]{x1D707}_{m}^{\otimes 2})_{V}$ is finite (Theorem 6.1) implies that $\ker \unicode[STIX]{x1D702}$ is finite. Then using the injectivity of $\unicode[STIX]{x1D6FF}_{K}$ , we conclude that $\ker \unicode[STIX]{x1D704}$ is finite as well, as needed.◻

We now turn to spinor groups. Let $q$ be a nondegenerate quadratic form of dimension at least three over a field $K$ of characteristic $\neq 2$ and let $\unicode[STIX]{x1D70B}:\operatorname{Spin}_{n}(q)\rightarrow \operatorname{SO}_{n}(q)$ be the canonical $K$ -isogeny. For a field extension $F/K$ , we let

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{F}:H^{1}(F,\operatorname{Spin}_{n}(q))\rightarrow H^{1}(F,\operatorname{SO}_{n}(q))\end{eqnarray}$$

be the map induced by $\unicode[STIX]{x1D70B}$ . Assume now that $K$ is equipped with a set $V$ of discrete valuations such that the map $\unicode[STIX]{x1D703}:H^{1}(K,\operatorname{SO}_{n}(q))\rightarrow \prod _{v\in V}H^{1}(K_{v},\operatorname{SO}_{n}(q))$ is proper. Then, analyzing the commutative diagram

where $\unicode[STIX]{x1D6F1}=\prod _{v\in V}\unicode[STIX]{x1D70B}_{K_{v}}$ , and using twisting, we see that in order to prove that $\tilde{\unicode[STIX]{x1D703}}$ is proper, it is enough to prove that for any nondegenerate $n$ -dimensional quadratic form $q^{\prime }$ over $K$ , the set

(5) $$\begin{eqnarray}\unicode[STIX]{x0428}(\operatorname{Spin}_{n}(q^{\prime }),V)\,\cap \,\ker \unicode[STIX]{x1D70B}_{K}^{\prime }\end{eqnarray}$$

(where $\unicode[STIX]{x1D70B}_{K}^{\prime }$ is the map analogous to $\unicode[STIX]{x1D70B}_{K}$ for the quadratic form $q^{\prime }$ ) is finite. For a field extension $F/K$ , combining the standard exact sequence of cohomology corresponding to the exact sequence

$$\begin{eqnarray}1\rightarrow \unicode[STIX]{x1D707}_{2}\longrightarrow \operatorname{Spin}_{n}(q^{\prime })\stackrel{\unicode[STIX]{x1D70B}^{\prime }}{\longrightarrow }\operatorname{SO}_{n}(q^{\prime })\rightarrow 1\end{eqnarray}$$

with the description of the spinor norm (cf. [KMRT98, 13.30, 13.31]), one easily obtains a (functorial) identification of $\ker \unicode[STIX]{x1D70B}_{F}^{\prime }$ with the quotient $F^{\times }/\operatorname{Sn}(q^{\prime },F)$ , where $\operatorname{Sn}(q^{\prime },F)$ denotes the image of the spinor norm $\operatorname{SO}_{n}(q^{\prime })_{F}$ . Furthermore, one observes that this identification induces an injection of the set (5) into the kernel of the map

$$\begin{eqnarray}\unicode[STIX]{x1D708}_{q^{\prime },V}:K^{\times }/\operatorname{Sn}(q^{\prime },K)\longrightarrow \mathop{\prod }_{v\in V}K_{v}^{\times }/\operatorname{Sn}(q^{\prime },K_{v}).\end{eqnarray}$$

Thus, if $\unicode[STIX]{x1D703}$ is known to be proper, then to establish the properness of $\tilde{\unicode[STIX]{x1D703}}$ , it is sufficient to prove that the kernel of $\unicode[STIX]{x1D708}_{q^{\prime },V}$ is finite for all nondegenerate $n$ -dimensional quadratic forms $q^{\prime }$ over $K$ . (Of course, $\ker \unicode[STIX]{x1D708}_{q^{\prime },V}$ is automatically trivial if $q^{\prime }$ is $K$ -isotropic, so the properness of $\unicode[STIX]{x1D703}$ immediately implies that of $\tilde{\unicode[STIX]{x1D703}}$ if every $n$ -dimensional quadratic form over $K$ is isotropic, i.e.  $n$ is greater than the $u$ -invariant of $K$ .)

Here we would like to point out the following partial result on the finiteness of the set (5).

Proposition 5.8. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be its divisorial set of places. If $q$ is a quadratic Pfister form over $K$ of dimension $n=2^{d}$ , then the intersection

$$\begin{eqnarray}\unicode[STIX]{x0428}(\operatorname{Spin}_{n}(q),V)\cap \ker \unicode[STIX]{x1D70B}_{K}\end{eqnarray}$$

is finite.

Proof. According to the discussion prior to the statement of the proposition, it is enough to show that the map

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{q,V}:K^{\times }/\operatorname{Sn}(q,K)\rightarrow \mathop{\prod }_{v\in V}K_{v}^{\times }/\operatorname{Sn}(q,K_{v})\end{eqnarray}$$

has finite kernel. By our assumption, $q$ is a Pfister form $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ for some $a_{1},\ldots ,a_{d}\in K^{\times }$ . For any field extension $F/K$ , using the fact that the nonzero values of $q$ over $F$ form a subgroup of $F^{\times }$ (see [Lam05, ch. X, Theorem 1.8]), it is easy to see that $a\in F^{\times }$ belongs to $\operatorname{Sn}(q,F)$ if and only if the Pfister form $q\bot (-aq)=\langle \!\langle a_{1},\ldots ,a_{d},a\rangle \!\rangle$ is isotropic, hence hyperbolic (cf. [Lam05, ch. X, Theorem 1.7]). The latter is equivalent to the symbol

$$\begin{eqnarray}(a_{1},\ldots ,a_{d},a)\in k_{d+1}(F)=K_{d+1}(F)/2\cdot K_{d+1}(F),\end{eqnarray}$$

or the cup-product

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{a_{1}}\cup \cdots \cup \unicode[STIX]{x1D712}_{a_{d}}\cup \unicode[STIX]{x1D712}_{a}\in H^{d+1}(F,\unicode[STIX]{x1D707}_{2}),\end{eqnarray}$$

being trivial. Thus, the map

$$\begin{eqnarray}F^{\times }\rightarrow H^{d+1}(F,\unicode[STIX]{x1D707}_{2}),\quad a\mapsto \unicode[STIX]{x1D712}_{a_{1}}\cup \cdots \cup \unicode[STIX]{x1D712}_{a_{d}}\cup \unicode[STIX]{x1D712}_{a},\end{eqnarray}$$

gives rise to an injective map

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{F}:F^{\times }/\operatorname{Sn}(q,F)\rightarrow H^{d+1}(F,\unicode[STIX]{x1D707}_{2}).\end{eqnarray}$$

We then have the following commutative diagram

where $\unicode[STIX]{x1D6E5}_{V}=\prod _{v\in V}\unicode[STIX]{x1D6FF}_{K_{v}}$ . As we already mentioned several times, the finiteness of the unramified cohomology group $H^{d+1}(K,\unicode[STIX]{x1D707}_{2})_{V}$ (see § 4.1), implies the finiteness of $\ker \unicode[STIX]{x1D704}_{d+1}$ . Thus, the injectivity of $\unicode[STIX]{x1D6FF}_{K}$ yields the finiteness of $\ker \unicode[STIX]{x1D702}_{q,V}$ , as required.◻

6 The finiteness of unramified $H^{3}$ for two-dimensional global fields: characteristic zero case

To complete the proof of Theorem 1.1 and its consequences, we need to prove the following.

Theorem 6.1. Let $K$ be a two-dimensional global field, let $n\geqslant 1$ be an integer prime to $\operatorname{char}K$ and let $V$ be a divisorial set of places such that $\operatorname{char}K^{(v)}$ is prime to $n$ for all $v\in V$ . Then the unramified cohomology group $H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V}$ is finite.

Corollary 6.2. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places such that $\operatorname{char}K^{(v)}\neq 2$ for all $v\in V$ . Then the unramified cohomology group $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}$ is finite.

In the current section, we treat the characteristic zero case, where $K$ is the function field of a smooth geometrically integral curve $C$ over a number field $k$ , following the referee’s suggestions; the positive characteristic case is handled at the end of § 7. We begin by considering the case of a projective curve. Thus, let $\widetilde{C}$ be a smooth projective geometrically integral curve over a number field $k$ with function field $K=k(\widetilde{C})$ , and let $V_{0}(\widetilde{C})$ be the set of geometric places of $K$ corresponding to the closed points of $\widetilde{C}$ . It was observed by Colliot-Thélène [Col96, proof of Theorem 6.2] and (independently) by Berhuy [Ber07, Example 6] that Kato’s cohomological Hasse principle [Kat86] implies the following finiteness result.

Theorem 6.3. In the above notation, the unramified cohomology group $H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V_{0}(\widetilde{C})}$ is finite, for any $n\geqslant 1$ .

Proof. Let $V^{k}$ be the set of (the equivalence classes of) all valuations of $k$ . Then according to [Kat86, Theorem 0.8], the map

$$\begin{eqnarray}H^{3}(k(\widetilde{C}),\unicode[STIX]{x1D707}_{n}^{\otimes 2})\longrightarrow \mathop{\prod }_{v\in V^{k}}H^{3}(k_{v}(\widetilde{C}_{v}),\unicode[STIX]{x1D707}_{n}^{\otimes 2}),\end{eqnarray}$$

where $\widetilde{C}_{v}=\widetilde{C}\times _{k}k_{v}$ , is injective. Clearly, for $v\in V^{k}$ , the restriction map takes $H^{3}(k(\widetilde{C}),\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V_{0}(\widetilde{C})}$ into $H^{3}(k_{v}(\widetilde{C}),\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V_{0}(\widetilde{C}_{v})}$ . However, it was shown in [Kat86, Corollary 2.9] that the latter group is finite for all $v$ (see also Theorem 7.1 below), and is in fact trivial if $v$ is nonarchimedean and $\widetilde{C}_{v}$ has good reduction at $v$ . As $\widetilde{C}$ has good reduction at almost all $v\in V^{k}$ , the required finiteness follows.◻

Now, given an arbitrary smooth geometrically integral curve $C$ over a number field $k$ , we let $\widetilde{C}$ denote the (unique) smooth geometrically integral projective curve over $k$ containing $C$ as an open subset. Then Theorem 6.3 immediately gives the finiteness of the unramified cohomology group $H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$ for any divisorial set of places $V$ of $K=k(C)=k(\widetilde{C})$ that contains $V_{0}(\widetilde{C})$ . To establish the finiteness for any divisorial set, we need to describe the relationship between the unramified cohomology of a scheme and that of the complement of a closed subscheme of codimension one. This relationship, which essentially follows from the construction of Kato’s complex in [Kat86], was first observed by Colliot-Thélène [Col96, § 2].

Let $X$ be an excellent Noetherian scheme and $n$ an integer invertible on $X$ . For a point $x\in X$ , we let $\unicode[STIX]{x1D705}(x)$ denote its residue field, and let $X_{p}$ be the set of points of dimension $p$ . For any integers $i,j$ , Kato [Kat86] constructs a homological complex $C_{n}^{i,j}(X)$

$$\begin{eqnarray}\cdots \rightarrow \bigoplus _{x\in X_{p}}H^{p+i}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (p+j)})\rightarrow \bigoplus _{x\in X_{p-1}}H^{p+i-1}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (p+j-1)})\rightarrow \cdots \rightarrow \bigoplus _{x\in X_{0}}H^{i}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes j}),\end{eqnarray}$$

where the term $\bigoplus _{x\in X_{p}}H^{p+i}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (p+j)})$ is placed in degree $p$ . The differentials

$$\begin{eqnarray}\unicode[STIX]{x2202}_{p}:\bigoplus _{x\in X_{p}}H^{p+i}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (p+j)})\rightarrow \bigoplus _{x\in X_{p-1}}H^{p+i-1}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (p+j-1)})\end{eqnarray}$$

are defined as follows. Let $x\in X_{p}$ and set $Z_{x}=\overline{\{x\}}$ to be the closure of $x$ in $X$ . Then each point $y$ of codimension one on $Z_{x}$ corresponds to a point in $X_{p-1}$ . Let $y_{1},\ldots ,y_{s}$ be the points on the normalization $\tilde{Z}_{x}$ lying above $y$ . The local ring at each $y_{k}$ is a discrete valuation ring, yielding a discrete valuation on the function field $\unicode[STIX]{x1D705}(x)$ of $\tilde{Z}_{x}$ . Let

$$\begin{eqnarray}\unicode[STIX]{x2202}_{y_{k}}^{x}:H^{p+i}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (p+j)})\rightarrow H^{p+i-1}(k(y_{k}),\unicode[STIX]{x1D707}_{n}^{\otimes (p+j-1)})\end{eqnarray}$$

be the corresponding residue map. One then defines

$$\begin{eqnarray}\unicode[STIX]{x2202}_{y}^{x}=\mathop{\sum }_{k=1}^{s}\operatorname{Cor}_{\unicode[STIX]{x1D705}(y_{k})/\unicode[STIX]{x1D705}(y)}\circ \unicode[STIX]{x2202}_{y_{k}}^{x},\end{eqnarray}$$

where $\operatorname{Cor}$ is the corestriction map. The differential $\unicode[STIX]{x2202}_{p}$ is the direct sum of all such maps. In [Kat86, Proposition 1.7], Kato verified that this construction produces a complex. Note that if $X$ is a Noetherian integral normal scheme of dimension $d$ with function field $K$ , then the construction of the differentials implies that the degree $d$ homology $H_{d}(C_{n}^{i,j}(X))$ of $C_{n}^{i,j}(X)$ is precisely the unramified cohomology $H^{d+j}(K,\unicode[STIX]{x1D707}_{n}^{\otimes (d+j)})_{V(X)}$ , where $V(X)$ is the set of discrete valuations of $K$ corresponding to the prime divisors of $X$ .

Suppose now that $X$ is an excellent integral Noetherian normal scheme with function field $K$ and $Y\subset X$ is a normal closed subscheme of pure codimension one. Thus, if $Y=\bigcup _{j\in J}Y_{j}$ is the decomposition into irreducible components, then each $Y_{j}$ has codimension one in $X$ , and we let $\unicode[STIX]{x1D705}_{j}$ denote the function field of $Y_{j}$ . Set $U=X\setminus Y$ , and let $n$ be a positive integer invertible on $X$ . Then, for any $i\geqslant 1$ and $\ell$ , we have a canonical embedding

$$\begin{eqnarray}s:H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })_{V(X)}\longrightarrow H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })_{V(U)}.\end{eqnarray}$$

On the other hand, for each $j\in J$ , one has the residue map $H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })\rightarrow H^{i-1}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})$ , and we let

$$\begin{eqnarray}r:H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })\longrightarrow \bigoplus _{j\in J}H^{i-1}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})\end{eqnarray}$$

denote the direct sum of the residue maps.

Proposition 6.4. The above maps $r$ and $s$ give the following exact sequence of unramified cohomology groups

(6) $$\begin{eqnarray}0\rightarrow H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })_{V(X)}\stackrel{s}{\longrightarrow }H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })_{V(U)}\stackrel{r}{\longrightarrow }\bigoplus _{j\in J}H^{i-1}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})_{V(Y_{j})}.\end{eqnarray}$$

Proof. Let $d=\dim X$ . The initial segment of the Kato complex for $X$ is

$$\begin{eqnarray}\unicode[STIX]{x2202}_{X}:H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })\rightarrow \biggl(\bigoplus _{x\in U_{d-1}}H^{i-1}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})\biggr)\,\oplus \,\biggl(\bigoplus _{j\in J}H^{i-1}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})\biggr)\end{eqnarray}$$

and for $U$ it is

$$\begin{eqnarray}\unicode[STIX]{x2202}_{U}:H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })\rightarrow \bigoplus _{x\in U_{d-1}}H^{i-1}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)}).\end{eqnarray}$$

As noted above, the normality of $X$ implies that

$$\begin{eqnarray}H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })_{V(X)}=\ker \unicode[STIX]{x2202}_{X}\quad \text{and}\quad H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })_{V(U)}=\ker \unicode[STIX]{x2202}_{U}.\end{eqnarray}$$

At the next stage of the Kato complex for $X$ , there are residue maps

$$\begin{eqnarray}\unicode[STIX]{x2202}_{Y_{j}}:H^{i-1}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})\rightarrow \bigoplus _{x\in (Y_{j})_{c-1}}H^{i-2}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -2)})\end{eqnarray}$$

for all $j\in J$ , where $c=\dim Y_{j}=d-1$ , and the assumption that $Y$ is normal gives

$$\begin{eqnarray}H^{i-1}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})_{V(Y_{j})}=\ker \unicode[STIX]{x2202}_{Y_{j}}.\end{eqnarray}$$

Now, the fact that we have a complex implies that the map $r$ takes $H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes \ell })_{V(U)}$ into

$$\begin{eqnarray}\bigoplus _{j\in J}H^{i-1}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n}^{\otimes (\ell -1)})_{V(Y_{j})},\end{eqnarray}$$

and the exactness of (6) then follows easily from the construction. ◻

Proof of Theorem 6.1.

First, let $C$ be any smooth geometrically integral curve over $k$ such that $K=k(C)$ , and let $\widetilde{C}$ be a smooth projective geometrically integral curve over $k$ containing $C$ . Suppose that we are given smooth models ${\mathcal{C}}\subset \widetilde{{\mathcal{C}}}$ of $C$ and $\tilde{C}$ , respectively, over a suitable open subsect $T\subset \operatorname{Spec}(\mathbb{Z})$ such that $\tilde{{\mathcal{C}}}\setminus {\mathcal{C}}$ is of pure codimension one. The inclusion

$$\begin{eqnarray}H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V(\widetilde{{\mathcal{C}}})}\subset H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V_{0}(\widetilde{C})}\end{eqnarray}$$

in conjunction with Theorem 6.3 implies that $H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V(\widetilde{{\mathcal{C}}})}$ is finite.

Now, let

$$\begin{eqnarray}\widetilde{{\mathcal{C}}}\setminus {\mathcal{C}}=\mathop{\bigcup }_{j\in J}Y_{j}\end{eqnarray}$$

be the decomposition into irreducible components, and let $\unicode[STIX]{x1D705}_{j}$ be the function field of $Y_{j}$ . By construction, each $\unicode[STIX]{x1D705}_{j}$ is a number field and $Y_{j}$ is the spectrum of the ring of $S_{j}$ -integers in $\unicode[STIX]{x1D705}_{j}$ for some finite subset $S_{j}\subset V^{\unicode[STIX]{x1D705}_{j}}$ containing $V_{\infty }^{\unicode[STIX]{x1D705}_{j}}$ . Set $T_{j}=V^{\unicode[STIX]{x1D705}_{j}}\setminus S_{j}$ . Then $H^{2}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n})_{V(Y_{j})}$ coincides with the unramified Brauer group $_{n\!}\operatorname{Br}(\unicode[STIX]{x1D705}_{j})$ , the finiteness of which easily follows from the Artin–Hasse–Brauer–Noether theorem (cf. [CRR13, 3.6] and [CRR16a, 3.5]). Let us consider now the exact sequence of Proposition 6.4 for $X=\widetilde{{\mathcal{C}}}$ and $U={\mathcal{C}}$ for $i=3$ and $\ell =2$ . As the finiteness of

$$\begin{eqnarray}H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V(\widetilde{{\mathcal{C}}})}\quad \text{and}\quad \bigoplus _{j\in J}H^{2}(\unicode[STIX]{x1D705}_{j},\unicode[STIX]{x1D707}_{n})_{V(Y_{j})}\end{eqnarray}$$

has already been established, we obtain the finiteness of $H^{3}(K,\unicode[STIX]{x1D707}_{n}^{\otimes 2})_{V({\mathcal{C}})}$ . This proves the Theorem for the divisorial set $V=V({\mathcal{C}})$ . However, it is easy to see that any divisorial set of valuations $V$ of $K$ contains a set of the form $V({\mathcal{C}})$ for a suitable curve $C$ over $k$ and its model ${\mathcal{C}}$ as above, completing the argument in the general case.◻

As we have already mentioned, in Appendix A we give an alternative (in fact, our initial) proof of Theorem 6.1 in characteristic zero, which does not use Kato’s cohomological Hasse principle, but rather is inspired by Jannsen’s [Jan89–90] approach to the latter.

7 Proofs of Theorem 1.4 and of Theorem 6.1 in positive characteristic

The reason we treat these two results in the same section is that both arguments ultimately depend on the same techniques, namely finiteness theorems in étale cohomology and Bloch–Ogus theory. We begin with the proof of Theorem 1.4, which relies on the following.

Theorem 7.1 [Rap18, Theorem 1.3(a)].

Let $C$ be a smooth geometrically integral curve over a field $k$ , let $V_{0}$ be the set of places of $K=k(C)$ associated with the closed points of $C$ and let $m\geqslant 1$ be an integer prime to $\operatorname{char}k$ . If $k$ satisfies condition $(\text{F}_{m}^{\prime })$ (see § 1), then for any $i\geqslant 1$ and any $j$ , the unramified cohomology group $H^{i}(K,\unicode[STIX]{x1D707}_{n}^{\otimes j})_{V_{0}}$ is finite.

We refer the reader to [Rap18] for a discussion of properties and examples of fields satisfying condition $(\text{F}_{m}^{\prime })$ . A sketch of the proof of Theorem 7.1 will be given below after briefly recalling some facts from Bloch–Ogus theory that are also needed for the proof of Theorem 6.1 in positive characteristic. Here we only mention that the proof of Theorem 7.1 makes use of the following statement, which is of independent interest and which we will also need in the proof of Theorem 1.4.

Proposition 7.2 [Rap18, Theorem 1.1].

Let $K$ be a field and $m\geqslant 1$ be an integer prime to $\operatorname{char}K$ . Assume that $K$ satisfies $(\text{F}_{m}^{\prime })$ . Then for any finite Galois module $A$ over $K$ such that $mA=0$ , the groups $H^{i}(K,A)$ are finite for all $i\geqslant 0$ .

Proof of Theorem 1.4.

We need to check the conditions of Theorem 2.1 for $V=V_{0}$ . Condition (2) follows immediately from Theorem 7.1. To verify (1), we observe that the argument given in the proof of Proposition 4.1 shows that there exists an embedding

$$\begin{eqnarray}\operatorname{Pic}(V_{0})/2\cdot \operatorname{Pic}(V_{0}){\hookrightarrow}H^{2}(C,\unicode[STIX]{x1D707}_{2}).\end{eqnarray}$$

Then, because the group $H^{2}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{2})$ is finite and of exponent 2 (see [SGA4, Exposé XVI, Théorème 5.2]), the finiteness of $H^{2}(C,\unicode[STIX]{x1D707}_{2})$ follows easily from Proposition 7.2 (with condition $(\text{F}_{2}^{\prime })$ ) and the Hochschild–Serre spectral sequence

$$\begin{eqnarray}E_{2}^{p,q}=H^{p}(k,H^{q}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{2}))\Rightarrow H^{p+q}(C,\unicode[STIX]{x1D707}_{2}).\end{eqnarray}$$

Alternatively, note that we may assume that $C$ is affine, in which case the fundamental sequence (2) yields the following exact sequence

$$\begin{eqnarray}H^{2}(k,\unicode[STIX]{x1D707}_{2})\longrightarrow H^{2}(C,\unicode[STIX]{x1D707}_{2})\longrightarrow H^{1}(k,H^{1}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{2})).\end{eqnarray}$$

The extreme terms are finite in view of condition $(\text{F}_{2}^{\prime })$ and Proposition 7.2, so the middle term is also finite, as required.◻

We expect a suitable analogue of Theorem 1.4 to be valid for all types, and would like to propose the following conjecture.

Conjecture 7.3. Let $K=k(C)$ be the function field of a smooth affine geometrically integral curve over a field $k$ and let $V_{0}$ be the set of discrete valuations associated with the closed points of $C$ . Furthermore, let $G$ be an absolutely almost simple simply connected algebraic $K$ -group and let $m$ be the order of the automorphism group of its root system. Assume that $\operatorname{char}k$ is prime to $m$ and that $k$ satisfies $(\text{F}_{m}^{\prime })$ . Then the set of $K$ -isomorphism classes of $K$ -forms of $G$ that have good reduction at all $v\in V_{0}$ is finite.

It is likely that the conclusion should be true under more lax assumptions, e.g. it is probably enough to require condition $(\text{F}_{p}^{\prime })$ for all primes $p\mid m$ or even a certain subset of the set of such primes.

Applying Theorem 3.4 in conjunction with Theorem 7.1, we obtain the following.

Theorem 7.4. Let $C$ be a smooth geometrically integral curve over a field $k$ of characteristic $\neq 2$ that satisfies condition $(\text{F}_{2}^{\prime })$ , and let $K=k(C)$ be its function field. Denote by $V_{0}$ the set of discrete valuations of $K$ associated with the closed points of $C$ . Then for a nondegenerate quadratic form $q$ of dimension $n\geqslant 5$ over $K$ and $G=\operatorname{SO}_{n}(q)$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V_{0}}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V_{0}}H^{1}(K_{v},G)\end{eqnarray}$$

is proper.

We expect that the conclusion of the theorem should be true for any absolutely almost simple group under the assumptions made in Conjecture 7.3. Here is one result for spinor groups in a more specialized situation. Keep the above notations but now assume that $k$ is a finite extension of the $p$ -adic field $\mathbb{Q}_{p}$ . Recall that $k$ satisfies Serre’s condition $(\text{F})$ (cf. [Ser97, ch. III, § 4]), hence condition $(\text{F}_{m}^{\prime })$ for all $m$ . Furthermore, Parimala and Suresh [PS10] showed that the $u$ -invariant of $K$ is $8$ , i.e. any quadratic form over $K$ of dimension ${\geqslant}9$ is isotropic. Thus, combining Theorem 7.4 with the discussion preceding Proposition 5.8, we obtain the following.

Corollary 7.5. In the above notation, for any quadratic form $q$ over $K$ in $n\geqslant 9$ variables and $\tilde{G}=\operatorname{Spin}_{n}(q)$ , the map $\unicode[STIX]{x1D703}_{\tilde{G},V_{0}}$ is proper.

We now recall several results from Bloch–Ogus theory. For any smooth algebraic variety $X$ over an arbitrary field $F$ , Bloch and Ogus [BO74] established the existence of the following cohomological first quadrant spectral sequence

(7) $$\begin{eqnarray}E_{1}^{p,q}(X/F,\unicode[STIX]{x1D707}_{m}^{\otimes b})=\bigoplus _{x\in X^{(p)}}H^{q-p}(\unicode[STIX]{x1D705}(x),\unicode[STIX]{x1D707}_{m}^{\otimes (b-p)})\Rightarrow H^{p+q}(X,\unicode[STIX]{x1D707}_{m}^{\otimes b}),\end{eqnarray}$$

where $X^{(p)}$ denotes the set of points of $X$ of codimension $p$ and the summands are the Galois cohomology groups of the residue fields $\unicode[STIX]{x1D705}(x)$ (the original statement of Bloch–Ogus was actually given in terms of étale homology, with the above version obtained via absolute purity; we refer the reader to [CHK97] for a derivation of this spectral sequence that avoids the use of étale homology, as well as to [Col95] for an extensive discussion of applications of this spectral sequence to the Gersten conjecture). The spectral sequence yields a complex

(8) $$\begin{eqnarray}E_{1}^{\bullet ,q}(X/F,\unicode[STIX]{x1D707}_{m}^{\otimes b}),\end{eqnarray}$$

and it is well-known (see, e.g., [JSS14, Remark 2.5.5]) that the differentials in (8) coincide up to sign with the differentials in Kato’s complex that was recalled in § 5. The fundamental result of Bloch and Ogus was the calculation of the $E_{2}$ -term of (7): they showed that

$$\begin{eqnarray}E_{2}^{p,q}(X/F,\unicode[STIX]{x1D707}_{m}^{\otimes b})=H^{p}(X,{\mathcal{H}}^{q}(\unicode[STIX]{x1D707}_{m}^{\otimes b})),\end{eqnarray}$$

where ${\mathcal{H}}^{q}(\unicode[STIX]{x1D707}_{m}^{\otimes j})$ denotes the Zariski sheaf on $X$ associated to the presheaf that assigns to an open $U\subset X$ the cohomology group $H^{i}(U,\unicode[STIX]{x1D707}_{m}^{\otimes j}).$ The resulting (first quadrant) spectral sequence

(9) $$\begin{eqnarray}E_{2}^{p,q}(X/F,\unicode[STIX]{x1D707}_{m}^{\otimes b})=H^{p}(X,{\mathcal{H}}^{q}(\unicode[STIX]{x1D707}_{m}^{\otimes b}))\Rightarrow H^{p+q}(X,\unicode[STIX]{x1D707}_{m}^{\otimes b})\end{eqnarray}$$

is usually referred to as the Bloch–Ogus spectral sequence and has the following key properties:

1. (a) $E_{2}^{p,q}=0$ for $p>\dim X$ and all $q$ ;

2. (b) $E_{2}^{p,q}=0$ for $p>q$ ; and

3. (c) $E_{2}^{0,q}=H^{0}(X,{\mathcal{H}}^{q}(\unicode[STIX]{x1D707}_{m}^{\otimes b}))$ coincides with the unramified cohomology $H^{q}(F(X),\unicode[STIX]{x1D707}_{m}^{\otimes b})_{V_{0}}$ with respect to the geometric places (associated with the prime divisors of $X$ ) of the function field $F(X)$ .

For our applications, we now state the following proposition, whose proof combines Proposition 7.2 with the Hochschild–Serre spectral sequence.

Proposition 7.6 [Rap18, Corollary 3.2].

Suppose $K$ is a field and $m\geqslant 1$ is integer prime to $\operatorname{char}K$ . If $K$ is of type $(F_{m}^{\prime })$ , then for any smooth geometrically integral algebraic variety $X$ over $K$ , the étale cohomology groups $H^{i}(X,\unicode[STIX]{x1D707}_{m}^{\otimes j})$ are finite for all $i\geqslant 0$ and all $j$ .

Sketch of proof of Theorem 7.1.

As $\dim C=1$ , property (a) of the Bloch–Ogus spectral sequence gives surjective edge maps $E^{i}\rightarrow E_{2}^{0,i}$ for all $i\geqslant 1$ . Thus, in view of property (c), we obtain surjections

$$\begin{eqnarray}H^{i}(C,\unicode[STIX]{x1D707}_{2}){\twoheadrightarrow}H^{i}(k(C),\unicode[STIX]{x1D707}_{2})_{V_{0}}\end{eqnarray}$$

for all $i\geqslant 1$ . The finiteness of the unramified cohomology groups then follows from Proposition 7.6.

Proof of Theorem 6.1 in positive characteristic.

Here $K$ is the function field $k(X)$ of a smooth geometrically integral surface $X$ over a finite field $k$ . Property (b) of the Bloch–Ogus spectral sequence yields an exact sequence

$$\begin{eqnarray}E^{3}\rightarrow E_{2}^{0,3}\rightarrow E_{2}^{2,2}.\end{eqnarray}$$

Hence, using the well-known isomorphism

$$\begin{eqnarray}H^{2}(X,{\mathcal{H}}^{2}(\unicode[STIX]{x1D707}_{m}^{\otimes 2}))\simeq CH^{2}(X)/m\cdot CH^{2}(X),\end{eqnarray}$$

where $CH^{2}(X)$ is the Chow group of codimension-two cycles on $X$ (see, e.g. the proof of [BO74, Theorem 7.7]), together with property (c) above, we obtain the exact sequence

$$\begin{eqnarray}H^{3}(X,\unicode[STIX]{x1D707}_{m}^{\otimes 2})\rightarrow H^{3}(K,\unicode[STIX]{x1D707}_{m}^{\otimes 2})_{V_{0}}\rightarrow CH^{2}(X)/m\cdot CH^{2}(X).\end{eqnarray}$$

The left-most term is finite by Proposition 7.6. On the other hand, because $X$ is a smooth surface over a finite field, the group $CH^{2}(X)$ is finitely generated (see [CSS83, Proposition 4 and Corollaire 7]). It follows that $H^{3}(K,\unicode[STIX]{x1D707}_{m}^{\otimes 2})_{V_{0}}$ is finite, as needed.◻

8 Special unitary groups of Hermitian forms over quadratic extensions

Let $K$ be a field of characteristic $\neq 2$ and let $L/K$ be a quadratic field extension with nontrivial automorphism $\unicode[STIX]{x1D70F}$ . A choice of a basis of $L$ over $K$ enables us to identify $L$ with $K^{2}$ . Then for any $n\geqslant 1$ the space $L^{n}$ gets identified with $K^{2n}$ , and under this identification, to any $\unicode[STIX]{x1D70F}$ -Hermitian form $h$ on $L^{n}$ , there naturally corresponds a quadratic form $q=q_{h}$ on $K^{2n}$ given by $q_{h}(x)=h(x)$ for $x\in K^{2n}=L^{n}$ . Recall that if $s(x,y)$ is the sesquilinear form on $L^{n}\times L^{n}$ that gives $h$ (i.e.  $h(x)=s(x,x)$ ), then the bilinear form $b$ on $K^{2n}\times K^{2n}$ associated with $q_{h}$ is given by

$$\begin{eqnarray}b(x,y)={\textstyle \frac{1}{2}}\operatorname{Tr}_{L/K}(s(x,y))\quad \text{for }x,y\in K^{2n}=L^{n}.\end{eqnarray}$$

The following result, attributed in [Ger08, § 2.9] to Jacobson, is well-known and easy to prove: two $n$ -dimensional Hermitian forms $h_{1}$ and $h_{2}$ are equivalent if and only if the corresponding $2n$ -dimensional quadratic forms $q_{h_{1}}$ and $q_{h_{2}}$ are equivalent. In terms of Galois cohomology, this means that the natural map

$$\begin{eqnarray}H^{1}(K,\text{U}_{n}(L/K,h))\longrightarrow H^{1}(K,\text{O}_{2n}(q_{h}))\end{eqnarray}$$

is injective. On the other hand, it is well-known that the map

$$\begin{eqnarray}H^{1}(K,\operatorname{SU}_{n}(L/K,h))\longrightarrow H^{1}(K,\text{U}_{n}(L/K,h))\end{eqnarray}$$

is also injective (the proof repeats verbatim the argument for the injectivity of the map

$$\begin{eqnarray}H^{1}(K,\operatorname{SO}_{m}(q))\longrightarrow H^{1}(K,\text{O}_{m}(q)),\end{eqnarray}$$

cf. [Ber10, proof of Corollary IV.11.3] or [KMRT98, 29.E]). It follows that the map

$$\begin{eqnarray}H^{1}(K,\operatorname{SU}_{n}(L/K,h))\longrightarrow H^{1}(K,\operatorname{SO}_{2n}(q_{h}))\end{eqnarray}$$

is injective (in fact, it remains injective also over any extension of $K$ ). In this section, we use these facts to prove the analogues of Theorems 1.1–1.4 for the special unitary groups $\operatorname{SU}_{n}(L/K,h)$ of nondegenerate $n$ -dimensional Hermitian forms $(n\geqslant 2)$ .

First, let $G=\operatorname{SU}_{n}(L/K,h)$ in the above notation and let $v$ be a discrete valuation of $K$ . There are two cases.

Case 1: $v$ splits in $L$ (i.e.  $L\,\otimes _{K}\,K_{v}\simeq K_{v}\oplus K_{v}$ ). In this case $G$ is $K_{v}$ -isomorphic to $\operatorname{SL}_{n}$ , hence has good reduction at $v$ . At the same time, the corresponding quadratic form $q_{h}$ becomes hyperbolic over $K_{v}$ and therefore (trivially) $[q_{h}]\in W_{0}(K_{v})$ in the notation of § 3.

Case 2: $v$ does not split in $L$ (i.e.  $L_{v}:=L\,\otimes _{K}\,K_{v}$ is a quadratic field extension of $K_{v}$ ). Then $G$ has good reduction at $v$ if and only if $L_{v}/K_{v}$ is unramified at $v$ and there exists $\unicode[STIX]{x1D706}\in K_{v}^{\times }$ such that the Hermitian $L_{v}/K_{v}$ -form $\unicode[STIX]{x1D706}h$ is equivalent to a Hermitian form given by

$$\begin{eqnarray}h^{\prime }(x_{1},\ldots ,x_{n})=a_{1}N_{L_{v}/K_{v}}(x_{1})+\cdots +a_{v}N_{L_{v}/K_{v}}(x_{n})\end{eqnarray}$$

with $a_{i}\in U(K_{v})$ . Then again $[\unicode[STIX]{x1D706}q_{h}]\in W_{0}(K_{v})$ .

Thus, in all cases, the fact that $G$ has a good reduction at $v$ implies that there exists $\unicode[STIX]{x1D706}\in K_{v}^{\times }$ such that $[\unicode[STIX]{x1D706}q_{h}]\in W_{0}(K_{v})$ , or equivalently the group $H=\operatorname{Spin}_{2n}(q_{h})$ has a good reduction at $v$ . It is now easy to derive a unitary analogue of Theorem 1.1.

Theorem 8.1. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places. Fix a quadratic extension $L/K$ , and let $n\geqslant 2$ . Then the number of $K$ -isomorphism classes of special unitary groups $G=\operatorname{SU}_{n}(L/K,h)$ of nondegenerate Hermitian $L/K$ -forms in $n$ variables that have good reduction at all $v\in V$ is finite.

Indeed, let $G_{i}=\operatorname{SU}_{n}(L/K,h_{i})$ $(i\in I)$ be an infinite family of pairwise nonisomorphic special unitary groups associated with the quadratic extension $L/K$ such that each $G_{i}$ has a good reduction at every $v\in V$ . Then $H_{i}=\operatorname{Spin}_{2n}(q_{h_{i}})$ $(i\in I)$ is a family of spinor groups each having good reduction at all $v\in V$ . Applying Theorem 1.1, we conclude that the groups $H_{i}$ and $H_{j}$ are $K$ -isomorphic for some $i,j\in I$ , $i\neq j$ . Then for some $\unicode[STIX]{x1D706}\in K^{\times }$ , the quadratic forms $q_{h_{i}}$ and $\unicode[STIX]{x1D706}q_{h_{j}}$ are equivalent. It now follows from Jacobson’s theorem that the Hermitian forms $h_{i}$ and $\unicode[STIX]{x1D706}h_{j}$ are equivalent, hence the groups $G_{i}$ and $G_{j}$ are isomorphic, a contradiction.

Remark 8.2. It follows, for example, from [CRR16a, Proposition 5.1] that $K$ has only finitely many quadratic extensions $L/K$ that are unramified at all $v\in V$ (note that this conclusion remains valid for any finitely generated field $K$ and a divisorial set of places $V$ ). Thus, in effect, Theorem 8.1 yields the finiteness of the set of $K$ -isomorphic classes of the special unitary groups with good reduction at all $v\in V$ of $n$ -dimensional nondegenerate Hermitian forms associated with all quadratic extensions $L/K$ .

Turning now to the genus of $G=\operatorname{SU}_{n}(L/K,h)$ , we observe that any $G^{\prime }\in \mathbf{gen}_{K}(G)$ is of the form $G^{\prime }=\operatorname{SU}_{n}(L/K,h^{\prime })$ . This is clear for $n=2$ , so we assume that $n\geqslant 3$ . The group $G$ possesses a maximal $K$ -torus $T$ of the form $\text{R}_{L/K}(\mathbb{G}_{m})^{n-1}$ , so the group $G^{\prime }$ also has such a maximal $K$ -torus. Note that the nontrivial automorphism $\unicode[STIX]{x1D70F}\in \operatorname{Gal}(L/K)$ acts on the character group $X(T)$ as multiplication by $(-1)$ , and because $-\text{id}$ is not in the Weyl group of the root system of type $A_{n-1}$ $(n\geqslant 3)$ , we see that $G^{\prime }$ is an outer form of the split group of this type and $L$ is the minimal extension of $K$ over which it becomes an inner form. Furthermore, because $G^{\prime }$ splits over $L$ , it cannot involve any noncommutative division $L$ -algebra in its description, and therefore it must be of the form $\operatorname{SU}_{n}(L/K,h^{\prime })$ (cf. [PR94, 2.3]). Now, arguing as in the proof of Theorem 1.2 (cf. § 4.4) on the basis of Theorem 8.1, we obtain the following statement, which is even more complete (in the sense that it has no exceptions) than Theorem 1.2.

Theorem 8.3. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $G=\operatorname{SU}_{n}(L/K,h)$ , where $L/K$ is a quadratic extension and $h$ is a nondegenerate Hermitian form of dimension $n\geqslant 2$ associated with $L/K$ . Then the genus $\mathbf{gen}_{K}(G)$ is finite.

Next, we have the following cohomological statement, which is analogous to Theorem 1.3.

Theorem 8.4. Notation as in Theorem 8.1, for $G=\operatorname{SU}_{n}(L/K,h)$ the map

$$\begin{eqnarray}H^{1}(K,G)\rightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G)\end{eqnarray}$$

is proper.

Indeed, let $H=\operatorname{SO}_{2n}(q_{h})$ . Then we have the following commutative diagram.

According to Theorem 1.3, the map $\unicode[STIX]{x1D6FF}$ is proper. On the other hand, as we pointed out earlier in this section, the map $\unicode[STIX]{x1D6FD}$ is injective, and the properness of $\unicode[STIX]{x1D6FC}$ follows.

Finally, we have the following result for special unitary groups over function fields of curves over fields satisfying $(\text{F}_{2}^{\prime })$ , which is analogous to Theorem 1.4 and which is actually derived from it just like Theorem 8.1 was derived from Theorem 1.1.

Theorem 8.5. Let $C$ be a smooth geometrically integral curve over a field $k$ of characteristic $\neq 2$ that satisfies condition $(\text{F}_{2}^{\prime })$ and let $K=k(C)$ . Denote by $V$ the set of discrete valuations of $K$ corresponding to the closed points of $C$ . Let $L/K$ be a quadratic extension. Then the number of $K$ -isomorphism classes of special unitary groups $G=\operatorname{SU}_{n}(L/K,h)$ of nondegenerate Hermitian $L/K$ -forms $h$ in $n\geqslant 2$ variables that have good reduction at all $v\in V$ is finite.

Remark 8.6. A similar approach can be used to obtain the analogues of Theorems 8.1 and 8.3–8.5 for the (special) unitary groups $G=\operatorname{SU}_{n}(D,h)$ of nondegenerate Hermitian forms $h$ of dimension $n\geqslant 2$ over a quaternion division algebra $D$ with center $K$ with the canonical involution. (We note that these are precisely the absolutely almost simple simply connected groups of type $\mathsf{C}_{n}$ that split over a quadratic extension of the base field.) More precisely, let us first fix a central quaternion division algebra $D$ over $K$ . Then fixing a basis of $D$ over $K$ enables us to identify $D^{n}$ with $K^{4n}$ , and in terms of this identification, the Hermitian form $h$ corresponds to a quadratic form $q_{h}$ on $K^{4n}$ . Furthermore, one has an analogue of Jacobson’s theorem: two nondegenerate $n$ -dimensional Hermitian forms $h_{1}$ and $h_{2}$ over $D$ if and only if the corresponding $4n$ -dimensional quadratic forms $q_{h_{1}}$ and $q_{h_{2}}$ are equivalent. Next, if $G=\operatorname{SU}_{n}(D,h)$ has good reduction at a discrete valuation $v$ of $K$ , then $D$ is unramified at $v$ and $H=\operatorname{Spin}_{4n}(b_{h})$ has good reduction at $v$ . Using these remarks, one can repeat the above arguments almost verbatim to establish the analogues of the theorems of the current section in this situation over the same two classes of fields, i.e. over two-dimensional global fields $K$ of characteristic $\neq 2$ equipped with a divisorial set of places $V$ and over the function fields $K=k(C)$ of smooth geometrically integral curves $C$ over a field $k$ of characteristic $\neq 2$ satisfying condition $(\text{F}_{2}^{\prime })$ equipped with the set $V$ of geometric places associated with the closed points of $C$ . Details will be published elsewhere. We now recall that for both classes, $_{2\!}\operatorname{Br}(K)_{V}$ is known to be finite, implying that the number of isomorphism classes of central quaternion division $K$ -algebras $D$ such that there exists a nondegenerate $n$ -dimensional Hermitian form $h$ for which $G=\operatorname{SU}_{n}(D,h)$ has good reduction at all $v\in V$ is finite. Eventually, this shows that the set of $K$ -isomorphism classes of absolutely almost simple simply connected $K$ -groups of type $\mathsf{C}_{n}$ that split over a quadratic extension of the base field and have good reduction at all $v\in V$ is finite (for $K$ and $V$ as above). Observe that Remark 8.2 yields a similar statement for type $\mathsf{A}_{n}$ , and the results of §9 do so for groups of type $\mathsf{G}_{2}$ . Thus, it would be interesting to see whether this statement in fact extends to forms of all types that split over a quadratic extension of the base field – this would be an important test for the general problem () from § 1.

9 Groups of type $\mathsf{G}_{2}$

Let $G_{0}$ be the split group of type $\mathsf{G}_{2}$ over a field $K$ of characteristic $\neq 2$ . Then the $K$ -isomorphism classes of $K$ -groups of type $\mathsf{G}_{2}$ are in a natural one-to-one correspondence with the elements of the (pointed) set $H^{1}(K,G_{0})$ . Furthermore, there is a natural map

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{K}:H^{1}(K,G_{0})\rightarrow H^{3}(K,\unicode[STIX]{x1D707}_{2})\end{eqnarray}$$

that has the following explicit description: if $\unicode[STIX]{x1D709}\in H^{1}(K,G_{0})$ and the twisted group $G=_{\unicode[STIX]{x1D709}}G_{0}$ is the automorphism group of the octonion algebra $\mathbb{O}=\mathbb{O}(a,b,c)$ corresponding to a triple $(a,b,c)\in (K^{\times })^{3}$ , then

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D712}_{a}\cup \unicode[STIX]{x1D712}_{b}\cup \unicode[STIX]{x1D712}_{c},\end{eqnarray}$$

where for $t\in K^{\times }$ we let $\unicode[STIX]{x1D712}_{t}\in H^{1}(K,\unicode[STIX]{x1D707}_{2})$ denote, by abuse of notation, the cohomology class of the usual 1-cocycle $\unicode[STIX]{x1D712}_{t}$ coming from Kummer theory. It is well-known that $\unicode[STIX]{x1D706}_{K}$ is injective (cf. [Ser97, ch. III, Appendix 2, 3.3]).

Now, suppose that $K$ is equipped with a discrete valuation $v$ such that $\operatorname{char}K^{(v)}\neq 2$ . Then the automorphism group $G$ of an octonian algebra $\mathbb{O}$ has good reduction at $v$ if and only if $\mathbb{O}$ can be represented by a triple $(a,b,c)\in (K^{\times })^{3}$ such that

$$\begin{eqnarray}v(a)=v(b)=v(c)=0.\end{eqnarray}$$

It follows that if $G=_{\unicode[STIX]{x1D709}}G_{0}$ has good reduction at $v$ , then the cocycle $\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709})\in H^{3}(K,\unicode[STIX]{x1D707}_{2})$ is unramified at $v$ . Now, using Corollary 6.2, we obtain the following.

Theorem 9.1. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ , let $V$ be a divisorial set of places and let $G$ be a simple algebraic $K$ -group of type $\mathsf{G}_{2}$ .

1. (i) The number of $K$ -isomorphism classes of $K$ -forms $G^{\prime }$ having good reduction at all $v\in V$ is finite.

2. (ii) The genus $\mathbf{gen}_{K}(G)$ is finite.

3. (iii) The map $\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \prod _{v\in V}H^{1}(K_{v},G)$ is proper.

Proof. Part (i) immediately follows from the remarks preceding the statement of the theorem and Corollary 6.2. Part (ii) is derived from part (i) and Theorem 5.2 just like Theorem 1.2 was derived from Theorem 1.1. Finally, part (iii) is derived from part (i) and the observation mentioned in § 5.2. (Alternatively, one can use the injectivity of $\unicode[STIX]{x1D706}_{K}$ and the fact that the kernel $\unicode[STIX]{x1D6FA}_{3}$ of the map

$$\begin{eqnarray}H^{3}(K,\unicode[STIX]{x1D707}_{2})\longrightarrow \mathop{\prod }_{v\in V}H^{3}(K_{v},\unicode[STIX]{x1D707}_{2})\end{eqnarray}$$

is finite, which immediately follows from the finiteness of $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}$ .)◻

Next, we describe the analogues for groups of type $\mathsf{G}_{2}$ of the results of § 7.

Theorem 9.2. Let $k$ be a field of characteristic $\neq 2$ that satisfies condition $(\text{F}_{2}^{\prime })$ , let $K=k(C)$ be the function field of a smooth affine geometrically integral curve $C$ over $k$ and let $V_{0}$ be the set of places of $K$ corresponding to the closed points of $C$ . If $G$ is a simple algebraic $K$ -group of type $\mathsf{G}_{2}$ , then:

1. (i) the number of $K$ -isomorphism classes of $K$ -forms $G^{\prime }$ having good reduction at all $v\in V_{0}$ is finite;

2. (ii) the map $\unicode[STIX]{x1D703}_{G,V_{0}}:H^{1}(K,G)\longrightarrow \prod _{v\in V_{0}}H^{1}(K_{v},G)$ is proper.

Proof. Again, part (i) follows from the discussion preceding the statement of Theorems 9.1 and 7.1. Then part (ii) follows from (i) and the observation made in [CRR15, Rap14] that we discussed at the beginning of § 5.2. (As in the proof of Theorem 9.1, one can alternatively use the injectivity of $\unicode[STIX]{x1D706}_{K}$ in conjunction with the fact that the homomorphism

$$\begin{eqnarray}H^{3}(K,\unicode[STIX]{x1D707}_{2})\longrightarrow \mathop{\prod }_{v\in V_{0}}H^{3}(K_{v},\unicode[STIX]{x1D707}_{2})\end{eqnarray}$$

has finite kernel, which is a consequence of the finiteness of $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V_{0}}$ .)◻

We now turn to a result that provides information about the genus of a group of type $\mathsf{G}_{2}$ over the fields of rational functions over global fields.

Theorem 9.3. Let $K=k(x_{1},\ldots ,x_{r})$ the field of rational functions in $r$ variables over a global field $k$ of characteristic $\neq 2$ and let $G$ be a simple $K$ -group of type $\mathsf{G}_{2}$ .

1. (i) If $r=1$ , then the genus $\mathbf{gen}_{K}(G)$ reduces to a single element.

2. (ii) The genus $\mathbf{gen}_{K}(G)$ is finite for any $r$ .

The proof requires more detailed information about the ‘residues’ of groups of type $\mathsf{G}_{2}$ at places of bad reduction. Thus, let $K$ be a field with a discrete valuation $v$ such that the residue field $\mathscr{K}=K^{(v)}$ is of characteristic $\neq 2$ and let

$$\begin{eqnarray}\unicode[STIX]{x2202}_{v}:H^{3}(K,\unicode[STIX]{x1D707}_{2})\rightarrow H^{2}(\mathscr{K},\unicode[STIX]{x1D707}_{2})\end{eqnarray}$$

be the corresponding residue map. It is well-known (cf. [GS06, §§ 7.1 and 7.5]) that any ‘decomposable’ element $\unicode[STIX]{x1D712}_{a}\,\cup \,\unicode[STIX]{x1D712}_{b}\,\cup \,\unicode[STIX]{x1D712}_{c}$ , where $a,b,c\in K^{\times }$ (and by the Bloch–Kato conjecture such elements generate $H^{3}(K,\unicode[STIX]{x1D707}_{2})$ ) can be written in the form $\unicode[STIX]{x1D712}_{a^{\prime }}\,\cup \,\unicode[STIX]{x1D712}_{b^{\prime }}\,\cup \,\unicode[STIX]{x1D712}_{c^{\prime }}$ with $v(a^{\prime })=v(b^{\prime })=0$ and $v(c^{\prime })=0$ or $1$ , and that on elements of this form the residue map is given by

$$\begin{eqnarray}\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D712}_{a}\cup \unicode[STIX]{x1D712}_{b}\cup \unicode[STIX]{x1D712}_{c})=\left\{\begin{array}{@{}ll@{}}0, & v(a)=v(b)=v(c)=0,\\ \unicode[STIX]{x1D712}_{\bar{a}}\cup \unicode[STIX]{x1D712}_{\bar{b}}, & v(a)=v(b)=0,v(c)=1,\end{array}\right.\end{eqnarray}$$

where $\bar{a},\bar{b}$ are the images of $a,b$ in $\mathscr{K}^{\times }$ . We define the quaternion algebra corresponding to the residue of such an element to be the matrix algebra $M_{2}(\mathscr{K})$ in the first case and the standard quaternion algebra $((\bar{a},\bar{b})/\mathscr{K})$ in the second. We will assume henceforth that $\mathscr{K}$ is finitely generated.

Lemma 9.4. Given $\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2}\in H^{1}(K,G_{0})$ , we denote by $G_{i}=_{\unicode[STIX]{x1D709}_{i}}G_{0}$ the corresponding group and set $\mathscr{D}_{i}$ to be the quaternion algebra over $\mathscr{K}$ corresponding to the residue $\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{i}))$ for $i=1,2$ . If $G_{1}$ and $G_{2}$ have the same isomorphism classes of maximal $K$ -tori, then $\mathscr{D}_{1}$ and $\mathscr{D}_{2}$ have the same isomorphism classes of maximal étale $\mathscr{K}$ -subalgebras.

Proof. Let $\mathbb{O}_{i}$ be an octonian $K$ -algebra such that $G_{i}=\operatorname{Aut}(\mathbb{O}_{i})$ for $i=1,2$ . Suppose $\mathbb{O}_{i}$ can be represented by a triple $(a_{i},b_{i},c_{i})\in (K^{\times })^{3}$ such that $v(a_{i})=v(b_{i})=v(c_{i})=0$ . Then $G_{i}$ has a good reduction at $v$ . As $G_{1}$ and $G_{2}$ have the same isomorphism classes of maximal $K$ -tori, by Theorem 5.2 the group $G_{3-i}$ also has good reduction at $v$ . This means that $\mathbb{O}_{3-i}$ can also be represented by a triple $(a_{3-i},b_{3-i},c_{3-i})\in (K^{\times })^{3}$ such that $v(a_{3-i})=v(b_{3-i})=v(c_{3-i})=0$ . In this case, both residues $\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{1}))$ and $\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{2}))$ are trivial, so our claim is obvious. It remains to consider the case where $\mathbb{O}_{i}$ is represented by a triple $(a_{i},b_{i},c_{i})\in (K^{\times })^{3}$ satisfying

$$\begin{eqnarray}v(a_{i})=v(b_{i})=0,\quad v(c_{i})=1\end{eqnarray}$$

for $i=1,2$ . Then $\mathscr{D}_{i}$ is the quaternion algebra $((\bar{a}_{i},\bar{b}_{i})/\mathscr{K})$ where $\bar{a}_{i},\bar{b}_{i}$ are the images of $a_{i},b_{i}$ in $\mathscr{K}^{\times }$ . To prove that $\mathscr{D}_{1}$ and $\mathscr{D}_{2}$ have the same maximal étale subalgebras, it suffices to prove the following:

Let $\mathscr{L}$ be an extension of $\mathscr{K}$ of degree ${\leqslant}2$ . Then $\mathscr{L}$ splits $\mathscr{D}_{1}$ if and only if it splits $\mathscr{D}_{2}$ .

Let $L$ be the unramified extension of $K_{v}$ with the residue field $\mathscr{L}$ . We have

$$\begin{eqnarray}\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{i}))=\hat{\unicode[STIX]{x2202}}_{v}(\unicode[STIX]{x1D706}_{K_{v}}(\hat{\unicode[STIX]{x1D709}}_{i})),\end{eqnarray}$$

where $\hat{\unicode[STIX]{x2202}}_{v}:H^{3}(K_{v},\unicode[STIX]{x1D707}_{2})\rightarrow H^{2}(\mathscr{K},\unicode[STIX]{x1D707}_{2})$ is the residue map and $\hat{\unicode[STIX]{x1D709}}_{i}$ is the image of $\unicode[STIX]{x1D709}_{i}$ under the restriction map $H^{1}(K,G_{0})\rightarrow H^{1}(K_{v},G_{0})$ . As residue maps commute with restriction maps for unramified extensions, we further obtain

(10) $$\begin{eqnarray}\operatorname{Res}_{\mathscr{L}/\mathscr{K}}(\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{i})))=\hat{\unicode[STIX]{x2202}}_{v}^{L}(\unicode[STIX]{x1D706}_{L}(\operatorname{Res}_{L/K_{v}}(\hat{\unicode[STIX]{x1D709}}_{i}))),\end{eqnarray}$$

where $\hat{\unicode[STIX]{x2202}}_{v}^{L}:H^{3}(L,\unicode[STIX]{x1D707}_{2})\rightarrow H^{2}(\mathscr{L},\unicode[STIX]{x1D707}_{2})$ is the residue map for $L$ (see [Col95, Proposition 3.3.1]).

Now, suppose that $\mathscr{L}$ splits $\mathscr{D}_{i}$ , i.e. the quaternion algebra $((\bar{a}_{i},\bar{b}_{i})/\mathscr{L})$ is trivial. Then it follows from Hensel’s lemma that the quaternion algebra $((a_{i},b_{i})/L)$ is also trivial, and therefore the group $G_{i}$ becomes split over $L$ . Define a two-dimensional $K_{v}$ -torus $T$ to be

$$\begin{eqnarray}\mathbb{G}_{m}\times \mathbb{G}_{m}\quad \text{if }L=K_{v}\quad \text{and}\quad \text{R}_{L/K_{v}}^{(1)}(\mathbb{G}_{m})\times \text{R}_{L/K_{v}}^{(1)}(\mathbb{G}_{m})\quad \text{if }[L:K_{v}]=2\end{eqnarray}$$

(where, as usual, $\text{R}_{L/K_{v}}^{(1)}(\mathbb{G}_{m})$ denotes the corresponding norm torus). Then $T$ is isomorphic to a maximal $K_{v}$ -torus of $G_{i}$ . As $G_{1}$ and $G_{2}$ have the same isomorphism classes of maximal $K$ -tori, they also have the same isomorphism classes of maximal $K_{v}$ -tori (see [RR10, Remark 2.2]). Thus, $T$ is isomorphic to a maximal $K_{v}$ -torus of $G_{3-i}$ as well. Then the cocycle $\operatorname{Res}_{L/K_{v}}(\hat{\unicode[STIX]{x1D709}}_{3-i})$ is trivial, so applying (10) with $i$ replaced by $3-i$ , we obtain that $\operatorname{Res}_{\mathscr{L}/\mathscr{K}}(\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{3-i})))$ is trivial, i.e.  $\mathscr{L}$ splits $\mathscr{L}_{3-i}$ , as required.◻

Corollary 9.5. Assume that the residue field $\mathscr{K}=K^{(v)}$ is finitely generated and has the following property:

$(\star )$

If $D_{1}$ and $D_{2}$ are central quaternion algebras over $\mathscr{K}$ that have the same maximal étale subalgebras, then $D_{1}\simeq D_{2}$ .

If $\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2}\in H^{1}(K,G_{0})$ are such that the corresponding groups $G_{i}=_{\unicode[STIX]{x1D709}_{i}}G_{0}$ for $i=1,2$ have the same isomorphism classes of maximal $K$ -tori, then $\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{1}))=\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}_{2}))$ .

Proof of Theorem 9.3(i).

We will prove a more general result. To formulate it, in addition to property $(\star )$ of a field $\mathscr{K}$ from Corollary 9.5, we need to introduce the following property of a field $k$ .

$(\star \star )$

If $G_{1}$ and $G_{2}$ are two $k$ -groups of type $\mathsf{G}_{2}$ having the same maximal tori, then

$$\begin{eqnarray}G_{1}\simeq G_{2}\quad \text{over }k.\end{eqnarray}$$

Theorem 9.6. Assume that a field $k$ of characteristic $\neq 2$ satisfies $(\star \star )$ and that any finite extension $\ell$ of $k$ satisfies $(\star )$ . Then the field of rational functions $K=k(x)$ satisfies $(\star \star )$ .

We note that the mere fact that $k$ satisfies $(\star )$ implies that $K=k(x)$ also satisfies $(\star )$ ; see [RR10, Theorem A]. Furthermore, it is well-known that global fields satisfy both conditions $(\star )$ (cf. [CRR13, 3.6]) and $(\star \star )$ (cf. [PR09, Theorem 7.5]), so Theorem 9.6 immediately yields the assertion of Theorem 9.3(i).

To prove Theorem 9.6, we let $V$ denote the set of all places of $K=k(x)$ that are trivial on $k$ (these of course correspond to the closed points of $\mathbb{P}_{k}^{1}$ ). It follows from the above remarks that we may assume the field $k$ to be infinite. Now, let $G^{\prime }\in \mathbf{gen}_{K}(G)$ , and let $\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}^{\prime }\in H^{1}(K,G_{0})$ be the corresponding cocycles (so that $G=_{\unicode[STIX]{x1D709}}G_{0}$ and $G^{\prime }=_{\unicode[STIX]{x1D709}^{\prime }}G_{0}$ ). For any $v\in V$ , since the residue field $K^{(v)}$ is a finite extension of $k$ , hence satisfies $(\star )$ by our assumption, we conclude from Corollary 9.5 that

$$\begin{eqnarray}\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}))=\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}^{\prime })).\end{eqnarray}$$

Thus,

(11) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}^{\prime })=\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709})\cdot \unicode[STIX]{x1D701}\quad \text{for some }\unicode[STIX]{x1D701}\in H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}.\end{eqnarray}$$

However, according to Faddeev’s sequence (cf. [GMS03, ch. III, § 9] and [GS06, 6.9]), the natural (inflation) map $\unicode[STIX]{x1D704}:H^{3}(k,\unicode[STIX]{x1D707}_{2})\rightarrow H^{3}(K,\unicode[STIX]{x1D707}_{2})$ identifies $H^{3}(k,\unicode[STIX]{x1D707}_{2})$ with the unramified group $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}$ . Then in terms of this identification, $\unicode[STIX]{x1D701}$ in (11) belongs to $H^{3}(k,\unicode[STIX]{x1D707}_{2})$ . We would like to show that actually $\unicode[STIX]{x1D701}=1$ , which will imply that $G^{\prime }\simeq G$ .

As $k$ is infinite, we can pick $v\in V$ so that $K^{(v)}=k$ and $G$ and $G^{\prime }$ have smooth reductions $\text{}\underline{G}^{(v)}$ and ${\text{}\underline{G}^{\prime }}^{(v)}$ at $v$ . We have the natural map

$$\begin{eqnarray}\unicode[STIX]{x1D716}_{v}:H^{3}(K,\unicode[STIX]{x1D707}_{2})_{\{v\}}\rightarrow H^{3}(k,\unicode[STIX]{x1D707}_{2})\end{eqnarray}$$

(defined on the $v$ -unramified part) such that $\unicode[STIX]{x1D716}_{v}\circ \unicode[STIX]{x1D704}=\operatorname{id}$ . Clearly, the images under $\unicode[STIX]{x1D706}_{k}$ of the cocycles that correspond to $\text{}\underline{G}^{(v)}$ and ${\text{}\underline{G}^{\prime }}^{(v)}$ coincide with $\unicode[STIX]{x1D716}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}))$ and $\unicode[STIX]{x1D716}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}^{\prime }))$ , respectively. Now, by the last assertion of Theorem 5.2, the groups $\text{}\underline{G}^{(v)}$ and ${\text{}\underline{G}^{\prime }}^{(v)}$ have the same isomorphism classes of maximal $k$ -tori, so because $k$ is assumed to satisfy $(\star \star )$ , we conclude that $\text{}\underline{G}^{(v)}\simeq {\text{}\underline{G}^{\prime }}^{(v)}$ . This means that

(12) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}))=\unicode[STIX]{x1D716}_{v}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}^{\prime })).\end{eqnarray}$$

Applying $\unicode[STIX]{x1D716}_{v}$ to (11) and comparing the result with (12), we obtain that $\unicode[STIX]{x1D716}_{v}(\unicode[STIX]{x1D701})=1$ , and consequently $\unicode[STIX]{x1D701}=1$ , as required.◻

Proof of Theorem 9.3(ii).

We view $K$ as the field of rational functions on $\mathbb{P}_{k}^{r}$ , and let $V$ be the set of discrete valuations on $K$ associated with the prime divisors. Pick a finite subset $V_{0}=\{v_{1},\ldots ,v_{d}\}$ of $V$ so that $G$ has a good reduction at all $v\in V\setminus V_{0}$ . For each $i=1,\ldots ,d$ , we let $\mathscr{D}_{i}$ denote the quaternion algebra over the residue field $\mathscr{K}_{i}=K^{(v_{i})}$ that corresponds to the residue $\unicode[STIX]{x2202}_{v_{i}}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}))$ , where $\unicode[STIX]{x1D709}\in H^{1}(K,G_{0})$ is such that $G=_{\unicode[STIX]{x1D709}}G_{0}$ . Given any other $K$ -group $G^{\prime }$ of type $\mathsf{G}_{2}$ , we set $\unicode[STIX]{x1D709}^{\prime }=\unicode[STIX]{x1D709}(G^{\prime })$ to be the cohomology class in $H^{1}(K,G_{0})$ such that $G^{\prime }=_{\unicode[STIX]{x1D709}^{\prime }}G_{0}$ , and for $i=1,\ldots ,d,$ denote by $\mathscr{D}_{i}^{\prime }=\mathscr{D}_{i}(G^{\prime })$ the quaternion algebra corresponding to the residue $\unicode[STIX]{x2202}_{v_{i}}(\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}^{\prime }))$ . Then Lemma 9.4 tells us that the correspondence $G^{\prime }\mapsto (\mathscr{D}_{i}^{\prime })$ gives a map

$$\begin{eqnarray}\unicode[STIX]{x1D70C}:\mathbf{gen}_{K}(G)\longrightarrow \mathop{\prod }_{i=1}^{r}\mathbf{gen}_{\mathscr{ K}_{i}}(\mathscr{D}_{i}),\end{eqnarray}$$

where the genus $\mathbf{gen}_{\mathscr{K}}(\mathscr{D})$ of a central quaternion algebra $\mathscr{D}$ over a field $\mathscr{K}$ (of characteristic $\neq 2$ ) is defined to be the set of the Brauer classes $[\mathscr{D}^{\prime }]$ of central quaternion $\mathscr{K}$ -algebras $\mathscr{D}^{\prime }$ having the same maximal étale subalgebras as $\mathscr{D}$ ; see [CRR15, CRR16a] for the details. As each genus $\mathbf{gen}_{\mathscr{K}_{i}}(\mathscr{D}_{i})$ is finite [CRR15], to prove that $\mathbf{gen}_{K}(G)$ is finite, it suffices to show that each fiber $\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(G^{\prime }))$ for $G^{\prime }\in \mathbf{gen}_{K}(G)$ is finite. For this we note that Theorem 5.2 implies that for any $G^{\prime }\in \mathbf{gen}_{K}(G)$ and any $v\in V\setminus V_{0}$ , the residue $\unicode[STIX]{x2202}_{v}(\unicode[STIX]{x1D706}_{K}(G^{\prime }))$ is trivial. Consequently, for any $G^{\prime }\in \mathbf{gen}_{K}(G)$ , any $G^{\prime \prime }\in \unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(G^{\prime }))$ , and corresponding cocycles $\unicode[STIX]{x1D709}^{\prime },\unicode[STIX]{x1D709}^{\prime \prime }$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}^{\prime \prime })=\unicode[STIX]{x1D706}_{K}(\unicode[STIX]{x1D709}^{\prime })\cdot \unicode[STIX]{x1D701}\quad \text{for some }\unicode[STIX]{x1D701}\in H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}.\end{eqnarray}$$

However, from Faddeev’s sequence, we again see that in our situation $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}=H^{3}(k,\unicode[STIX]{x1D707}_{2})$ (cf. [GMS03, Theorem 10.1]). As $k$ is a global field, the latter group is finite by Poitou–Tate, so, from the injectivity of $\unicode[STIX]{x1D706}_{K}$ , we obtain the finiteness of the fibers $\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(G^{\prime }))$ for $G^{\prime }\in \mathbf{gen}_{K}(G)$ , and hence the finiteness of $\mathbf{gen}_{K}(G)$ .◻