---
abstract: 'Let $K={\mathbb{F}}_q(C)$ be the global function field of rational functions over a smooth and projective curve $C$ defined over a finite field ${\mathbb{F}}_q$. The ring of regular functions on $C-S$ where $S \neq \emptyset$ is any finite set of closed points on $C$ is a Dedekind domain ${\mathcal{O}_S}$ of $K$. For a semisimple ${\mathcal{O}_S}$-group ${\underline}{G}$ with a smooth fundamental group ${\underline}{F}$, we aim to describe both the set of genera of ${\underline}{G}$ and its principal genus (the latter if ${\underline}{G} \otimes_{{\mathcal{O}_S}} K$ is isotropic at $S$) in terms of abelian groups depending on ${\mathcal{O}_S}$ and ${\underline}{F}$ only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain ${\underline}{G}$. We also use it to express the Tamagawa number $\tau(G)$ of a semisimple $K$-group $G$ by the Euler Poincaré invariant. This facilitates the computation of $\tau(G)$ for twisted $K$-groups.'
author:
- 'Rony A. Bitan'
title: On the genera of semisimple groups defined over an integral domain of a global function field
---

[^1]

Introduction {#Introduction}
============

Let $C$ be a projective algebraic curve defined over a finite field ${\mathbb{F}}_q$, assumed to be geometrically connected and smooth. Let $K={\mathbb{F}}_q(C)$ be the global field of rational functions over $C$, and let ${\Omega}$ be the set of all closed points of $C$. For any point ${\mathfrak{p}}\in {\Omega}$, let $v_{\mathfrak{p}}$ be the induced discrete valuation on $K$, $\hat{{\mathcal{O}}}_{\mathfrak{p}}$ the complete valuation ring with respect to $v_{\mathfrak{p}}$, and $\hat{K}_{\mathfrak{p}}, k_{\mathfrak{p}}$ its fraction field and residue field at ${\mathfrak{p}}$, respectively. Any *Hasse set* of $K$, namely, a non-empty finite set $S \subset {\Omega}$, gives rise to an integral domain of $K$ called a *Hasse domain*: $${\mathcal{O}_S}:= \{x \in K: v_{\mathfrak{p}}(x) \geq 0 \ \forall {\mathfrak{p}}\notin S\}.$$ This is a regular and one dimensional Dedekind domain. Group schemes defined over ${\text{Spec} \,}{\mathcal{O}_S}$ are underlined, being omitted in the notation of their generic fibers.

Let ${\underline}{G}$ be an affine, smooth and of finite type group scheme defined over ${\text{Spec} \,}{\mathcal{O}_S}$. We define $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ to be the set of isomorphism classes of ${\underline}{G}$-torsors over ${\text{Spec} \,}{\mathcal{O}_S}$ relative to the étale or the flat topology (the classification for the two topologies coincide when ${\underline}{G}$ is smooth; cf. [@SGA4 VIII Cor. 2.3]). The sets $H^1(K,G)$ and $H^1_{\text{\'et}}(\hat{{\mathcal{O}}}_{\mathfrak{p}},{\underline}{G}_{\mathfrak{p}})$, for every ${\mathfrak{p}}\notin S$, are defined similarly. All these three sets are naturally pointed: the distinguished point of $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ (resp., $H^1(K,G)$, $H^1_{\text{\'et}}(\hat{{\mathcal{O}}}_{\mathfrak{p}},{\underline}{G}_{\mathfrak{p}})$) is the class of the trivial ${\underline}{G}$-torsor ${\underline}{G}$ (resp. trivial $G$-torsor $G$, trivial ${\underline}{G}_{\mathfrak{p}}$-torsor ${\underline}{G}_{\mathfrak{p}}$). There exists a canonical map of pointed-sets (mapping the distinguished point to the distinguished point): $$\label{lm}
{\lambda}: H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \to H^1(K,G) \times \prod\limits_{{\mathfrak{p}}\notin S} H^1_{\text{\'et}}(\hat{{\mathcal{O}}}_{\mathfrak{p}},{\underline}{G}_{\mathfrak{p}})$$ which is defined by mapping a class in $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ represented by $X$ to the class represented by $(X \otimes_{{\mathcal{O}_S}} {\text{Spec} \,}K) \times \prod_{{\mathfrak{p}}\notin S} X \otimes_{{\mathcal{O}_S}} {\text{Spec} \,}\hat{{\mathcal{O}}}_{\mathfrak{p}}$. Let $[\xi_0] := {\lambda}([{\underline}{G}])$. The *principal genus* of ${\underline}{G}$ is then ${\lambda}^{-1}([\xi_0])$, i.e., the classes of ${\underline}{G}$-torsors over ${\text{Spec} \,}{\mathcal{O}_S}$ that are generically and locally trivial at all points of ${\mathcal{O}_S}$. More generally, a *genus* of ${\underline}{G}$ is any fiber ${\lambda}^{-1}([\xi])$ where $[\xi] \in {\operatorname{Im}}({\lambda})$. The *set of genera* of ${\underline}{G}$ is then: $$\text{gen}({\underline}{G}) := \{ {\lambda}^{-1}([\xi]) \ : \ [\xi] \in {\operatorname{Im}}({\lambda}) \},$$ whence $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ is a disjoint union of its genera.

Given a representative $P$ of a class in $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$, by referring also to ${\underline}{G}$ as a ${\underline}{G}$-torsor acting on itself by conjugations, the quotient of $P \times_{{\mathcal{O}_S}} {\underline}{G}$ by the ${\underline}{G}$-action $(p,g) \mapsto (ps^{-1},sgs^{-1})$ is an affine ${\mathcal{O}_S}$-group scheme ${^P}{\underline}{G}$, called the *twist* of ${\underline}{G}$ by $P$. It is an inner form of ${\underline}{G}$, thus is locally isomorphic to ${\underline}{G}$ in the étale topology, namely, every fiber of it at a prime of ${\mathcal{O}_S}$ is isomorphic to ${\underline}{G}_{\mathfrak{p}}:= {\underline}{G} \otimes_{{\mathcal{O}_S}} \hat{{\mathcal{O}}}_{\mathfrak{p}}$ over some finite étale extension of $\hat{{\mathcal{O}}}_{\mathfrak{p}}$. The map ${\underline}{G} \mapsto {^P}{\underline}{G}$ defines a bijection of pointed-sets $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \to H^1_{\text{\'et}}({\mathcal{O}_S},{^P}{\underline}{G})$ (e.g., [@Sko §2.2, Lemma 2.2.3, Examples 1,2]).

A group scheme defined over ${\text{Spec} \,}{\mathcal{O}_S}$ is said to be *reductive* if it is affine and smooth over ${\text{Spec} \,}{\mathcal{O}_S}$, and each geometric fiber of it at a prime ${\mathfrak{p}}$ is (connected) reductive over $k_{\mathfrak{p}}$ ([@SGA3 Exp. XIX Def. 2.7]). It is *semisimple* if it is reductive, and the rank of its root system equals that of its lattice of weights ([@SGA3 Exp. XXI Def. 1.1.1]). Suppose ${\underline}{G}$ is semisimple and that its fundamental group ${\underline}{F}$ is of order prime to $\text{char}(K)$. Being finite, of multiplicative type ([@SGA3 XXII, Cor. 4.1.7]), commutative and smooth, ${\underline}{F}$ decomposes into finitely many factors of the form $\text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$ or $\text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$ where ${\underline}{\mu}_{m} := {\text{Spec} \,}{\mathcal{O}_S}[t]/(t^{m}-1)$ and $R$ is some finite (possibly trivial) étale extension of ${\mathcal{O}_S}$. Consequently, $H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{F})$ are abelian groups for all $r \geq 0$. The following two ${\mathcal{O}_S}$-invariants of ${\underline}{F}$ will play a major role in the description of $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$:

\[i\] Let $R$ be a finite étale extension of ${\mathcal{O}_S}$. We define: $$\begin{aligned}
i({\underline}{F}) := \left \{ \begin{array}{l l}
{{\mathrm{Br}}}(R)[m]                        &  {\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m) \\
\ker({{\mathrm{Br}}}(R)[m] \xrightarrow{N^{(2)}} {{\mathrm{Br}}}({\mathcal{O}_S})[m])  &  {\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m) 
\end{array}\right.  \end{aligned}$$ where $N^{(2)}$ is induced by the norm map $N_{R/{\mathcal{O}_S}}$ and for a group $*$, $*[m]$ stands for its $m$-torsion part. For ${\underline}{F} = \prod_{k=1}^r {\underline}{F}_k$ where each ${\underline}{F}_k$ is one of the above, $i({\underline}{F})$ is the direct product $\prod_{k=1}^r i({\underline}{F}_k)$.

We also define for such $R$: $$\begin{aligned}
j({\underline}{F}) := \left \{ \begin{array}{l l}
{\text{Pic~}}(R)/m                                                          &  {\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m) \\
\ker \left( {\text{Pic~}}(R)/m \xrightarrow{N^{(1)}/m} {\text{Pic~}}({\mathcal{O}_S})/m \right)  &  {\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m) \\
\end{array}\right.  \end{aligned}$$ where $N^{(1)}$ is induced by $N_{R/{\mathcal{O}_S}}$, and again $j(\prod_{k=1}^r {\underline}{F}_k) := \prod_{k=1}^r j({\underline}{F}_k)$.

\[admissible\] We call ${\underline}{F}$ *admissible* if it is a finite direct product of the following factors:

-   $\text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$,

-   $\text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m), [R:{\mathcal{O}_S}]$ is prime to $m$,

where $R$ is any finite étale extension of ${\mathcal{O}_S}$.

After computing in Section \[Section: class set\] the cohomology sets of some related ${\mathcal{O}_S}$-groups, we observe in Section \[Section genera\] Proposition \[sequence of wG\], that if ${\underline}{F}$ is admissible then there exists an exact sequence of pointed sets: $$1 \to {\text{Cl}_S}({\underline}{G}) \xhookrightarrow{h} H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \xrightarrow{w_{{\underline}{G}}} i({\underline}{F}) \to 1.$$ We deduce in Corollary \[genera\] that $\text{gen}({\underline}{G})$ bijects to $i({\underline}{F})$. In Section \[Section genus\], Theorem \[genus isotropic\], we show that ${\text{Cl}_S}({\underline}{G})$ surjects onto $j({\underline}{F})$. If $G_S := \prod_{s \in S} G(\hat{K}_s)$ is non-compact, then this is a bijection. This leads us to formulate in Corollary \[criterion\] a necessary and sufficient condition for the *Hasse local-global principle* to hold for ${\underline}{G}$. In Section \[Section application\], we use the above results to express in Theorem \[tau G 2\] the Tamagawa number $\tau(G)$ of an almost simple $K$-group $G$ with an admissible fundamental group $F$, using the (restricted) Euler-Poincaré characterstic of some ${\mathcal{O}_S}$-model of $F$ and a local invariant, and show how this new description facilitates the computation of $\tau(G)$ when $G$ is a twisted group.

Étale cohomology {#Section: class set}
================

The class set
-------------

Consider the ring of $S$-integral adèles ${\mathbb{A}}_S := \prod_{{\mathfrak{p}}\in S} \hat{K}_{\mathfrak{p}}\times \prod_{{\mathfrak{p}}\notin S} \hat{{\mathcal{O}}}_{\mathfrak{p}}$, being a subring of the adèles ${\mathbb{A}}$. The $S$-*class set* of an affine and of finite type ${\mathcal{O}_S}$-group ${\underline}{G}$ is the set of double cosets: $${\text{Cl}_S}({\underline}{G}) := {\underline}{G}({\mathbb{A}}_S) \backslash {\underline}{G}({\mathbb{A}}) / G(K)$$ (when over each $\hat{{\mathcal{O}}}_{\mathfrak{p}}$ the above local model ${\underline}{G}_{\mathfrak{p}}$ is taken). It is finite (cf. [@BP Proposition 3.9]), and its cardinality, called the $S$-*class number* of ${\underline}{G}$, is denoted by $h_S({\underline}{G})$. According to Nisnevich ([@Nis Thm. I.3.5]) if ${\underline}{G}$ is smooth, the map ${\lambda}$ introduced in applied to it forms the following exact sequence of pointed-sets (when the trivial coset is considered as the distinguished point in ${\text{Cl}_S}({\underline}{G})$): $$\label{Nis} 
1 \to {\text{Cl}_S}({\underline}{G}) \to H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \xrightarrow{{\lambda}} H^1(K,G) \times \prod_{{\mathfrak{p}}\notin S} H^1_{\text{\'et}}(\hat{{\mathcal{O}}}_{\mathfrak{p}},{\underline}{G}_{\mathfrak{p}}).$$ The left exactness reflects the fact that ${\text{Cl}_S}({\underline}{G})$ can be identified with the principal genus of ${\underline}{G}$.

If, furthermore, ${\underline}{G}$ has the property: $$\label{property}
\forall {\mathfrak{p}}\notin S : \ \ H^1_{\text{\'et}}(\hat{{\mathcal{O}}}_{\mathfrak{p}},{\underline}{G}_{\mathfrak{p}}) \hookrightarrow H^1_{\text{\'et}}(\hat{K}_{\mathfrak{p}},G_{\mathfrak{p}}),$$ then sequence is simplified to (cf. [@Nis Cor. 3.6]): $$\label{Nis simple}
1 \to {\text{Cl}_S}({\underline}{G}) \to H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \xrightarrow{{\lambda}_K} H^1(K,G),$$ which indicates that any two ${\underline}{G}$-torsors share the same genus if and only if they are $K$-isomorphic. If ${\underline}{G}$ has connected fibers, then by Lang’s Theorem $H^1_{\text{\'et}}(\hat{{\mathcal{O}}}_{\mathfrak{p}},{\underline}{G}_{\mathfrak{p}})$ vanishes for any prime ${\mathfrak{p}}$ (see [@Ser Ch.VI, Prop.5] and recall that all residue fields are finite), thus ${\underline}{G}$ has property .

\[Picard group is finite\] The multiplicative ${\mathcal{O}_S}$-group ${\underline}{{\mathbb{G}}}_m$ admits property thus sequence , in which the rightmost term vanishes by Hilbert 90 Theorem. Hence the class set ${\text{Cl}_S}({\underline}{{\mathbb{G}}}_m)$, being finite as previously mentioned, is bijective as a pointed-set to $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{{\mathbb{G}}}_m)$, which is identified with ${\text{Pic~}}({\mathcal{O}_S})$ (cf. [@Mil1 Cha.III,§4]) thus being finite too. This holds true for any finite étale extension $R$ of ${\mathcal{O}_S}$.

\[disconnected\] If ${\underline}{G}$ (locally of finite presentation) is disconnected but its connected component ${\underline}{G}^0$ is reductive and ${\underline}{G}/{\underline}{G}^0$ is a finite representable group, then it admits again property (see the proof of Proposition 3.14 in [@CGP]), thus sequence as well. If, furthermore, for any $[{\underline}{G}'] \in {\text{Cl}_S}({\underline}{G})$, the map $G'(K) \to (G'/(G')^0)(K)$ is surjective, then ${\text{Cl}_S}({\underline}{G}) = {\text{Cl}_S}({\underline}{G}^0)$ (cf. [@Bit3 Lemma 3.2]).

\[H1=1 sc\] Let ${\underline}{G}$ be a smooth and affine ${\mathcal{O}_S}$-group scheme with connected fibers. Suppose that its generic fiber $G$ is almost simple, simply connected and $G_S$ is non-compact. Then $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})=1$.

The proof, basically relying on the strong approximation property related to $G$, is the one of Lemma 3.2 in [@Bit1], replacing $\{{\infty}\}$ by $S$.

The fundamental group: the quasi-split case
-------------------------------------------

The following is the Shapiro Lemma for the étale cohomology:

\[Shapiro\] Let $f:R \to S$ be a finite étale extension of schemes and ${\Gamma}$ a smooth $R$-module. Then $\forall p: \ H^p_{\text{\'et}}(S,\text{Res}_{R/S}({\Gamma})) \cong H^p_{\text{\'et}}(R,{\Gamma})$.

(See [@SGA4 VIII, Cor. 5.6] in which the Leray spectral sequence for $R/S$ degenerates, whence the edge morphism $H^p_{\text{\'et}}(S,\text{Res}_{R/S}({\Gamma})) \to H^p_{\text{\'et}}(R,{\Gamma})$ is an isomorphism.)

\[finite etale extension is embedded in generic fiber\] As $C$ is smooth, ${\text{Spec} \,}{\mathcal{O}_S}$ is normal, i.e., is integrally closed locally everywhere, thus any finite étale covering of ${\mathcal{O}_S}$ arises by its normalization in some separable unramified extension of $K$ (e.g., [@Len Theorem 6.13]).

Assume ${\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$, $R$ is finite étale over ${\mathcal{O}_S}$. Then the Shapiro Lemma (\[Shapiro\]) with $p=2$ gives $H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \cong H^2_{\text{\'et}}(R,{\underline}{\mu}_m)$. Étale cohomology applied to the Kummer sequence over $R$ $$\label{R Kummer} 
1 \to {\underline}{\mu}_m \to {\underline}{{\mathbb{G}}}_m \xrightarrow{x \mapsto x^m} {\underline}{{\mathbb{G}}}_m \to 1$$ gives rise to the exact sequences of abelian groups: $$\begin{aligned}
 \label{Kummer mu2 H1} 
1 &\to H^0_{\text{\'et}}(R,{\underline}{\mu}_m)  \to  R^\times                         \xrightarrow{\times m}   (R^\times)^m  \to 1,  \\ \nonumber
1 &\to R^\times/(R^\times)^m  \to  H^1_{\text{\'et}}(R,{\underline}{\mu}_m)            \to                    {\text{Pic~}}(R)[m] \to 1,  \\ \nonumber
1 &\to {\text{Pic~}}(R)/m              \to  H^2_{\text{\'et}}(R,{\underline}{\mu}_m)            \xrightarrow{i_*}      {{\mathrm{Br}}}(R)[m]  \to 1,  
$$ in which as above ${\text{Pic~}}(R)$ is identified with $H^1_{\text{\'et}}(R,{\underline}{{\mathbb{G}}}_m)$, and the Brauer group ${{\mathrm{Br}}}(R)$ – classifying Azumaya $R$-algebras – is identified with $H^2_{\text{\'et}}(R,{\underline}{{\mathbb{G}}}_m)$ (cf. [@Mil1 Cha.IV, §2]).

The fundamental group: the non quasi-split case {#subsection nqs}
-----------------------------------------------

The group ${\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$ fits into the short exact sequence of smooth ${\mathcal{O}_S}$-groups (recall ${\underline}{\mu}_m$ is assumed to be smooth as $m$ is prime to $\text{char}(K)$): $$1 \to {\underline}{F} \to \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m) \xrightarrow{N_{R/{\mathcal{O}_S}}} {\underline}{\mu}_m \to 1$$ which yields by étale cohomology together with Shapiro’s isomorphism the long exact sequence: $$\label{LES nqs} 
 ... \to H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{I^{(r)}} H^r_{\text{\'et}}(R,{\underline}{\mu}_m) \xrightarrow{N^{(r)}} H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m) \to H^{r+1}_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \to ... \ .$$

\[\[m\] and /m\] For a group homomorphism $f:A \to B$, we denote by $f/m:A/m \to B/m$ and $f[m]:A[m] \to B[m]$ the canonical maps induced by $f$.

\[N surjective\] If $[R:{\mathcal{O}_S}]$ is prime to $m$, then $N^{(r)},N^{(r)}[m]$ and $N^{(r)}/m$ are surjective for all $r \geq 0$. In particular, if ${\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}^{(1)}({\underline}{\mu}_m)$, then sequence \[LES nqs\] induces an exact sequence of abelian groups for every $r \geq 0$: $$\label{degree is prime to m}
1 \to H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{I^{(r)}} H^r_{\text{\'et}}(R,{\underline}{\mu}_m) \xrightarrow{N^{(r)}} H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m) \to 1.$$

The composition of the induced norm $N_{R/{\mathcal{O}_S}}$ with the diagonal morphism coming from the Weil restriction $$\label{composition} 
{\underline}{\mu}_{m,{\mathcal{O}_S}} \to \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_{m,R}) \xrightarrow{N_{R/{\mathcal{O}_S}}} {\underline}{\mu}_{m,{\mathcal{O}_S}}$$ is the multiplication by $n := [R:{\mathcal{O}_S}]$. It induces for every $r \geq 0$ the maps: $$\label{N} 
H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m) \to H^r_{\text{\'et}}(R,{\underline}{\mu}_m) \xrightarrow{N^{(r)}} H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m)$$ whose composition is again the multiplication by $n$ on $H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m)$, being an automorphism when $n$ is prime to $m$. Hence $N^{(r)}$ is surjective for all $r \geq 0$.

Replacing ${\underline}{\mu}_m$ with ${\underline}{{\mathbb{G}}}_m$ in sequence and taking the $m$-torsion subgroups of the resulting cohomology sets, we get the group maps: $$H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{{\mathbb{G}}}_m)[m] \to H^r_{\text{\'et}}(R,{\underline}{{\mathbb{G}}}_m)[m] \xrightarrow{N^{(r)}[m]} H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{{\mathbb{G}}}_m)[m]$$ whose composition is multiplication by $n$ on $H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{{\mathbb{G}}}_m)[m]$, being an automorphism again as $n$ is prime to $m$, whence $N^{(r)}[m]$ is an epimorphism for every $r \geq 0$. The same argument applied to $N^{(r)}/m$ shows it is surjective for every $r \geq 0$ as well.

Back to the general case ($[R:{\mathcal{O}_S}]$ does not have to be prime to $m$), applying the Snake lemma to the exact and commutative diagram of abelian groups: $$\label{N^2 diagram} 
\xymatrix{                     
1 \ar[r] & {\text{Pic~}}(R)/m  \ar[r] \ar[d]^{N^{(1)}/m} & H^2_{\text{\'et}}(R,{\underline}{\mu}_m) \ar[r]^{i_*} \ar[d]^{N^{(2)}} & {{\mathrm{Br}}}(R)[m]   \ar[r] \ar[d]^{N^{(2)}[m]} & 1 \\ 
1 \ar[r] & {\text{Pic~}}({\mathcal{O}_S})/m \ar[r]       & H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m) \ar[r]                      & {{\mathrm{Br}}}({\mathcal{O}_S})[m] \ar[r]        & 1
}$$ yields an exact sequence of $m$-torsion abelian groups: $$\begin{aligned}
 \label{i_*'} 
1 &\to \ker({\text{Pic~}}(R)/m \xrightarrow{N^{(1)}/m} {\text{Pic~}}({\mathcal{O}_S})/m) \to \ker(N^{(2)}) \xrightarrow{i_*'} \ker({{\mathrm{Br}}}(R)[m] \xrightarrow{N^{(2)}[m]} {{\mathrm{Br}}}({\mathcal{O}_S})[m]) \\ \nonumber 
  &\to \operatorname{coker}({\text{Pic~}}(R)/m \xrightarrow{N^{(1)}/m} {\text{Pic~}}({\mathcal{O}_S})/m),  \end{aligned}$$ where $i_*'$ is the restriction of $i_*$ to $\ker(N^{(2)})$. Together with the surjection $I^{(2)}:H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) {\twoheadrightarrow}\ker(N^{(2)})$ coming from sequence , we get the commutative diagram: $$\label{nqs diagram}
\xymatrix{                     
                                                                 & \ker \left({\text{Pic~}}(R)/m \to {\text{Pic~}}({\mathcal{O}_S})/m \right) \ar@{^{(}->}[d]      \\     
H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \ar@{->>}[r]^-{I^{(2)}} \ar[rd]_-{i_*^{(1)}} & \ker \left(H^2_{\text{\'et}}(R,{\underline}{\mu}_m) \xrightarrow{N^{(2)}} H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m) \right) \ar[d]^{i_*'}   \\ 
                                                                 & \ker \left({{\mathrm{Br}}}(R)[m] \to {{\mathrm{Br}}}({\mathcal{O}_S})[m] \right).   
}$$

\[i\_\*’ surjective\] If $[R:{\mathcal{O}_S}]$ is prime to $m$, then there exists a canonical exact sequence of abelian groups $$1 \to \ker \left({\text{Pic~}}(R)/m \to {\text{Pic~}}({\mathcal{O}_S})/m \right) \to \ker(N^{(2)}) \xrightarrow{i_*'} \ker \left({{\mathrm{Br}}}(R)[m] \to {{\mathrm{Br}}}({\mathcal{O}_S})[m] \right) \to 1.$$

This sequence is the column in diagram since Lemma \[N surjective\] shows the surjectivity of $N^{(1)}/m$, which in turn implies the surjectivity of $i_*'$ by the exactness of sequence .

Recall the definition of $i({\underline}{F})$ (Def. \[i\]), and of the maps $i_*$ and $i_*^{(1)}$ (sequences and ).

\[i\*\] Let ${\underline}{F}$ be one of the basic factors of an admissible fundamental group (see Def. \[admissible\]). The map ${\overline}{i}_*:H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \to i({\underline}{F})$ is defined as: $$\begin{aligned}
{\overline}{i}_* := \left \{ \begin{array}{l l}
i_*        &  {\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m), \\
i_*^{(1)}  &  {\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m) \ \text{and} \ ([R:{\mathcal{O}_S}],m)=1.   
\end{array}\right.   \end{aligned}$$ More generally, if ${\underline}{F} = \prod_{k=1}^r {\underline}{F}_k$ where each ${\underline}{F}_k$ is one of the above, we set it to be the composition: $${\overline}{i}_*:H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{\sim} \bigoplus_{k=1}^r H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}_k) \xrightarrow{\bigoplus_{k=1}^r ({\overline}{i}_*)_k} i({\underline}{F}) = \prod_{k=1}^r i({\underline}{F}_k).$$

\[admissible surjective\] If ${\underline}{F}$ is admissible, then there exists a short exact sequence $$\label{exact sequence} 
1 \to j({\underline}{F}) \to H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{{\overline}{i}_*} i({\underline}{F}) \to 1.$$

If ${\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$ then the sequence of the corollary is simply a restatement of the last sequence in by the definitions of $i({\underline}{F})$ and $j({\underline}{F})$ (see Definition \[i\]). On the other hand, if ${\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}^{(1)}({\underline}{\mu}_m)$ with $[R:{\mathcal{O}_S}]$ is prime to $m$, then $I^{(2)}$ induces an isomorphism of abelian groups $H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \cong \ker(N^{(2)})$ by the exactness of for $r=2$. Thus the sequence of the corollary is isomorphic to the sequence in Proposition \[i\_\*’ surjective\] again by the definitions of $j({\underline}{F})$ and $i({\underline}{F})$. The two cases considered above suffice to establish the corollary by the definition of admissible (see Def. \[admissible\]) and the definition of ${\overline}{i}_*$ (see Def. \[i\*\]).

\[Euler\] Let ${\underline}{X}$ be a constructible sheaf defined over ${\text{Spec} \,}{\mathcal{O}_S}$ and let $h_i({\underline}{X}) := |H^i_{\text{\'et}}({\mathcal{O}_S},{\underline}{X})|$. The (restricted) *Euler-Poincaré characteristic* of ${\underline}{X}$ is defined to be (cf. [@Mil2 Ch.II §2]): $$\chi_S({\underline}{X}) := \prod_{i=0}^2 h_i({\underline}{X})^{(-1)^i}.$$

\[l\] Let $R$ be a finite étale extension of ${\mathcal{O}_S}$. We define: $$\begin{aligned}
l({\underline}{F}) := \left \{ \begin{array}{l l}
{\frac}{|R^\times[m]|}{[ R^{\times}:(R^{\times})^m ]}     &  {\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m) \\ \\
{\frac}{|\ker(N^{(0)}[m])|}{|\ker(N^{(0)}/m)|}            &  {\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m).  
\end{array}\right.  \end{aligned}$$ As usual, for ${\underline}{F} = \prod_{k=1}^r {\underline}{F}_k$ where each ${\underline}{F}_k$ is one of the above, we put $l({\underline}{F})=\prod_{k=1}^r l({\underline}{F}_k)$.

\[abs almost simple non qs\] If ${\underline}{F}$ is admissible then $\chi_S({\underline}{F}) = l({\underline}{F}) \cdot |i({\underline}{F})|$.

It is sufficient to check the assertion for the two basic types of (direct) factors:\
Suppose ${\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$. Then sequences together with Shapiro’s Lemma give $$\begin{aligned}
h_i({\underline}{F}) = |H^i_{\text{\'et}}(R,{\underline}{\mu}_m)| =
\left \{ 
\begin{array}{l l}
|R^\times[m]|,                              & i=0 \\ 
\left[R^{\times}:(R^{\times})^m \right] \cdot |{\text{Pic~}}(R)[m]|, & i=1 \\
|{\text{Pic~}}(R)/m| \cdot |{{\mathrm{Br}}}(R)[m]|               & i=2.  
\end{array}\right. \end{aligned}$$ So as ${\text{Pic~}}(R)$ is finite (see Remark \[Picard group is finite\]), $|{\text{Pic~}}(R)[m]| = |{\text{Pic~}}(R)/m|$ and we get: $$\begin{aligned}
\chi_S({\underline}{F}) := {\frac}{h_0({\underline}{F}) \cdot h_2({\underline}{F})}{h_1({\underline}{F})} = {\frac}{|R^\times[m]| \cdot |{\text{Pic~}}(R)/m| \cdot |{{\mathrm{Br}}}(R)[m]|}{[ R^{\times}:(R^{\times})^m ] \cdot |{\text{Pic~}}(R)[m]|} 
              = l({\underline}{F}) \cdot |i({\underline}{F})|. \end{aligned}$$

Now suppose ${\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$ such that $[R:{\mathcal{O}_S}]$ is prime to $m$. By Lemma \[N surjective\] $N^{(r)},N^{(r)}[m]$ and $N^{(r)}/m$ are surjective for all $r \geq 0$, so the long sequence is cut into short exact sequences: $$\label{for all r}
\forall r \geq 0: \ 1 \to H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{I^{(r)}} H^r_{\text{\'et}}(R,{\underline}{\mu}_m) \xrightarrow{N^{(r)}} H^r_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m) \to 1$$ from which we see that (notice that $N^{(0)}[m]$ coincides with $N^{(0)}$): $$\label{H0(F)}
h_0({\underline}{F}) = |\ker(R^\times[m] \xrightarrow{N^{(0)}[m]} {\mathcal{O}_S}^\times[m])|.$$

The Kummer exact sequences for ${\underline}{\mu}_m$ defined over both ${\mathcal{O}_S}$ and $R$ yield the exact diagram: $$\label{N2}
\xymatrix{
1 \ar[r] & R^\times   /(R^\times)^m   \ar[r] \ar@{->>}[d]^{N^{(0)}/m} & H^1_{\text{\'et}}(R,{\underline}{\mu}_m)  \ar@{->>}[d]^{N^{(1)}} \ar[r] & {\text{Pic~}}(R)[m]  \ar@{->>}[d]^{N^{(1)}[m]} \ar[r] & 1\\ 
1 \ar[r] & {\mathcal{O}_S}^\times /({\mathcal{O}_S}^\times)^m \ar[r]                          & H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_m)                       \ar[r] & {\text{Pic~}}({\mathcal{O}_S})[m]                          \ar[r] & 1
}$$ from which we see together with sequence that: $$h_1({\underline}{F}) = |\ker(N^{(1)})| = |\ker(R^\times/(R^\times)^m \xrightarrow{N^{(0)}/m} {\mathcal{O}_S}^\times /({\mathcal{O}_S}^\times)^m)| \cdot |\ker({\text{Pic~}}(R)[m] \xrightarrow{N^{(1)}[m]} {\text{Pic~}}({\mathcal{O}_S})[m])|.$$ Similarly, by sequence and Proposition \[i\_\*’ surjective\] we find that: $$h_2({\underline}{F}) = |\ker(N^{(2)})| = |\ker({\text{Pic~}}(R)/m \xrightarrow{N^{(1)}/m} {\text{Pic~}}({\mathcal{O}_S})/m)| \cdot |\ker({{\mathrm{Br}}}(R)[m] \xrightarrow{N^{(2)}[m]} {{\mathrm{Br}}}({\mathcal{O}_S})[m])|.$$ Altogether we get: $$\begin{aligned}
\chi_{S}({\underline}{F}) = {\frac}{h_0({\underline}{F}) \cdot h_2({\underline}{F})}{h_1({\underline}{F})} = {\frac}{|\ker(N^{(0)}[m])|}{|\ker(N^{(0)}/m)|} \cdot {\frac}{|\ker(N^{(1)}[m])|}{|\ker(N^{(1)}/m)|} \cdot |\ker(N^{(2)}[m])|.                               $$ The group of units $R^\times$ is a finitely generated abelian group (cf. [@Ros Prop. 14.2]), thus the quotient $R^\times / (R^\times)^m$ is a finite group. Since ${\text{Pic~}}(R)[m]$ is also finite, $\ker(N^{(1)})$ in diagram is finite, thus $|\ker(N^{(1)})[m]| = |\ker(N^{(1)})/m|$, and we are left with: $$\chi_{S}({\underline}{F}) = {\frac}{|\ker(N^{(0)}[m])|}{|\ker(N^{(0)}/m)|} \cdot |\ker(N^{(2)}[m])|  = l({\underline}{F}) \cdot |i({\underline}{F})|.  \hfill \qedhere                              $$

The computation of $l({\underline}{F})$, for specific choices of $R$, ${\mathcal{O}_S}$ and $m$, is an interesting (and probably open) problem. For example, when ${\underline}{F}$ is not quasi-split, the denominator of this number is the order of the group of units of $R$ whose norm down to ${\mathcal{O}_S}$ is an $m$-th power of a unit in ${\mathcal{O}_S}$, modulo $(R^\times)^m$. Such computations are hard to find in the literature, if they exist at all.

The set of genera {#Section genera}
=================

From now and on we assume ${\underline}{G}$ is semisimple and that its fundamental group ${\underline}{F}$ is of order prime to $\text{char}(K)$, thus smooth. Étale cohomology applied to the universal covering of ${\underline}{G}$ $$\label{universal covering} 
1 \to {\underline}{F} \to {\underline}{G}^{\text{sc}}\to {\underline}{G} \to 1,$$ gives rise to the exact sequence of pointed-sets: $$\label{universal covering cohomology} 
H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}^{\text{sc}}) \to H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \xrightarrow{{\delta}_{{\underline}{G}}} H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F})$$ in which the co-boundary map ${\delta}_{{\underline}{G}}$ is surjective, as the domain ${\mathcal{O}_S}$ is of Douai-type, implying that $H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}^{\text{sc}})=1$ (see Definition 5.2 and Example 5.4 (iii) in [@Gon]).

\[sequence of wG\] There exists an exact sequence of pointed-sets: $$1 \to {\text{Cl}_S}({\underline}{G}) \xrightarrow{h} H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \xrightarrow{w_{{\underline}{G}}} i({\underline}{F})$$ in which $h$ is injective. If ${\underline}{F}$ is admissible, then $w_{{\underline}{G}}$ is surjective.

It is shown in [@Nis Thm. 2.8 and proof of Thm. 3.5] that there exist a canonical bijection ${\alpha}_{{\underline}{G}} : H^1_{\text{Nis}}({\mathcal{O}_S},{\underline}{G}) \cong {\text{Cl}_S}({\underline}{G})$ and a canonical injection $i_{{\underline}{G}} : H^1_{\text{Nis}}({\mathcal{O}_S},{\underline}{G}) \hookrightarrow H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ of pointed-sets (as Nisnevich’s covers are étale). Then the map $h$ of the statement is the composition $i_{{\underline}{G}} \circ {\alpha}_{{\underline}{G}}^{-1}$.

Assume ${\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$. The composition of the surjective map ${\delta}_{{\underline}{G}}$ from with Shapiro’s isomorphism and the surjective morphism $i_*$ from , is a surjective $R$-map: $$\label{witt invariant}
w_{{\underline}{G}}: H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) {  \xrightarrow[]{{\delta}_{{\underline}{G}}}\mathrel{\mkern-14mu}\rightarrow
} H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{\sim} H^2_{\text{\'et}}(R,{\underline}{\mu}_m) {  \xrightarrow[]{i_*}\mathrel{\mkern-14mu}\rightarrow
} {{\mathrm{Br}}}(R)[m].$$ On the generic fiber, since $G^{\text{sc}}:={\underline}{G}^{\text{sc}}\otimes_{{\mathcal{O}_S}} K$ is simply connected, $H^1(K,G^{\text{sc}})$ vanishes due to Harder (cf. [@Har Satz A]), as well as its other $K$-forms (this would not be true, however, if $K$ were a number field with real places). So Galois cohomology applied to the universal $K$-covering $$\label{K universal covering} 
1 \to F \to G^{\text{sc}}\to G \to 1$$ yields an embedding of pointed-sets ${\delta}_{G}:H^1(K,G) \hookrightarrow H^2(K,F)$, which is also surjective as $K$ is of Douai-type as well. The extension $R$ of ${\mathcal{O}_S}$ arises from an unramified Galois extension $L$ of $K$ by Remark \[finite etale extension is embedded in generic fiber\], and Galois cohomology applied to the Kummer exact sequence of $L$-groups $$1 \to \mu_m \to {\mathbb{G}}_m \xrightarrow{x \mapsto x^m} {\mathbb{G}}_m \to 1$$ yields, together with Shapiro’s Lemma $H^2(K,F)\cong H^2(L,{\underline}{\mu}_m)$ and Hilbert 90 Theorem, the identification $(i_*)_{L}: H^2(K,F) \cong {{\mathrm{Br}}}(L)[m]$, whence the composition $(i_*)_{L} \circ {\delta}_{G}$ is an injective $L$-map: $$w_{G}:  H^1(K,G) \xhookrightarrow{{\delta}_G} H^2(K,F) \stackrel{(i_*)_{L}}{\cong} {{\mathrm{Br}}}(L)[m].$$ Now we know due to Grothendieck that ${{\mathrm{Br}}}(R)$ is a subgroup of ${{\mathrm{Br}}}(L)$ (see [@Gro Prop. 2.1] and [@Mil1 Example 2.22, case (a)]). Altogether we retrieve the commutative diagram of pointed-sets: $$\label{Witt diagram qs}
\xymatrix{
H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})   \ar@{->>}[r]^{w_{{\underline}{G}}} \ar[d]^{{\lambda}_K}  & {{\mathrm{Br}}}(R)[m] \ar@{^{(}->}[d]^{j}  \\
H^1(K,G)              \ar@{^{(}->}[r]^{w_G}                     & {{\mathrm{Br}}}(L)[m],  
}$$ from which, together with sequence (recall ${\underline}{G}$ has connected fibers), we may observe that: $${\text{Cl}_S}({\underline}{G}) = \ker({\lambda}_K) = \ker(w_{{\underline}{G}}).$$

When ${\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$, we define the map $w_{{\underline}{G}}$ using diagram \[nqs diagram\] to be the composition $$\label{w_G nqs} 
w_{{\underline}{G}} : H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \xrightarrow{{\delta}_{{\underline}{G}}} H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{i_*^{(1)}} \ker \left({{\mathrm{Br}}}(R)[m] \xrightarrow{N^{(2)}[m]} {{\mathrm{Br}}}({\mathcal{O}_S})[m] \right)$$ being surjective by Corollary \[admissible surjective\] given that $[R:{\mathcal{O}_S}]$ is prime to $m$. On the generic fiber, Galois cohomology with Hilbert 90 Theorem give: $$w_{G}: H^1(K,G) \xhookrightarrow{{\delta}_G} H^2(K,F) \stackrel{(i^{(1)}_*)_K}{\cong} \ker \left({{\mathrm{Br}}}(L)[m] \xrightarrow{N^{(2)}_L[m]} {{\mathrm{Br}}}(K)[m] \right).$$ This time we get the commutative diagram of pointed sets: $$\label{Witt diagram nqs}
\xymatrix{
H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})   \ar[r]^-{w_{{\underline}{G}}} \ar[d]^{{\lambda}_K}  & \ker \left({{\mathrm{Br}}}(R)[m]  \xrightarrow{N^{(2)}[m]}     {{\mathrm{Br}}}({\mathcal{O}_S})[m] \right) \ar@{^{(}->}[d]^{j}  \\
H^1(K,G)              \ar@{^{(}->}[r]^-{w_G}               & \ker \left({{\mathrm{Br}}}(L)[m]  \xrightarrow{(N^{(2)}[m])_L} {{\mathrm{Br}}}(K)[m]   \right),  
}$$ from which we may deduce again that: $${\text{Cl}_S}({\underline}{G}) \stackrel{\eqref{Nis simple}}{=} \ker({\lambda}_K) = \ker(w_{{\underline}{G}}).$$

More generally, if ${\underline}{F}$ is a direct product of such basic factors, then as the cohomology sets commute with direct products, the target groups of $w_{{\underline}{G}}$ and $w_G$ become the product of the target groups of their factors, and the same argument gives the last assertion.

\[genera\] There is an injection of pointed sets $w_{{\underline}{G}}': \text{gen}({\underline}{G}) \hookrightarrow i({\underline}{F})$.\
If ${\underline}{F}$ is admissible then $w_{{\underline}{G}}'$ is a bijection. In particular if ${\underline}{F}$ is split, then $|\text{gen}({\underline}{G})| = |F|^{|S|-1}$.

The commutativity of diagrams and and the injectivity of the map $j$ in them show that $w_{{\underline}{G}}$ is constant on each fiber of ${\lambda}_K$, i.e., on the genera of ${\underline}{G}$. Thus $w_G$ induces a map (see Proposition \[sequence of wG\]): $$w_{{\underline}{G}}' : \text{gen}({\underline}{G}) \to {\operatorname{Im}}(w_{{\underline}{G}}) \subseteq i({\underline}{F}).$$ These diagrams commutativity together with the injectivity of $w_G$ imply the injectivity of $w_{{\underline}{G}}'$.

If ${\underline}{F}$ is admissible then ${\operatorname{Im}}(w_{{\underline}{G}}) = i({\underline}{F})$. In particular if ${\underline}{F}$ is split, then ${\text{gen}}({\underline}{G}) \cong \prod_{i=1}^r {{\mathrm{Br}}}({\mathcal{O}_S})[m_i]$. It is shown in the proof of [@Bit1 Lemma 2.2] that ${{\mathrm{Br}}}({\mathcal{O}_S}) = \ker \left({\mathbb{Q}}/{\mathbb{Z}}\xrightarrow{\sum_{{\mathfrak{p}}\in S}\text{Cor}_{\mathfrak{p}}} {\mathbb{Q}}/{\mathbb{Z}}\right)$ where $\text{Cor}_{\mathfrak{p}}$ is the corestriction map at ${\mathfrak{p}}$. So $|{{\mathrm{Br}}}({\mathcal{O}_S})[m_i]| = m_i^{|S|-1}$ for all $i$ and the last assertion follows.

The following table refers to ${\mathcal{O}_S}$-group schemes whose generic fibers are split, absolutely almost simple and adjoint. The right column is Corollary  \[genera\]:

[|c | c | c | ]{} Type of ${\underline}{G}$ & ${\underline}{F}$ & \# $\text{gen}({\underline}{G})$\
${^1}\text{A}_{n-1}$ & ${\underline}{\mu}_n$ & $n^{|S|-1}$\
$\text{B}_n,\text{C}_n,\text{E}_7$ & ${\underline}{\mu}_2$ & $2^{|S|-1}$\
${^1}\text{D}_n$ &

  ----------------------------------------------------------
               ${\underline}{\mu}_4, \ n=2k+1$
   ${\underline}{\mu}_2 \times {\underline}{\mu}_2, \ n=2k$
  ----------------------------------------------------------

& $4^{|S|-1}$\
${^1}\text{E}_6$ & ${\underline}{\mu}_3$ & $3^{|S|-1}$\
$\text{E}_8,\text{F}_4,\text{G}_2$ & $1$ & $1$\

\[H1G iso H2F\] Let ${\underline}{G}$ be a semisimple and almost simple ${\mathcal{O}_S}$-group not of (absolute) type $\text{A}$, then $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ bijects as a pointed-set to the abelian group $H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F})$.

Since $G^{\text{sc}}$ is not of (absolute) type $\text{A}$, it is locally isotropic everywhere ([@BT 4.3 and 4.4]), whence $\ker({\delta}_{{\underline}{G}}) \subseteq H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}^{\text{sc}})$ vanishes due to Lemma \[H1=1 sc\]. Moreover, for any ${\underline}{G}$-torsor $P$, the base-point change: ${\underline}{G} \mapsto {^P}{\underline}{G}$ defines a bijection of pointed-sets: $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \to H^1_{\text{\'et}}({\mathcal{O}_S},{^P}{\underline}{G})$ (see in Section \[Introduction\]). But ${^P}{\underline}{G}$ is an inner form of ${\underline}{G}$, thus not of type $\text{A}$ as well, hence also $H^1_{\text{\'et}}({\mathcal{O}_S},({^P}{\underline}{G})^{\text{sc}})=1$. We get that all fibers of ${\delta}_{{\underline}{G}}$ in are trivial, which together with the surjectivity of ${\delta}_{{\underline}{G}}$ amounts to the asserted bijection.

In other words, the fact that ${\underline}{G}$ is not of (absolute) type $\text{A}$ guarantees that not only $G^{\text{sc}}$, but also the universal covering of the generic fiber of inner forms of ${\underline}{G}$ of other genera are locally isotropic everywhere. This provides $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ the structure of an abelian group.

\[the same cardinality\] If ${\underline}{G}$ is not of (absolute) type $\text{A}$, then all its genera share the same cardinality.

The map $w_{{\underline}{G}}$ factors through ${\delta}_{{\underline}{G}}$ (see and ) which is a bijection of pointed-sets in this case by Lemma \[H1G iso H2F\]. So writing: $w_{{\underline}{G}} = {\overline}{w}_{{\underline}{G}} \circ {\delta}_{{\underline}{G}}$. we get due to Proposition \[sequence of wG\] the exact sequence of pointed-sets (a-priory, abelian groups): $$1 \to {\text{Cl}_S}({\underline}{G}) \to H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \xrightarrow{{\overline}{w}_{{\underline}{G}}} i({\underline}{F})$$ in which all genera, corresponding to the fibers of $w_{{\underline}{G}}$, are of the same cardinality.

Following E. Artin in [@Art], we shall say that a Galois extension $L$ of $K$ is *imaginary* if no prime of $K$ is decomposed into distinct primes in $L$.

\[imaginary\] If ${\underline}{G}$ is of (absolute) type $\text{A}$, but $S=\{{\infty}\}$, $G$ is $\hat{K}_{\infty}$-isotropic, and $F$ splits over an imaginary extension of $K$, then $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ still bijects as a pointed-set to $H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F})$.

As aforementioned, removing one closed point of a projective curve, the resulting Hasse domain has a trivial Brauer group. Thus ${{\mathrm{Br}}}({\mathcal{O}_S}={\mathcal{O}_{\{{\infty}\}}})=1$, and as $F$ splits over an imaginary extension $L = {\mathbb{F}}_q(C')$, corresponding to an étale extension $R = {\mathbb{F}}_q[C'- \{ {\infty}' \}]$ of ${\mathcal{O}_{\{{\infty}\}}}$ (see Remark \[finite etale extension is embedded in generic fiber\]) where ${\infty}'$ is the unique prime of $L$ lying above ${\infty}$, ${{\mathrm{Br}}}(R)$ remains trivial. This implies by Corollary \[genera\] that ${\underline}{G}$ has only one genus, namely, the principal one, in which the generic fibers of all representatives (being $K$-isomorphic to $G$) are isotropic at ${\infty}$. Then the resulting vanishing of $\ker({\delta}_{{\underline}{G}}) \subseteq H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}^{\text{sc}})$ due to Lemma \[H1=1 sc\] is equivalent to the injectivity of ${\delta}_{{\underline}{G}}$.

The following general framework due to Giraud (see [@CF §2.2.4]), gives an interpretation of the ${\underline}{G}$-torsors which may help us describe $w_{{\underline}{G}}$ more concretely.

\[flat classification\] Let $R$ be a scheme and $X_0$ be an $R$-form, namely, an object of a fibered category of schemes defined over $R$. Let $\textbf{Aut}_{X_0}$ be its $R$-group of automorphisms. Let $\mathfrak{Forms}(X_0)$ be the category of $R$-forms that are locally isomorphic for some topology to $X_0$ and let $\mathfrak{Tors}(\text{Aut}_{X_0})$ be the category of $\text{Aut}_{X_0}$-torsors in that topology. The functor $${\varphi}:\mathfrak{Forms}(X_0) \to \mathfrak{Tors}(\textbf{Aut}_{X_0}): \ X \mapsto \textbf{Iso}_{X_0,X}$$ is an equivalence of fibered categories.

Let $(V,q)$ be a regular quadratic ${\mathcal{O}_S}$-space of rank $n \geq 3$ and let ${\underline}{G}$ be the associated *special orthogonal group* ${{\underline}{\textbf{SO}}_q}$ (see [@Con1 Definition 1.6]). It is smooth and connected (cf. [@Con1 Theorem 1.7]), and its generic fiber is of type $\text{B}_n$ if $\text{rank}(V)$ is odd, and of type ${^1}\text{D}_n$ otherwise. In both cases ${\underline}{F} = {\underline}{\mu}_2$, so we assume $\text{char}(K)$ is odd. Any such quadratic regular ${\mathcal{O}_S}$-space $(V',q')$ of rank $n$ gives rise to a ${\underline}{G}$-torsor $P$ by $$V' \mapsto P = \textbf{Iso}_{V,V'}$$ where an isomorphism $A:V \to V'$ is a *proper* $q$-isometry, i.e., such that $q' \circ A = q$ and $\det(A)=~1$. So $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ properly classifies regular quadratic ${\mathcal{O}_S}$-spaces that are locally isomorphic to $(V,q)$ in the étale topology. Then ${\delta}_{{\underline}{G}}([P])$ is the *second Stiefel-Whitney* class of $P$ in $H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{\mu}_2)$, classifying ${\mathcal{O}_S}$-Azumaya algebras with involutions (see Def. 1, Remark 3.3 and Prop. 4.5 in [@Bit2]), and $$w_{{\underline}{G}}([{\underline}{\textbf{SO}}_{q'}]) = 
\left \{ \begin{array}{l l}
[\textbf{C}_0(q')] - [\textbf{C}_0(q)] \in {{\mathrm{Br}}}({\mathcal{O}_S})[2]  \ &  \ n \ \text{is odd} \\ \notag  
[\textbf{C}(q')] - [\textbf{C}(q)] \in {{\mathrm{Br}}}({\mathcal{O}_S})[2] &  \ n \ \text{is even} 
\end{array}\right.,$$ where $\textbf{C}(q)$ and $\textbf{C}_0(q)$ are the Clifford algebra of $q$ and its even part, respectively.

Let ${\underline}{G}={\underline}{\textbf{PGL}}_n$ for $n \geq 2$. It is smooth and connected ([@Con2 Lemma 3.3.1]) with ${\underline}{F} = {\underline}{\mu}_n$, so we assume $(\text{char}(K),n)=1$. For any projective ${\mathcal{O}_S}$-space of rank $n$, by the Skolem-Noether Theorem for unital rings (see [@Knus p.145]) ${\underline}{\textbf{PGL}}(V) = \textbf{Aut}(\text{End}_{{\mathcal{O}_S}}(V))$. It is an inner form of ${\underline}{G}$ obtained for $V = {\mathcal{O}_S}^n$. So the pointed-set $H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G})$ classifies the projective ${\mathcal{O}_S}$-modules of rank $n$ up to invertible ${\mathcal{O}_S}$-modules. Given such a projective ${\mathcal{O}_S}$-module $V$, the Azumaya ${\mathcal{O}_S}$-algebra $A = {\text{End}}_{{\mathcal{O}_S}}(V)$ of rank $n^2$ corresponds to a ${\underline}{G}$-torsor by (see [@Gir V,Remarque 4.2]): $$A \mapsto P = \textbf{Iso}_{{\underline}{M}_n,A}$$ where ${\underline}{M}_n$ is the ${\mathcal{O}_S}$-sheaf of $n \times n$ matrices. Here $w_{{\underline}{G}}([P])=[A]$ in ${{\mathrm{Br}}}({\mathcal{O}_S})[n]$.

The principal genus {#Section genus}
===================

In this section, we study the structure of the principal genus ${\text{Cl}_S}({\underline}{G})$.

\[genus isotropic\] If ${\underline}{F}$ is admissible then there exists a surjection of pointed-sets $$\psi_{{\underline}{G}} : {\text{Cl}_S}({\underline}{G}) \twoheadrightarrow j({\underline}{F}),$$ being a bijection provided that $G_S$ is non-compact (e.g., $G$ is not anisotropic of type $\text{A}$).

Combining the two epimorphisms – $w_{{\underline}{G}}$ defined in Prop. \[sequence of wG\] and ${\delta}_{{\underline}{G}}$ described in Section \[Section genera\] – together with the exact sequence , yields the exact and commutative diagram: $$\label{genus diagram}
\xymatrix{
          & 1 \ar[r]  \ar[d]             & H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \ar@{->>}[d]^{{\delta}_{{\underline}{G}}} \ar@{=}[r]  & H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{G}) \ar@{->>}[d]^{w_{{\underline}{G}}} \ar[r] & 1   \\   
1  \ar[r] & j({\underline}{F}) \ar[r]^-{\partial} & H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F}) \ar[r]^-{{\overline}{i}_*}                      & i({\underline}{F})           \ar[r]                           & 1   \\   
}$$ in which $\ker(w_{{\underline}{G}}) = {\text{Cl}_S}({\underline}{G})$. We imitate the Snake Lemma argument (the diagram terms are not necessarily all groups): for any $[H] \in {\text{Cl}_S}({\underline}{G})$ one has ${\overline}{i}_*({\delta}_{{\underline}{G}}([H]))=[0]$, i.e., ${\delta}_{{\underline}{G}}([H])$ has a $\partial$-preimage in $j({\underline}{F})$ which is unique as $\partial$ is a monomorphism of groups. This constructed map denoted $\psi_{{\underline}{G}}$ gives rise to an exact sequence of pointed-sets: $$1 \to \mathfrak{K} \to {\text{Cl}_S}({\underline}{G}) \xrightarrow{\psi_{{\underline}{G}}} j({\underline}{F}) \to 1.$$ If $G_S$ is non-compact, then for any $[{\underline}{H}] \in {\text{Cl}_S}({\underline}{G})$ the generic fiber $H$ is $K$-isomorphic to $G$ thus $H_S$ is non-compact as well, thus $\ker(H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{H}) \xrightarrow{{\delta}_{{\underline}{H}}} H^2_{\text{\'et}}({\mathcal{O}_S},{\underline}{F})) \subseteq H^1_{\text{\'et}}({\mathcal{O}_S},{\underline}{H}^{\text{sc}})$ vanishes by Lemma \[H1=1 sc\]. This means that ${\delta}_{{\underline}{G}}$ restricted to ${\text{Cl}_S}({\underline}{G})$ is an embedding, so $\mathfrak{K}=1$ and $\psi_{{\underline}{G}}$ is a bijection.

The description of ${\text{Cl}_S}({\underline}{G})$ in Theorem \[genus isotropic\] holds true also for a disconnected group ${\underline}{G}$ (where ${\underline}{F}$ is the fundamental group of ${\underline}{G}^0$), under the hypotheses of Remark \[disconnected\].

We say that the *local-global Hasse principle* holds for ${\underline}{G}$ if $h_S({\underline}{G})=1$.

This property means (when ${\underline}{G}$ is connected) that a ${\underline}{G}$-torsor is ${\mathcal{O}_S}$-isomorphic to ${\underline}{G}$ if and only if its generic fiber is $K$-isomorphic to $G$. Recall the definition of $j({\underline}{F})$ from Def. .

\[criterion\] Suppose ${\underline}{F} \cong \prod_{i=1}^r \text{Res}_{R_i/{\mathcal{O}_S}}({\underline}{\mu}_{m_i})$ where $R_i$ are finite étale extensions of ${\mathcal{O}_S}$. If $G_S$ is non-compact, then the Hasse principle holds for ${\underline}{G}$ if and only if $\forall i: (|{\text{Pic~}}(R_i)|,m_i)=1$. Otherwise ($G_S$ is compact), this principle holds for ${\underline}{G}$ only if $\forall i:(|{\text{Pic~}}(R_i)|,m_i)=1$.\
More generally, if ${\underline}{F}$ is admissible and $G_S$ is non-compact, then this principle holds for ${\underline}{G}$ provided that for each factor of the form $\text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$ or $\text{Res}^{(1)}_{R/{\mathcal{O}_S}}({\underline}{\mu}_m)$ one has: $(|{\text{Pic~}}(R)|,m)=1$.

If $C^{\text{af}}$ is an affine non-singular ${\mathbb{F}}_q$-curve of the form $y^2=x^3+ax+b$, i.e., obtained by removing some ${\mathbb{F}}_q$-rational point ${\infty}$ from an elliptic (projective) ${\mathbb{F}}_q$-curve $C$, then ${\text{Pic~}}(C^{\text{af}}) = {\text{Pic~}}({\mathcal{O}_{\{{\infty}\}}}) \cong C({\mathbb{F}}_q)$ (cf. e.g., [@Bit1 Example 4.8]). Let again ${\underline}{G} = {\underline}{\textbf{PGL}}_n$ such that $(\text{char}(K),n)=~1$. As $|S|=1$ and ${\underline}{F}$ is split, ${\underline}{G}$ admits a single genus (Corollary \[genera\]), which means that all projective ${\mathcal{O}_{\{{\infty}\}}}$-modules of rank $n$ are $K$-isomorphic. If ${\underline}{G}$ is $K$-isotropic, according to Theorem \[genus isotropic\], there are exactly $|C^{\text{af}}({\mathbb{F}}_q)/2|$ ${\mathcal{O}_{\{{\infty}\}}}$-isomorphism classes of such modules, so the Hasse principle fails for ${\underline}{G}$ if and only if $|C^{\text{af}}({\mathbb{F}}_q)|$ is even. This occurs exactly when $C^{\text{af}}$ has at least one ${\mathbb{F}}_q$-point on the $x$-axis (thus of order $2$).

On the other hand, take ${\mathcal{O}_S}={\mathbb{F}}_3[t,t^{-1}]$ obtained by removing $S=\{t,t^{-1}\}$ from the projective ${\mathbb{F}}_3$-line, and ${\underline}{G}={\underline}{\textbf{PGL}}_n$ to be rationally isotropic over ${\mathcal{O}_S}$: for example for $n=2$, it is isomorphic to the special orthogonal group of the standard split ${\mathcal{O}_S}$-form $q_3(x_1,x_2,x_3)=x_1x_2+x_3^2$. Then as $q_3$ is rationally isotropic over ${\mathcal{O}_S}$ (e.g., $q_3(1,2,1)=0$) and ${\mathcal{O}_S}$ is a UFD, according to Corollary \[criterion\] the Hasse-principle holds for ${\underline}{G}$ and there are two genera as $|F|=|S|=2$ (Cor. \[genera\]).

\[non-split D\] Let $(V,q)$ be an ${\mathcal{O}_S}$-regular quadratic form of even rank $n=2k \geq 4$ and let ${\underline}{G} = \text{Res}_{R/{\mathcal{O}_S}}({{\underline}{\textbf{SO}}_q})$ where $R$ is finite étale over ${\mathcal{O}_S}$. Then ${\underline}{F} = \text{Res}_{R/{\mathcal{O}_S}}({\underline}{\mu}_2)$, whence according to Corollary \[genera\], $\text{gen}({\underline}{G}) \cong {{\mathrm{Br}}}(R)[2]$. As $G$ and its twisted $K$-forms are $K$-isotropic (e.g., [@PR p.352]), each genus of $q$ contains exactly ${\text{Pic~}}(R)/2$ elements.

\[non-split A\] Let $C'$ be an elliptic ${\mathbb{F}}_q$-curve and $(C')^{\text{af}}:= C' - \{{\infty}'\}$. Then $R := {\mathbb{F}}_q[(C')^{\text{af}}]$ is a quadratic extension of ${\mathcal{O}_{\{{\infty}\}}}= {\mathbb{F}}_q[x]$ where ${\infty}= (1/x)$ and ${\infty}'$ is the unique prime lying above ${\infty}$, thus $L:= R \otimes_{{\mathcal{O}_{\{{\infty}\}}}} K$ is imaginary over $K$. Let ${\underline}{G} = \text{Res}_{R/{\mathcal{O}_{\{{\infty}\}}}}({\underline}{\textbf{PGL}}_m)$, $m$ is odd and prime to $q$. Then ${\underline}{F} = \text{Res}^{(1)}_{R/{\mathcal{O}_{\{{\infty}\}}}}({\underline}{\mu}_m)$ is smooth, and ${\underline}{G}$ is smooth and quasi-split as well as its generic fiber, thus is $K$-isotropic. By Remark \[imaginary\] and sequence , we get (notice that ${\mathcal{O}_{\{{\infty}\}}}$ is a PID and that ${{\mathrm{Br}}}(R)=1$): $${\text{Cl}_S}({\underline}{G}) = H^1_{\text{\'et}}({\mathcal{O}_{\{{\infty}\}}},{\underline}{G}) \cong H^2_{\text{\'et}}({\mathcal{O}_{\{{\infty}\}}},{\underline}{F}) \cong \ker({\text{Pic~}}(R)/m \to {\text{Pic~}}({\mathcal{O}_{\{{\infty}\}}})/m) = {\text{Pic~}}(R)/m.$$ Hence the Hasse-principle holds for ${\underline}{G}$ if and only if $|{\text{Pic~}}(R)|=|C'({\mathbb{F}}_q)|$ is prime to $m$.

The Tamagawa number of twisted groups {#Section application}
=====================================

In this section we start with the generic fiber. Let $G$ be a semisimple group defined over a global field $K = {\mathbb{F}}_q(C)$ with fundamental group $F$. The *Tamagawa number* $\tau(G)$ of $G$ is defined as the covolume of the group $G(K)$ in the adelic group $G(\mathbb{A})$ (embedded diagonally as a discrete subgroup), with respect to the Tamagawa measure (see [@Weil]). T. Ono has established in [@Ono] a formula for the computation of $\tau(G)$ in case $K$ is an algebraic number field, which was later proved by Behrend and Dhillon in [@BD Theorem 6.1] also in the function field case: $$\label{Ono}
\tau(G) = {\frac}{|{\widehat}{F}^{\mathfrak{g}}|}{|\Sh^1({\widehat}{F})|}$$ where ${\widehat}{F} := {\text{Hom}}(F \otimes K^s,{\mathbb{G}}_m)$, ${\mathfrak{g}}$ is the absolute Galois group of $K$, and $\Sh^1({\widehat}{F})$ is the first Shafarevitch–Tate group assigned to ${\widehat}{F}$ over $K$. As a result, if $F$ is split, then $\tau(G)=|F|$. So our main innovation, based on the above results and the following ones, would be simplifying the computation of $\tau(G)$ in case $F$ is not split, as may occur when $G$ is a twisted group.

The following construction, as described in [@BK] and briefly revised here, expresses the global invariant $\tau(G)$ using some local data. Suppose $G$ is almost simple defined over the above $K={\mathbb{F}}_q(C)$, not anisotropic of type $\text{A}$, such that $(|F|,\text{char}(K))=1$. We remove one arbitrary closed point ${\infty}$ from $C$ and refer as above to the integral domain ${\mathcal{O}_S}= {\mathcal{O}_{\{{\infty}\}}}$. At any prime ${\mathfrak{p}}\neq {\infty}$, we consider the Bruhat-Tits ${\mathcal{O}}_{\mathfrak{p}}$-model of $G_{\mathfrak{p}}$ corresponding to some special vertex in its associated building. Patching all these ${\mathcal{O}}_{\mathfrak{p}}$-models along the generic fiber results in an affine and smooth ${\mathcal{O}_{\{{\infty}\}}}$-model ${\underline}{G}$ of $G$ (see [@BK §5]). It may be locally disconnected only at places that ramify over a minimal splitting field $L$ of $G$ (cf. [@BT 4.6.22]).

Denote ${\mathbb{A}}_{\infty}:= {\mathbb{A}}_{\{{\infty}\}} =  \hat{K}_{\infty}\times \prod_{{\mathfrak{p}}\neq {\infty}} \hat{{\mathcal{O}}}_{\mathfrak{p}}\subset {\mathbb{A}}$. Then ${\underline}{G}({\mathbb{A}}_{\infty}) G(K)$ is a normal subgroup of ${\underline}{G}({\mathbb{A}})$ (cf. [@Tha Thm. 3.2 3]). The set of places $\text{Ram}_G$ that ramify in $L$ is finite, thus by the Borel density theorem (e.g., [@CM Thm. 2.4, Prop. 2.8]), ${\underline}{G}({\mathcal{O}}_{\{{\infty}\} \cup \text{Ram}_G})$ is Zariski-dense in $\prod_{{\mathfrak{p}}\in \text{Ram}_G \backslash \{ {\infty}\}} {\underline}{G}_{\mathfrak{p}}$. This implies that ${\underline}{G}({\mathbb{A}}_{\infty})G(K) = {\underline}{G}^0({\mathbb{A}}_{\infty})G(K)$, where ${\underline}{G}^0$ is the connected component of ${\underline}{G}$.

Since all fibers of $\varphi$ are isomorphic to $\ker(\varphi)$, we get a bijection of measure spaces $$\begin{aligned}
 \label{decompositionfirst}
{\underline}{G}({\mathbb{A}})/G(K) &\cong \left( {\underline}{G}({\mathbb{A}}) / {\underline}{G}({\mathbb{A}}_{\infty})G(K) \right) \times \left( {\underline}{G}({\mathbb{A}}_{\infty})G(K) / G(K) \right) \\ \notag
                &= \left( {\underline}{G}^0({\mathbb{A}}) / {\underline}{G}^0({\mathbb{A}}_{\infty})G(K) \right) \times \left( {\underline}{G}^0({\mathbb{A}}_{\infty})G(K) / G(K) \right) \\ \notag
                &\cong \text{Cl}_{\{{\infty}\}}({\underline}{G}^0) \times \left( {\underline}{G}^0({\mathbb{A}}_{\infty}) / {\underline}{G}^0({\mathbb{A}}_{\infty}) \cap G(K) \right)  \end{aligned}$$ in which the left factor cardinality is the finite index $h_{\infty}(G):= h_{\{{\infty}\}}({\underline}{G}^0)$ (see Section \[Section: class set\]), and in the right factor ${\underline}{G}^0({\mathbb{A}}_{\infty}) \cap G(K) = {\underline}{G}^0({\mathcal{O}_{\{{\infty}\}}})$. Due to the Weil conjecture stating that $\tau(G^{\text{sc}})=1$, as was recently proved in the function field case by Gaistgory and Lurie (see [@Lur (2.4)]), applying the Tamagawa measure $\tau$ on these spaces results in the Main Theorem in [@BK]:

\[tau G\] Let ${\mathfrak{g}}_{\infty}= \text{Gal}(\hat{K}_{\infty}^s/\hat{K}_{\infty})$ be the Galois absolute group, $F_{\infty}:=\ker(G^{\text{sc}}_{\infty}\to G_{\infty})$, ${\underline}{F}:= \ker({\underline}{G}^{\text{sc}}\to {\underline}{G})$ whose order is prime to $\text{char}(K)$, and ${\widehat}{F_{\infty}} := {\text{Hom}}(F_{\infty}\otimes \hat{K}_{\infty}^s,{\mathbb{G}}_{m,\hat{K}_{\infty}^s})$. Then $$\tau(G) =  h_{\infty}(G) \cdot {\frac}{t_{\infty}(G)}{j_{\infty}(G)},$$ where $t_{\infty}(G) = |{\widehat}{F_{\infty}}^{{\mathfrak{g}}_{\infty}}|$ is the number of types in one orbit of a special vertex, in the Bruhat–Tits building associated to $G_{\infty}(\hat{K}_{\infty})$, and $j_{\infty}(G) = h_1({\underline}{F}) / h_0({\underline}{F})$.

We adopt Definition \[admissible\] of being admissible to $F$, with a Galois extension $L/K$ replacing $R/{\mathcal{O}_S}$. If ${\underline}{G}$ is not of (absolute) type $\text{A}$ and $F$ is admissible, then due to the above results Theorem \[tau G\] can be reformulated involving the fundamental group data only:

\[tau G 2\] Let $G$ be an almost-simple group not of (absolute) type $\text{A}$ defined over $K={\mathbb{F}}_q(C)$ with an admissible fundamental group $F$ whose order is prime to $\text{char}(K)$. Then for any choice of a prime ${\infty}$ of $K$ one has: $$\tau(G) = {\frac}{\chi_{\{{\infty}\}}({\underline}{F})}{|i({\underline}{F})|} \cdot |{\widehat}{F_{\infty}}^{{\mathfrak{g}}_{\infty}}| = l({\underline}{F}) \cdot |{\widehat}{F_{\infty}}^{{\mathfrak{g}}_{\infty}}|,$$ where $\chi_{\{{\infty}\}}({\underline}{F})$ is the (restricted) Euler-Poincaré characteristic (cf. Definition \[Euler\]), $i({\underline}{F})$ and $l({\underline}{F})$ are as in Definitions \[i\] and \[l\], respectively, and the right factor is a local invariant.

If $G$ is not of (absolute) type $\text{A}$, according to Corollary \[the same cardinality\] all genera of ${\underline}{G}$ have the same cardinality. By Lemma \[H1G iso H2F\] and Corollary \[genera\] (${\underline}{F}$ is admissible as $F$ is, see Remark \[finite etale extension is embedded in generic fiber\]) we then get $$h_{\infty}(G) = |\text{Cl}_{\{{\infty}\}}({\underline}{G})| = {\frac}{|H^1_{\text{\'et}}({\mathcal{O}_{\{{\infty}\}}},{\underline}{G})|}{|\text{gen}({\underline}{G})|} = {\frac}{h_2({\underline}{F})}{|i({\underline}{F})|}.$$ Now the first asserted equality follows from Theorem \[tau G\] together with Definition \[Euler\]: $$\tau(G) = 1/j_{\infty}({\underline}{G}) \cdot h_{\infty}({\underline}{G}) \cdot t_{\infty}(G) 
        = {\frac}{h_0({\underline}{F})}{h_1({\underline}{F})} \cdot {\frac}{h_2({\underline}{F})}{|i({\underline}{F})|} \cdot |{\widehat}{F_{\infty}}^{{\mathfrak{g}}_{\infty}}|
        = {\frac}{\chi_{\{{\infty}\}}({\underline}{F})}{|i({\underline}{F})|} \cdot |{\widehat}{F_{\infty}}^{{\mathfrak{g}}_{\infty}}|. 
$$ The rest is Lemma \[abs almost simple non qs\].

\[density\] By the geometric version of Čebotarev’s density theorem (see in [@Jar]), there exists a closed point ${\infty}$ on $C$ at which $G_{\infty}$ is split. We shall call such a point a *splitting point* of $G$.

\[quasi split group\] Let $G$ be an adjoint group defined over $K={\mathbb{F}}_q(C)$ with fundamental group $F$ whose order is prime to $\text{char}(K)$ and whose splitting field is $L$. Choose some splitting point ${\infty}$ of $G$ on $C$ and let $R$ be a minimal étale extension of ${\mathcal{O}_{\{{\infty}\}}}:= {\mathbb{F}}_q[C-\{{\infty}\}]$ such that $R \otimes_{{\mathcal{O}_{\{{\infty}\}}}} K = L$. Let $N^{(0)}:R^\times \to {\mathcal{O}_{\{{\infty}\}}}^\times$ be the induced norm. Then:

-   If $G$ is of type ${^2}\text{D}_{2k}$ then $\tau(G) = {\frac}{|R^\times[2]|}{[R^\times:(R^\times)^2]} \cdot |F|$.

-   If $G$ is of type ${^{3,6}}\text{D}_4$ or ${^2}\text{E}_6$ then $\tau(G) = {\frac}{|\ker(N^{(0)}[m])|}{|\ker(N^{(0)}/m)|} \cdot |F|$ (see Notation \[\[m\] and /m\]).

In both cases if $L$ is imaginary over $K$, then $\tau(G) = |F|$.

All groups under consideration are almost simple. When $G$ is adjoint of type ${^2}\text{D}_{2k}$ then $F$ is quasi-split, and when it is adjoint both of type ${^{3,6}}\text{D}_4$ or ${^2}\text{E}_6$ then $F = \text{Res}^{(1)}_{L/K}(\mu_m)$ where $m$ is prime to $[L:K]$ (e.g., [@PR p.333]), thus $F$ is admissible. So the assertions $(1),(2)$ are just Theorem \[tau G 2\] in which as $F_{\infty}$ splits, $|{\widehat}{F_{\infty}}^{{\mathfrak{g}}_{\infty}}| = |F_{\infty}| = |F|$.

As $C$ is projective, removing a single point ${\infty}$ from it implies that ${\mathcal{O}_{\{{\infty}\}}}^\times = {\mathbb{F}}_q^\times$ (an element of ${\mathcal{O}_{\{{\infty}\}}}$ is regular at ${\infty}^{-1}$, thus its inverse is irregular there, hence not invertible in ${\mathcal{O}_{\{{\infty}\}}}$, unless it is a unit). If $L$ is imaginary, then in particular $R = {\mathbb{F}}_q[C'-\{{\infty}'\}]$ where $C'$ is a finite étale cover of $C$ and ${\infty}'$ is the unique point lying over ${\infty}$, thus still $R^\times = {\mathbb{F}}_q^\times$ being finite, whence $|R^\times[2]| = [R^\times:(R^\times)^2]$. In the cases $F$ is not quasi-split the equality $R^\times = {\mathcal{O}_{\{{\infty}\}}}^\times = {\mathbb{F}}_q^\times$ means that $N^{(0)}$ is trivial, and we are done.

[**Acknowledgements:**]{} I thank P. Gille, B. Kunyavskiĭ and U. Vishne for valuable discussions concerning the topics of the present article. I would like also to thank the anonymous referee for a careful reading and many constructive remarks.

[\[groups\]]{} E. Artin, [*Quadratische Körper im Gebiete der höheren Kongruenzen*]{}, I. Math. Z., [**19**]{} (1927), 153–206.

M. Artin, A.  Grothendieck, J.-L. Verdier, [*Théorie des Topos et Cohomologie Étale des Schémas*]{} (SGA 4) LNM, Springer (1972/1973). K. Behrend, A. Dhillon, [*Connected components of moduli stacks of torsors via Tamagawa numbers*]{}, Canad. J. Math. [**61**]{} (2009), 3–28.

R. A. Bitan, [*The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field*]{}, J. Number Theory, [**168**]{} (2016), 346–359.

R. A. Bitan, [*On the classification of quadratic forms over an integral domain of a global function field*]{}, J. Number Theory, [**180**]{} (2017), 26–44.

R. A. Bitan, [*Between the genus and the ${\Gamma}$-genus of an integral quadratic ${\Gamma}$-form*]{}, Acta Arithemtica Journal, [**181.2**]{} (2017).

R. A. Bitan, R. Köhl [*A building-theoretic approach to relative Tamagawa numbers of semisimple groups over global function fields*]{}, Funct. Approx. Comment. Math. [**53**]{}, Number 2 (2015), 215–247.

A. Borel, G. Prasad, [*Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups*]{}, Publ. Math. IHES [**69**]{} (1989), 119–171. F. Bruhat, J. Tits, [*Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée*]{}, Inst. Hautes Etudes Sci. Publ. Math. [**60**]{} (1984), 197–376.

B. Calmès, J. Fasel, [*Groupes Classiques*]{}, Panorama et Synthéses (2015).

P.-E. Caprace, N. Monod, [*Isometry groups of non-positively curved spaces: discrete subgroups*]{}, J. Topology [**2**]{} (2009), 701–746.

V. Chernousov, P. Gille, A. Pianzola, [*A classification of torsors over Laurent polynomial rings*]{}, Math. Helv. [**92**]{} (2017), no. 1, 37–55.

B. Conrad, [*Math 252. Properties of orthogonal groups*]{},\
http://math.stanford.edu/\~conrad/252Page/handouts/O(q).pdf

B. Conrad, [*Math 252. Reductive group schemes*]{}, http://math.stanford.edu/\~conrad/papers/luminysga3.pdf

M. Demazure, A. Grothendieck, [*Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes*]{}, Tome II, Réédition de SGA3, P. Gille, P. Polo (2011).

J. Giraud, [*Cohomologie non abélienne*]{}, Grundlehren math. Wiss., Springer-Verlag Berlin Heidelberg New York (1971). C. D. González-Avilés, [*Quasi-abelian crossed modules and nonabelian cohomology*]{}, J. of Algebra [**369**]{} (2012), 235–255. A. Grothendieck, [*Le groupe de Brauer III: Exemples et compléments*]{}, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam (1968), 88–188. G. Harder, [*Über die Galoiskohomologie halbeinfacher algebraischer Gruppen, III*]{}, J. Reine Angew. Math. [**274/275**]{} (1975), 125–138. M. Jarden, [*The Čebotarev density theorem for function fields: An elementary approach*]{}, Math. Ann., 261 [**4**]{} (1982), 467–475. M. A.  Knus, [*Quadratic and hermitian forms over rings, Grundlehren*]{}, der mat. Wissenschaften [**294**]{} (1991), Springer.

H. W. Lenstra, [*Galois theory for schemes*]{}, http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf J. Lurie [*Tamagawa Numbers of Algebraic Groups Over Function Fields*]{}.

J. S. Milne, [*Étale Cohomology*]{}, Princeton University Press, Princeton (1980).

J. S. Milne, [*Arithmetic Duality Theorems*]{}, Second Edition (electronic version) (2006).

Y. Nisnevich, [*Étale Cohomology and Arithmetic of Semisimple Groups*]{}, PhD thesis, Harvard University (1982). T. Ono, [*On the Relative Theory of Tamagawa Numbers*]{}, Ann. of Math. [**82**]{} (1965), 88–111. V. Platonov, A. Rapinchuk, [*Algebraic Groups and Number Theory*]{}, Academic Press, San Diego (1994). M. Rosen, [*Number Theory in Function Fields*]{}, Graduate texts in mathematics, Springer, (2000). J.-P. Serre, [*Algebraic Groups and Class Fields*]{}, Springer, Berlin (1988). A. N. Skorobogatov, [*Torsors and Rational Points*]{}, Cambridge Univ. Press, [**144**]{} (2001). N. Q. Thang, [*A Norm Principle for class groups of reductive group schemes over Dedekind rings*]{}, Vietnam J. Math., [**43**]{} (2015), Issue 2, 257–281.

A. Weil, [*Adèles and Algebraic Groups*]{}, Progress in Mathematics, Birkhauser, Basel (1982).

[^1]: The research was partially supported by the ERC grant 291612.
