Obstructions to weak approximation for reductive groups over $p$-adic function fields

TL;DR

This paper develops duality theorems for reductive groups over p-adic function fields and uses them to identify obstructions to weak approximation, linking these to unramified Galois cohomology.

Contribution

It introduces new duality results for complexes associated with reductive groups over p-adic function fields and connects obstructions to weak approximation with Galois cohomology.

Findings

Established arithmetic duality theorems for reductive groups over p-adic function fields.

Identified obstructions to weak approximation for certain reductive groups.

Linked these obstructions to unramified Galois cohomology groups.

Abstract

We establish arithmetic duality theorems for short complexes associated to reductive groups over $p$-adic function fields. Using dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split ones) and relate this obstruction to an unramified Galois cohomology group.

Equations

$${}^1(C)\times {}^1(C^{\prime })\to \mathbb{Q}/\mathbb{Z}.$$

$$\begin{array}[]{c}1\to\overline{G(K)}\to\prod_{v\in X^{(1)}}G(K_{v})\to{}^{1}_{\omega}(C^{\prime})^{D}\to{}^{1}(C)\to 1.\end{array}$$

$$(-,-):\prod _{v\in X^{(1)}}G(K_v)\times H_{\mathrm{nr}}^3(K(G),\mathbb{Q}/\mathbb{Z}(2))\to \mathbb{Q}/\mathbb{Z}.$$

$$u:{}_{\omega }^1(C^{\prime })\to H_{\mathrm{nr}}^3(K(G),\mathbb{Q}/\mathbb{Z}(2))$$

$$1\to R\to H\to G\to 1$$

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$${}^{1}(C)\colonequals\operatorname{Ker}\big{(}\mathbb{H}^{1}(K,C)\to\textstyle\prod\limits_{v\in X^{(1)}}\mathbb{H}^{1}(K_{v},C)\big{)}.$$

$${}^{1}_{S}(C)\colonequals\operatorname{Ker}\big{(}\mathbb{H}^{1}(K,C)\to\textstyle\prod\limits_{v\notin S}\mathbb{H}^{1}(K_{v},C)\big{)}.$$

$$C{\otimes }^{\mathbf{L}}{C}^{\prime }\to \mathbb{Z}(2)[3]\text{and}\mathcal{C}{\otimes }^{\mathbf{L}}{\mathcal{C}}^{\prime }\to \mathbb{Z}(2)[3].$$

$$0\to {\mathbb{H}}^i(U,\mathcal{C})/n\to {\mathbb{H}}^i(U,\mathcal{C}{\otimes }^{\mathbf{L}}\mathbb{Z}/n)\to {}_n{\mathbb{H}}^{i+1}(U,\mathcal{C})\to 0$$

$${\mathbb{H}}^i(U,\mathcal{C})\{\ell \}\times {\mathbb{H}}_c^{2-i}(U,{\mathcal{C}}^{\prime }{)}^{(\ell )}\to \mathbb{Q}/\mathbb{Z}.$$

$${\mathbb{H}}^i(U,\mathcal{C}{\otimes }^{\mathbf{L}}\mathbb{Z}/n)\times {\mathbb{H}}_c^{1-i}(U,{\mathcal{C}}^{\prime }{\otimes }^{\mathbf{L}}\mathbb{Z}/n)\to \mathbb{Q}/\mathbb{Z}$$

$${}_{{\ell }^n}{\mathbb{H}}^i(U,\mathcal{C})\times {\mathbb{H}}_c^{2-i}(U,{\mathcal{C}}^{\prime })/{\ell }^n\to \mathbb{Q}/\mathbb{Z}\text{and}{\mathbb{H}}^i(U,\mathcal{C})/{\ell }^n\times {}_{{\ell }^n}{\mathbb{H}}^{2-i}(U,{\mathcal{C}}^{\prime })\to \mathbb{Q}/\mathbb{Z}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbb{K}_{n}(U)\simeq\operatorname{Coker}\big{(}\mathbb{H}^{i-1}(U,\mathcal{C})/\ell^{n}\to\big{(}{{}_{\ell^{n}}}\mathbb{H}^{3-i}_{c}(U,\mathcal{C}^{\prime})\big{)}^{D}\big{)},$$

$$\varinjlim_{n}\mathbb{K}_{n}(U)\simeq\varinjlim_{n}\operatorname{Coker}\big{(}\mathbb{H}^{i}(U,\mathcal{C})/\ell^{n}\to\big{(}{{}_{\ell^{n}}}\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})\big{)}^{D}\big{)}.$$

$$0\to\varinjlim_{n}\mathbb{K}_{n}(U)\to\mathbb{H}^{i}(U,\mathcal{C})\{\ell\}\to\big{(}\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})^{(\ell)}\big{)}^{D}$$

$${\mathbb{H}}^1(K_v,C)\times {\mathbb{H}}^0(K_v,C^{\prime }{)}^{\wedge }\to \mathbb{Q}/\mathbb{Z}.$$

$${}_{{\ell }^n}{\mathbb{H}}^1(K_v,C)\times {\mathbb{H}}^0(K_v,C^{\prime })/{\ell }^n\to \mathbb{Q}/\mathbb{Z}.$$

$$H^0(K_v,T_2^{\prime })\to H^0(K_v,T_1^{\prime })\to {\mathbb{H}}^0(K_v,C^{\prime })\to H^1(K_v,T_2^{\prime })\to H^1(K_v,T_1^{\prime }).$$

$$H^0(K_v,T_1^{\prime }{)}^{\wedge }\to {\mathbb{H}}^0(K_v,C^{\prime }{)}^{\wedge }\to H^1(K_v,T_2^{\prime })\to H^1(K_v,T_1^{\prime })$$

$$H^0(K_v,T_2^{\prime }{)}^{\wedge }\to H^0(K_v,T_1^{\prime }{)}^{\wedge }\to {\mathbb{H}}^0(K_v,C^{\prime }{)}^{\wedge }.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${}_{{\ell }^n}{\mathbb{H}}^1(K_v,C)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathbb{H}}^0(K_v,C{)}_{\wedge }\times {\mathbb{H}}^1(K_v,C^{\prime })\to \mathbb{Q}/\mathbb{Z}.$$

$${\mathbb{H}}^1(K_v,C)\{\ell \}\times {\mathbb{H}}^0(K_v,C^{\prime }{)}^{(\ell )}\to \mathbb{Q}/\mathbb{Z}.$$

$$\big{(}\textstyle\prod\limits_{v\in X^{(1)}}\mathbb{H}^{1}(K_{v},C)\big{)}\{\ell\}\times\big{(}\textstyle\bigoplus\limits_{v\in X^{(1)}}\mathbb{H}^{0}(K_{v},C^{\prime})\big{)}^{(\ell)}\to\mathbb{Q}/\mathbb{Z}.$$

$${{}_{\ell^{n}}}\big{(}\textstyle\prod\limits_{v}\mathbb{H}^{1}(K_{v},C)\big{)}\simeq\textstyle\prod\limits_{v}\big{(}\mathbb{H}^{0}(K_{v},C^{\prime})/\ell^{n}\big{)}^{D}\simeq\big{(}\big{(}\textstyle\bigoplus\limits_{v}\mathbb{H}^{0}(K_{v},C^{\prime})\big{)}/\ell^{n}\big{)}^{D}.$$

$$\big{(}\textstyle\prod\limits_{v\in X^{(1)}}\mathbb{H}^{0}(K_{v},C)\big{)}\{\ell\}\times\big{(}\textstyle\bigoplus\limits_{v\in X^{(1)}}\mathbb{H}^{1}(K_{v},C^{\prime})\big{)}^{(\ell)}\to\mathbb{Q}/\mathbb{Z}.$$

Full text

Introduction

This article is a subsequent work of [HSSz15] on the investigation of weak approximation for a connected reductive group under certain assumptions that hold for quasi-split connected reductive groups and tori. Let $X$ be a smooth projective and geometrically integral curve defined over some $p$-adic field $k$ and let $K$ be the function field of $X$. Since each closed point $v\in X^{(1)}$ induces a discrete valuation of $K$ (recall that the local ring $\mathcal{O}_{X,v}$ is a $1$-dimensional regular local ring, hence a discrete valuation ring), the completion $K_{v}$ of $K$ with respect to $v$ makes sense and hence we can ask typical questions concerning the arithmetic of algebraic $K$-groups. For example, Harari and Szamuely studied the cohomological obstruction to the kernel of $H^{1}(K,G)\to\prod_{v\in X^{(1)}}H^{1}(K_{v},G)$ being trivial in [HSz16]*Section 6 for connected linear reductive groups. Also, Harari, Scheiderer and Szamuely described the closure of $T(K)$ in $\prod_{v\in X^{(1)}}T(K_{v})$ with respect to the product of $v$-adic topologies for a $K$-torus $T$ in [HSSz15]. We want to extend the results of [HSSz15] to a certain class of connected linear reductive groups and to homogeneous spaces over $K$ under such groups.

The first main step is to establish global duality for the Tate–Shafarevich group of short complexes of $K$-tori.

Theorem**.**

Let $C=[T_{1}\to T_{2}]$ be an arbitrary complex of $K$-tori concentrated in degree $-1$ and [math]. Let $T_{1}^{\prime}$ and $T_{2}^{\prime}$ be the respective dual torus of $T_{1}$ and $T_{2}$, and let $C^{\prime}=[T_{2}^{\prime}\to T_{1}^{\prime}]$. Let {}^{1}(C)\colonequals\operatorname{Ker}\big{(}\mathbb{H}^{1}(K,C)\to\prod_{v\in X^{(1)}}\mathbb{H}^{1}(K_{v},C)\big{)} be the Tate–Shafarevich group of the complex $C$. There is a perfect, functorial in $C$, pairing of finite groups:

[TABLE]

The development of the global duality theorem is parallel to Izquierdo’s work [Diego-these] where he considered duality theorems for groups of multiplicative type over higher local fields. Recall that a group $M$ of multiplicative type may be identified with the kernel of an epimorphism $T_{1}\to T_{2}$ of tori over the base field. As a consequence, one may use the complex $[T_{1}\to T_{2}]$ to describe the arithmetic of $M$ and part of the dualities established by Izquierdo [Diego-these] is based on the surjectivity of $T_{1}\to T_{2}$. In our context, we get rid of the surjectivity assumption on $T_{1}\to T_{2}$ but the price is to restrict ourselves to $p$-adic function fields (which are of cohomological dimension $3$). This refinement is important in our situation since the short complex of tori associated with a connected reductive group (see the paragraph below) is not an epimorphism in general. Finally, we recall that just analogous to global duality results between ${}^{1}(C)$ and ${}^{1}(\widehat{C})$ (where $\widehat{C}=[\widehat{T_{2}}\to\widehat{T_{1}}]$) over number fields [Dem11]*Théorème 5.7, we do not need the finiteness of $\operatorname{Ker}(T_{1}\to T_{2})$.

Now we consider a connected reductive linear group $G$ over the function field $K$. Following Deligne [Deligne79]*2.4.7 and Borovoi [Bor98], we consider the composite $\rho:G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}\to G$, where $G^{{\mathrm{ss}}}=\mathscr{D}G$ is the derived subgroup of $G$ (it is semi-simple) and $G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}$ is the simply connected covering of $G^{{\mathrm{ss}}}$ (it is simply connected). For a maximal torus $T$ of $G$, its inverse image $T^{\mathrm{sc}}\colonequals\rho^{-1}(T)$ is a maximal torus of $G^{\mathrm{sc}}$. To each reductive group $G$, we associate a short complex $C=[T^{\mathrm{sc}}\to T]$ of $K$-tori concentrated in degree $-1$ and [math]. The next result says that in general there is an obstruction to weak approximation for $G$ which is controlled by some sort of Tate–Shafarevich group of $C^{\prime}$.

We fix here a technical condition which is satisfied by quasi-split reductive $K$-groups (see Corollary 2.3) and $K$-tori.

Definition**.**

We say $G^{\mathrm{sc}}$ satisfies $(*)$ if it satisfies weak approximation and it contains a quasi-trivial maximal torus.

Theorem**.**

Let ${}^{1}_{\omega}(C^{\prime})$ be the group of elements in $\mathbb{H}^{1}(K,C^{\prime})$ being trivial in $\mathbb{H}^{1}(K_{v},C^{\prime})$ for all but finitely many $v\in X^{(1)}$ and let ${}^{1}_{\omega}(C^{\prime})^{D}$ be the group of homomorphisms ${}^{1}_{\omega}(C^{\prime})\to\mathbb{Q}/\mathbb{Z}$. Suppose $G$ is a connected reductive group such that $G^{\mathrm{sc}}$ satisfies $(*)$. There is an exact sequence of groups

[TABLE]

Here $\overline{G(K)}$ denotes the closure of the diagonal image of $G(K)$ in $\prod_{v\in X^{(1)}}G(K_{v})$ for the product of $v$-adic topologies.

Note that unlike the case of number field $($see Kneser [Kneser62, Kneser65-SA], Harder [Harder68] and [PR94]*Proposition 7.9$)$, currently it is not known that semi-simple simply connected groups verify weak approximation over $p$-adic function fields. However, thanks to a theorem of Thăńg [Tha96]*Theorem 1.4, quasi-split semi-simple simply connected groups do have weak approximation.

We shall see in the sequel that condition $(*)$ enables us to conclude certain maps are surjective (see Lemma 2.6 and Lemma 2.8, diagram (10) as well). Let us briefly indicate other reasons why we did not manage to drop the assumption that $G^{\mathrm{sc}}$ satisfies $(*)$.

• •

Consider a quasi-trivial connected reductive group $H$ (its derived subgroup $H^{\mathrm{sc}}\colonequals\mathscr{D}H$ is simply connected and the maximal toric quotient $H^{\mathrm{tor}}$ is quasi-trivial) over a field $L$ of arithmetic type such that $H^{\mathrm{sc}}$ satisfies weak approximation. If $L$ is a number field, then we know $H^{1}(L_{v},H^{\mathrm{sc}})=1$ for non-Archimedean places $v$ and $H^{1}(L,H^{\mathrm{sc}})\simeq\prod_{v|\infty}H^{1}(L_{v},H^{\mathrm{sc}})$ by [PR94]*Theorem 6.4 and 6.6. Subsequently the method of Sansuc [San81] (see also the proof of Lemma 2.8) implies that $H$ satisfies weak approximation as well. But if $L$ is a $p$-adic function field, we do not have the vanishing of $H^{1}(L_{v},H^{\mathrm{sc}})$ for all but finitely many places and so it is not clear that $H$ satisfies weak approximation.

• •

Suppose $G$ is semi-simple, then there is an exact sequence $1\to F\to G^{\mathrm{sc}}\to G\to 1$ with $F$ being a commutative finite étale group scheme. In this case, Sansuc’s method does not give a defect to weak approximation for $G$ for the same reason.

The exact sequence (1) tells us that the group ${}^{1}_{\omega}(C^{\prime})^{D}$ can be viewed as a defect of weak approximation for the group $G$. Actually, we may rephrase the exact sequence (1) in terms of the reciprocity obstruction to weak approximation. More precisely, there is a pairing which annihilates the closure of the diagonal image of $G(K)$ on the left:

[TABLE]

See [CT95SBB]*§4.1 for general definitions and properties of $H_{{\mathrm{nr}}}^{3}(K(G),\mathbb{Q}/\mathbb{Z}(2))$ and see [HSSz15]*pp. 18, pairing (17) for the construction of the pairing (2) above.

Theorem**.**

Let $G$ be a connected reductive group over $K$ such that $G^{\mathrm{sc}}$ satisfies $(*)$. There exists a homomorphism

[TABLE]

such that each family $(g_{v})\in\prod_{v\in X^{(1)}}G(K_{v})$ satisfying $((g_{v}),\operatorname{Im}u)=0$ under the pairing $(\ref{pairing: reciprocity obs pairing})$ lies in the closure $\overline{G(K)}$ with respect to the product topology.

More precisely, the obstruction is given by \operatorname{Im}\big{(}H^{3}(G^{c},\mu_{n}^{\operatorname{\otimes}2})\to H_{{\mathrm{nr}}}^{3}(K(G),\mathbb{Q}/\mathbb{Z}(2))\big{)} for some sufficiently large $n$.

As some sort of complement to the present paper, we will establish global duality between ${}^{0}(C)$ and ${}^{2}(C^{\prime})$, which enables one to construct a $12$-term (resp. $15$-term) Poitou–Tate exact sequence of topological abelian groups associated to the complex $C=[T_{1}\to T_{2}]$ (resp. $C\operatorname{\otimes^{\mathbf{L}}}\mathbb{Z}/n$) in an upcoming paper.

Acknowledgements. This work was done under the supervision of David Harari. I thank him for many helpful discussions and suggestions, and also for his support and patience. I thank Jean-Louis Colliot-Thélène for valuable comments. I appreciate the EDMH doctoral program for support and Université Paris-Sud for excellent conditions for research.

Notation and conventions

Unless otherwise stated, all (hyper)cohomology groups will be taken with respect to the étale topology. In particular, (hyper)cohomology groups over fields are identified with Galois (hyper)cohomology groups. We fix once and for all an algebraic closure $\overline{L}$ of a field $L$ of characteristic zero. In the sequel, varieties over $L$ will always mean separated schemes of finite type over $L$.

Abelian groups. Let $A$ be an abelian group. We shall denote by ${}_{n}A$ (resp. $A\{\ell\}$) for the $n$-torsion subgroup (resp. $\ell$-primary subgroup with $\ell$ prime) of $A$. Moreover, let $A_{\mathrm{tors}}$ be the torsion subgroup of $A$, so $A_{\mathrm{tors}}=\varinjlim{{}_{n}}A$ is the direct limit of $n$-torsion subgroups of $A$. We write $A^{\wedge}$ for the profinite completion of $A$ (that is, the inverse limit of its finite quotients), $A_{\wedge}\colonequals\varprojlim A/nA$ and $A^{(\ell)}\colonequals\varprojlim A/\ell^{n}$ for the $\ell$-adic completion with $\ell$ a prime number. A torsion abelian group $A$ is of cofinite type if ${}_{n}A$ is finite for each $n\geq 1$. If $A$ is $\ell$-primary torsion of cofinite type, then $A/\operatorname{Div}A\simeq A^{(\ell)}$ where the former group is the quotient of $A$ by its maximal divisible subgroup. For a topological abelian group $A$, we write $A^{D}\colonequals\operatorname{Hom}_{\mathrm{cont}}(A,\mathbb{Q}/\mathbb{Z})$ for the group of continuous homomorphisms.

Tori. We write $\widehat{T}$ (resp.

$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}$

$\textstyle T$ ) for the character module (resp. cocharacter module) of a $L$-torus $T$. These are finitely generated free abelian groups endowed with a $\operatorname{Gal}(\overline{L}|L)$-action, and moreover

$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}$

$\textstyle T$ is the $\mathbb{Z}$-linear dual of $\widehat{T}$. The dual torus $T^{\prime}$ of $T$ is the torus with character group

$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}$

$\textstyle T$ , that is, \widehat{T^{\prime}}={\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\displaystyle T}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\textstyle T}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.06285pt}}}}}\cr\hbox{\scriptstyle T}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.61632pt}}}}}\cr\hbox{\scriptscriptstyle T}}}}} as Galois modules. We say a torus $T$ is flasque if H^{1}(L^{\prime},{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\displaystyle T}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\textstyle T}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.06285pt}}}}}\cr\hbox{\scriptstyle T}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.61632pt}}}}}\cr\hbox{\scriptscriptstyle T}}}}})=0 for each finite extension $L^{\prime}|L$ contained in $\overline{L}$. A torus $T$ is quasi-trivial if $\widehat{T}$ admits a $\operatorname{Gal}(\overline{{L}}|{L})$-invariant $\mathbb{Z}$-basis. Equivalently, $T$ is quasi-trivial if $T\simeq\prod R_{L_{i}|L}\mathbb{G}_{m}$ for some finite extensions $L_{i}|L$ contained in $\overline{L}$, where the $R_{L_{i}|L}$ are various Weil restrictions.

Linear algebraic groups and homogeneous spaces. Let $H$ be an algebraic group defined over $L$ and let $Y$ be a smooth $L$-scheme endowed with an $H$-action. Then $Y$ is called a *homogeneous space * under $H$ if the $H(\overline{L})$-action on $Y(\overline{L})$ is transitive. A homogeneous space $Y$ under $H$ is a $K$-torsor if the action is simply transitive. By definition, reductive algebraic groups will mean connected reductive groups. If $H$ is reductive, then we denote $H^{\mathrm{ss}}\colonequals\mathscr{D}H$ for its derived subgroup (it is semi-simple). Let $H^{\mathrm{sc}}\to H^{\mathrm{ss}}$ be the universal covering of $H^{{\mathrm{ss}}}$ (it is a finite covering) with $H^{\mathrm{sc}}$ being simply connected. Finally, we say $H$ is quasi-split if it contains a Borel subgroup defined over the base field $L$. Equivalently, $H$ is quasi-split if and only if some parabolic subgroup is solvable (see [SGA3III]*Section 3.9, or [Milne17]*Chapter 17, Section I).

Fundamental diagram associated to a flasque resolution. Let $G$ be a connected reductive group over $L$. By [CT08-resolution-flasque], there is an exact sequence of connected reductive groups:

[TABLE]

called a flasque resolution of $G$, where $H$ is a quasi-trivial connected reductive group which is an extension of a quasi-trivial torus by $G^{\mathrm{sc}}$, and $R$ is a flasque $L$-torus which is central in $H$. By [CT08-resolution-flasque]*0.3, $H$ is an extension of the quasi-trivial torus $H^{\mathrm{tor}}$ by $G^{\mathrm{sc}}$, so we obtain an identification $H^{\mathrm{ss}}\simeq G^{\mathrm{sc}}$.

Recall [CT08-resolution-flasque]*pp. 94 that there is a commutative diagram associated with such a flasque resolution

[TABLE]

with exact rows and columns. In the bottom row of diagram (3), $G^{\mathrm{tor}}$ and $H^{\mathrm{tor}}$ are respective maximal toric quotient of $G$ and $H$, and kernel $M$ of $H^{\mathrm{tor}}\to G^{\mathrm{tor}}$ is a group of multiplicative type. Finally, $F$ is the kernel of $R\to H^{\mathrm{tor}}$. As the kernel of $G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}$, $F$ is finite and central in $G^{\mathrm{sc}}$. It follows that $M$ is a torus, as a quotient of the torus $R$ by the finite group $F$.

Let $\rho:G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}\to G$ be the composition and let $T\subset G$ be a maximal $L$-torus. Then $T^{\mathrm{sc}}\colonequals\rho^{-1}(T)$ is a maximal torus of $G^{\mathrm{sc}}$. Applying [CT08-resolution-flasque]*Appendice A to the maximal torus $T$, we obtain a commutative diagram

[TABLE]

with exact rows and columns, where $T_{H}\subset H$ is a maximal torus of $H$. Recall that $H^{\mathrm{tor}}$ is a quasi-trivial torus.

Special coverings. An isogeny $G_{0}\to G$ of connected reductive $L$-groups is called a special covering (see [San81]) if $G_{0}$ is the product of a semi-simple simply connected group and a quasi-trivial torus. For each reductive $L$-group $G$, there exist an integer $m\geq 1$ and a quasi-trivial $L$-torus $Q$ such that $G^{m}\times_{L}Q$ admits a special covering by [San81]*Lemme 1.10.

Motivic complexes. Let $Y$ be a variety over $L$. Bloch introduced a so-called cycle complex $z^{i}(Y,\bullet)$ in [Bloch86]. When $Y$ is smooth, we denote the étale motivic complex over $Y$ by the complex of sheaves $\mathbb{Z}(i)\colonequals z^{i}(-,\bullet)[-2i]$ on the small étale site of $Y$. For example, we have quasi-isomorphisms $\mathbb{Z}(0)\simeq\mathbb{Z}$ and $\mathbb{Z}(1)\simeq\mathbb{G}_{m}[-1]$ by [Bloch86]*Corollary 6.4. We write $A(i)\colonequals A\operatorname{\otimes^{\mathbf{L}}}\mathbb{Z}(i)$ for any abelian group $A$. Finally, [GL01:Bloch-Kato]*Theorem 1.5 gives a quasi-isomorphism $\mathbb{Z}/n\mathbb{Z}(i)\simeq\mu_{n}^{\operatorname{\otimes}i}$ where $\mu_{n}$ is concentrated in degree [math]. We shall write $\mathbb{Q}/\mathbb{Z}(i)\colonequals\varinjlim\mu_{n}^{\operatorname{\otimes}i}$ for the direct limit of the sheaves $\mu_{n}^{\operatorname{\otimes}i}$ for all $n\geq 1$.

Function fields. Throughout this article, $K$ will be the function field of a smooth proper and geometrically integral curve $X$ over a $p$-adic field. For $v\in X$, we write $\mathcal{O}_{X,v}$ for the local ring at $v$ and $\kappa(v)$ for its residue field. Since $\mathcal{O}_{X,v}$ is a discrete valuation ring for each $v\in X^{(1)}$, closed points will also be referred to places in the sequel. Moreover, $K_{v}$ (resp. $K_{v}^{h}$) will be the completion (resp. Henselization) of $K$ with respect to $v$ and $\mathcal{O}_{v}$ (resp. $\mathcal{O}_{X,v}^{h}$) will be the ring of integers in $K_{v}$ (resp. $K_{v}^{h}$). Note that the fields $K$ and $K_{v}$ have cohomological dimension $3$.

Tate–Shafarevich groups. Let $C=[T_{1}\stackrel{{\scriptstyle\rho}}{{\rightarrow}}T_{2}]$ be a short complex of $K$-tori concentrated in degree $-1$ and [math] and let $C^{\prime}=[T_{2}^{\prime}\to T_{1}^{\prime}]$ be its dual (again it is concentrated in degree $-1$ and [math]). We put

[TABLE]

For example, if $C=[0\to T]$, then $C$ is quasi-isomorphic to $T$ and we have ${}^{1}(T)\simeq{}^{1}(C)$. If $C=[T\to 0]$, then $C\simeq T[1]$ and hence ${}^{2}(T)\simeq{}^{1}(C)$. For a finite set $S$ of places, we put

[TABLE]

We write ${}^{1}_{\omega}(C)$ for locally trivial elements for all but finitely many $v$, so ${}^{1}_{\omega}(C)=\bigcup_{S}{}^{1}_{S}(C)$ with $S$ running through all finite set of places.

1 Dualities for short complexes of tori

Let $C=[T_{1}\stackrel{{\scriptstyle\rho}}{{\rightarrow}}T_{2}]$ and let $C^{\prime}=[T_{2}^{\prime}\to T_{1}^{\prime}]$. We fix some sufficiently small non-empty open subset $X_{0}$ of $X$ such that $T_{1}$ and $T_{2}$ extends to $X_{0}$-tori (in the sense of [SGA3II]) $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, respectively. The complexes $\mathcal{C}=[\mathcal{T}_{1}\to\mathcal{T}_{2}]$ and $\mathcal{C}^{\prime}=[\mathcal{T}_{2}^{\prime}\to\mathcal{T}_{1}^{\prime}]$ over $X_{0}$ are defined analogously (these short complexes are concentrated in degree $-1$ and [math]). By [Diego-these]*Lemme 1.4.3 there are respective natural pairings of complexes over $K$ and $X_{0}$:

[TABLE]

If $C=[0\to T]$ consists a single torus, then $C\operatorname{\otimes^{\mathbf{L}}}C^{\prime}\simeq T\operatorname{\otimes^{\mathbf{L}}}T^{\prime}[1]\to\mathbb{Z}(2)[3]$ is constructed in [HSz16]*pp. 4. Finally, we write $\mathbb{H}^{i}_{c}(X_{0},\mathcal{C})\colonequals\mathbb{H}^{i}(X_{0},j_{0!}\mathcal{C})$ for the compact support cohomology where $j_{0}:X_{0}\to X$ denotes the open immersion.

We begin with a list of properties of the groups under consideration in the sequel.

Lemma 1.1**.**

Let $T$ be a $K$-torus. Let $C$ and $\mathcal{C}$ as above. Let $U\subset X_{0}$ be a non-empty open subset.

1. (1)

The groups $\mathbb{H}^{i}(U,\mathcal{C}\operatorname{\otimes^{\mathbf{L}}}\mathbb{Z}/n)$ and $\mathbb{H}^{i}_{c}(U,\mathcal{C}\operatorname{\otimes^{\mathbf{L}}}\mathbb{Z}/n)$ are finite for $i\in\mathbb{Z}$. 2. (2)

The torsion groups $\mathbb{H}^{1}(U,\mathcal{C})_{\mathrm{tors}}$ and $\mathbb{H}^{1}_{c}(U,\mathcal{C})_{\mathrm{tors}}$ are of cofinite type. 3. (3)

The groups $\mathbb{H}^{i}(U,\mathcal{C})$ and $\mathbb{H}^{i}_{c}(U,\mathcal{C})$ are torsion of cofinite type for $i\geq 2$. 4. (4)

The group $H^{1}(K_{v},T)$ is finite and the groups ${}^{i}(T)$ are finite for $i=1,2$.

Proof.

1. (1)

This is proved in [Diego-these]*Proposition 1.4.4. 2. (2)

The first statement is a consequence of the sequence

[TABLE]

induced by the distinguished triangle $\mathcal{C}\to\mathcal{C}\to\mathcal{C}\operatorname{\otimes^{\mathbf{L}}}\mathbb{Z}/n\to\mathcal{C}[1]$. The same argument works for $\mathbb{H}^{1}_{c}(U,\mathcal{C})_{\mathrm{tors}}$. 3. (3)

By [HSz16]*Corollary 3.3 and Proposition 3.4(1), the groups $H^{i}(U,\mathcal{T}_{2})$ and $H^{i+1}(U,\mathcal{T}_{1})$ are torsion of cofinite type for $i\geq 2$. Now we deduce that $\mathbb{H}^{i}(U,\mathcal{C})$ is torsion by the exactness of $H^{i}(U,\mathcal{T}_{2})\to\mathbb{H}^{i}(U,\mathcal{C})\to H^{i+1}(U,\mathcal{T}_{1})$. The group $\mathbb{H}^{i}(U,\mathcal{C})$ is of cofinite type thanks to the short exact sequence $0\to\mathbb{H}^{i-1}(U,\mathcal{C})/n\to\mathbb{H}^{i-1}(U,\mathcal{C}\operatorname{\otimes^{\mathbf{L}}}\mathbb{Z}/n)\to{{}_{n}}\mathbb{H}^{i}(U,\mathcal{C})\to 0$. The same argument works for $\mathbb{H}^{i}_{c}(U,\mathcal{C})$. 4. (4)

The group $H^{1}(K_{v},T)$ is finite because it has finite exponent and is of cofinite type (see [HSz16]*Proposition 2.2 for more details). The second statement is part of [HSz16]*Proposition 3.4(2).∎

1.1 An Artin–Verdier style duality

The following result is some sort of variation of the classical Artin–Verdier duality theorem, which provides a more precise statement concerning the $\ell$-primary part.

Proposition 1.2**.**

Let $U\subset X_{0}$ be any non-empty open subset. There is a pairing with divisible left kernel for each prime number $\ell$,

[TABLE]

Proof.

First, recall [Diego-these]*Proposition 1.4.4 that there is a perfect pairing of finite groups

[TABLE]

for $i\in\mathbb{Z}$. The pairing $\mathcal{C}\operatorname{\otimes^{\mathbf{L}}}\mathcal{C}^{\prime}\to\mathbb{Z}(2)[3]$ induces a pairing $\mathbb{H}^{i}(U,\mathcal{C})\times\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})\to\mathbb{Q}/\mathbb{Z}$ by [HSz16]*Lemma 1.1. In particular, we obtain pairings

[TABLE]

which fit into the following commutative diagram with exact rows (it commutes by functoriality of the cup-product analogous to [CTH16]*Lemme 3.1):

[TABLE]

Since the middle vertical arrow is an isomorphism by the pairing (5), we obtain an isomorphism by snake lemma

[TABLE]

where \mathbb{K}_{n}(U)=\operatorname{Ker}\big{(}{{}_{\ell^{n}}}\mathbb{H}^{i}(U,\mathcal{C})\to(\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})/\ell^{n})^{D}\big{)}. Taking direct limit over all $n$ yields an isomorphism

[TABLE]

The latter limit is a quotient of the divisible group \varinjlim\big{(}{{}_{\ell^{n}}}\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})\big{)}^{D}\simeq\big{(}\varprojlim{{}_{\ell^{n}}}\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})\big{)}^{D}, so it is also divisible. Indeed, since $\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})\{\ell\}$ is a torsion group of cofinite type, so it is of the form \big{(}\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}\big{)}^{\operatorname{\otimes}r}\operatorname{\oplus}F_{\ell} where $F_{\ell}$ is a finite $\ell$-group. Thus the dual of its Tate module $\varprojlim{{}_{\ell^{n}}}\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})$ is a direct sum of copies of $\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}$, i.e. \big{(}\varprojlim{{}_{\ell^{n}}}\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})\big{)}^{D} is divisible. Being isomorphic to a quotient of the divisible group \big{(}\varprojlim{{}_{\ell^{n}}}\mathbb{H}^{2-i}_{c}(U,\mathcal{C}^{\prime})\big{)}^{D}, we see that $\varinjlim_{n}\mathbb{K}_{n}(U)$ is divisible as well. Passing to the direct limit over all $n$ yield an exact sequence (by definition of $\mathbb{K}_{n}(U)$ and exactness of direct limit) of abelian groups

[TABLE]

which guarantees the required pairing having divisible left kernel. ∎

Remark 1.3**.**

We shall see later in Theorem 1.18 that there exists a non-empty open subset $U_{0}$ of $X_{0}$ such that the induced map \mathbb{H}^{1}(U,\mathcal{C})\{\ell\}\to\big{(}\mathbb{H}^{1}_{c}(U,\mathcal{C}^{\prime})^{(\ell)}\big{)}^{D} is an isomorphism for each $U\subset U_{0}$, because the direct limit $\varinjlim_{n}\mathbb{K}_{n}(U)$ is contained in a finite group (so it vanishes as it a finite divisible group).

1.2 Local dualities for short complex of tori

In this subsection, we prove local dualities for the completion $K_{v}$ and the Henselization $K_{v}^{h}$ with respect to $v$.

Proposition 1.4** (Local dualities).**

Let $\ell$ be a prime number.

1. (1)

There is a perfect pairing functorial in $C$ between discrete and profinite groups:

[TABLE] 2. (2)

There is a perfect pairing functorial in $C$ between finite groups:

[TABLE]

Proof.

1. (1)

The distinguished triangle $T_{2}^{\prime}\to T_{1}^{\prime}\to C^{\prime}\to T_{2}^{\prime}[1]$ induces an exact sequence

[TABLE]

Since $H^{1}(K_{v},T_{2}^{\prime})$ is finite by Lemma 1.1(4), by [HSz05]*Appendix, Proposition there is an exact sequence

[TABLE]

and a complex

[TABLE]

Now the statement follows by exactly the same argument as [Diego-these]*Proposition 1.4.9(ii). 2. (2)

Consider the following exact commutative diagram

[TABLE]

with the middle vertical arrow being an isomorphism of finite groups (see the proof of [Diego-these]*Proposition 1.4.9(i)). It follows that the right vertical arrow is surjective. Moreover, we have a commutative diagram

[TABLE]

where the lower horizontal arrow is injective by (1). Therefore the upper horizontal arrow is also injective and hence it is an isomorphism. ∎

Remark 1.5**.**

By a similar argument as Proposition 1.4(2), we obtain a perfect pairing between profinite and discrete groups

[TABLE]

So we can identify $\mathbb{H}^{0}(K_{v},C)_{\wedge}$ with the profinite completion $\mathbb{H}^{0}(K_{v},C)^{\wedge}$ by Proposition 1.4(1).

Corollary 1.6**.**

Let $\ell$ be a prime number.

1. (1)

There is a perfect pairing between discrete and profinite groups:

[TABLE] 2. (2)

There is a perfect pairing between locally compact groups:

[TABLE]

More precisely, the former group is a direct limit of profinite groups and the latter is a projective limit of discrete torsion groups.

Proof.

We apply the local duality Proposition 1.4(2), i.e. the isomorphism {{}_{\ell^{n}}}\mathbb{H}^{1}(K_{v},C)\simeq\big{(}\mathbb{H}^{0}(K_{v},C^{\prime})/\ell^{n}\big{)}^{D}.

1. (1)

Passing to the direct limit over all $n$ yields the isomorphism \mathbb{H}^{1}(K_{v},C)\{\ell\}\simeq\big{(}\mathbb{H}^{0}(K_{v},C^{\prime})^{(\ell)}\big{)}^{D}. 2. (2)

Taking product over all places gives isomorphisms

[TABLE]

Thus the desired perfect pairing follows by passing to the direct limit over all $n\geq 1$.∎

Remark 1.7**.**

Analogously, there is a perfect pairing between locally compact groups

[TABLE]

More precisely, the former group is a direct limit of profinite groups and the latter is a projective limit of discrete torsion groups.

The next lemma is probably well-known:

Lemma 1.8**.**

Let $A_{1}\to A_{2}\to A_{3}\to 0$ be an exact sequence of abelian groups. If ${{}_{\ell}}A_{3}$ is finite, then $A_{1}^{(\ell)}\to A_{2}^{(\ell)}\to A_{3}^{(\ell)}\to 0$ is exact for each prime number $\ell$.

Proof.

Let’s say $f:A_{1}\to A_{2}$, $g:A_{2}\to A_{3}$ and $g_{n}:A_{2}/\ell^{n}\to A_{3}/\ell^{n}$. Thus there is a short exact sequence $0\to\operatorname{Ker}g_{n}\to A_{2}/\ell^{n}\to A_{3}/\ell^{n}\to 0$. Since $\operatorname{Ker}g_{n}$ is a quotient of $A_{1}/\ell^{n}$, $\{\operatorname{Ker}g_{n}\}$ forms a surjective system in the sense of [AM69]*Proposition 10.2 and it follows that $0\to\varprojlim\operatorname{Ker}g_{n}\to A_{2}^{(\ell)}\to A_{3}^{(\ell)}\to 0$ is exact. By the snake lemma, there is an exact sequence $0\to{{}_{\ell^{n}}}\operatorname{Ker}g\to{{}_{\ell^{n}}}A_{2}\to{{}_{\ell^{n}}}A_{3}\to(\operatorname{Ker}g)/\ell^{n}\to\operatorname{Ker}g_{n}\to 0$. But ${{}_{\ell^{n}}}A_{3}$ is finite by assumption, we conclude that $(\operatorname{Ker}g)^{(\ell)}\to\varprojlim\operatorname{Ker}g_{n}$ is surjective by Mittag–Leffler condition. Finally, let \operatorname{Ker}f_{n}\colonequals\operatorname{Ker}\big{(}A_{1}/\ell^{n}\to(\operatorname{Ker}g)/\ell^{n}\big{)}. Then $\{\operatorname{Ker}f_{n}\}$ is a surjective system (because $\operatorname{Ker}f/\ell^{n}\to\operatorname{Ker}f_{n}$ is surjective), and hence $A_{1}^{(\ell)}\to(\operatorname{Ker}g)^{(\ell)}$ is surjective. Summing up, the sequence $A_{1}^{(\ell)}\to A_{2}^{(\ell)}\to A_{3}^{(\ell)}\to 0$ is exact. ∎

Lemma 1.9**.**

Let $\ell$ be a prime number. Let $T$ be a $K$-torus and let $C=[T_{1}\to T_{2}]$ be as above.

1. (1)

The natural map $H^{0}(K_{v}^{h},T)\to H^{0}(K_{v},T)$ induces an isomorphism $H^{0}(K_{v}^{h},T)^{(\ell)}\simeq H^{0}(K_{v},T)^{(\ell)}$. Moreover, there is an isomorphism $\mathbb{H}^{0}(K_{v}^{h},C)^{(\ell)}\simeq\mathbb{H}^{0}(K_{v},C)^{(\ell)}$. 2. (2)

For $i\geq 1$, there is an isomorphism $\mathbb{H}^{i}(K_{v}^{h},C)\to\mathbb{H}^{i}(K_{v},C)$.

Proof.

1. (1)

The same argument as [Dem11]*Lemme 3.7 yields an isomorphism $H^{0}(K_{v}^{h},T)/\ell^{n}\simeq H^{0}(K_{v},T)/\ell^{n}$. Therefore the first assertion follows by passing to the inverse limit over all $n$. For the second statement, since $H^{1}(K_{v}^{h},T_{i})\simeq H^{1}(K_{v},T_{i})$ is finite for $i=1,2$, there is a commutative diagram of complexes with rows exact at the last four terms by Lemma 1.8:

[TABLE]

Now all the vertical arrows except the middle one are isomorphisms, and hence the middle one is also an isomorphism by the $5$-lemma. 2. (2)

By [HSz16]*Corollary 3.2, we know that $H^{i}(K_{v}^{h},T)\simeq H^{i}(K_{v},T)$ for each $i\geq 1$ and for each $K$-torus $T$. Thus the isomorphism $\mathbb{H}^{i}(K_{v}^{h},C)\simeq\mathbb{H}^{i}(K_{v},C)$ for each $i\geq 1$ follows after applying dévissage to the distinguished triangle $T_{1}\to T_{2}\to C\to T_{1}[1]$. ∎

Corollary 1.10**.**

There is a perfect pairing between direct limit of profinite groups and projective limit of discrete torsion groups:

[TABLE]

Proof.

The same argument as Proposition 1.4 yields a perfect pairing ${{}_{\ell^{n}}}\mathbb{H}^{1}(K_{v}^{h},C)\times\mathbb{H}^{0}(K_{v}^{h},C^{\prime})/\ell^{n}\to\mathbb{Q}/\mathbb{Z}$ of finite groups. Therefore Lemma 1.9(2) implies that $\mathbb{H}^{0}(K_{v}^{h},C^{\prime})/\ell^{n}\simeq\mathbb{H}^{0}(K_{v},C^{\prime})/\ell^{n}$. The desired perfect pairing is an immediate consequence by the same argument as Corollary 1.6(2). ∎

1.3 Global dualities for short complex of tori

The goal of this subsection is to establish a perfect pairing ${}^{1}(C)\times{}^{1}(C^{\prime})\to\mathbb{Q}/\mathbb{Z}$ between finite groups. We first prove the finiteness of ${}^{1}(C)$ and ${}^{1}(C^{\prime})$.

Lemma 1.11**.**

Let $C=[T_{1}\to T_{2}]$ be a short complex of tori. The group ${}^{1}_{\omega}(C)$ is of finite exponent.

Proof.

Let $L|K$ be a finite Galois extension that splits both $T_{1}$ and $T_{2}$. Then for $i=1,2$, the $L$-tori $T_{i,L}=T_{i}\times_{K}L$ are products of $\mathbb{G}_{m}$, and $H^{1}(L,T_{2,L})=0$ by Hilbert’s theorem $90$. The distinguished triangle $T_{1}\to T_{2}\to C\to T_{1}[1]$ induces a commutative diagram

[TABLE]

where $C_{L}=[T_{1,L}\to T_{2,L}]$. Recall [HSSz15]*Lemma 3.2(2) that ${}^{2}_{\omega}(T_{1})$ is of finite exponent, and hence a restriction-corestriction argument shows that ${}^{1}_{\omega}(C)$ is of finite exponent. ∎

The following statement on compact support hypercohomology is the analogue of [HSz16]*Proposition 3.1.

Proposition 1.12**.**

Let $U\subset X_{0}$ be a non-empty open subset.

1. (1)

Let $V\subset U$ be a further non-empty open subset. There is an exact sequence

[TABLE]

where $i_{v}:\operatorname{Spec}\kappa(v)\to U$ is the closed immersion. 2. (2)

There is an exact sequence of hypercohomology groups

[TABLE]

where $K_{v}^{h}$ is the Henselization of $K$ with respect to the place $v$ and by abuse of notation we write $C$ for the pull-back of $\mathcal{C}$ by the natural morphism $\operatorname{Spec}K_{v}^{h}\to U$. 3. (3)

There is an exact sequence for $i\geq 1$:

[TABLE] 4. (4)

(Three Arrows Lemma).* There is a commutative diagram*

[TABLE]

Proof.

Actually the proofs follow from [HSz16]*Proposition 3.1 after replacing cohomology by hypercohomology.

1. (1)

Applying [MilneEC]*III. Remark 1.30 to the open immersion $V\to U$ and the closed immersion $U\setminus V\to U$ yields the required long exact sequence for hypercohomology. 2. (2)

The long exact sequence for hypercohomology associated to the open immersion $j:U\to X$ reads as

[TABLE]

Now the same argument as [MilneADT]*II. Lemma 2.4 implies the required long exact sequence. 3. (3)

Applying Lemma 1.9(2) to the previous long exact sequence yields the desired long exact sequence. 4. (4)

There is an isomorphism of hypercohomologies $\mathbb{H}^{i}_{c}(U,\mathcal{C})=\mathbb{H}^{i}(X,j_{!}\mathcal{C})\simeq\operatorname{\mathbf{Ext}}^{i}_{X}(\mathbb{Z},j_{!}\mathcal{C})$ by [MilneEC]*III. Remark 1.6(e) (where $j:U\to X$ denotes the open immersion), thus the same argument as [HSz16]*Proposition 3.1(3) completes the proof. ∎

The following result provides both the finiteness of ${}^{1}(C)$ and a crucial point in the proof of global duality.

Lemma 1.13**.**

Let $U\subset X_{0}$ be a non-empty open subset, and put \mathrm{D}^{2}_{K}(U,\mathcal{T})=\operatorname{Im}\big{(}H^{2}_{c}(U,\mathcal{T})\to H^{2}(K,T)\big{)} for any $U$-torus $\mathcal{T}$. Then there exists a non-empty open subset $U_{2}\subset X_{0}$ such that $\mathrm{D}^{2}_{K}(U,\mathcal{T})$ is of finite exponent for any non-empty open subset $U\subset U_{2}$.

Proof.

By a restriction-corestriction argument, it will be sufficient to show that $\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})$ is of finite exponent. By [Gro68BrauerIII]*pp. 96, (2.9), there is an exact sequence

[TABLE]

Since $k$ is a $p$-adic local field, we conclude $\operatorname{Br}X/\operatorname{Br}_{0}X\simeq H^{1}(k,\operatorname{\underline{\operatorname{Pic}}}_{X/k})$ for cohomological dimension reasons. Recall that there is a canonical short exact sequence $0\to\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k}\to\operatorname{\underline{\operatorname{Pic}}}_{X/k}\to\underline{\mathbb{Z}}\to 0$, thus $H^{1}(k,\operatorname{\underline{\operatorname{Pic}}}_{X/k})$ is a quotient of $H^{1}(k,\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k})$ where $\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k}$ is an abelian variety. But $H^{1}(k,\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k})$ is dual to $\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k}(k)$111Note that $\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k}$ is the Jacobian of a curve, so it is isomorphic to its dual. by Tate duality over local fields [MilneADT]*Chapter I, Corollary 3.4, we deduce that $H^{1}(k,\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k})\simeq F_{0}\bigoplus(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\oplus r}$ with $F_{0}$ a finite abelian group by Mattuck’s theorem (see [Mattuck55] and [MilneADT]*pp. 41).

Suppose first there is a rational point $e\in X(k)$ on $X$. Let $e^{*}:\operatorname{Br}X\to\operatorname{Br}k\simeq\mathbb{Q}/\mathbb{Z}$ be the induced map and put $\operatorname{Br}_{e}X=\{\alpha\in\operatorname{Br}X\ |\ e^{*}(\alpha)=0\}$. Note that in this case the map $\operatorname{Br}k\to\operatorname{Br}X$ induced by the structural morphism $X\to\operatorname{Spec}k$ is injective and there is an isomorphism $H^{1}(k,\operatorname{\underline{\operatorname{Pic}}}^{\circ}_{X/k})\simeq H^{1}(k,\operatorname{\underline{\operatorname{Pic}}}_{X/k})$. Thus there is a split short exact sequence $0\to\operatorname{Br}_{e}X\to\operatorname{Br}X\to\operatorname{Br}k\to 0$ and consequently $\operatorname{Br}_{e}X\simeq\operatorname{Br}X/\operatorname{Br}k$. Moreover, if $e\notin U$, then \mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})\subset\operatorname{Ker}\big{(}\operatorname{Br}U\to\bigoplus_{v\notin U}\operatorname{Br}K_{v}\big{)}\subset\operatorname{Br}_{e}X. It follows that there is an injective map $\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})\to F_{0}\bigoplus(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\oplus r}$. Next, we show that there is a non-empty open subset $U_{2}\subset X_{0}$ such that the decreasing sequence $\{\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})\{\ell\}\}$ is stable for $U\subset U_{2}$. By [HSz16]*Proposition 3.4, the group $H^{2}_{c}(U,\mathbb{G}_{m})$ is of cofinite type and hence so is $\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})$. Since $F_{0}$ is finite, there exists only finitely many $\ell\neq p$ such that $\ell$ divides the order of $F_{0}$. As a consequence, there exists a non-empty open subset $U_{1}\subset X_{0}$ (which is independent of $\ell$) such that $\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})\{\ell\}=\mathrm{D}^{2}_{K}(U_{1},\mathbb{G}_{m})\{\ell\}$ holds for any non-empty open subset $U\subset U_{1}$ and for each $\ell\neq p$ by [HSz16]*Lemma 3.7. Again the decreasing sequence $\{\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})\{p\}\}_{U\subset U_{1}}$ stabilizes, so there exists some $U_{2}\subset U_{1}$ such that $\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})=\mathrm{D}^{2}_{K}(U_{2},\mathbb{G}_{m})$ for all $U\subset U_{2}$. Note that ${}^{2}(\mathbb{G}_{m})$ is the direct limit of $\mathrm{D}^{2}(U,\mathbb{G}_{m})$. Letting $U$ run through all non-empty open subsets of $U_{2}$ yields $\mathrm{D}^{2}_{K}(U_{2},\mathbb{G}_{m})={}^{2}(\mathbb{G}_{m})=0$, where the vanishing of ${}^{2}(\mathbb{G}_{m})$ is a consequence of [HSSz15]*Lemma 3.2.

In general, there exists a finite Galois extension $k^{\prime}|k$ such that $X(k^{\prime})\neq\varnothing$. Put $X_{k^{\prime}}\colonequals X\times_{k}k^{\prime}$ and $U_{k^{\prime}}\colonequals U\times_{k}k^{\prime}$. Let $K^{\prime}$ be the function field of $X_{k^{\prime}}$. For any sufficiently small non-empty open subset $U$ of $X$, we know that $\mathrm{D}^{2}_{K^{\prime}}(U_{k^{\prime}},\mathbb{G}_{m})\subset\operatorname{Br}K^{\prime}$ vanishes by the previous paragraph. Therefore a restriction-corestriction argument implies that $\mathrm{D}^{2}_{K}(U,\mathbb{G}_{m})\subset\operatorname{Br}K$ has finite exponent. ∎

Proposition 1.14**.**

We put \mathbb{D}_{K}^{1}(U,\mathcal{C})\colonequals\operatorname{Im}\big{(}\mathbb{H}^{1}_{c}(U,\mathcal{C})\to\mathbb{H}^{1}(K,C)\big{)}. Then there exists a non-empty open subset $U_{0}$ of $X_{0}$ such that

[TABLE]

for each non-empty open subset $U\subset U_{0}$. Moreover, the group ${}^{1}(C)$ is finite.

Proof.

By Lemma 1.13, the group $\mathrm{D}^{2}_{K}(U,\mathcal{T}_{1})$ is of finite exponent for $U$ sufficiently small. Since $H^{1}(K,T_{2})$ is of finite exponent, it follows that $\mathbb{D}^{1}_{K}(U,\mathcal{C})$ is of finite exponent (say $N$) by dévissage. In particular, the epimorphism $\mathbb{H}^{1}_{c}(U,\mathcal{C})\to\mathbb{D}^{1}_{K}(U,\mathcal{C})$ factors through $\mathbb{H}^{1}_{c}(U,\mathcal{C})\to\mathbb{H}^{1}_{c}(U,\mathcal{C})/N$. Recall that $\mathbb{H}^{1}_{c}(U,\mathcal{C})/N$ is a subgroup of the finite group $\mathbb{H}^{1}_{c}(U,\mathcal{C}\operatorname{\otimes^{\mathbf{L}}}\mathbb{Z}/N)$, hence its quotient $\mathbb{D}^{1}_{K}(U,\mathcal{C})$ is finite.

For non-empty open subsets $V\subset U\subset X_{0}$ of $X_{0}$, we have $\mathbb{D}_{K}^{1}(V,\mathcal{C})\subset\mathbb{D}_{K}^{1}(U,\mathcal{C})$ by covariant functoriality of $\mathbb{H}^{1}_{c}(-,\mathcal{C})$. The decreasing sequence $\{\mathbb{D}_{K}^{1}(U,\mathcal{C})\}_{U\subset X_{0}}$ of finite abelian groups must be stable, hence there exists a non-empty open subset $U_{0}$ of $X_{0}$ such that $\mathbb{D}_{K}^{1}(U,\mathcal{C})=\mathbb{D}_{K}^{1}(U_{0},\mathcal{C})$ for each non-empty open subset $U\subset U_{0}$. Note that \mathbb{D}^{1}_{K}(U,\mathcal{C})\subset\operatorname{Ker}\big{(}\mathbb{H}^{1}(K,C)\to\prod_{v\notin U}\mathbb{H}^{1}(K_{v},C)\big{)} by Proposition 1.12(3). Letting $U$ run through all non-empty open subset of $U_{0}$ implies that $\mathbb{D}_{K}^{1}(U,\mathcal{C})=\mathbb{D}_{K}^{1}(U_{0},\mathcal{C})={}^{1}(C)$. Since the former two groups are finite, so is ${}^{1}(C)$. ∎

Applying Proposition 1.14 to both $C$ and $C^{\prime}$, we obtain a non-empty open subset $U_{0}$ of $X$ such that (6) holds for both $C$ and $C^{\prime}$. In the sequel, we fix such a non-empty open subset $U_{0}$ of $X$.

We will need an auxiliary Grothendieck–Serre conjecture style result (for example, see [CSan87a]*Theorem 4.1). Let $\mathcal{O}$ be a Henselian regular local integral domain with fraction field $\Omega$ and residue field $\kappa$. Let $\mathcal{J}=[\mathcal{T}_{1}\stackrel{{\scriptstyle\rho}}{{\rightarrow}}\mathcal{T}_{2}]$ be a complex of $\mathcal{O}$-tori concentrated in degree $-1$ and [math]. Let $T_{i}=\mathcal{T}_{i}\times_{\mathcal{O}}\Omega$ be the generic fibre of $\mathcal{T}_{i}$ for $i=1,2$ and let $J=[T_{1}\stackrel{{\scriptstyle\rho}}{{\rightarrow}}T_{2}]$ be the associated complex in degree $-1$ and [math].

Proposition 1.15**.**

The natural homomorphism $\mathbb{H}^{1}(\mathcal{O},\mathcal{J})\to\mathbb{H}^{1}(\Omega,J)$ induced by $\mathcal{O}\subset\Omega$ is injective.

Proof.

Let $q:\mathcal{Q}\to\mathcal{T}_{2}$ be an epimorphism of $\mathcal{O}$-tori with $\mathcal{Q}$ being quasi-trivial (for example, we may take a flasque resolution of $\mathcal{T}_{2}$, see [CSan87a]*(1.3.3)). Let $\mathcal{M}\colonequals\mathcal{Q}\times_{\mathcal{T}_{2}}\mathcal{T}_{1}$ and let $\mathcal{Q}\times_{\mathcal{T}_{2}}\mathcal{T}_{1}\to\mathcal{T}_{2}$ be the map $(r,t_{1})\mapsto q(r)\rho(t_{1})^{-1}$. Let $\operatorname{pr}_{1}:\mathcal{M}\to\mathcal{Q}$ and $\operatorname{pr}_{2}:\mathcal{M}\to\mathcal{T}_{1}$ be the respective canonical projections. By construction of $\mathcal{M}$, we have $q\circ\operatorname{pr}_{1}=\rho\circ\operatorname{pr}_{2}$. A direct verification yields isomorphisms $\operatorname{Ker}\operatorname{pr}_{1}\simeq\operatorname{Ker}\rho$ and $\operatorname{Coker}\operatorname{pr}_{1}\simeq\operatorname{Coker}\rho$, that is, $\mathcal{J}_{0}=[\mathcal{M}\to\mathcal{Q}]$ is quasi-isomorphic to $\mathcal{J}=[\mathcal{T}_{1}\to\mathcal{T}_{2}]$. Note that $Q=\mathcal{Q}\times_{\mathcal{O}}\Omega$ is a quasi-trivial $\Omega$-torus and being faithfully flat is stable under base change, the same argument as above yields that $J_{0}=[M\to Q]$ with $M=\mathcal{M}\operatorname{\otimes}_{\mathcal{O}}\Omega$ is quasi-isomorphic to the complex $J=[T_{1}\to T_{2}]$. Thus it suffices to show $\mathbb{H}^{1}(\mathcal{O},\mathcal{J}_{0})\to\mathbb{H}^{1}(\Omega,J_{0})$ is injective.

By construction $\mathcal{Q}$ is a quasi-trivial $\mathcal{O}$-torus, thus $H^{1}(\mathcal{O},\mathcal{Q})=H^{1}(\kappa,\mathcal{Q})=0$ and $H^{1}(\Omega,Q)=0$ by Shapiro’s lemma and Hilbert’s theorem $90$. Consequently it will be sufficient to show $H^{2}(\mathcal{O},\mathcal{M})\to H^{2}(\Omega,M)$ is injective by applying dévissage to the distinguished triangle $\mathcal{M}\to\mathcal{Q}\to\mathcal{J}_{0}\to\mathcal{M}[1]$. Take an exact sequence $1\to\mathcal{M}\to\mathcal{Q}_{1}\to\mathcal{Q}_{2}\to 1$ over $\mathcal{O}$ with $\mathcal{Q}_{1}$ a quasi-trivial $\mathcal{O}$-torus and $\mathcal{Q}_{2}$ an $\mathcal{O}$-torus (for example, see [CSan87a]*pp. 158, (1.3.1)). It induces the commutative diagram below with exact rows

[TABLE]

where $Q_{i}=\mathcal{Q}_{i}\times_{\mathcal{O}}\Omega$ for $i=1,2$. The left vertical arrow is injective by [CSan87a]*Theorem 4.1 and the right one is injective by Shapiro’s lemma and the injectivity for Brauer groups by [MilneEC]*IV, Corollary 2.6, therefore the middle one is also injective. ∎

Corollary 1.16**.**

The homomorphism $\Phi_{v}:\mathbb{H}^{1}(\mathcal{O}_{U,v}^{h},\mathcal{C})\to\mathbb{H}^{1}(K_{v}^{h},C)$ induced by the inclusion $\mathcal{O}^{h}_{U,v}\to K_{v}^{h}$ is injective for $v\in U^{(1)}$.

Proof.

Taking $\mathcal{O}=\mathcal{O}_{U,v}^{h}$, $\Omega=K_{v}^{h}$ and $\mathcal{J}=\mathcal{C}$ in Proposition 1.15 yields the desired injectivity. ∎

To state a key step, we first construct a map $\bigoplus_{v\in X^{(1)}}\mathbb{H}^{0}(K_{v}^{h},C)\to\mathbb{H}^{1}_{c}(U,\mathcal{C})$ for some non-empty open subset $U$ of $X_{0}$. Take $\alpha\in\bigoplus_{v\in X^{(1)}}\mathbb{H}^{0}(K_{v}^{h},C)$ supported outside some non-empty open subset $V$ of $U$, i.e. $\alpha_{v}=0$ for $v\in V$. Applying Proposition 1.12(2) to $V$ sends $\alpha$ to $\mathbb{H}^{1}_{c}(V,\mathcal{C})$, and so $\alpha$ is sent to $\mathbb{H}^{1}_{c}(U,\mathcal{C})$ by the covariant functoriality of $\mathbb{H}^{1}_{c}(-,\mathcal{C})$. The construction is independent of the choice of $V$ by the same argument as [HSz16]*pp. 11, (12).

Proposition 1.17**.**

There is an exact sequence

[TABLE]

Proof.

The sequence is a complex by exactly the same argument of [HSz16]*Proposition 4.2. The surjectivity of the last arrow is just the definition of $\mathbb{D}_{K}^{1}(U,\mathcal{C})$. Take \alpha\in\operatorname{Ker}\big{(}\mathbb{H}^{1}_{c}(U,\mathcal{C})\to\mathbb{D}_{K}^{1}(U,\mathcal{C})\big{)} and a non-empty open subset $V\subset U$. Consider the diagram

[TABLE]

where the right vertical arrow is constructed as the composite

[TABLE]

The first isomorphism is a consequence of $H^{1}(\kappa(v),i_{v}^{*}\mathcal{T}_{i})\simeq H^{1}(\mathcal{O}_{U,v}^{h},\mathcal{T}_{i})$ for $i=1,2$ (see [MilneADT]*Chapter II, Proposition 1.1(b)) and a dévissage argument. The diagram commutes for the same reason as in the proof of [HSz16]*Proposition 4.2. Since the right vertical arrow is injective by Corollary 1.16, a diagram chasing shows that $\alpha$ comes from $\mathbb{H}^{1}_{c}(V,\mathcal{C})$. But $\alpha$ goes to zero in $H^{1}(K,C)$, we may take $V$ sufficiently small such that $\alpha$ already goes to zero in $\mathbb{H}^{1}(V,\mathcal{C})$ by Three Arrows Lemma, i.e. $\alpha$ comes from $\bigoplus_{v\notin V}\mathbb{H}^{0}(K_{v}^{h},C)$ by Proposition 1.12(2) and hence the desired sequence is indeed exact. ∎

Recall that $\mathbb{D}^{1}_{K}(U,\mathcal{C})$ is finite by Proposition 1.14, by Lemma 1.8 there is an exact sequence

[TABLE]

Now we arrive at the global duality of the short complex $C$.

Theorem 1.18**.**

There is a perfect, functorial in $C$, pairing of finite groups:

[TABLE]

Proof.

We proceed by constructing a perfect pairing of finite groups ${}^{1}(C)\{\ell\}\times{}^{1}(C^{\prime})\{\ell\}\to\mathbb{Q}/\mathbb{Z}$ for a fixed prime $\ell$. Define $\mathbb{D}^{1}_{\mathrm{sh}}(U,\mathcal{C})$ by the exact sequence $0\to\mathbb{D}^{1}_{\mathrm{sh}}(U,\mathcal{C})\to\mathbb{H}^{1}(U,\mathcal{C})\to\prod_{v\in X^{(1)}}\mathbb{H}^{1}(K_{v},C)$ for each $U\subset U_{0}$. Dualizing the exact sequence (7) yields the following commutative diagram (by a similar argument as [CTH15]*Proposition 4.3(f)) with exact rows

[TABLE]

The first vertical arrow is induced by \mathbb{H}^{1}(U,\mathcal{C})\{\ell\}\to\big{(}\mathbb{H}^{1}_{c}(U,\mathcal{C}^{\prime})^{(\ell)}\big{)}^{D} in view of the commutativity of the right square. By local duality Corollary 1.10, the right vertical arrow is an isomorphism, and it follows that the kernels of the first two vertical arrows are identified. Passing to the direct limit of the dashed arrow induces an exact sequence of abelian groups

[TABLE]

Recall that $\varinjlim\mathbb{K}_{n}(U)$ is the divisible kernel of the middle vertical arrow introduced in Proposition 1.2. Note that the second limit is just ${}^{1}(C)\{\ell\}$ by definition of $\mathbb{D}^{1}_{\mathrm{sh}}(U,\mathcal{C})$. In particular, the first limit is trivial being a divisible subgroup of a finite abelian group. Now we can conclude the following isomorphisms of finite abelian groups

[TABLE]

where the central equality holds by Proposition 1.14. Summing up, there is an injection ${}^{1}(C)\{\ell\}\to{}^{1}(C^{\prime})\{\ell\}^{D}$ and therefore the desired isomorphism ${}^{1}(C)\{\ell\}\simeq{}^{1}(C^{\prime})\{\ell\}^{D}$ follows by exchanging the role of $C$ and $C^{\prime}$. ∎

2 Obstruction to weak approximation via special covering

Let $G$ be a connected reductive group. Let $G^{{\mathrm{ss}}}$ be the derived subgroup of $G$ and let $G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}$ be the universal cover of $G^{{\mathrm{ss}}}$. We consider the composition $\rho:G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}\to G$. Let $T\subset G$ be a maximal torus over $K$ and let $T^{\mathrm{sc}}=\rho^{-1}(T)$. Recall that $T^{\mathrm{sc}}$ is a maximal torus of $G^{\mathrm{sc}}$. We apply the above dualities to the morphism $\rho:T^{\mathrm{sc}}\to T$, i.e. to the complexes $C=[T^{\mathrm{sc}}\to T]$ and $C^{\prime}=[T^{\prime}\to(T^{\mathrm{sc}})^{\prime}]$ concentrated in degree $-1$ and [math]. Recall $G^{\mathrm{sc}}$ satisfies $(*)$ if it has weak approximation and contains a quasi-trivial maximal torus.

Proposition 2.1**.**

Let $H$ be a quasi-split semi-simple simply connected group over $K$. Then $H$ satisfies weak approximation with respect to any finite set $S\subset X^{(1)}$ of places.

Proof.

Let $B$ be a Borel subgroup of $H$ defined over $K$ and let $P$ be a maximal $K$-torus contained in $B$. Applying [HSz16]*Lemma 6.7 and its proof implies that $\widehat{P}\simeq\widehat{B}\simeq\operatorname{Pic}(\overline{H/B})$ is a permutation module, i.e. $P$ is a quasi-trivial torus. Moreover, $P$ is a Levi subgroup of $B$ by [BT65]*Corollaire 3.14. Now [Tha96]*Corollary 1.5 yields a bijection from the defect of weak approximation $\prod_{v\in S}H(K_{v})/\overline{H(K)}_{S}$ for $H$ to that $\prod_{v\in S}P(K_{v})/\overline{P(K)}_{S}$ for $P$ with respect to any finite set $S$ of places. Here $\overline{H(K)}_{S}$ (resp. $\overline{P(K)}_{S}$) denotes the closure of $H(K)$ (resp. $P(K)$) in $\prod_{v\in S}H(K_{v})$ (resp. $\prod_{v\in S}P(K_{v})$) with respect to the product of $v$-adic topologies. Since $P$ is a quasi-trivial torus, by construction of Weil restriction it is an open subscheme of some affine space. Hence $P$ satisfies weak approximation and so is $H$. ∎

Proposition 2.2**.**

Let $H$ be a connected reductive group over $K$. The following are equivalent: $(1)$ $H$ is quasi-split; $(2)$ $H^{\mathrm{ss}}$ is quasi-split; $(3)$ $H^{\mathrm{sc}}$ is quasi-split.

Proof.

• •

By [SGA3III]*Proposition 6.2.8(ii), there is a one-to-one correspondence between Borel subgroups of $H$ and that of $H^{\mathrm{ss}}$, so $H$ is quasi-split if and only if $H^{\mathrm{ss}}$ is quasi-split.

• •

Suppose $H^{\mathrm{sc}}$ is quasi-split. Since the universal covering $q:H^{\mathrm{sc}}\to H^{\mathrm{ss}}$ is faithfully flat, [Milne17]*Proposition 17.68 implies that $q$ sends Borel subgroups of $H^{\mathrm{sc}}$ to Borel subgroups of $H^{\mathrm{ss}}$. In particular, $H^{\mathrm{ss}}$ is quasi-split.

• •

Suppose $H^{\mathrm{ss}}$ is quasi-split and let $B^{\mathrm{ss}}$ be a Borel subgroup of $H^{\mathrm{ss}}$. Then \big{(}q^{-1}(B^{\mathrm{ss}})\big{)}\times_{K}\overline{K} is a Borel subgroup of $H^{\mathrm{sc}}\times_{K}\overline{K}$ by [Milne17]*Proposition 17.20, i.e. $H^{\mathrm{sc}}$ is quasi-split. ∎

The following corollary says that our technical assumption $(*)$ holds if $G$ is quasi-split.

Corollary 2.3**.**

If $H$ is a quasi-split connected reductive group over $K$, then $H^{\mathrm{sc}}$ satisfies $(*)$.

Proof.

Indeed, $H^{\mathrm{sc}}$ is quasi-split by Proposition 2.2 and so it satisfies weak approximation by Proposition 2.1. Moreover, $H^{\mathrm{sc}}$ contains a quasi-trivial maximal torus by [HSz16]*Lemma 6.7. ∎

Suppose $G^{\mathrm{sc}}$ contains a quasi-trivial maximal torus $T^{\mathrm{sc}}\subset G^{\mathrm{sc}}$. Because $G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}$ is an epimorphism, the image $T^{{\mathrm{ss}}}$ of $T^{\mathrm{sc}}$ in $G^{\mathrm{ss}}$ is again a maximal torus by [Humphreys-alg-group]*§21.3, Corollary C. Therefore we may choose a maximal torus $T\subset G$ of $G$ such that $T\cap G^{{\mathrm{ss}}}=T^{{\mathrm{ss}}}$, i.e. $T^{\mathrm{sc}}=\rho^{-1}(T)$. By [Bor98]*Section 2.4, different choices of $[T^{\mathrm{sc}}\to T]$ give rise to the same hypercohomology group and thus we are allowed to chose a quasi-trivial maximal torus $T^{\mathrm{sc}}$.

Theorem 2.4**.**

Let $G$ be a connected reductive group such that $G^{\mathrm{sc}}$ satisfies $(*)$.

1. (1)

Let $S\subset X^{(1)}$ be a finite set of places. There is an exact sequence

[TABLE]

Here $\overline{G(K)}_{S}$ denotes the closure of the diagonal image of $G(K)$ in $\prod_{v\in S}G(K_{v})$ for the product topology. 2. (2)

There is an exact sequence

[TABLE]

Here $\overline{G(K)}$ denotes the closure of the diagonal image of $G(K)$ in $\prod_{v\in X^{(1)}}G(K_{v})$ for the product topology.

Example 2.5**.**

Let us first look at two special cases of the sequence (8).

1. (1)

If $G$ is semi-simple, then there is an exact sequence $1\to F\to G^{\mathrm{sc}}\to G\to 1$ with $F$ finite and central in $G^{\mathrm{sc}}$. In particular, there are exact sequences

[TABLE]

Here $F^{\prime}=\operatorname{Hom}(F,\mathbb{Q}/\mathbb{Z}(2))$ and the latter sequence is obtained from the dual isogeny of $T^{\mathrm{sc}}\to T$. Consequently there are quasi-isomorphisms $C\simeq F[1]$ and $C^{\prime}\simeq F^{\prime}[1]$, and hence ${}^{1}_{S}(C)\simeq{}^{2}_{S}(F)$ and ${}^{1}(C^{\prime})\simeq{}^{2}(F^{\prime})$. Therefore the exact sequence (8) reads as

[TABLE]

Here the second arrow is given by the composite of the coboundary map $G(K_{v})\to H^{1}(K_{v},F)$ and the local duality $H^{1}(K_{v},F)\times H^{2}(K_{v},F^{\prime})\to\mathbb{Q}/\mathbb{Z}$, and the last one is given by the global duality ${}^{2}(F)\times{}^{2}(F^{\prime})\to\mathbb{Q}/\mathbb{Z}$ for finite Galois modules (see [HSz16]*(10) and Theorem 4.4 for details). 2. (2)

If $G=T$ is a torus, then $C=[T^{\mathrm{sc}}\to T]$ is quasi-isomorphic to the complex $[0\to T]\simeq T$ and its dual $C^{\prime}=[T^{\prime}\to(T^{\mathrm{sc}})^{\prime}]$ is quasi-isomorphic to the complex $[T^{\prime}\to 0]\simeq T^{\prime}[1]$. So we have ${}^{1}(C)\simeq{}^{1}(T)$ and ${}^{1}_{S}(C^{\prime})\simeq{}^{2}_{S}(T^{\prime})$. Now the exact sequence (8) is of the following form

[TABLE]

This is the obstruction to weak approximation for tori given by Harari, Scheiderer and Szamuely in [HSSz15].

The rest of this section is devoted to the proof of Theorem 2.4. For a field $L$ and $i=0,1$, we shall denote by $\mathrm{ab}^{i}:H^{i}(L,G)\to\mathbb{H}^{i}(L,C)$ the abelianization map in the sequel. The construction of the abelianization map is set forth in [Bor98]*Section 3.

Lemma 2.6**.**

Let $L$ be a field of characteristic zero and let $G$ be a connected reductive group over $L$ such that $G^{\mathrm{sc}}$ satisfies $(*)$. Then the canonical map $\mathrm{ab}^{0}:H^{0}(L,G)\to\mathbb{H}^{0}(L,C)$ is surjective.

Proof.

Let $A\to B$ be a crossed module (see [Bor98]*Section 3) of (not necessarily abelian) $\operatorname{Gal}(\overline{L}|L)$-groups concentrated in degree $-1$ and [math]. We write $\mathbb{H}_{\mathrm{rel}}^{i}(L,[A\to B])$ for the non-abelian hypercohomology in the sense of [Bor98]*Section 3 for $-1\leq i\leq 1$. Now we view $\rho:T^{\mathrm{sc}}\to T$ and $\rho:G^{\mathrm{sc}}\to G$ as crossed modules of $\operatorname{Gal}(\overline{L}|L)$-groups concentrated in degree $-1$ and [math]. Recall $T^{\mathrm{sc}}$ is chosen to be quasi-trivial. In particular, we have a commutative diagram of crossed modules of $\operatorname{Gal}(\overline{L}|L)$-groups

[TABLE]

Applying the functor $\mathbb{H}^{0}_{\mathrm{rel}}(\operatorname{Gal}(\overline{L}|L),-)$ with values in the category of pointed sets and taking into account the identification $\mathbb{H}^{0}_{\mathrm{rel}}(\operatorname{Gal}(\overline{L}|L),[A\to 1])\simeq H^{1}(\operatorname{Gal}(\overline{L}|L),A)$ (see [Bor98]*Example 3.1.2(2)), we obtain the following commutative diagram of pointed sets

[TABLE]

By [Bor98]*Lemma 3.8.1, there are isomorphisms of pointed sets

[TABLE]

Since $T^{\mathrm{sc}}$ is quasi-trivial, we conclude that $\mathbb{H}^{0}(L,C)\to H^{1}(L,G^{\mathrm{sc}})$ is the trivial map, that is, the canonical map $\mathrm{ab}^{0}:G(L)\to\mathbb{H}^{0}(L,C)$ is surjective (see [Bor98]*3.10). ∎

We now proceed as in [San81]. The first step is to show the following:

Lemma 2.7**.**

*Let $m\geq 1$ be an integer and let $Q$ be a quasi-trivial $K$-torus. If the sequence *(8) is exact for $G^{m}\times_{K}Q$, then it is also exact for $G$.

Proof.

If the sequence (8) is exact for some finite direct product $G^{m}$, then (8) is exact for $G$ as well. We claim if (8) is exact for the product $G\times_{K}Q$ of $G$ by some quasi-trivial $K$-torus $Q$, then (8) is also exact for $G$. Since $T\subset G$ is a maximal torus of $G$, $T\times_{K}Q$ is a maximal torus of $G\times_{K}Q$. Moreover, $\mathscr{D}(G\times_{K}Q)=G^{{\mathrm{ss}}}$, so we have a composite $\rho_{Q}:G^{\mathrm{sc}}\to G^{{\mathrm{ss}}}\to G\times_{K}Q$. We introduce the complex $C_{Q}=[T^{\mathrm{sc}}\to T\times_{K}Q]$ which is concentrated in degree $-1$ and [math]. Consider the following commutative diagram

[TABLE]

where the second vertical map is an isomorphism since $Q$ is a quasi-trivial $K$-torus. Applying [HSSz15]*Lemma 3.2(a) yields isomorphisms ${}^{1}(C_{Q})\simeq{}^{1}(T)\simeq{}^{1}(C)$. Similarly, ${}^{1}_{S}(C_{Q}^{\prime})^{D}\simeq{}^{1}_{S}(C^{\prime})^{D}$ holds for any finite subset $S$ of places. Finally, recall that quasi-trivial $K$-tori are $K$-rational, hence in particular $Q$ satisfies weak approximation. It follows that the cokernel of the first map in (8) is stable under multiplying $G$ by a quasi-trivial torus and hence the exactness of (8) for $G\times_{K}Q$ yields the exactness of (8) for $G$. Summing up, to prove the exactness of (8), we are free to replace $G$ by $G^{m}\times_{K}Q$ for some integer $m$ and some quasi-trivial $K$-torus $Q$. ∎

Suppose $G^{\mathrm{sc}}$ satisfies $(*)$. By [BT65]*Proposition 2.2 and Ono’s lemma [San81]*Lemme 1.7, there exist an integer $m\geq 1$, quasi-trivial $K$-tori $Q$ and $Q_{0}$ such that $G^{\mathrm{sc},m}\times_{K}Q\to G^{m}\times_{K}Q_{0}$ is a central $K$-isogeny. By Lemma 2.7, we may therefore assume that $G$ has a special covering $1\to F_{0}\to G_{0}\to G\to 1$ where $G_{0}$ satisfies weak approximationand has derived subgroup $\mathscr{D}G_{0}=G^{\mathrm{sc}}$. Moreover, $G_{0}$ contains a quasi-trivial maximal torus $T_{0}$ over $K$ such that $T_{0}\cap G^{\mathrm{sc}}=T^{\mathrm{sc}}$ and that the sequence $1\to F_{0}\to T_{0}\to T\to 1$ is exact by construction. Therefore we may assume $G$ admits a special covering in the sequel.

The second step is to show the exactness at the first three terms:

Lemma 2.8**.**

There is an exact sequence $1\to\overline{G(K)}_{S}\to\prod_{v\in S}G(K_{v})\to{}^{1}_{S}(C^{\prime})^{D}$.

Proof.

After passing to the dual isogeny of the exact sequence $1\to F_{0}\to T_{0}\to T\to 1$, we obtain ${}^{2}_{S}(F_{0}^{\prime})\simeq{}^{2}_{S}(T^{\prime})$ since $T_{0}^{\prime}$ is quasi-trivial and ${}^{2}_{\omega}(T_{0}^{\prime})=0$ by [HSSz15]*Lemma 3.2(a). Moreover, the distinguished triangle $T^{\prime}\to(T^{\mathrm{sc}})^{\prime}\to C^{\prime}\to T^{\prime}[1]$ induces an isomorphism ${}^{1}_{S}(C^{\prime})\simeq{}^{2}_{S}(T^{\prime})$ for the same reason. In particular, we obtain an isomorphism ${}^{2}_{S}(F_{0}^{\prime})\simeq{}^{1}_{S}(C^{\prime})$ which fits into the commutative diagram with $F_{0}^{\prime}=\operatorname{Hom}(F_{0},\mathbb{Q}/\mathbb{Z}(2))$

[TABLE]

Recall the third column is exact by [HSSz15]*Lemma 3.1. We claim the coboundary map $\operatorname{\partial}$ is surjective. Since $F_{0}$ is finite and central in $G_{0}$, $F_{0}$ is contained in one, hence every, maximal torus of $G_{0}$. In particular, $F_{0}$ is contained in the quasi-trivial maximal torus $T_{0}$ of $G_{0}$ and consequently the map $H^{1}(K,F_{0})\to H^{1}(K,G_{0})$ factors through $H^{1}(K,F_{0})\to H^{1}(K,T_{0})$. By construction $T_{0}$ is quasi-trivial, so the vanishing $H^{1}(K,T_{0})=0$ implies that $H^{1}(K,F_{0})\to H^{1}(K,G_{0})$ is the trivial map, i.e. $\operatorname{\partial}$ is surjective. Similarly, the coboundary map $\operatorname{\partial}_{v}:G(K_{v})\to H^{1}(K_{v},F_{0})$ is surjective for each place $v\in X^{(1)}$. Since $G_{0}$ satisfies weak approximation by assumption, $G_{0}(K)$ has dense image in $\prod_{v\in S}G_{0}(K_{v})$. A diagram chasing now yields the desired exact sequence.∎

In order to prove Theorem 2.4(1), the last step is to show the exactness of the last three terms. By definition there is an exact sequence

[TABLE]

of discrete abelian groups. Dualizing the sequence, we obtain an exact sequence of profinite groups:

[TABLE]

by Proposition 1.4 and Theorem 1.18. Since ${}^{1}_{S}(C^{\prime})\simeq{}^{1}_{S}(T^{\prime})$ is a finite group, the groups $\prod_{v\in S}\mathbb{H}^{0}(K_{v},C)$ and $\prod_{v\in S}\mathbb{H}^{0}(K_{v},C)^{\wedge}$ have the same image in ${}^{1}_{S}(C^{\prime})^{D}$. By Lemma 2.6, the canonical abelianization map $\prod_{v\in S}G(K_{v})\to\prod_{v\in S}\mathbb{H}^{0}(K_{v},C)$ is surjective which guarantees the desired exactness.

Proof of Theorem 2.4(2).

Passing to the projective limit of (8) over all finite subset $S\subset X^{(1)}$ yields an exact sequence of groups

[TABLE]

Dualizing the exact sequence of discrete groups

[TABLE]

yields an exact sequence of profinite groups

[TABLE]

Since $G(K_{v})\to\mathbb{H}^{0}(K_{v},C)$ is surjective and $\prod_{v\in X^{(1)}}\mathbb{H}^{0}(K_{v},C)$ is dense in $\prod_{v\in X^{(1)}}\mathbb{H}^{0}(K_{v},C)^{\wedge}$, it will be sufficient to show the image of $\prod_{v\in X^{(1)}}G(K_{v})$ is closed in ${}^{1}_{\omega}(C^{\prime})^{D}$. In view of diagram (10), the quotient of $\prod_{v\in X^{(1)}}G(K_{v})$ by $\overline{G(K)}$ is isomorphic to the quotient of the profinite group $\prod_{v\in X^{(1)}}H^{1}(K_{v},F_{0})$ by the closure of the image of $H^{1}(K,F_{0})$. Consequently, the quotient of $\prod_{v\in X^{(1)}}G(K_{v})$ by $\overline{G(K)}$ is compact and hence the image of $\prod_{v\in X^{(1)}}G(K_{v})$ in ${}^{1}_{\omega}(C^{\prime})^{D}$ is closed, as required. ∎

Actually the defect of weak approximation for $G$ can also be given by a simpler group ${}^{2}_{\omega}(G^{*})$ where $G^{*}$ is the group of multiplicative type whose character module is $\widehat{G^{*}}=\pi_{1}^{\mathrm{alg}}(\overline{G})$. Here $\pi_{1}^{\mathrm{alg}}$ (see [Bor98]§1 or [CT08-resolution-flasque]§6 for more details) denotes the algebraic fundamental group of a connected reductive group $G$ defined as \pi_{1}^{\mathrm{alg}}(\overline{G})\colonequals{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\displaystyle T}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\textstyle T}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.06285pt}}}}}\cr\hbox{\scriptstyle T}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.61632pt}}}}}\cr\hbox{\scriptscriptstyle T}}}}}/\rho_{*}{\mathchoice{{\ooalign{\hbox{\raise 7.49443pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.49443pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=8.03886pt\vrule height=0.0pt,width=11.93044pt}}}}}\cr\hbox{\displaystyle T^{\mathrm{sc}}}}}}{{\ooalign{\hbox{\raise 7.49443pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.49443pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=8.03886pt\vrule height=0.0pt,width=11.93044pt}}}}}\cr\hbox{\textstyle T^{\mathrm{sc}}}}}}{{\ooalign{\hbox{\raise 6.69629pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.69629pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=5.64444pt\vrule height=0.0pt,width=8.4184pt}}}}}\cr\hbox{\scriptstyle T^{\mathrm{sc}}}}}}{{\ooalign{\hbox{\raise 6.24074pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.24074pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=4.27777pt\vrule height=0.0pt,width=6.97186pt}}}}}\cr\hbox{\scriptscriptstyle T^{\mathrm{sc}}}}}}} where \rho_{*}:{\mathchoice{{\ooalign{\hbox{\raise 7.49443pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.49443pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=8.03886pt\vrule height=0.0pt,width=11.93044pt}}}}}\cr\hbox{\displaystyle T^{\mathrm{sc}}}}}}{{\ooalign{\hbox{\raise 7.49443pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.49443pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=8.03886pt\vrule height=0.0pt,width=11.93044pt}}}}}\cr\hbox{\textstyle T^{\mathrm{sc}}}}}}{{\ooalign{\hbox{\raise 6.69629pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.69629pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=5.64444pt\vrule height=0.0pt,width=8.4184pt}}}}}\cr\hbox{\scriptstyle T^{\mathrm{sc}}}}}}{{\ooalign{\hbox{\raise 6.24074pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.24074pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=4.27777pt\vrule height=0.0pt,width=6.97186pt}}}}}\cr\hbox{\scriptscriptstyle T^{\mathrm{sc}}}}}}}\to{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\displaystyle T}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.23265pt}}}}}\cr\hbox{\textstyle T}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.06285pt}}}}}\cr\hbox{\scriptstyle T}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.61632pt}}}}}\cr\hbox{\scriptscriptstyle T}}}}} is induced by $\rho:G^{\mathrm{sc}}\to G$.

Proposition 2.9**.**

Let $G$ be a connected reductive group such that $G^{\mathrm{sc}}$ satisfies $(*)$. Let $G^{*}$ be the group of multiplicative type whose character module is $\pi_{1}^{\mathrm{alg}}(\overline{G})$. There is a surjective map of groups ${}^{1}_{\omega}(G^{*})\to{}^{1}_{\omega}(C^{\prime})$.

Proof.

Thanks to the distinguished triangle $T^{\prime}\to(T^{\mathrm{sc}})^{\prime}\to C^{\prime}\to T^{\prime}[1]$, there is a commutative diagram

[TABLE]

with exact upper row. Since $(T^{\mathrm{sc}})^{\prime}$ is quasi-trivial, ${}^{2}_{\omega}((T^{\mathrm{sc}})^{\prime})=0$ by [HSSz15]*Lemma 3.2(a) and it follows that ${}^{1}_{\omega}(C^{\prime})\simeq{}^{2}_{\omega}(T^{\prime})$ after a diagram chasing.

Let $1\to R\to H\to G\to 1$ be a flasque resolution of $G$ and consider the associated fundamental diagrams (3) and (4). Since $T^{\mathrm{sc}}$ and $H^{\mathrm{tor}}$ are quasi-trivial tori, we obtain $H^{1}(K,(T_{H})^{\prime})=0$ and ${}^{2}_{\omega}((T_{H})^{\prime})=0$ by the associated long exact sequence of $1\to(H^{\mathrm{tor}})^{\prime}\to(T_{H})^{\prime}\to(T^{\mathrm{sc}})^{\prime}\to 1$. Dualizing the middle row of the diagram (4) and considering the associated long exact sequence, we obtain

[TABLE]

Next, we relate ${}^{2}_{\omega}(G^{*})$ with ${}^{2}_{\omega}(T^{\prime})$. Recall [CT08-resolution-flasque]*Proposition 6.8 that $\pi_{1}^{\mathrm{alg}}$ is an exact functor from the category of connected reductive $K$-groups to that of $\operatorname{Gal}(\overline{K}|K)$-modules of finite type. Recall also that $\pi_{1}^{\mathrm{alg}}(\overline{R})$ of a $K$-torus $R$ is its module of cocharacters

$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.67015pt}$

$\textstyle R$ . Thus there is a commutative diagram

[TABLE]

of Galois modules of finite type with exact lower row by the exactness of $\pi_{1}^{\mathrm{alg}}$. Recall there is an anti-equivalence from the category of groups of multiplicative type to that of Galois modules of finite type which respects exact sequences, thus diagram (12) corresponds to a commutative diagram

[TABLE]

of groups of multiplicative type over $K$. Taking Galois cohomology gives rise to commutative diagrams

[TABLE]

with ${}^{1}_{\omega}(R^{\prime})\simeq{}^{2}_{\omega}(T^{\prime})$ by (11). It follows that $\gamma\circ\delta$ is an isomorphism, so $\gamma$ is surjective. Hence ${}^{2}_{\omega}(G^{*})\to{}^{2}_{\omega}(T^{\prime})\simeq{}^{1}_{\omega}(C^{\prime})$ is surjective. ∎

By [HSz05]*Appendix, Proposition (2), ${}^{1}_{\omega}(C^{\prime})^{D}\to{}^{2}_{\omega}(G^{*})^{D}$ is injective (${}^{1}_{\omega}(C^{\prime})$ and ${}^{2}_{\omega}(G^{*})$ are discrete). Thus there is an exact sequence of groups by Theorem 2.4(2)

[TABLE]

Consequently, the defect of weak approximation may also be given by the group ${}^{2}_{\omega}(G^{*})$.

3 Reciprocity obstruction to weak approximation

The next theorem is the promised generalization of [HSSz15]Theorem 4.2 to the non-commutative case. Let us briefly recall the construction of the pairing [HSSz15](17) concerning unramified cohomology groups. Let $Y$ be a smooth integral variety over $K$ with function field $K(Y)$. Let $y_{v}:\operatorname{Spec}K_{v}\to Y$ be a $K_{v}$-point on $Y$. Take $\alpha\in H_{{\mathrm{nr}}}^{3}(K(Y),\mu_{n}^{\operatorname{\otimes}{2}})$ and lift it uniquely to $\alpha_{v}\in H^{3}(\mathcal{O}_{Y,y_{v}},\mu_{n}^{\operatorname{\otimes}{2}})$ (here the uniqueness follows by the injective property for $\mu_{n}^{\operatorname{\otimes}{j}}$ over discrete valuation rings, see [CT95SBB]*§3.6). Now $\alpha_{v}$ goes to $H^{3}(K_{v},\mu_{n}^{\operatorname{\otimes}{2}})$ via $H^{3}(\mathcal{O}_{Y,y_{v}},\mu_{n}^{\operatorname{\otimes}{2}})\to H^{3}(K_{v},\mu_{n}^{\operatorname{\otimes}{2}})$. Summing up, we obtain an evaluation pairing

[TABLE]

Taking the isomorphism $H^{3}(K_{v},\mathbb{Q}/\mathbb{Z}(2))\simeq\mathbb{Q}/\mathbb{Z}$ for each $v\in X^{(1)}$ into account, we can construct a pairing

[TABLE]

Theorem 3.1**.**

Let $G$ be a connected reductive group such that $G^{\mathrm{sc}}$ satisfies $(*)$. There exists a homomorphism

[TABLE]

such that each family $(g_{v})\in\prod_{v\in X^{(1)}}G(K_{v})$ satisfying $((g_{v}),\operatorname{Im}u)=0$ under the pairing $(\ref{pairing: unramified H 3})$ lies in the closure $\overline{G(K)}$ with respect to the product topology.

More precisely, the obstruction is given by \operatorname{Im}\big{(}H^{3}(G^{c},\mu_{n}^{\operatorname{\otimes}2})\to H_{{\mathrm{nr}}}^{3}(K(G),\mathbb{Q}/\mathbb{Z}(2))\big{)} for some sufficiently large $n$.

Proof.

We first construct the homomorphism $u:{}^{1}_{\omega}(C^{\prime})\to H^{3}_{{\mathrm{nr}}}(K(G),\mathbb{Q}/\mathbb{Z}(2))$. Let $\rho:G^{\mathrm{sc}}\to G$ be as before. Let $T^{\mathrm{sc}}\subset G^{\mathrm{sc}}$ be a quasi-trivial maximal torus and let $T\subset G$ be a maximal torus such that $T^{\mathrm{sc}}=\rho^{-1}(T)$. Recall that there is a fundamental diagram (4) associated with the flasque resolution $1\to R\to H\to G\to 1$, and recall also that $H^{\mathrm{tor}}$ is a quasi-trivial torus. Moreover, there is a homomorphism ${}^{1}_{\omega}(C^{\prime})\to H^{1}(K,R^{\prime})$ via the inclusion ${}^{1}_{\omega}(R^{\prime})\to H^{1}(K,R^{\prime})$ in view of (11).

Because $R$ is a flasque $K$-torus, applying [CSan87a]*Theorem 2.2(i) implies that the class $[H]\in H^{1}(G,R)$ comes from a class $[Y]\in H^{1}(G^{c},R)$, where $G^{c}$ is a smooth compactification of $G$. The pairing $R\operatorname{\otimes}^{\mathbf{L}}R^{\prime}\to\mathbb{Z}(2)[2]$ now induces a homomorphism

[TABLE]

with $a_{G^{c}}$ denoting the image of $a$ under $H^{1}(K,R^{\prime})\to H^{1}(G^{c},R^{\prime})$. The same argument as in [HSSz15]*Theorem 4.2 shows that there is a natural map

[TABLE]

fitting into a commutative diagram

[TABLE]

Now take an element $(g_{v})\in\prod_{v\in X^{(1)}}G(K_{v})$. By Theorem 2.4(2), $(g_{v})$ lies in the closure $\overline{G(K)}$ if and only if $(g_{v})$ is orthogonal to ${}^{1}_{\omega}(C^{\prime})$. We consider the commutative diagram (up to sign) of various cup-products

[TABLE]

Since $H^{1}(K_{v},T^{\mathrm{sc}})=0$ by the quasi-trivialness of $T^{\mathrm{sc}}$, $H^{0}(K_{v},T)\to\mathbb{H}^{0}(K_{v},C)$ is surjective. In particular, there exists $t_{v}\in H^{0}(K_{v},T)$ such that its image in $\mathbb{H}^{0}(K_{v},C)$ equals $\mathrm{ab}^{0}_{v}(g_{v})$. The diagram together with Theorem 2.4 imply that $(g_{v})\in\overline{G(K)}$ if and only if $(t_{v})$ is orthogonal to ${}^{1}_{\omega}(R^{\prime})$. Recall we have isomorphisms (11). More explicitly, it means that

[TABLE]

for each $a\in{}^{1}_{\omega}(C^{\prime})\simeq{}^{1}_{\omega}(R^{\prime})$ (here the first two $a_{v}$ lie in $\mathbb{H}^{1}(K_{v},C^{\prime})$ while the last lies in $H^{1}(K_{v},R^{\prime})$). Note that $\delta_{v}t_{v}$ is given by $t_{v}^{*}:H^{1}(T,R)\to H^{1}(K_{v},R)$, $[T_{H}]\mapsto[T_{H}](t_{v})=[Y](g_{v})$. Let $a_{T}$ be the image of $a\in H^{1}(K,R^{\prime})$ in $H^{2}(K,T^{\prime})$ and let $a_{G^{c}}$ be the image of $a\in H^{1}(K,R^{\prime})$ in $H^{1}(G^{c},R^{\prime})$. It follows that

[TABLE]

holds thanks to the commutative diagram

[TABLE]

Note that the vanishing of the last term in (14) means that $(g_{v})$ is orthogonal to the image of $u$ under the pairing $(-,-)$ which completes the proof.

Recall that $H^{1}(K,R^{\prime})$ has finite exponent (say $N$). Then $[Y]\in H^{1}(G^{c},R)$ goes to $\operatorname{\partial}_{N}([Y])\in H^{2}(G^{c},{{}_{N}}R)$ induced a Kummer sequence, and we can lift $a\in H^{1}(K,R^{\prime})$ to $a_{N}\in H^{1}(K,{{}_{N}}R^{\prime})$. Now we obtain a class $(a_{N})_{G^{c}}\cup\operatorname{\partial}_{N}([Y])\in H^{3}(G^{c},\mu_{N}^{\operatorname{\otimes}2})$ induced by ${{}_{N}}R\operatorname{\otimes^{\mathbf{L}}}{{}_{N}}R^{\prime}\to\mu_{N}^{\operatorname{\otimes}2}$ and it restricts to a class in $H_{{\mathrm{nr}}}^{3}(K(G),\mu_{N}^{\operatorname{\otimes}2})$. ∎

Remark 3.2**.**

The following argument was pointed out to the author by Colliot-Thélène. Let $G$ be a quasi-split reductive group over $K$. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$ and let $B^{-}$ be the unique Borel subgroup of $G$ such that $B\cap B^{-}=T$. Let $U^{+}=\operatorname{rad}^{u}(B)$ and $U^{-}=\operatorname{rad}^{u}(B^{-})$ be respective unipotent radicals of $B$ and $B^{-}$. Then the big cell of $G$ is $U^{-}\times T\times U^{+}$ by [BCnotes-red]*Proposition 1.4.11 (which is dense in $G$). But $U^{\pm}$ are isomorphic to some affine spaces as varieties, the big cell of $G$ is thus isomorphic to $T\times\mathbb{A}^{N}$ for suitable $N$, i.e. $G$ is stably birational to its maximal torus $T$. In this point of view, one sees that $H_{{\mathrm{nr}}}^{3}(K(G),\mathbb{Q}/\mathbb{Z}(2))\simeq H_{{\mathrm{nr}}}^{3}(K(T),\mathbb{Q}/\mathbb{Z}(2))$, and that weak approximation for $G$ is equivalent to that for $T$.

4 Appendix

In this appendix, we compare cohomological obstructions to the Hasse principle and weak approximation for tori constructed in [HSz16, HSSz15], respectively. Let $T$ be a $K$-torus and let $T^{\prime}$ be its dual torus. There is a canonical map constructed in [HSz16]*pp. 15, (20)

[TABLE]

via the Hochschild–Serre spectral sequence $H^{p}(K,H^{q}(\overline{T},\mathbb{Q}/\mathbb{Z}(2)))\Rightarrow H^{p+q}(T,\mathbb{Q}/\mathbb{Z}(2))$. Then we send ${}^{2}_{\omega}(T^{\prime})$ to $H^{3}(K(T),\mathbb{Q}/\mathbb{Z}(2))/H^{3}(K,\mathbb{Q}/\mathbb{Z}(2))$ through $H^{3}(T,\mathbb{Q}/\mathbb{Z}(2))/H^{3}(K,\mathbb{Q}/\mathbb{Z}(2))$. Our goal is to show the image of ${}^{2}_{\omega}(T^{\prime})$ lies in $H_{{\mathrm{nr}}}^{3}(K(T),\mathbb{Q}/\mathbb{Z}(2))/H^{3}(K,\mathbb{Q}/\mathbb{Z}(2))$, the unramified part of $H^{3}(K(T),\mathbb{Q}/\mathbb{Z}(2))/H^{3}(K,\mathbb{Q}/\mathbb{Z}(2))$.

Let $1\to R\to Q\to T\to 1$ be a flasque resolution with $R$ a flasque $K$-torus and $Q$ a quasi-trivial $K$-torus.

Proposition 4.1**.**

There is a commutative diagram $($up to sign$)$

[TABLE]

*where the upper horizontal map is defined by the cup-product $H^{1}(K,R^{\prime})\times H^{1}(T^{c},R)\to H^{4}(T^{c},\mathbb{Z}(2))$ $($see [HSSz15]pp. 19 for details$)$, and the right vertical map is induced by the exact sequence $0\to\mathbb{Z}(2)\to\mathbb{Q}(2)\to\mathbb{Q}/\mathbb{Z}(2)\to 0$.

Corollary 4.2**.**

The image of ${}^{2}_{\omega}(T^{\prime})$ in $H^{3}(K(T),\mathbb{Q}/\mathbb{Z}(2))/H^{3}(K,\mathbb{Q}/\mathbb{Z}(2))$ lies in the unramified part.

Proof.

The map $H^{1}(K,R^{\prime})\to H^{4}(T,\mathbb{Z}(2))$ factors through $H^{4}(T^{c},\mathbb{Z}(2))\to H^{4}(T,\mathbb{Z}(2))$ by construction [HSSz15]*Theorem 4.2, so the image of ${}^{1}_{\omega}(R^{\prime})\simeq{}^{2}_{\omega}(T^{\prime})$ lies in the unramified part $H^{3}_{{\mathrm{nr}}}(K(T),\mathbb{Q}/\mathbb{Z}(2))/H^{3}(K,\mathbb{Q}/\mathbb{Z}(2))$. ∎

The rest of the appendix is devoted to the proof of Proposition 4.1. We begin with some observations on torsion groups under consideration. Let $L|K$ be a finite Galois extension splits both $R$ and $T$. The vanishing of $H^{1}(T_{L},R_{L})=0$ implies that the class $[Q]\in H^{1}(T,R)$ is torsion by a restriction-corestriction argument. The spectral sequence $H^{p}(K,\operatorname{Ext}^{q}_{\overline{K}}(\overline{R}^{\prime},\overline{T}^{\prime}))\Rightarrow\operatorname{Ext}^{p+q}_{K}(R^{\prime},T^{\prime})$ together with the vanishing $\operatorname{Ext}^{1}_{\overline{K}}(\overline{R}^{\prime},\overline{T}^{\prime})=0$ implies that $H^{1}(K,\operatorname{Hom}_{\overline{K}}(\overline{R}^{\prime},\overline{T}^{\prime}))\to\operatorname{Ext}^{1}_{K}(R^{\prime},T^{\prime})$ is an isomorphism. Thus the group $\operatorname{Ext}^{1}_{K}(R^{\prime},T^{\prime})$ is torsion. We choose a suitable integer $n$ such that the classes $[Q]\in H^{1}(T,R)$ and $[Q^{\prime}]\in\operatorname{Ext}^{1}_{K}(R^{\prime},T^{\prime})$ are both $n$-torsion.

Step 1: We verify the commutativity of diagram (15) with a different construction of the left vertical arrow as in diagram (18).

The Kummer sequence $1\to{{}_{n}}R\to R\to R\to 1$ yields a surjection $H^{1}(T,{{}_{n}}R)\to{{}_{n}}H^{1}(T,R)$, i.e. $[Q]=\iota_{n}([Q_{n}])$ for some class $[Q_{n}]\in H^{1}(T,{{}_{n}}R)$ with $\iota_{n}:H^{1}(T,{{}_{n}}R)\to H^{1}(T,R)$ induced by the Kummer sequence.

Let $p:T\to\operatorname{Spec}K$ be the structural morphism. Let $\mathbf{D}(K)$ be the derived category of bounded complexes of Galois modules. We consider the object $ND(T)=(\tau_{\leq 1}\mathbf{R}p_{*}\mu_{n}^{\operatorname{\otimes}{2}})[1]$ in $\mathbf{D}(K)$ which fits into a distinguished triangle

[TABLE]

We will follow [HSk13descent]*Proposition 1.1 to construct a map

[TABLE]

The pairing ${{}_{n}}R\operatorname{\otimes^{\mathbf{L}}}{{}_{n}}R^{\prime}\to\mu_{n}^{\operatorname{\otimes}{2}}$ yields a map $H^{1}(T,{{}_{n}}R)\to H^{1}(T,\operatorname{\underline{\operatorname{Hom}}}({{}_{n}}R^{\prime},\mu_{n}^{\operatorname{\otimes}{2}}))$. Moreover, we obtain a map $H^{1}(T,\operatorname{\underline{\operatorname{Hom}}}({{}_{n}}R^{\prime},\mu_{n}^{\operatorname{\otimes}{2}}))\to\operatorname{Ext}_{T}^{1}({{}_{n}}R^{\prime},\mu_{n}^{\operatorname{\otimes}{2}})$ from the exact sequence in low degrees associated to the local-to-global spectral sequence

[TABLE]

Since $\mathbf{R}\operatorname{Hom}_{T}({{}_{n}}S^{\prime},-)=\mathbf{R}\operatorname{Hom}_{K}({{}_{n}}S^{\prime},-)\circ\mathbf{R}p_{*}(-)$ is a composition, formally there is a canonical isomorphism

[TABLE]

Because $\tau_{\geq 2}\mathbf{R}p_{*}\mu_{n}^{\operatorname{\otimes}{2}}$ is acyclic in degrees [math] and $1$, we obtain an isomorphism

[TABLE]

from the distinguished triangle $\tau_{\leq 1}\mathbf{R}p_{*}\mu_{n}^{\operatorname{\otimes}{2}}\to\mathbf{R}p_{*}\mu_{n}^{\operatorname{\otimes}{2}}\to\tau_{\geq 2}\mathbf{R}p_{*}\mu_{n}^{\operatorname{\otimes}{2}}\to ND(T)$. Now $\chi$ is just the composition

[TABLE]

All the above constructions yield a diagram of cup-products

[TABLE]

where the upper diagram commutes by functoriality of the cup product pairing (see [HSz16]*diagram (26) for more details), and the lower two diagrams commute by [MilneEC]*Proposition V.1.20. Diagram (17) gives the commutativity of the left two squares of the following diagram (where the second square comes from the lower three rows):

[TABLE]

The right two squares in diagram (18) commute by construction of the Hochschild–Serre spectral sequences. Passing to the quotient by respective subgroup of constants and taking limits over all $n$ imply the commutativity of diagram (15). Consequently, we are done if the upper row of diagram (18) gives the coboundary map $H^{1}(K,R^{\prime})\to H^{2}(K,T^{\prime})$ induced by the short exact sequence $1\to T^{\prime}\to Q^{\prime}\to R^{\prime}\to 1$ of tori.

Step 2: We check that composite of arrows in the upper row of diagram (18) is just the desired coboundary map in diagram (15).

The Kummer sequence $1\to{{}_{n}}T^{\prime}\to T^{\prime}\to T^{\prime}\to 1$ induces a surjection $\operatorname{Ext}^{1}_{K}(R^{\prime},{{}_{n}}T^{\prime})\to{{}_{n}}\operatorname{Ext}^{1}_{K}(R^{\prime},T^{\prime})$, so the class $[Q^{\prime}]$ lifts to a class $[M_{n}]\in\operatorname{Ext}^{1}_{K}(R^{\prime},{{}_{n}}T^{\prime})$. Similarly, the Kummer sequence $1\to{{}_{n}}R^{\prime}\to R^{\prime}\to R^{\prime}\to 1$ induces an isomorphism $\operatorname{Hom}_{K}({{}_{n}}R^{\prime},{{}_{n}}T^{\prime})\to\operatorname{Ext}^{1}_{K}(R^{\prime},{{}_{n}}T^{\prime})$ by the vanishing of $\operatorname{Hom}_{K}(R^{\prime},{{}_{n}}T^{\prime})=0$. Hence there is a commutative diagram

[TABLE]

Applying the functor $H^{n}(K,-)$ to diagram (19) yields a commutative diagram

[TABLE]

which tells us the composite $H^{1}(K,R^{\prime})\to H^{2}(K,{{}_{n}}R^{\prime})\to H^{2}(K,{{}_{n}}T^{\prime})\to H^{2}(K,T^{\prime})$ is exact the coboundary map $H^{1}(K,R^{\prime})\to H^{2}(K,T^{\prime})$ induced by the bottom row of diagram (19).

It remains to show the map $H^{2}(K,{{}_{n}}R^{\prime})\to H^{2}(K,{{}_{n}}T^{\prime})$ obtained from $\operatorname{Ext}^{1}_{K}(R^{\prime},{{}_{n}}T^{\prime})\simeq\operatorname{Hom}_{K}({{}_{n}}R^{\prime},{{}_{n}}T^{\prime})$ coincides with the composition $H^{2}(K,{{}_{n}}R^{\prime})\to H^{2}(K,ND(T))\to H^{2}(K,{{}_{n}}T^{\prime})$. The cup-product pairing

[TABLE]

defines a map $H^{1}(\overline{T},{{}_{n}}R)\to\operatorname{Hom}_{\overline{K}}({{}_{n}}R^{\prime}(\overline{K}),H^{1}(\overline{T},\mu_{n}^{\operatorname{\otimes}{2}}))$. Again there is a commutative diagram by [MilneEC]*Proposition V.1.20:

[TABLE]

which may be rewritten into the following commutative diagram

[TABLE]

The arrow $\alpha$ is induced by the distinguished triangle (16). Now $\alpha\circ\chi=\Phi(-,-)$ says that ${{}_{n}}R^{\prime}(\overline{K})\to ND(\overline{T})\to H^{1}(\overline{T},\mu_{n}^{\operatorname{\otimes}{2}})$ is the same as $\Phi([Q_{n}],-)$. In particular, $H^{2}(K,{{}_{n}}R^{\prime})\to H^{2}(K,ND(T))\to H^{2}(K,{{}_{n}}T^{\prime})$ is the same as $H^{2}(K,{{}_{n}}R^{\prime})\to H^{2}(K,{{}_{n}}T^{\prime})$ obtained from the identification $\operatorname{Ext}^{1}_{K}(T,{{}_{n}}R)\simeq\operatorname{Hom}_{K}({{}_{n}}R^{\prime},{{}_{n}}T^{\prime})$.