# Integral Points for Groups of Multiplicative Type

Dasheng Wei1 and  Fei Xu1,2 1 Academy of Mathematics and System Science, CAS,
Beijing 100190, P.R.China<EMAIL_ADDRESS>2 School of Mathematics, Capital
Normal University, Beijing 100048, P.R.China<EMAIL_ADDRESS>
To Professor Kezheng Li on his 65-th birthday

## Abstract

We construct a finite subgroup of Brauer-Manin obstruction for detecting the
existence of integral points on integral models of homogeneous spaces of
linear algebraic groups of multiplicative type. As application, the strong
approximation theorem for linear algebraic groups of multiplicative type is
established. Moreover, the sum of two integral squares over some quadratic
fields is discussed.

MSC classification : 11E72, 11G35, 11R37, 14F22, 14G25, 20G30, 20G35

Keywords : integral point, linear algebraic group of multiplicative type,
Galois cohomology, Brauer-Manin obstruction, strong approximation, sum of two
squares.

## Introduction

The integral points on homogeneous spaces of semi-simple and simply connected
linear algebraic groups of non-compact type were studied by Borovoi and
Rudnick in [BR95] and homogeneous spaces of connected semi-simple linear
algebraic groups of non-compact type by Colliot-Thélène and the second named
author in [CTX] using the strong approximation theorem and the Brauer-Manin
obstruction. Recently, Harari [Ha08] showed that the Brauer-Manin obstruction
accounts for the nonexistence of integral points. Colliot-Thélène noticed that
a finite subgroup of the Brauer group is enough to account for the
nonexistence of integral points by the compactness arguments. These results
are nonconstructive: they do not say which finite subgroup to use. In our
previous paper [WX], we construct such finite groups for multi-norm tori
explicitly. In this paper, we’ll extend this construction for linear algebraic
groups of multiplicative type.

The paper is organized as follows. In Section 1, we’ll give an explicit
construction of the so called admissible groups (see Definition 1.9) and
establish the criterion for existence of the integral points in terms of
admissible groups. In Section 2, we translate such admissible groups into the
finite Brauer-Manin obstruction. As an application, we establish the strong
approximation theorem for the group of multiplicative type in Section 3. In
Section 4, we discussed the sum of two integral squares over some quadratic
fields.

Notation and terminology are standard if not explained. Let $F$ be a number
field, $\mathfrak{o}_{F}$ be the ring of integers of $F$, $\Omega_{F}$ be the
set of all primes in $F$ and $\infty_{F}$ be the set all infinite primes in
$F$. For simplicity, we write $\mathfrak{p}<\infty_{F}$ for
$\mathfrak{p}\in\Omega_{F}\setminus\infty_{F}$. For any finite set
$S_{0}\subset\Omega_{F}\setminus\infty_{F}$, the ring of $S_{0}$-integers of
$F$ is defined as

$\mathfrak{o}_{S_{0}}=\\{x\in F:\ \lvert x\rvert_{\mathfrak{p}}\leq 1\
\text{for all}\ \mathfrak{p}<\infty_{F}\ \text{and}\ \mathfrak{p}\not\in
S_{0}\\}.$

Let $F_{\mathfrak{p}}$ be the completion of $F$ at $\mathfrak{p}$ and
$\mathfrak{o}_{F_{\mathfrak{p}}}$ be the local completion of
$\mathfrak{o}_{F}$ at $\mathfrak{p}$ for each $\mathfrak{p}\in\Omega_{F}$.
Write $\mathfrak{o}_{F_{\mathfrak{p}}}=F_{\mathfrak{p}}$ for
$\mathfrak{p}\in\infty_{F}$. We also denote the adele (resp. the idele) of $F$
by $\mathbb{A}_{F}$ (resp. $\mathbb{I}_{F}$) and

$F_{\infty}=\prod_{\mathfrak{p}\in\infty_{F}}F_{\mathfrak{p}}.$

A group $G$ of multiplicative type over $F$ is defined as a closed subgroup of
a torus over $F$. Let $\hat{G}=Hom_{\bar{F}}(G,\mathbb{G}_{m})$ be the
character of $G$. Then the functor $G\mapsto\hat{G}$ is an anti-equivalence
between the category of $F$-groups of multiplicative type with the category of
finitely generated abelian groups with the continuous action of
$Gal(\bar{F}/F)$ (see [Sko]). Moreover, $G$ is a torus if and only if
$\hat{G}$ is free. For any positive integer $k$, the finite group scheme
$G[k]$ over $F$ stands for the kernel of the multiplication by $k$ over $G$.
We also use $\bar{F}$ to denote the algebraic closure of $F$.

Let $\bf X$ be a separated $\mathfrak{o}_{F}$-scheme of finite type whose
generic fiber $X_{F}$ is a trivial torsor of $G$. The obvious necessary
condition for ${\bf X}(\mathfrak{o}_{F})\neq\emptyset$ is

$\prod_{\mathfrak{p}\in\Omega_{F}}{\bf
X}(\mathfrak{o}_{F_{\mathfrak{p}}})\neq\emptyset$ (0.1)

which is assumed throughout this paper. The Brauer group $Br(X_{F})$ of
$X_{F}$ is defined as

$Br(X_{F})=H^{2}_{et}(X_{F},\mathbb{G}_{m})\ \ \ \text{and}\ \ \
Br_{1}(X_{F})=ker[Br(X_{F})\rightarrow Br(\bar{X})]$

where $\bar{X}=X_{F}\times_{F}\bar{F}$. Since the image of $Br(F)$ induced by
the structure morphism lies in $Br_{1}(X_{F})$, one defines

$Br_{a}(X_{F})=coker[Br(F)\rightarrow Br_{1}(X)].$

For any subgroup $\mathfrak{s}$ of $Br_{a}(X_{F})$, one can define the
integral Brauer-Manin set with respect to $\mathfrak{s}$ as (see [CTX])

$(\prod_{\mathfrak{p}\in\Omega_{F}}{\bf
X}(\mathfrak{o}_{F_{\mathfrak{p}}}))^{\mathfrak{s}}=\\{(x_{\mathfrak{p}})\in\prod_{\mathfrak{p}\in\Omega_{F}}{\bf
X}(\mathfrak{o}_{F_{\mathfrak{p}}}):\ \
\sum_{\mathfrak{p}\in\Omega_{F}}inv_{\mathfrak{p}}(s(x_{\mathfrak{p}}))=0,\ \
\ \forall s\in\mathfrak{s}\\}.$

##  1\. Construction of X-admissible groups

In this section, we’ll extend the construction of ${\bf X}$-admissible groups
in [WX] for multi-norm tori to the general groups of multiplicative type. By
using the anti-equivalence between the category of $F$-tori with the category
of free $\mathbb{Z}$-modules of finite rank with the continuous action of
$Gal(\bar{F}/F)$, one can first show the following lemma.

######  Lemma 1.1.

Let $T_{1}$ and $T_{2}$ be two tori over $F$ and $\phi:T_{1}\rightarrow T_{2}$
be a surjective morphism of tori. Then there is a morphism of tori

$\psi:T_{2}\rightarrow T_{1}\ \ \ \text{such that}\ \ \ \phi\circ\psi=[l]$

for some positive integer $l$.

###### Proof.

Since the category of $F$-tori is anti-equivalent to the category of free
$\mathbb{Z}$-modules of finite rank with the continuous action of
$Gal(\bar{F}/F)$, one has the injective $Gal(\bar{F}/F)$-module homomorphism
$\hat{\phi}:\hat{T}_{2}\rightarrow\hat{T}_{1}$. Extending this map over
$\mathbb{Q}$, one obtains the injective $Gal(\bar{F}/F)$-module homomorphism

$\hat{\phi}_{\mathbb{Q}}:\hat{T}_{2}\otimes_{\mathbb{Z}}\mathbb{Q}\longrightarrow\hat{T}_{1}\otimes_{\mathbb{Z}}\mathbb{Q}.$

Since the representation of compact group over a field of characteristic 0 is
semi-simple (Maschke’s Theorem), there is a $Gal(\bar{F}/F)$-module
homomorphism

$\hat{\psi^{\prime}}_{\mathbb{Q}}:\hat{T}_{1}\otimes_{\mathbb{Z}}\mathbb{Q}\longrightarrow\hat{T}_{2}\otimes_{\mathbb{Z}}\mathbb{Q}$

such that $\hat{\psi^{\prime}}_{\mathbb{Q}}\circ\hat{\phi}_{\mathbb{Q}}=1$.
Since $\hat{T}_{1}$ is of finite rank, there is a positive integer $l$ such
that $l\hat{\psi^{\prime}}_{\mathbb{Q}}(\hat{T}_{1})\subseteq\hat{T}_{2}$.
Then the homomorphism from $T_{2}$ to $T_{1}$ associated to
$l\hat{\psi^{\prime}}_{\mathbb{Q}}$ is as required. ∎

Fix a finite Galois extension $E/F$ such that $Gal(\bar{F}/E)$ acts on
$\hat{G}$ trivially. Let $M$ be a free $\mathbb{Z}$-module of finite rank such
that $f:M\rightarrow\hat{G}$ is a surjective homomorphism. One can extend this
map to the $Gal(\bar{F}/F)$-module surjective homomorphism

$\mathbb{Z}[Gal(E/F)]\otimes_{\mathbb{Z}}M\rightarrow\hat{G},\ \ \
\sigma\otimes m\mapsto\sigma(f(m))$

where $Gal(\bar{F}/F)$ acts on $\mathbb{Z}[Gal(E/F)]\otimes_{\mathbb{Z}}M$ as
follows

$\tau\circ(\sigma\otimes m)=\bar{\tau}\sigma\otimes m$

for $\tau\in Gal(\bar{F}/F)$ and $\sigma\otimes
m\in\mathbb{Z}[Gal(E/F)]\otimes_{\mathbb{Z}}M$. By the anti-equivalence
between the category of multiplicative groups over $F$ with the category of
finitely generated $\mathbb{Z}$-modules with the continuous action of
$Gal(\bar{F}/F)$, there is an $F$-torus $T$ such that

$0\rightarrow
G\xrightarrow{\lambda}\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})\xrightarrow{\nu}T\rightarrow
0$ (1.2)

where $n=rank_{\mathbb{Z}}(M)$.

Applying Lemma 1.1 to (1.2), one can fix a homomorphism

$\mu:T\rightarrow\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})$

such that $\nu\circ\mu=[l]$ for some positive integer $l$.

Fix a finite subset $S_{0}\subset\Omega_{F}\setminus\infty_{F}$ such that

1) the groups $G$, $\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})$ and $T$ over $F$
extend to the group schemes $\mathbb{G}$, $\mathbb{R}_{E/F}(\mathbb{G}_{m})$
and $\mathbb{T}$ over $\mathfrak{o}_{S_{0}}$ such that (1.2) extends to

$0\longrightarrow\mathbb{G}\longrightarrow\prod_{1}^{n}\mathbb{R}_{E/F}(\mathbb{G}_{m})\longrightarrow\mathbb{T}\longrightarrow
0$

over $\mathfrak{o}_{S_{0}}$ and

$\mathbb{R}_{E/F}(\mathbb{G}_{m})(\mathfrak{o}_{F_{\mathfrak{p}}})=\mathfrak{o}_{E_{\mathfrak{p}}}^{\times}$

for $\mathfrak{p}\not\in S_{0}$.

2) the trivial torsor $X_{F}$ of $G$ over $F$ extend to the trivial torsor
$\mathbb{X}_{S_{0}}=\mathbb{X}\times_{\mathfrak{o}_{F}}\mathfrak{o}_{S_{0}}$
of $\mathbb{G}$ over $\mathfrak{o}_{S_{0}}$.

3) the homomorphism $\mu$ also extends to
${\mathbb{T}}\rightarrow\prod_{1}^{n}\mathbb{R}_{E/F}(\mathbb{G}_{m})$ over
$\mathfrak{o}_{S_{0}}$.

4) any $l$-torsion point of $T(F_{\mathfrak{p}})$ is contained in
$\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})$ for all $\mathfrak{p}\not\in
S_{0}$.

For the above torus $T$, the class group of $T$ with respect to $S_{0}$ can be
defined as follows

$cls(T,S_{0})=T(\mathbb{A}_{F})/[T(F)+(\prod_{\mathfrak{p}\in
S_{0}}T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in
S_{0}}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}}))]$

and this group $cls(T,S_{0})$ is finite by Theorem 5.1 in [PR94]. Denote
$h(T,S_{0})=\sharp cls(T,S_{0})$.

Let $A$ be a finite group scheme over $F$. One defines

$\cyr{X}_{F}^{1}(A)=ker[H^{1}(F,A)\longrightarrow\prod_{\mathfrak{p}<\infty_{F}}H^{1}(F_{\mathfrak{p}},A)].$

######  Lemma 1.3.

For any torus $T$ over $F$, there is a positive integer $c=c(F,T)$ such that
$\cyr{X}_{F}^{1}(T[k])$ is killed by $c$ for all positive integer $k$.

###### Proof.

Let $E$ be a finite Galois extension over $F$ such that

$T_{E}:=T\times_{F}E\cong\prod_{1}^{n}\mathbb{G}_{m,E}$

for some positive integer $n$. Therefore $T_{E}[k]=\prod_{1}^{n}\mu_{k}$. By
the inf-res sequence, one has the following commutative diagram

$\begin{array}[]{ccccccc}H^{1}(E/F,T(E)[k])&\longrightarrow&H^{1}(F,T[k])&\longrightarrow&H^{1}(E,T_{E}[k])\\\
&&\downarrow&&\downarrow\\\
&&\prod_{\mathfrak{p}<\infty_{F}}H^{1}(F_{\mathfrak{p}},T[k])&\longrightarrow&\prod_{\mathfrak{P}<\infty_{E}}H^{1}(E_{\mathfrak{P}},T_{E}[k]).\end{array}$

By (9.1.3) Theorem in [NSW], one knows that $\cyr{X}_{E}^{1}(T_{E}[k])$ is
killed by 2. If $\mu(E)$ is the order of the set of all roots of unity
contained in $E$, then $H^{1}(E/F,T(E)[k])$ is killed by the maximal common
divisor $\mu(E)$ and $[E:F]$. One can take $c=2(\mu(E),[E:F])$. ∎

From the above proof, one can further obtain a uniform bound of
$\cyr{X}_{F}^{1}(T[k])$ for all positive integer $k$. The above result is good
enough for our application.

######  Lemma 1.4.

If $T\cong\prod_{1}^{d}\mathbb{G}_{m}$ for some positive integer $d$, then

$T(F)\cap[\prod_{\mathfrak{p}\in\infty_{F}}T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in\infty_{F}}[2k]T(F_{\mathfrak{p}})]\subseteq[k]T(F)$

for any positive integer $k$.

###### Proof.

Any element

$x\in
T(F)\cap[\prod_{\mathfrak{p}\in\infty_{F}}T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in\infty_{F}}[2k]T(F_{\mathfrak{p}})]$

can be written as $x=(x_{1},\cdots,x_{d})$ where $x_{i}\in F^{\times}$ and
$x_{i}\in(F_{\mathfrak{p}}^{\times})^{2k}$ for all $\mathfrak{p}<\infty_{F}$
with $1\leq i\leq d$. Applying (9.1.3) Theorem in [NSW] for $\mu_{2k}$, one
obtains that $x_{i}^{2}\in(F^{\times})^{2k}$ for $1\leq i\leq d$. There is
$y_{i}\in F^{\times}$ such that $x_{i}=y_{i}^{k}$ or $x_{i}=-y_{i}^{k}$ with
$1\leq i\leq d$.

Suppose $x_{i}=-y_{i}^{k}$ for some $1\leq i\leq d$. Let $k=2^{s}k_{1}$ with
$2\nmid k_{1}$ and $\zeta_{2^{s+1}}$ be a primitive $2^{s+1}$-th roots of
unity. Then $\zeta_{2^{s+1}}\in F_{\mathfrak{p}}^{\times}$ for all
$\mathfrak{p}<\infty_{F}$. By the Chebotarev density theorem, one concludes
that $\zeta_{2^{s+1}}\in F$. Therefore
$-1=(\zeta_{2^{s+1}})^{k}\in(F^{\times})^{k}$ and the proof is complete. ∎

For a general torus $T$, one can fix the splitting field $K/F$ of $T$ such
that

$T_{K}=T\times_{F}K\cong\prod_{1}^{d}\mathbb{G}_{m,K}$

for some positive integer $d$. Since $K/F$ is a finite extension, the
$K$-rational torsion points of $T_{K}$ is finite. Fix a positive integer $t$
such that $t$ kills all $K$-rational torsion points of $T_{K}$.

######  Lemma 1.5.

Let $T$ be a torus over $F$ as above and $k$ be a positive integer. Then there
is a finite subset $S_{1}$ of $\Omega_{F}\setminus\infty_{F}$ containing
$S_{0}$ such that

$T(F)\cap[\prod_{\mathfrak{p}\in S}[c\cdot 2t\cdot h(T,S_{0})\cdot
k]T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in
S}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})]\subseteq[k]T(F)$

for any finite subset $S\supseteq S_{1}$, where $c=c(F,T)$ is as in Lemma 1.3
and $t$ is as above.

###### Proof.

Let $U_{0}$ be a finite subset of $\Omega_{K}\setminus\infty_{K}$ such that
the ring $\mathfrak{o}_{U_{0}}$ of $U_{0}$-integers of $K$ is the integral
closure of $\mathfrak{o}_{S_{0}}$ inside $K$. Let $U$ be a finite subset of
$\Omega_{K}\setminus\infty_{K}$ containing $U_{0}$ such that

$\mathbb{T}_{U_{0}}(\mathfrak{o}_{U})=\prod_{1}^{n}\mathfrak{o}_{U}^{\times}$

where
$\mathbb{T}_{U_{0}}=\mathbb{T}\times_{\mathfrak{o}_{S_{0}}}\mathfrak{o}_{U_{0}}$.
By the Dirichlet unit theorem, one obtains that
$\mathbb{T}_{U_{0}}(\mathfrak{o}_{U})$ is a finitely generated abelian group.
Since
$\mathbb{T}(\mathfrak{o}_{U_{0}})=\mathbb{T}_{U_{0}}(\mathfrak{o}_{U_{0}})$ is
a subgroup of $\mathbb{T}_{U_{0}}(\mathfrak{o}_{U})$ and
$\mathbb{T}(\mathfrak{o}_{S_{0}})$ is a subgroup of
$\mathbb{T}(\mathfrak{o}_{U_{0}})$, one concludes that
$\mathbb{T}(\mathfrak{o}_{S_{0}})$ is a finitely generated abelian group.
Therefore $\mathbb{T}(\mathfrak{o}_{S_{0}})/[c\cdot 2t\cdot
k]\mathbb{T}(\mathfrak{o}_{S_{0}})$ is finite.

If the coset $\alpha+[c\cdot 2t\cdot k]\mathbb{T}(\mathfrak{o}_{S_{0}})$ of
$\mathbb{T}(\mathfrak{o}_{S_{0}})/[c\cdot 2t\cdot
k]\mathbb{T}(\mathfrak{o}_{S_{0}})$ satisfies that

$\alpha\not\in[c\cdot 2t\cdot k]T(F_{\mathfrak{p}})$

for some prime $\mathfrak{p}<\infty_{F}$, we fix such a prime
$\mathfrak{p}_{\alpha}$. Let $\mathfrak{S}$ be the set consisting of all such
$\mathfrak{p}_{\alpha}$ and $S_{1}=S_{0}\cup\mathfrak{S}$.

For any finite set $S\supseteq S_{1}$ and

$\alpha\in T(F)\cap[\prod_{\mathfrak{p}\in S}[c\cdot 2t\cdot h(T,S_{0})\cdot
k]T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in
S}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})],$

there is $\beta_{\mathfrak{p}}\in T(F_{\mathfrak{p}})$ such that
$\alpha=[c\cdot 2t\cdot h(T,S_{0})\cdot k]\beta_{\mathfrak{p}}$ for each
$\mathfrak{p}\in S$. Since $\beta_{\mathfrak{p}}$ can be viewed as an element
in $T(\mathbb{A}_{F})$ whose $\mathfrak{p}$ component is
$\beta_{\mathfrak{p}}$ and the others are 1, there is
$\varpi_{\mathfrak{p}}\in T(F)$ such that

$\varpi_{\mathfrak{p}}+[h(T,S_{0})]\beta_{\mathfrak{p}}\in[\prod_{\mathfrak{p}\in
S_{0}}T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in
S_{0}}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})]$

for each finite $\mathfrak{p}\in S\setminus S_{0}$. This implies that

$\gamma=\alpha+[c\cdot 2t\cdot k](\sum_{\mathfrak{p}\in
S\setminus(S_{0}\cup\infty_{F})}\varpi_{\mathfrak{p}})\in\mathbb{T}(\mathfrak{o}_{S_{0}}).$

Therefore the coset of $\gamma+[c\cdot 2t\cdot
k]\mathbb{T}(\mathfrak{o}_{S_{0}})$ of
$\mathbb{T}(\mathfrak{o}_{S_{0}})/[c\cdot 2t\cdot
k]\mathbb{T}(\mathfrak{o}_{S_{0}})$ is not one of the above mentioned cosets.
One concludes that

$\alpha\in
T(F)\cap[\prod_{\mathfrak{p}\in\infty_{F}}T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in\infty_{F}}[c\cdot
2t\cdot k]T(F_{\mathfrak{p}})].$

Hence

$\displaystyle T(F)\cap[\prod_{\mathfrak{p}\in S}[c\cdot 2t\cdot
h(T,S_{0})\cdot k]T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in
S}\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})]$ $\displaystyle\subseteq$
$\displaystyle
T(F)\cap[\prod_{\mathfrak{p}\in\infty_{F}}T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in\infty_{F}}[c\cdot
2t\cdot k]T(F_{\mathfrak{p}})].$

For any element

$x\in
T(F)\cap[\prod_{\mathfrak{p}\in\infty_{F}}T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in\infty_{F}}[c\cdot
2t\cdot k]T(F_{\mathfrak{p}})],$

there is $y\in T(F)$ such that $[c]x=[c\cdot 2t\cdot k]y$ by Lemma 1.3. Let
$z=x-[2t\cdot k]y$. Then $z\in T(F)[c]$. By applying Lemma 1.4 to
$T_{K}=T\times_{F}K$ over $K$, one obtains that there is $w\in T(K)$ such that
$z=[t\cdot k]w$. Since $z$ is a torsion point of $T$, one has $w$ is also a
torsion point of $T_{K}$. Since $t$ kills all $K$-torsion points of $T_{K}$,
one concludes that $z=0$. The proof is complete. ∎

######  Lemma 1.6.

If $C_{\mathfrak{p}}$ is an open subgroup of $G(F_{\mathfrak{p}})$ and $k$ is
a positive integer, then

$\lambda(C_{\mathfrak{p}})\cdot\mu([k]T(F_{\mathfrak{p}}))$

is an open subgroup of

$\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})(F_{\mathfrak{p}})=\prod_{1}^{n}E_{\mathfrak{p}}^{\times}$

for $\mathfrak{p}$-adic topology with $\mathfrak{p}<\infty_{F}$.

###### Proof.

Since $C_{\mathfrak{p}}$ is an open subgroup of $G(F_{\mathfrak{p}})$ and
$\lambda[G(F_{\mathfrak{p}})]$ is a closed subgroup of

$\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})(F_{\mathfrak{p}})=\prod_{1}^{n}E_{\mathfrak{p}}^{\times},$

there is a positive integer $a$ such that

$[\prod_{1}^{n}(1+\mathfrak{p}^{a}\mathfrak{o}_{E_{\mathfrak{p}}})]\cap\lambda[G(F_{\mathfrak{p}})]\subseteq\lambda(C_{\mathfrak{p}}).$

Since $\mu\circ\nu$ is a continuous endomorphism of

$\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})(F_{\mathfrak{p}})=\prod_{1}^{n}E_{\mathfrak{p}}^{\times},$

there is a positive integer $a_{1}>a$ such that

$\mu\circ\nu[\prod_{1}^{n}(1+\mathfrak{p}^{a_{1}}\mathfrak{o}_{E_{\mathfrak{p}}})]\subseteq\prod_{1}^{n}(1+\mathfrak{p}^{a}\mathfrak{o}_{E_{\mathfrak{p}}}).$

By Hensel’s lemma, there is a positive integer $a_{2}>a_{1}$ such that

$(1+\mathfrak{p}^{a_{2}}\mathfrak{o}_{E_{\mathfrak{p}}})\subseteq(1+\mathfrak{p}^{a_{1}}\mathfrak{o}_{E_{\mathfrak{p}}})^{lk}.$

For any
$x\in\prod_{1}^{n}(1+\mathfrak{p}^{a_{2}}\mathfrak{o}_{E_{\mathfrak{p}}})$,
there is

$y\in\prod_{1}^{n}(1+\mathfrak{p}^{a_{1}}\mathfrak{o}_{E_{\mathfrak{p}}})\ \ \
\text{such that}\ \ \ x=y^{lk}.$

Then

$x=y^{lk}=[y^{lk}(\mu\circ\nu(y^{k}))^{-1}](\mu\circ\nu(y^{k})).$

Since

$\nu[y^{lk}(\mu\circ\nu(y^{k}))^{-1}]=[l]\nu(y^{k})-\nu\circ\mu\circ\nu(y^{k})=0,$

one concludes

$y^{lk}(\mu\circ\nu(y^{k}))^{-1}\in\lambda(C_{\mathfrak{p}}).$

Therefore

$\prod_{1}^{n}(1+\mathfrak{p}^{a_{2}}\mathfrak{o}_{E_{\mathfrak{p}}})\subseteq\lambda(C_{\mathfrak{p}})\cdot\mu([k]T(F_{\mathfrak{p}}))$

and the proof is complete. ∎

Since $\bf X$ is separated over $\mathfrak{o}_{F}$, one can view ${\bf
X}(\mathfrak{o}_{F_{\mathfrak{p}}})$ as an open subset of
$X_{F}(F_{\mathfrak{p}})$ by the natural map for any
$\mathfrak{p}\in\Omega_{F}$.

######  Definition 1.7.

Define

$Stab({\bf X}(\mathfrak{o}_{F_{\mathfrak{p}}}))=\\{g\in G(F_{\mathfrak{p}}):\
\ g{\bf X}(\mathfrak{o}_{F_{\mathfrak{p}}})={\bf
X}(\mathfrak{o}_{F_{\mathfrak{p}}})\\}$

for $\mathfrak{p}\in\Omega_{F}$.

It is clear that $Stab({\bf X}(\mathfrak{o}_{F_{\mathfrak{p}}}))$ is an open
subgroup of $G(F_{\mathfrak{p}})$. Since $X_{F}$ is a trivial torsor of $G$
over $F$, one has that

$Stab({\bf X}(\mathfrak{o}_{F_{\mathfrak{p}}}))={\bf
G}(\mathfrak{o}_{F_{\mathfrak{p}}})$

for almost all $\mathfrak{p}$ in $\Omega_{F}$.

######  Definition 1.8.

Define

$Stab_{\mathbb{A}}({\bf X})=\prod_{\mathfrak{p}\in\Omega_{F}}Stab({\bf
X}(\mathfrak{o}_{F_{\mathfrak{p}}})).$

It is clear that $Stab_{\mathbb{A}}({\bf X})$ is an open subgroup of
$G(\mathbb{A}_{F})$. The map given by (1.2)

$\lambda:G\longrightarrow\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})$

induces the homomorphism

$\lambda:\ \ G(\mathbb{A}_{F})\longrightarrow\prod_{1}^{n}\mathbb{I}_{E}.$

######  Definition 1.9.

An open subgroup $\Xi$ of $\prod_{1}^{n}\mathbb{I}_{E}$ is called $\bf
X$-admissible if

$\lambda[Stab_{\mathbb{A}}({\bf X})]\subseteq\Xi$

and the induced map

$\lambda:\ \ \ G(\mathbb{A}_{F})/G(F)Stab_{\mathbb{A}}({\bf
X})\longrightarrow\prod_{1}^{n}\mathbb{I}_{E}/(\prod_{1}^{n}E^{\times})\cdot\Xi$

is injective.

The main result of this section is to show the existence of the admissible
subgroups.

######  Theorem 1.10.

If $\bf X$ is a separated scheme over $\mathfrak{o}_{F}$ of finite type such
that the generic fiber $X_{F}$ is a trivial torsor of a multiplicative group
$G$ satisfying (1.2), then the $\bf X$-admissible subgroups of
$\prod_{1}^{n}\mathbb{I}_{E}$ always exist.

###### Proof.

By the condition 1) and 2) for the choice of $S_{0}$, one has

$\lambda[Stab({\bf X}(\mathfrak{o}_{F_{\mathfrak{p}}}))]=\lambda[{\bf
G}(\mathfrak{o}_{F_{\mathfrak{p}}})]=ker[\nu:(\prod_{1}^{n}\mathfrak{o}_{E_{\mathfrak{p}}}^{\times}){\longrightarrow}{\bf
T}(\mathfrak{o}_{F_{\mathfrak{p}}})]$

for all $\mathfrak{p}\not\in S_{0}$. For each $\mathfrak{p}\in S_{0}$, one can
fix a positive integer $r_{\mathfrak{p}}$ such that $r_{\mathfrak{p}}$ kills
all torsion points of $T(F_{\mathfrak{p}})$. Let

$r=\prod_{\mathfrak{p}\in S_{0}}r_{\mathfrak{p}}.$

Let $S_{1}$ be a finite subset of $\Omega_{F}$ outside $\infty_{F}$ such that
Lemma 1.5 holds for $k=l\cdot r$. Define

$\Xi=[\prod_{\mathfrak{p}\in S_{1}}\lambda(Stab({\bf
X}(\mathfrak{o}_{F_{\mathfrak{p}}})))\cdot\mu([c\cdot 2t\cdot h(T,S_{0})\cdot
r]T(F_{\mathfrak{p}}))]\times(\prod_{\mathfrak{p}\not\in
S_{1}}\prod_{1}^{n}\mathfrak{o}_{E_{\mathfrak{p}}}^{\times}).$

By Lemma 1.6, one has that $\Xi$ is an open subgroup of
$\prod_{1}^{n}\mathbb{I}_{E}$ and

$\lambda[Stab_{\mathbb{A}}({\bf X})]\subseteq\Xi.$

Consider $\sigma\in G({\mathbb{A}}_{F})$ such that $\lambda(\sigma)=a\cdot i$
with

$a\in\prod_{1}^{n}E^{\times}\ \ \ \text{and}\ \ \ i\in\Xi.$

Then

$\nu(a)\in[\prod_{\mathfrak{p}\in S_{1}}[c\cdot 2t\cdot h(T,S_{0})\cdot l\cdot
r]T(F_{\mathfrak{p}})\times\prod_{\mathfrak{p}\not\in S_{1}}{\bf
T}(\mathfrak{o}_{F_{\mathfrak{p}}})].$

By Lemma 1.5, there is $u\in T(F)$ such that $\nu(a)=[l\cdot r](u)$. Since

$\nu(a\cdot(\mu([r](u)))^{-1})=\nu(a)-[l\cdot r](u)=0,$

one obtains that

$a\cdot(\mu([r](u)))^{-1}\in\lambda(G(F)).$

Let $i=(i_{\mathfrak{p}})_{\mathfrak{p}\in\Omega_{F}}$. One can write

$i_{\mathfrak{p}}=s_{\mathfrak{p}}\cdot\mu([c\cdot 2t\cdot h(T,S_{0})\cdot
r](n_{\mathfrak{p}}))$

with

$s_{\mathfrak{p}}\in\lambda(Stab({\bf X}(\mathfrak{o}_{F_{\mathfrak{p}}})))\ \
\ \text{and}\ \ \ n_{\mathfrak{p}}\in T(F_{\mathfrak{p}})$

for $\mathfrak{p}\in S_{1}$. This implies that

$[l\cdot r](u+[c\cdot 2t\cdot h(T,S_{0})](n_{\mathfrak{p}}))=0$

for $\mathfrak{p}\in S_{1}$. Therefore $u+[c\cdot 2t\cdot
h(T,S_{0})](n_{\mathfrak{p}})$ is a torsion point of $T(F_{\mathfrak{p}})$ for
each $\mathfrak{p}\in S_{1}$.

If $\mathfrak{p}\in S_{0}$, then $r$ kills $u+[c\cdot 2t\cdot
h(T,S_{0})](n_{\mathfrak{p}})$. One concludes that

$\mu([r](u))\cdot\mu([c\cdot 2t\cdot h(T,S_{0})\cdot r](n_{\mathfrak{p}}))=1.$

If $\mathfrak{p}\in S_{1}\setminus S_{0}$, one has

$[r]u+[c\cdot 2t\cdot h(T,S_{0})\cdot
r](n_{\mathfrak{p}})\in\mathbb{T}(\mathfrak{o}_{F_{\mathfrak{p}}})$

by the condition 4) for the choice of $S_{0}$. By the condition 1) and 3) for
the choice of $S_{0}$, one obtains that

$\mu([r]u)\cdot\mu([c\cdot 2t\cdot h(T,S_{0})\cdot
r](n_{\mathfrak{p}}))\in\prod_{1}^{n}\mathfrak{o}_{E_{\mathfrak{p}}}^{\times}.$

Therefore one concludes that

$\mu([r]u)\cdot i\in\lambda[Stab_{\mathbb{A}}({\bf X})]\ \ \ \text{and}\ \ \
\sigma\in G(F)Stab_{\mathbb{A}}({\bf X}).$

The proof is complete. ∎

Since $X_{F}$ is a trivial $G$-torsor over $F$, one can fix a rational point
$P\in X_{F}(F)$ which induces the isomorphism $X_{F}\cong G$ as $F$-varieties.
Since $\bf X$ is separated over $\mathfrak{o}_{F}$, the natural morphism

$\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})\longrightarrow
X_{F}(\mathbb{A}_{F})$

is injective. Define

$f:\ \
\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})\rightarrow
X_{F}({\mathbb{A}}_{F})\cong
G({\mathbb{A}}_{F})\xrightarrow{\lambda}\prod_{1}^{n}\mathbb{I}_{E}.$

######  Corollary 1.11.

Let $\Xi$ be an $\bf X$-admissible subgroup of $\prod_{1}^{n}\mathbb{I}_{E}$.
Then

$\mathbb{X}(\mathfrak{o}_{F})\neq\emptyset\ \ \ \ \ \text{if and only if}\ \ \
\ \
f[\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})]\cap[(\prod_{1}^{n}E^{\times})\cdot\Xi]\neq\emptyset.$

###### Proof.

Since $\mathbb{X}$ is separated over $\mathfrak{o}_{F}$, one has
$\mathbb{X}(\mathfrak{o}_{F})\subseteq\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})$
for all $\mathfrak{p}\in\Omega_{F}$ and

$\mathbb{X}(\mathfrak{o}_{F})\subseteq\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})$

by the diagonal map. If $\mathbb{X}(\mathfrak{o}_{F})\neq\emptyset$, then

$f[\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})]\cap[(\prod_{1}^{n}E^{\times})\cdot\Xi]\supseteq
f_{E}[\mathbb{X}(\mathfrak{o}_{F})]\cap(\prod_{1}^{n}E^{\times})\neq\emptyset$

and the necessity follows.

Conversely, there is

$y_{A}\in\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})\
\ \ \text{ such that }\ \ \ f(y_{A})\in(\prod_{1}^{n}E^{\times})\cdot\Xi.$

By Definition 1.9, there are $\varrho\in G(F)$ and $\sigma_{A}\in
Stab_{A}(\mathbb{X})$ such that $y_{A}=\varrho\sigma_{A}(P)$. This implies
that

$\varrho(P)=\sigma_{A}^{-1}(y_{A})\in\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}}).$

Therefore $\varrho(P)\in\mathbb{X}(\mathfrak{o}_{F})\neq\emptyset$ and the
proof is complete. ∎

If $\Xi$ is an $\bf X$-admissible subgroup of $\prod_{1}^{n}\mathbb{I}_{E}$,
there is an open subgroup $\Xi_{i}$ of $\mathbb{I}_{E}$ for each $1\leq i\leq
n$ such that

$\prod_{i=1}^{n}\Xi_{i}\subseteq\Xi.$

By the class field theory, there is a finite abelian extension $K_{\Xi_{i}}/E$
such that the Artin map

$\psi_{K_{\Xi_{i}}/E}:\ \ \ \mathbb{I}_{E}/E^{\times}\Xi_{i}\cong
Gal(K_{\Xi_{i}}/E)$ (1.12)

gives the isomorphism for $1\leq i\leq n$. Projecting the image of $f$ to the
$i$-th component, one can define

$f_{i}:\ \
\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})\longrightarrow\prod_{1}^{n}\mathbb{I}_{E}\longrightarrow\mathbb{I}_{E}$

for $1\leq i\leq n$.

######  Corollary 1.13.

With the notation as above, $\mathbb{X}(\mathfrak{o}_{F})\neq\emptyset$ if and
only if there is

$x_{A}\in\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})\
\ \ \text{such that}\ \ \ \psi_{K_{\Xi_{i}}/E}(f_{i}(x_{A}))=1$

in $Gal(K_{\Xi_{i}}/E)$ for all $1\leq i\leq n$.

###### Proof.

Since $\psi_{K_{\Xi_{i}}/E}(f_{i}(x_{A}))=1$ for $1\leq i\leq n$, one has

$f(x_{A})\in[\prod_{i=1}^{n}(E^{\times}\cdot\Xi_{i})]\subseteq(\prod_{1}^{n}E^{\times})\cdot\Xi.$

Then the result follows from the same argument as those in Corollary 1.11. ∎

##  2\. Brauer-Manin obstruction

In this section, we will interpret $\mathbb{X}$-admissible subgroups in terms
of Brauer-Manin obstruction. This translation has been done for one dimension
tori in [WX]. For general case, the argument is similar. We keep the same
notation as the previous section.

Since $X_{F}$ is a trivial torsor of $G$ over $F$, the fixed rational
$F$-point $P$ gives $Gal(\bar{F}/F)$-module homomorphism
$\hat{G}\longrightarrow\bar{F}[X]^{\times}$, where $\bar{F}[X]^{\times}$ is
the global sections of $\bar{X}$. Then one obtains the homomorphism

$H^{2}(E,M)\cong H^{2}(F,\mathbb{Z}[Gal(E/F)]\otimes_{\mathbb{Z}}M)\rightarrow
H^{2}(F,\hat{G})\rightarrow H^{2}(F,\bar{F}[X]^{\times})$ (2.1)

by Shapiro’s lemma. Applying Hochschild-Serre spectral sequence (see Theorem
2.20 of Chapter III in [Milne80]) in étale cohomology

$H^{p}(F,H^{q}(\bar{X},\mathbb{G}_{m}))\Rightarrow
H^{p+q}(X_{F},\mathbb{G}_{m}),$

one has

$\phi_{P}:\ \ \ H^{2}(E,M)\longrightarrow
H^{2}(F,\bar{F}[X]^{\times})\longrightarrow Br_{1}(X_{F}).$

Fix the basis $\\{e_{1},\cdots,e_{n}\\}$ of $M$ such that the projection

$p_{i}:\ \ \ \prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})\longrightarrow
R_{E/F}(\mathbb{G}_{m})$

to the $i$-th component is given by $\mathbb{Z}e_{i}\subseteq M$ for each
$1\leq i\leq n$. Then the evaluation of the following morphism by using the
fixed rational point $P$

$X_{F}\xrightarrow{P}G\xrightarrow{\lambda}\prod_{1}^{n}R_{E/F}(\mathbb{G}_{m})\xrightarrow{p_{i}}R_{E/F}(\mathbb{G}_{m})$
(2.2)

at
$\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})$
is $f_{i}$ in §1.

Let $\Xi$ be an $\bf X$-admissible subgroup of $\prod_{1}^{n}\mathbb{I}_{E}$
and $\Xi_{i}$ be an open subgroup of $\mathbb{I}_{E}$ such that

$\prod_{i=1}^{n}\Xi_{i}\subseteq\Xi$

for each $1\leq i\leq n$. Let $K_{\Xi_{i}}/E$ be a finite abelian extension
satisfying (1.12) for $1\leq i\leq n$. Since

$H^{2}(Gal(K_{\Xi_{i}}/E),\mathbb{Z}e_{i})\cong
H^{1}(Gal(K_{\Xi_{i}}/E),\mathbb{Q}/\mathbb{Z})=Hom(Gal(K_{\Xi_{i}}/E),\mathbb{Q}/\mathbb{Z})$

is finite, one has the image of the inflation map composed with $\phi_{P}$

$H^{2}(Gal(K_{\Xi_{i}}/E),\mathbb{Z}e_{i})\rightarrow
H^{2}(E,\mathbb{Z}e_{i})\subseteq
H^{2}(E,M)\xrightarrow{\phi_{P}}Br_{1}(X_{F})$

is finite and denoted by $b(\Xi_{i})$.

######  Definition 2.3.

For any $\bf X$-admissible subgroup $\Xi$ of $\prod_{1}^{n}\mathbb{I}_{E}$ and
an open subgroup $\Xi_{i}$ of $\mathbb{I}_{E}$ with

$\prod_{i=1}^{n}\Xi_{i}\subseteq\Xi$

for each $1\leq i\leq n$, one defines the finite group $b(\Xi)$ of
$Br_{1}(X_{F})$ to be generated by $b(\Xi_{i})$ for $1\leq i\leq n$.

One can reformulate Corollary 1.11 in terms of Brauer-Manin obstruction.

######  Theorem 2.4.

Let $\Xi$ be an $\bf X$-admissible subgroup of $\prod_{1}^{n}\mathbb{I}_{E}$.
Then

$\mathbb{X}(\mathfrak{o}_{F})\neq\emptyset\ \ \ \text{ if and only if}\ \ \ \
[\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})]^{b(\Xi)}\neq\emptyset.$

###### Proof.

The necessity follows from the class field theory and one only needs to show
the sufficiency. Applying Galois cohomology to the short exact sequence

$0\longrightarrow\mathbb{Z}\longrightarrow\mathbb{Q}\longrightarrow\mathbb{Q}/\mathbb{Z}\longrightarrow
0$

with the trivial action, one obtains

$\delta_{i}\ \ Hom(Gal(K_{\Xi_{i}}/E),\mathbb{Q}/\mathbb{Z})\cong
H^{2}(Gal(K_{\Xi_{i}}/E),\mathbb{Z})$ (2.5)

with $1\leq i\leq n$. For any $\chi\in
Hom(Gal(K_{\Xi_{i}}/E),\mathbb{Q}/\mathbb{Z})$, the cup product gives

$\xi_{i}=e_{i}\cup\delta_{i}(\chi)\in
H^{2}(Gal(K_{\Xi_{i}}/E),{\mathbb{Z}}e_{i})\ \ \ \text{and}\ \ \
\beta_{i}=\phi_{P}(\xi_{i})\in b(\Xi_{i})$

with $1\leq i\leq n$.

Let

$(x_{\mathfrak{p}})_{\mathfrak{p}\in\Omega_{F}}\in[\prod_{\mathfrak{p}\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{\mathfrak{p}}})]^{b(\Xi)}$

and evaluate $\beta_{i}$ at $(x_{\mathfrak{p}})_{\mathfrak{p}\in\Omega_{F}}$.
Then

$inv_{\mathfrak{p}}(\beta_{i}(x_{\mathfrak{p}}))=\sum_{\mathfrak{P}|\mathfrak{p}}inv_{\mathfrak{P}}(f_{i}(x_{\mathfrak{p}})\cup\delta_{i}(\chi))$

by (2.1), (2.2), (8.1.4) Proposition and (7.1.4) Corollary in [NSW], where
$\mathfrak{P}$’s are all primes in $E$ above $\mathfrak{p}$. One has

$\sum_{\mathfrak{P}}inv_{\mathfrak{P}}(f_{i}(x_{\mathfrak{p}})\cup\delta_{i}(\chi))=0$

where $\mathfrak{P}$ runs over all primes of $E$. This implies

$\chi(\psi_{K_{\Xi_{i}}/E}(f_{i}[(x_{\mathfrak{p}})_{\mathfrak{p}\in\Omega_{F}}]))=0$

for all $\chi\in Hom(Gal(K_{\Xi_{i}}/E),\mathbb{Q}/\mathbb{Z})$ by (8.1.11)
Proposition in [NSW] and (1.12). Therefore

$\psi_{K_{\Xi_{i}}/E}(f_{i}[(x_{\mathfrak{p}})_{\mathfrak{p}\in\Omega_{F}}])=0$

for $1\leq i\leq n$. The result follows from Corollary 1.13. ∎

##  3\. Strong approximation

As an application, we’ll prove the strong approximation theorem for groups of
multiplicative type. Such strong approximation theorem has been established
for algebraic tori by Harari in [Ha08] and generalized to connected reductive
groups by Demarche in [Dem]. We give the different proof for groups of
multiplicative type where $G$ is not assumed to be connected. We keep the same
notation as that in the previous sections.

By (8.1.16) Corollary in [NSW], the evaluation gives the pair

$G(\mathbb{A}_{F})\times
H^{2}(F,\hat{G})\xrightarrow{inv}\mathbb{Q}/\mathbb{Z}$ (3.1)

such that $G(F)$ is lying in the left kernel of the above pair. Let
$G(F_{\infty})^{0}$ be the set of elements in $G(F_{\infty})$ which are lying
in the left kernel of (3.1).

######  Theorem 3.2.

$G(F)\cdot G(F_{\infty})^{0}$ is dense in the left kernel of (3.1).

###### Proof.

By the definition of $G(F_{\infty})^{0}$, one only needs to show that
$G(F)\cdot U$ contains the left kernel of (3.1) for any open subgroup $U$ of
$G(\mathbb{A}_{F})$ which contains $G(F_{\infty})$. Without loss of
generality, one can assume that

$U=G(F_{\infty})\times\prod_{\mathfrak{p}<\infty_{F}}U_{\mathfrak{p}}$

where $U_{\mathfrak{p}}$ is an open subgroup of $G(F_{\mathfrak{p}})$ for all
$\mathfrak{p}<\infty_{F}$ such that there is a finite subset $S_{0}$ of
$\Omega_{F}$ outside $\infty_{F}$ satisfying condition 1)-4) in §1 and

$U_{\mathfrak{p}}={\bf G}(\mathfrak{o}_{F_{\mathfrak{p}}})$

for $\mathfrak{p}\not\in S_{0}$. For each $\mathfrak{p}\in S_{0}$, one can fix
a positive integer $r_{\mathfrak{p}}$ such that $r_{\mathfrak{p}}$ kills all
torsion points of $T(F_{\mathfrak{p}})$. Let $r=\prod_{\mathfrak{p}\in
S_{0}}r_{\mathfrak{p}}$ and $S_{1}$ be a finite subset of $\Omega_{F}$ outside
$\infty_{F}$ such that Lemma 1.5 holds for $k=l\cdot r$. Define

$\Xi=[\prod_{\mathfrak{p}\in S_{1}}\lambda(U_{\mathfrak{p}})\cdot\mu([c\cdot
2t\cdot h(T,S_{0})\cdot
r]T(F_{\mathfrak{p}}))]\times(\prod_{\mathfrak{p}\not\in
S_{1}}\prod_{1}^{n}\mathfrak{o}_{E_{\mathfrak{p}}}^{\times}).$

By Lemma 1.6, one has that $\Xi$ is an open subgroup of
$\prod_{1}^{n}\mathbb{I}_{E}$ and $\lambda(U)\subseteq\Xi$. Moreover the
induced map

$\lambda:\ \ \ G(\mathbb{A}_{F})/G(F)\cdot
U\longrightarrow\prod_{1}^{n}\mathbb{I}_{E}/(\prod_{1}^{n}E^{\times})\cdot\Xi$
(3.3)

is injective by the exact same arguments in Theorem 1.10.

Let $\Xi_{i}$ be an open subgroup of $\mathbb{I}_{E}$ for each $1\leq i\leq n$
such that

$\prod_{i=1}^{n}\Xi_{i}\subseteq\Xi$

and $K_{\Xi_{i}}/E$ be a finite abelian extension such that $(\ref{art})$
holds for $1\leq i\leq n$. Then

$0\rightarrow\prod_{i=1}^{n}Hom(Gal(K_{\Xi_{i}}/E),\mathbb{Q}/\mathbb{Z})\rightarrow\prod_{1}^{n}Hom(Gal(\bar{F}/E),\mathbb{Q}/\mathbb{Z})\cong
H^{2}(E,M).$

Let

$g=(g_{\mathfrak{p}})\in G(\mathbb{A}_{F})\ \ \ \text{ and }\ \ \
\lambda(g)=(\sigma_{1},\cdots,\sigma_{n})$

with $\sigma_{i}\in\mathbb{I}_{E}$ for $1\leq i\leq n$. By the functoriality
of evaluation and (8.1.11) Proposition in [NSW], one obtains

$\sum_{i=1}^{n}\chi_{i}(\psi_{K_{\Xi_{i}}/E}(\sigma_{i}))=inv(g,\lambda^{*}(\chi_{1},\cdots,\chi_{n}))$

by (3.1) for all

$(\chi_{1},\cdots,\chi_{n})\in\prod_{i=1}^{n}Hom(Gal(K_{\Xi_{i}}/E),\mathbb{Q}/\mathbb{Z}).$

If $g$ is in the left kernel of (3.1), then $\sigma_{i}\in E^{\times}\Xi_{i}$
for $1\leq i\leq n$. Therefore

$\lambda(g)\in(\prod_{1}^{n}E^{\times})\cdot\Xi.$

The injectivity of (3.3) implies that $g\in G(F)\cdot U$. The proof is
complete. ∎

When $G$ is a finite commutative group scheme over $F$, the above result
follows from Poitou-Tate (see (8.6.13) in [NSW]).

######  Remark 3.4.

When $F_{\mathfrak{p}}=\mathbb{R}$, one defines

$N_{G}(F_{\mathfrak{p}})=\\{x+\bar{x}:x\in G(\mathbb{C})\ \text{and $\bar{x}$
is the conjugate point of $x$}\\}.$

When $F_{\mathfrak{p}}=\mathbb{C}$, one defines

$N_{G}(F_{\mathfrak{p}})=G(F_{\mathfrak{p}}).$

Let

$N_{G}(F_{\infty})=\prod_{\mathfrak{p}\in\infty_{F}}N_{G}(F_{\mathfrak{p}}).$

By the local duality over $\mathbb{R}$ by (7.2.17) Theorem in [NSW], one has
that

$N_{G}(F_{\infty})\subseteq G(F_{\infty})^{0}$

with the finite index. One can expect that $G(F)\cdot N_{G}(F_{\infty})$ is
dense in the left kernel of (3.1) by more careful study on archimedean places.

##  4\. Sum of two squares

The natural extension of Fermat-Gauss’ theorem about the sum of two squares
over ${\mathbb{Z}}$ to the ring of integers of quadratic field $F$ was already
studied by Niven for $F=\mathbb{Q}(\sqrt{-1})$ in [Ni] and by Nagell for
$F=\mathbb{Q}(\sqrt{d})$ where

$d=\pm 2,\pm 3,\pm 5,\pm 7,\pm 11,\pm 13,\pm 19,\pm 43,\pm 67,\pm 163$

in [Na53] and [Na61]. Both followed Gauss’ original idea. Therefore the class
number one is assumed and the results obtained there always satisfy the local-
global principle. In this section, we will apply the method developed in the
previous sections to study this question and gives an example of which the
local-global principle is no longer true.

It should be pointed out that our method only produces the idelic class groups
of $\mathbb{Q}(\sqrt{d},\sqrt{-1})$ for solving the problem of sum of two
squares. In order to get the explicit conditions for the sum of two squares,
one needs further to construct the explicit abelian extensions of
$\mathbb{Q}(\sqrt{d},\sqrt{-1})$ corresponding to the idelic class groups by
class field theory. Such explicit construction is a wide open problem
(Hilbert’s 12-th problem) in general but ad hoc method. In his series papers
[W1] and [W2], the first named author gives explicit construction for
infinitely many $d$ and solves the sum of two squares over infinitely many
quadratic fields.

Let $l$ be a prime with $l\equiv-1\mod 8$ and $F={\mathbb{Q}}(\sqrt{-2l})$. We
will study the sum of two squares over $\mathfrak{o}_{F}$. Let

$E=F(\sqrt{-1})\ \ \ \text{and}\ \ \ \Theta=E(\sqrt[4]{l}).$

######  Lemma 4.1.

The field $\Theta/E$ is unramified over all primes except the prime above $l$.

###### Proof.

Since $2$ is totally ramified in $E/\mathbb{Q}$, there is a unique prime $v$
of $E$ over $2$. One only needs to show that $v$ is unramified in $\Theta/E$.

Since $\sqrt{l}\not\in E$, one obtains that $x^{4}-l$ is irreducible over $E$
by the Kummer theory. By Hensel’s Lemma, there is $s\in(1+4{\mathbb{Z}}_{2})$
such that $s^{2}=-l$ over ${\mathbb{Z}}_{2}$. Since

$E_{v}={\mathbb{Q}}_{2}(\sqrt{-1},\sqrt{2})={\mathbb{Q}}_{2}(\zeta_{8})$

where $\zeta_{8}$ is the primitive $8-th$ root of unity, one has

$x^{4}-l=x^{4}+s^{2}=x^{4}-s^{2}(\sqrt{-1})^{2}=(x^{2}-s\sqrt{-1})(x^{2}+s\sqrt{-1})$

over $E_{v}$. By $\zeta_{8}^{2}=\sqrt{-1}$ and $-1=(\sqrt{-1})^{2}$, one
concludes that $\Theta/E$ is unramified over $v$. ∎

It is clear that $l$ is ramified in $F/{\mathbb{Q}}$ and the unique prime of
$F$ over $l$ is denoted by $\mathfrak{l}$. Since $l\equiv 3\mod 4$, one has
that $\mathfrak{l}$ is inert in $E/F$. The unique prime of $E$ above
$\mathfrak{l}$ is denoted by $\mathfrak{L}$ and

$E_{\mathfrak{L}}={\mathbb{Q}}_{l}(\sqrt{-1},\sqrt{l}).$

Since

$x^{4}-l=(x^{2}-\sqrt{l})(x^{2}+\sqrt{l})$

over $E_{\mathfrak{L}}$, there are only two primes $\mathfrak{L}_{1}$ and
$\mathfrak{L}_{2}$ of $\Theta$ over $\mathfrak{L}$ and

$\Theta_{\mathfrak{L}_{1}}=E_{\mathfrak{L}}(\sqrt{\sqrt{l}})\ \ \ \text{and}\
\ \ \Theta_{\mathfrak{L}_{2}}=E_{\mathfrak{L}}(\sqrt{-\sqrt{l}}).$

######  Lemma 4.2.

If $N_{E_{\mathfrak{L}}/F_{\mathfrak{l}}}(\xi)=1$, then the Hilbert symbols
over $E_{\mathfrak{L}}$

$(\xi,\sqrt{l})_{\mathfrak{L}}=(\xi,-\sqrt{l})_{\mathfrak{L}}=1.$

If $N_{E_{\mathfrak{L}}/F_{\mathfrak{l}}}(\xi)=-1$, then the Hilbert symbols
over $E_{\mathfrak{L}}$

$(\xi,\sqrt{l})_{\mathfrak{L}}=(\xi,-\sqrt{l})_{\mathfrak{L}}=-1.$

###### Proof.

If $N_{E_{\mathfrak{L}}/F_{\mathfrak{l}}}(\xi)=1$, then $\xi$ is a unit of
$E_{\mathfrak{L}}$. Therefore

$(\xi,\pm\sqrt{l})_{\mathfrak{L}}=(\xi,\sqrt{-2l})_{\mathfrak{L}}\cdot(\xi,\pm\sqrt{-2})_{\mathfrak{L}}=(N_{E_{\mathfrak{L}}/F_{\mathfrak{l}}}(\xi),\sqrt{-2l})_{\mathfrak{l}}=1$

by 63:11a in [OM] and (1.5.3) Proposition and (7.1.4) Corollary in [NSW].

If $N_{E_{\mathfrak{L}}/F_{\mathfrak{l}}}(\xi)=-1$, then

$(\xi,\pm\sqrt{l})_{\mathfrak{L}}=(\xi,\sqrt{-2l})_{\mathfrak{L}}\cdot(\xi,\pm\sqrt{-2})_{\mathfrak{L}}=(N_{E_{\mathfrak{L}}/F_{\mathfrak{l}}}(\xi),\sqrt{-2l})_{\mathfrak{l}}=(-1,2l)_{l}=-1$

by 63:11a in [OM] and (1.5.3) Proposition and (7.1.4) Corollary in [NSW]. ∎

Let $L=\mathfrak{o}_{F}+\mathfrak{o}_{F}\sqrt{-1}$ be the order of $E$ and
$K_{L}$ be the ring class field of $E$ defined by $L$.

######  Proposition 4.3.

Let $\mathbb{X}$ be the scheme defined by $x^{2}+y^{2}=n$ for some non-zero
integer $n$ over $\mathfrak{o}_{F}$. Then
$\mathbb{X}(\mathfrak{o}_{F})\neq\emptyset$ if and only if there is

$\prod_{w\in\Omega_{F}}(x_{w},y_{w})\in\prod_{w\in\Omega_{F}}\mathbb{X}(\mathfrak{o}_{F_{w}})$

such that

$\psi_{K_{L}/E}(\tilde{f}_{E}[\prod_{w\in\Omega_{F}}(x_{w},y_{w})])=1\ \ \
\text{and}\ \ \
\psi_{\Theta/E}(\tilde{f}_{E}[\prod_{w\in\Omega_{F}}(x_{w},y_{w})])=1$

where $K_{L}$ and $\Theta$ are defined as above, both $\psi_{K_{L}/E}$ and
$\psi_{\Theta/E}$ are the Artin maps and

$\tilde{f}_{E}[(x_{w},y_{w})]=\begin{cases}(x_{w}+y_{w}\sqrt{-1},x_{w}-y_{w}\sqrt{-1})\
\ \ &\text{if $w$ splits in $E/F$}\\\ x_{w}+y_{w}\sqrt{-1}\ \ \
&\text{otherwise.}\end{cases}$

###### Proof.

Define $E_{w}=E\otimes_{F}F_{w}$ for any $w\in\Omega_{F}$. Let $L_{w}$ be the
completion of $L$ inside $E_{w}$ for $w<\infty_{F}$ and $L_{w}=E_{w}$ for
$w\in\infty_{F}$. Let

$SO_{\mathbb{A}}(L)=\\{\sigma\in
R_{E/F}^{1}({\mathbb{G}}_{m})({\mathbb{A}}_{F}):\ \sigma L=L\\}.$

By Lemma 4.1 and Lemma 4.2, the natural group homomorphism

$\lambda_{E}:R_{E/F}^{1}({\mathbb{G}}_{m})({\mathbb{A}}_{F})/R_{E/F}^{1}({\mathbb{G}}_{m})(F)SO_{\mathbb{A}}(L)\rightarrow[\mathbb{I}_{E}/(E^{\times}N_{\Theta/E}(\mathbb{I}_{\Theta}))]\times[\mathbb{I}_{E}/(E^{\times}\prod_{w\in\Omega_{F}}L_{w}^{\times})]$

is well-defined.

If $u\in ker\lambda_{E}$, there are

$\alpha\in E^{\times}\ \ \ \text{ and }\ \ \
i\in\prod_{w\in\Omega_{F}}L_{w}^{\times}$

with $\lambda_{E}(u)=\alpha i$. Therefore

$N_{E/F}(\alpha)=N_{E/F}(i)^{-1}\in
F\cap(\prod_{w\in\Omega_{F}}\mathfrak{o}_{F_{w}}^{\times})=\\{\pm 1\\}.$

Suppose $N_{E/F}(\alpha)=N_{E/F}(i)=-1$. Write

$i=(i_{w})_{w}\in\prod_{w\in\Omega_{F}}L_{w}^{\times}.$

Since $\Theta/E$ is unramified over all primes of $E$ except $\mathfrak{L}$ by
Lemma 4.1, one has $\psi_{\Theta/E}(i_{w})$ is trivial for all primes
$w\neq\mathfrak{l}$, where $i_{w}$ is regarded as an idele whose $w$-component
is $i_{w}$ and 1 otherwise. Since

$N_{E/F}(i_{\mathfrak{l}})=N_{E_{\mathfrak{L}}/F_{\mathfrak{l}}}(i_{\mathfrak{l}})=-1,$

one gets

$\psi_{\Theta/E}(\alpha
i)=\psi_{\Theta/E}(i)=\psi_{\Theta/E}(i_{\mathfrak{l}})=-1\in\mu_{4}$

by Lemma 4.2, where $\mu_{4}$ is the set of 4-th roots of unity and
$Gal(\Theta/E)\cong\mu_{4}$. This contradicts to $u\in ker\lambda_{E}$.

Therefore

$N_{E/F}(\alpha)=N_{E/F}(i)=1.$

This implies that

$\alpha\in R_{E/F}^{1}({\mathbb{G}}_{m})(F)\ \ \ \text{ and }\ \ \ i\in
SO_{\mathbb{A}}(L).$

One concludes that $\lambda_{E}$ is injective. The result follows from
Corollary 1.13. ∎

Finally we will give an explicit example by Proposition 4.3 such that the
local-global principle is not true.

Let

$N_{F/{\mathbb{Q}}}(\delta)=2^{s_{1}}7^{s_{2}}p_{1}^{e_{1}}\cdots
p_{g}^{e_{g}}\ \ \ \text{ and }\ \ \ D(\delta)=\\{p_{1},\cdots,p_{g}\\}$

where $\delta=a+b\sqrt{-14},a,b\in{\mathbb{Z}}$ is an integer of
$F={\mathbb{Q}}(\sqrt{-14})$. Denote $a=7^{s_{3}}a_{1}$ with $7\nmid a_{1}$
and

$\displaystyle D_{1}=\\{p\in D(\delta):(\frac{-1}{p})=(\frac{14}{p})=1\text{
and }(\frac{7}{p})=-1\ \\}$ $\displaystyle D_{2}=\\{p\in
D(\delta):(\frac{-1}{p})=-(\frac{14}{p})=1\text{ and }(\frac{7}{p})=-1\ \\}$
$\displaystyle D_{3}=\\{p\in D(\delta):(\frac{-1}{p})=(\frac{14}{p})=1\text{
and }x^{4}\equiv 7\mod p\text{ is not solvable}\\}.$

It is clear that $e_{i}$ is even if $p_{i}\in D_{2}$.

######  Example 4.4.

Let $F={\mathbb{Q}}(\sqrt{-14})$ and $\delta\in\mathfrak{o}_{F}$ as above.
Then $\delta$ can be written as a sum of two squares over $\mathfrak{o}_{F}$
if and only if

(1) $\delta$ can be written a sum of two squares over $\mathfrak{o}_{F_{w}}$
at every place $w$ of $F$.

(2) $D_{1}\neq\emptyset$; or

$(\frac{a_{1}}{7})=(-1)^{s_{1}+s_{2}+\sum_{p_{i}\in
D_{2}}\frac{e_{i}}{2}+\sum_{p_{i}\in D_{3}}e_{i}}$

for $D_{1}=\emptyset$.

Acknowledgment  The authors would like to thank Colliot-Thélène for helpful
comments on the early version of the paper. The work is supported by the
Morningside Center of Mathematics. The first author is supported by NSFC,
grant # 10671104 and the second author is supported by NSFC, grant # 10325105
and # 10531060.

## References

@article{BR95}

  * author=M.Borovoi, author=Z.Rudnick, title=Hardy-Littlewood varieties and semisimple groups, journal=Invent. Math., volume=119, date=1995, number=, Pages=37 –66,
@article{CTX}

  * author=J-L.Colliot-Thélène, author=F. Xu, title=Brauer-Manin obstruction for integral points of homogeneous spaces and representations by integral quadratic forms, journal=Compositio Math., volume=145 date=2009, Pages=309–363,
@article{Dem}

  * author=C. Demarche, title=Théorèmes de dualité pour les complexes de tores et approximation forte dans les groupes réductifs, journal=preprint, volume=, number= date=, pages=,
@article{Ha08}

  * author=D. Harari, title=Le défaut d’approximation forte pour les groupes algébriques commutatifs, journal=Algebra and Number Theory, volume=2, date=2008, pages=595–611, number=5
@article{HS05}

  * author=D. Harari, author=T. Szamuely, title=Arithmetic duality theorems for 1-motives, journal=J. Reine Angew. Math. (Crelle), volume=578 , date=2005, pages=93–128, number=
@article{HS08}

  * author=D. Harari, author=T. Szamuely, title=Local-global principles for 1-motives, journal=Duke Math. J. , volume=143, number=3 date=2008, pages=531-557,
MilneJ. S.Etale cohomology Princeton Press{}1980 @book{Milne80, author = { J.
S. Milne}, title = {Etale cohomology}, volume = { }, publisher = {Princeton
Press}, place = {{}}, date = {1980}, journal = { }, series = {{}}, volume =
{}, number = { }} MilneJ. S.Arithmetic duality theorems Academic Press{}1986
Perspectives in Mathematics1 @book{Milne86, author = { J. S. Milne}, title =
{Arithmetic duality theorems}, volume = { }, publisher = {Academic Press},
place = {{}}, date = {1986}, journal = { }, series = {Perspectives in
Mathematics}, volume = {1}, number = { }} T.NagellOn the representations of
integers as the sum of two integral squares in algebraic, mainly quadratic
fieldsNova Acta Soc.Sci.Upsal.1511195377pp@article{Na53, author = {T.Nagell},
title = {On the representations of integers as the sum of two integral squares
in algebraic, mainly quadratic fields}, journal = {Nova Acta Soc.Sci.Upsal.},
volum = {15}, number = {11}, date = {1953}, pages = {77pp}} T.NagellOn the sum
of two integral squares in certain quadratic fieldsArk.
Mat.41961267–286@article{Na61, author = {T.Nagell}, title = {On the sum of two
integral squares in certain quadratic fields}, journal = {Ark. Mat.}, volum =
{4}, number = {{}}, date = {1961}, pages = {267-286}} NivenI.Integers of
quadratic fields as sums of
squaresTrans.Amer.Math.Soc.4831940405–417@article{Ni, author = {I. Niven},
title = {Integers of quadratic fields as sums of squares}, journal =
{Trans.Amer.Math.Soc.}, volume = {48}, number = {3}, date = {1940}, pages =
{405-417}} J.NeukirchA.SchmidtK.WingbergCohomology of number
fields323SpringerGrundlehren2000@book{NSW, author = { J.Neukirch}, author =
{A.Schmidt}, author = {K.Wingberg}, title = {Cohomology of Number Fields},
volume = {323}, publisher = {Springer}, series = {Grundlehren}, date = {2000}}
O.T.O’MearaIntroduction to quadratic formsSpringer place=1973@book{OM, author
= {O.T.O'Meara}, title = {Introduction to Quadratic Forms}, publisher =
{{Springer} place={}}, date = {1973}, volume = {{}}, number = {{}}} PlatonovV.
P.RapinchukA. S.Algebraic groups and number theoryAcademic Press{}1994
@book{PR94, author = {V. P. Platonov}, author = {A. S. Rapinchuk}, title =
{Algebraic groups and number theory}, publisher = {Academic Press}, place =
{{}}, date = {1994}, volume = { }, number = { }} SkorobogatovA. N.Torsors and
rational pointsCambridge University Press{} Cambridge Tracts in
Mathematics1442001 @book{Sko, author = {A. N. Skorobogatov}, title = {Torsors
and rational points}, publisher = {Cambridge University Press}, place = {{}},
journal = { }, series = {Cambridge Tracts in Mathematics}, volume = {144},
date = {2001}, number = { }, pages = {{}}} D.WeiOn the sum of two squares in
quadratic fields $\mathbb{Q}(\sqrt{\pm p})$ journal=preprint date=@article{W1,
author = {D.Wei}, title = {{On the sum of two squares in quadratic fields
$\Bbb Q(\sqrt{\pm p})$} journal={preprint}}, volume = {{}}, number = {{}
date={}}, pages = {{}}} D.WeiOn the sum of two integral squares in certain
imaginary quadratic fields journal=preprint date=@article{W2, author =
{D.Wei}, title = {{On the sum of two integral squares in certain imaginary
quadratic fields} journal={preprint}}, volume = {{}}, number = {{} date={}},
pages = {{}}} @article{WX}

  * author=D. Wei, author=F. Xu, title=Integral points for multi-norm tori, journal=preprint, volume=, number= date=, pages=,