Table Of ContentARITHMETIC PURITY OF STRONG APPROXIMATION
7 YANGCAO,YONGQILIANG,ANDFEI XU
1
0
2 Abstract. We prove that any open subset U of a semi-simple simply
connectedquasi-splitlinearalgebraicgroupGwithcodim(G\U,G)≥2
r
p over a number field satisfies strong approximation by establishing a fi-
A brationofGoveratoricvariety. Wealsoproveasimilarresultofstrong
approximationwith Brauer–Manin obstructionforapartialequivariant
1 smooth compactification of a homogeneous space where all invertible
functions are constant and the semi-simple part of the linear algebraic
]
G group is quasi-split. Some semi-abelian varieties of any given dimen-
sion where the complements of a rational point do not satisfy strong
A
approximation with Brauer–Manin obstruction are given.
.
h
t
a
m
[ Contents
2 1. Introduction 1
v
2. Notation and terminologies 4
9
5 3. Arithmetic purity for semi-simple simply connected quasi-split
2 linear algebraic groups 8
7
4. Some fibration arguments with Brauer–Manin obstruction 11
0
. 5. Zariski open strong approximation with Brauer–Manin
1
obstruction for a connected linear algebraic group 18
0
7 6. Arithmetic purity with Brauer–Manin obstruction for connected
1 linear algebraic groups 24
:
v 7. Arithmetic purity with Brauer–Manin obstruction for partial
i equivariant smooth compactifications of homogeneous spaces 29
X
8. Examples 34
r
a References 36
1. Introduction
It is well known that weak approximation over a number field is bira-
tionally invariant among smooth varieties by the implicit function theorem.
By using Manin’s idea, one can further study weak approximation with
Date: April 4, 2017.
Keywordsandphrases. strongapproximation,purity,Brauer–Maninobstruction,(lin-
ear) algebraic groups, Bruhat decomposition, homogeneous spaces.
MSC 2010 : 14G05 11G35 14G25.
1
2 YANGCAO,YONGQILIANG,ANDFEIXU
Brauer–Maninobstruction. Ifavarietyoveranumberfieldsatisfiesweakap-
proximation withBrauer–Manin obstruction whichis conjectured to be true
for rationally connected smooth varieties by Colliot-Th´el`ene (cf. [CT03]),
so are its open subsets with complement of codimension 2 by the purity
≥
theorem for ´etale cohomology.
Under a stronger topology – the adelic topology – instead of the product
topology, one can study strong approximation which is related to integral
points on a variety. Indeed, Eichler [Eic38]-[Eic52], Weil [Weil], Shimura
[Shi64], Kneser [Kne65], Platonov [Plat1]-[Plat2], Prasad [Pra77] and others
established strong approximation for semi-simple simply connected linear
algebraic groups. More recently, Browning and Schindler [BS] established
strong approximation for certain norm varieties by using analytic methods.
Minˇchev in [Min89] pointed out that strong approximation cannot be true
for varieties which are not simply connected. Therefore strong approxima-
tion is not birationally invariant among smooth varieties. However, one can
expect that it is invariantamong smooth varieties up to a closed sub-variety
of codimension at least 2 as Zariski–Nagata purity theorem. Indeed, Wei
[Wei14, Lemma 1.1] and the first and the third author [CX, Proposition
3.6] proved that this is true for the affine space An independently, which
was applied to show strong approximation with Brauer–Manin obstruction
for toric varieties. Harpaz and Wittenberg gave another application of this
resultin[HW]. Based onthisevidence, Wittenbergproposedsuchaninvari-
ance problemin [AIM14, Problem 6] andhis recentsurvey[Wit16, Question
2.11]. Colliot-Th´el`ene asked the same question in his talk in 2016 Indo-
FrenchconferenceinChennai. Inthispaperwegiveanaffirmativeanswerfor
quasi-splitsemi-simplesimplyconnectedlinear algebraicgroups. Inparticu-
lar, we give a new proof of strong approximation for quasi-split semi-simple
simply connected groups by establishing fibrations over toric varieties.
Theorem 1.1 (Theorem 3.6). Let G be a semi-simple simply connected and
quasi-split group over a number field k and S = be a finite set of places of
6 ∅
k. Then any open subset U of G with codim(G U,G) 2 satisfies strong
\ ≥
approximation off S.
In [CTX09], Colliot-Th´el`ene and the third author first studied strong ap-
proximation with Brauer–Manin obstruction andestablished strongapprox-
imation with Brauer–Manin obstruction for homogeneous spaces of semi-
simple simply connected groups. Since then, Harari [Har08], Demarche
[Dem] and Wei and the third author [WX] proved strong approximation
with Brauer–Manin obstruction for various linear algebraic groups. Borovoi
and Demarche [BD13] showed strong approximation with Brauer–Manin
obstruction for homogeneous spaces with geometrically connected stabi-
lizer. Colliot-Th´el`ene and the third author [CTX13], Colliot-Th´el`ene and
Harari [CTH16], the third author [Xu15] and Derenthal and Wei [DW] es-
tablished strong approximation for certain families of homogeneous spaces.
Thefirstandthethirdauthor[CX],[CX1]provedstrongapproximationwith
ARITHMETIC PURITY 3
Brauer–Maninobstructionforapartialequivariantsmoothcompactification
of a connected linear algebraic group and the first author [C16] further es-
tablishedstrongapproximationwithBrauer–Maninobstructionforapartial
equivariant smooth compactification of a homogeneous space by improving
the descent method with combination of the fibration method.
On the other hand, many examples do not satisfy strong approximation
with Brauer–Manin obstruction. We mention the following one which is
closely related to the topic of the present paper. In [CX, Example 5.2], the
first and the third author consider the Q-variety X = G G (1,0) .
m a
× \{ }
They show that for k = Q or an imaginary quadratic field, X does not
k
satisfy strong approximation with Brauer–Manin obstruction off . How-
k
∞
ever, for all other number fields k, X does satisfy strong approximation
k
with Brauer–Manin obstruction off . Such an example shows that the
k
∞
property of strong approximation with Brauer–Manin obstruction
- does not hold in general even for k-rational varieties;
- may become valid over some finite extension of the ground field;
- is not invariant under removed of a codimension 2 closed subset.
≥
Keeping such an example in mind, we may ask a similar question for
strong approximation with Brauer–Manin obstruction.
Question 1.2. Let X be a smooth geometrically integral variety overa num-
berfield k and S be a finite setof places of k. Suppose that Pic(X )isfinitely
k¯
generated and k¯[X]× = k¯× where k¯ isan algebraic closure of k. If X satisfies
strong approximation with Brauer–Manin obstruction off S, does any open
sub-variety of X with complement of codimension 2 satisfies the same
≥
property?
We call such phenomena arithmetic purity of strong approximation
(with Brauer–Manin obstruction) off S. D. Wei gave an affirmative answer
to this question for smooth toric varieties in [Wei14, Theorem 0.2]. In the
present paper, we extend this result to partial smooth equivariant compact-
ifications of homogeneous spaces of connected linear algebraic groups.
Theorem 1.3 (Theorem 7.5). Let G be a connected linear algebraic group,
X be a smooth and geometrically integral variety with an action of G over
a number field k and S = be a finite set of places of k. Suppose that
6 ∅
X contains a Zariski open dense G-orbit with the connected stabilizer of a
k-rational point. If k¯[X]× = k¯× and a simply connected covering Gsc of the
semi-simple part of G over k satisfies the arithmetic purity of strong approx-
imation off S, then X satisfies the arithmetic purity of strong approximation
with Brauer–Manin obstruction off S.
The purity assumption on Gsc holds when Gsc is quasi-split according to
Theorem 1.1.
Furthermore, in 8 we produce some examples which do not verify arith-
§
metic purity of strong approximation even for arbitrarily large codimen-
sion. These examples show that the geometric assumptions in Question
4 YANGCAO,YONGQILIANG,ANDFEIXU
1.2 are necessary. The proof is based on the combination of an extension
of Poonen’s argument [Po10] which was further explained and generalised
by Colliot-Th´el`ene, Pa´l and Skorobogatov in [CPS] with an extension of
Harari–Voloch’s argument [HV10].
Example 1.4 (Theorem 8.1 and Corollary 8.2). Let E be a semi-abelian
variety with dim(E) = 1 over a number field k such that E(k) is not discrete
in E(A∞k).
k
If E is an elliptic curve over k, then the complement of any k-rational
point in E does not satisfy strong approximation with Brauer–Manin ob-
struction off .
k
∞
If A is a non-trivial semi-abelian variety over k such that A(k) is discrete
in A(A∞k), then the complement of any k-rational point in A E does not
k ×k
satisfy strong approximation with Brauer–Manin obstruction off .
k
∞
Our examples, viewed as fibrations over A, have the following two new
features in contrast to Poonen’s construction in [Po10]
- the Brauer group of the total space does not come from the base
variety;
- the Brauer–Manin obstruction does not control the failure of strong
approximation on a“bad”fibre.
Compared to [CX, Example 5.2] mentioned above, these examples provide
someevidencethatthepropertyofstrongapproximationwithBrauer–Manin
obstruction may get lost over some finite field extensions.
2. Notation and terminologies
Inthepresentpaper,thebasefieldkwillbeanumberfieldifnototherwise
specified. We denote by Ω the set of places of k and denote by the set
k k
∞
of Archimedean places. For each place v Ω , the local field k is the
k v
∈
completion of k with respect to the absolute value of v and k = k .
∞ v∈∞k v
Wedenoteby theringofintegersofk forv Ω . Theringofad`eles
v v k k
O ∈ \∞ Q
is denoted by A and the ring of ad`eles without S-components is denoted
k
by AS for a finite set S of Ω . Moreover, we denote by prS : A AS the
k k k → k
natural projection and the induced projections on adelic points of varieties.
For any finite subset S of Ω containing , the ring of S-integers of k is
k k
∞
denoted by . When S = , we simply write for .
S k k S
O ∞ O O
For any abelian group A, we write AD = Hom(A,Q/Z). For any ring R,
we write R× for the set of invertible elements with respect to multiplication
of R.
Let G be a connected linear algebraic group over k. Denote by Gu the
unipotent radical of G. Then Gred = G/Gu is a reductive group. The
semi-simple part Gss of G is the commutator subgroup of Gred. The quo-
tient Gtor = Gred/Gss is an algebraic torus. Let Gssu = Ker(G Gtor)
→
which is an extension of Gss by Gu. Let Gsc be a simply connected cov-
ering of Gss over k. Fix an algebraic closure k¯ of k, the character group
ARITHMETIC PURITY 5
of a (not necessarily connected) linear algebraic group H is denoted by
H∗ = Hom (H ,G ). Then H∗ = k¯[H]×/k¯× by Rosenlicht Lemma.
k¯−gp k¯ m,k¯
Let X be an algebraic variety (separated geometrically connected scheme
of finite type) over k with Brauer group Br(X) = H2 (X,G ). The Brauer
´et m
group has a natural filtration given by
Br (X) = Ker[Br(X) Br(X )] and Br (X) = Im[Br(k) Br(X)].
1 → k¯ 0 →
The Brauer–Manin set X(A )B consists of adelic points that are orthogonal
k
to a subgroup B Br(X) under the Brauer–Manin pairing. It is a closed
⊂
subset of X(A ) with the adelic topology and contains the diagonal image
k
of the set of rational points X(k) by class field theory. When B = Br(X),
we simply write X(A )Br for X(A )Br(X). For a finite set S containing ,
k k k
∞
an integral model of X is defined to be a separated scheme of finite type
X
over satisfying k = X. For any extension K of k, we write
OS X ×OS
X =X K and denote by K[X] the ring of regular functions of X over
K k K
⊗
K. An algebraic variety X with an action of a linear algebraic group G is
called a G-variety.
Definition 2.1. Let X be a variety over a number field k, S be a finite
subset of Ω and B be a subgroup of Br(X).
k
We say that X satisfies strong approximation off S if the diagonal image
of X(k) is dense in prS(X(A )).
k
We say that X satisfies strong approximation with respect to B off S if
the diagonal image of X(k) is dense in prS(X(A )B). When B = Br(X),
k
we say X satisfies strong approximation with Brauer–Manin obstruction off
S.
We say that X satisfies Zariski open strong approximation with respect
to B off S if the diagonal image of U(k) is dense in prS(X(A )B) for any
k
ZariskiopendensesubsetU ofX. WhenB = Br(X), wesaythatX satisfies
Zariski open strong approximation with Brauer–Manin obstruction off S.
When X is proper, theadelic topology on X(A )coincides with theprod-
k
ucttopologyon X(k ),thenstrongapproximationpropertiescoincide
v∈Ωk v
with weak approximation properties.
Q
Itis clear thatZariskiopen strongapproximationwithBrauer–Manin ob-
struction is more restrictive than strong approximation with Brauer–Manin
obstruction by definition. If X satisfies Zariski open strong approxima-
tion with Brauer–Manin obstruction and X(k) = , then X(k) is Zariski
6 ∅
dense in X. For example, an elliptic curve E over a number field k satisfies
strong approximation with Brauer–Manin obstruction off if X(E) is
k
∞
finite. However, E does not satisfy Zariski open strong approximation with
Brauer–Manin obstruction off if E(k) is finite.
k
∞
In the following situation, these two definitions are equivalent.
Proposition 2.2. Let X be a smooth and geometrically integral variety over
a number field k, B be a finite subgroup of Br(X) and S be a finite set of Ω .
k
6 YANGCAO,YONGQILIANG,ANDFEIXU
If X satisfies strong approximation with respect to B off S, then X satisfies
Zariski open strong approximation with respect to B off S.
Proof. For a Zariski open dense subset U of X, there are a finite subset P
0
of Ω containing S and an integral model of X over such that
k ∪∞k X OP0
( ) B for all v P . For any open subset
v 0
X O ⊥ 6∈
W = W ( ) X(A )
v v k
× X O ⊂
v∈P v∈/P
Y Y
with P P and W = X(k ) for v S such that W X(A )B = , one
0 v v k
⊃ ∈ ∩ 6 ∅
can choose v P and define
0
6∈
W = W (U(k ) ( )) ( ) W.
1 v × v0 ∩X Ov0 × X Ov ⊂
vY∈P v∈/(PY∪{v0})
Since X is smooth over k, then U(k ) is dense in X(k ) by [PR, Lemma
v0 v0
3.2]. This implies that
U(k ) ( ) = and W X(A )B = .
v0 ∩X Ov0 6 ∅ 1∩ k 6 ∅
Therefore U(k) W = X(k) W = as required. (cid:3)
1 1
∩ ∩ 6 ∅
InSection5,wewillgiveacompletedescriptionofZariskiopenstrongap-
proximation with Brauer–Manin obstruction for connected linear algebraic
groups.
Definition 2.3. Let X be a smooth variety over a number field k, B be a
subgroup of Br(X), S be a finite subset of Ω and c be an integer bigger
k
than 1.
We say that X satisfies the arithmetic purity off S of codimension c if
any open subvariety with complement of codimension c satisfies strong
≥
approximation off S. In particular, when c = 2, we simply say that X
satisfies the arithmetic purity off S.
We say that X satisfies the arithmetic purity with respect to B off S of
codimension c if any open subvariety with complement of codimension c
≥
satisfies strong approximation with respect to B off S. In particular, when
c= 2, we simply say that X satisfies the arithmetic purity with respect to B
off S.
It is clear that the similar arithmetic purity for weak approximation with
Brauer–Manin obstruction holds for smooth varieties by the cohomological
purity theorem.
It is well known that a smooth variety X will satisfy strong approxima-
tion if an open dense subset of X satisfies strong approximation. Such a
result cannot be extended to the situation of strong approximation with
Brauer–Manin obstruction because any smooth geometrically integral vari-
ety over Q or an imaginary quadratic field contains an open dense subset
satisfying strong approximation with Brauer–Manin obstruction off by
k
∞
[LX15, Theorem 1.3]. However, if the Brauer groups of the open subsets are
ARITHMETIC PURITY 7
finite modulo the Brauer group of the base field, such a result still holds by
[CTX13, Proposition 2.6]. Under the assumption of Question 1.2, we have
the following result.
Proposition 2.4. Let X be a smooth and geometrically integral variety over
a number field k with k¯[X]× = k¯× and S be a finite subset of Ω . Suppose
k
that Pic(X ) is finitely generated and
k¯
Br (X) B Br(X) such that [B : Br (X)] < .
1 1
⊆ ⊆ ∞
(1) If a Zariski open dense subset U of X satisfies strong approximation
with respect to B off S, then X satisfies strong approximation with respect
to B off S.
(2) If X satisfies strong approximation with respect to B off S, then X
satisfies Zariski open strong approximation with respect to B off S.
Proof. One can assume that X(A )B = . For any
k
6 ∅
(x ) X(A )B with B Br (X),
v k 1
∈ ⊇
f
there is a universal torsor Y X with (y ) Y(A ) such that
v k
−→ ∈
f((y )) = (x )
v v
by [Sk99, Theorem 3].
Let U be a Zariski open dense subset of X and
V = U Y and Λ Br(X)
X
× ⊂
be a finite set of representatives of B/Br (X). Let P be a sufficiently large
1 0
finite set containing S and such that
k
∞
f f
(a) The morphism Y X extends to a morphism of integral
−→ Y −→ X
models over and y ( ) for v P .
OP0 v ∈ Y Ov 6∈ 0
(b) Both open immersions U X and V Y extend to open immer-
→ →
sions and of integral models over respectively.
U → X V → Y OP0
(c) = over and the projection = is an
V U ×X Y OP0 V U ×X Y → U
extension of V f|V U.
−−→
(d) ( ) Λ and ( ) = by the Lang–Weil estimation for all
v v
X O ⊥ V O 6 ∅
v P .
0
6∈
For an open set
(x ) W = W ( ) X(A )
v v v k
∈ × X O ⊂
v∈P v∈/P
Y Y
with P P and W = X(k ) for v S, we choose y′ V(k ) f−1(W )
⊃ 0 v v ∈ v ∈ v ∩ v
sufficiently close to y by [PR, Lemma 3.2] such that
v
b(f(y′)) = b(f(y )) = b(x )
v v v
for all b Λ and v P.
∈ ∈
8 YANGCAO,YONGQILIANG,ANDFEIXU
For v P, we choose y′ ( ) by (d). Then (y′) V(A ) Y(A )
6∈ v ∈ V Ov v ∈ k ⊆ k
with
f((y′)) W U(A )Br1(X)
v ∈ ∩ k
by [Sk99, Theorem 3]. For v P, both f(y′) and x are in ( ) by (a),
6∈ v v X Ov
(b) and (c). This implies that b(f(y′))= b(x ) are trivial for all b Λ with
v v ∈
v P by (d). Therefore f((y′)) W U(A )B.
6∈ v ∈ ∩ k
Since
(X(k) W) = (U(k) W)=
∩ ∩ 6 ∅
forbothcase(1)and(2)byassumptionofU andX respectively, oneobtains
the desired results. (cid:3)
Remark 2.5. In this paper we restrict ourselves to varieties over number
fields. IfX satisfiesstrongapproximationoveranumberfieldkandX(A ) =
k
6
, then X is simply connected by [Min89, Theorem 1] (see also [Rap14,
∅ k¯
Proposition 2.2]). The analogue is not true over a global function field.
For example, G satisfies strong approximation off a finite non-empty set
a
of primes over a global function field but G is not simply connected over
a
an algebraic closure of this field. This is because G admits Artin–Schreier
a
´etale coverings over a field of positive characteristic.
3. Arithmetic purity for semi-simple simply connected
quasi-split linear algebraic groups
In this section we study arithmetic purity for semi-simple simply con-
nectedquasi-splitlinearalgebraicgroupsbyestablishingfibrationsovertoric
varieties.
LetGbeaquasi-splitsemi-simplesimplyconnectedlinearalgebraicgroup
and B be a Borel subgroup of G over a field k of characteristic 0. By the
Bruhat decomposition (see [Sp98, (8.3.11) Corollary, (8.2.4) Proposition (ii)
and (8.3.2) Lemma (ii)]), there is a root w G(k) (which is the longest
0
∈
one) such that the big cell V = Bw B is a Zariski open dense subset of G
0 0
over k by Galois descent. If Bu is the unipotent radical of B and T is a
maximal torus of B over k, one has the short exact sequence
1 Bu B p T 1
→ → −→ →
of algebraic groups over k where the map p is T-equivariant. Moreover, one
has an isomorphism of varieties over k
(3.0.1) B Bu ∼= V ; (b,u) b w u
k 0 0
× −→ 7→ · ·
by [Sp98, (8.3.5) Lemma (ii) and (8.3.6) Lemma (ii)]. Combining (3.0.1)
with the map p, one obtains a T-equivariant morphism over k
(3.0.2) φ: V T with k¯[T]×/k¯× ∼= k¯[V ]×/k¯×
0 0
→ −→
as Gal(k¯/k)-modules by (3.0.1).
ARITHMETIC PURITY 9
Recall atorus T overk is called quasi-trivial if k¯[T]×/k¯× is a permutation
Gal(k¯/k)-module (see [CT08, Definition 1.1]). This is equivalent to that
T = Res (G ) for some finite ´etale k-algebra K.
K/k m
Lemma 3.1. Suppose G is a quasi-split semi-simple simply connected linear
algebraic group over a field k of characteristic 0 and B is a Borel subgroup
of G over k. If V = Bw B is the big cell of G over k as above, then the
0 0
map
k¯[V ]×/k¯× ∼= Div (G ); f div(f)
0 −→ Gk¯\(V0)k¯ k¯ 7→
is a canonical isomorphism of Gal(k¯/k)-modules. Moreover, any maximal
torus of B over k is quasi-trivial.
Proof. Since G is semi-simple and simply connected, one has k¯[G]× = k¯×
and Pic(G ) = 0 by [CTX09, Proposition 2.6]. These imply the first part.
k¯
The second part follows from (3.0.2). (cid:3)
Remark 3.2. Let G be the group of norm 1 elements of the division quater-
nionalgebraoverR. ItiswellknownthatGissemi-simplesimplyconnected
overR. SincethesetofrealpointsofanymaximaltorusofGiscompactbut
the set of real points of a quasi-trivial torus is not compact, one concludes
that there are no quasi-trivial maximal tori of G over R.
Since T is quasi-trivial, there is a finite ´etale k-algebra K such that T =
Res (G ). Then the morphism φ in (3.0.2) can be extended uniquely to
K/k m
a T-equivariant morphism
φ : G Res (G )
K/k a
→
by Lemma 3.1 and [C16, Proposition 2.3]. The following definition of stan-
dard toric varieties already appeared in [CX1, Definition 3.11].
Definition 3.3. Let K/k be a finite ´etale k-algebra. The unique minimal
open toric sub-variety R of Res (G ) over k with respect to Res (G )
K/k a K/k m
such that
codim(Res (G ) R,Res (G )) 2
K/k a K/k a
\ ≥
is called the standard toric variety of Res (G ).
K/k a
The following proposition is a variant of [CX1, Proposition 3.12].
Proposition 3.4. With the above notations, there is an open T-subvariety
φ
Y of G containing V such that the extension Y R of φ in (3.0.2) is
0
−→
smooth with non-empty geometrically integral fibres.
Proof. This follows from [C16, Proposition 2.2]. (cid:3)
The following proposition should be well known. We provide the proof
for completeness.
10 YANGCAO,YONGQILIANG,ANDFEIXU
Proposition 3.5. Let f : Y X be a faithfully flat morphism between
→
geometrically integral varieties over a field k and c be a positive integer. If
V is an open subset of Y such that codim(Y V,Y) c, then there is an
\ ≥
open dense subset U of X such that
codim(f−1(x) V,f−1(x)) c
\ ≥
for any closed point x U.
∈
Proof. Consider Z = Y V as a reduced closed sub-scheme of Y. Let
\
Z be the set of all irreducible components of Z and f = f for
{ i}1≤i≤l i |Zi
1 i l. For any given 1 i l, we consider the following two cases.
≤ ≤ ≤ ≤
If f (Z ) = X, there is an open dense subset W of Z such that f :W
i i i i i i
→
X is flat by [EGA IV, 11.1.1]. Let U = f (W ). Then U is an open dense
i i i i
subset of X. For any closed point x U , one concludes that
i
∈
dim(f−1(x)) = dim(Z ) dim(X) dim(Z) dim(X).
i i − ≤ −
Otherwise f (Z ) is not dense in X. Then U = X f (Z ) is an open
i i i i i
\
dense subset of X and f−1(x) is empty for any closed point x U .
i ∈ i
Let
l
U = U .
i
i=1
\
It is clear that U is an open dense subset of X and
l
codim(f−1(x) V,f−1(x)) = dim(f−1(x)) dim( f−1(x)) c
\ − i ≥
i=1
[
for any closed point x U as desired. (cid:3)
∈
The main result of this section is the following arithmetic purity theorem
for quasi-split semi-simple simply connected groups.
Theorem 3.6. Let G be a semi-simple simply connected and quasi-split
group over a number field k and S be a non-empty finite subset of Ω . Then
k
any open subset U of G with codim(G U,G) 2 satisfies strong approxi-
\ ≥
mation off S.
Proof. Let B be a Borel subgroup of G over k and V = Bw B be the big
0 0
cell of G where w G(k) be the longest root. Suppose T is a maximal
0
∈
torus of B. Then T = Res (G ) for some ´etale k-algebra K by Lemma
K/k m
3.1. Let R be the standard toric variety of Res (G ) in sense of (3.3).
K/k a
Then there are an open variety Y of G containing V and a T-equivariant
0
smooth surjective morphism φ : Y R with geometrically integral fibres
→
which extends (3.0.2) by Proposition 3.4.
Withoutlossof generality, weassumethatU Y withcodim(Y U,Y)
⊆ \ ≥
2. InordertoproveU satisfiesstrongapproximationoffS,oneonlyneedsto
verify three conditions of [CTX13, Proposition 3.1]. Let R = φ(U). Then
1
R is an open subset of R with codim(R R ,R) 2 by the faithful flatness
1 1
\ ≥
