Université Paris Sud

Faculté des Sciences d’Orsay

Département de Mathématiques

M2 Arithmétique, Analyse, Géométrie

Mémoire Master 2

presented by

Marco D’Addezio

Descent theory for strong approximation for varietiescontaining a torsor under a torus

directed by

Olivier Wittenberg

Academic year 2015/2016

M2 Arithmétique, Analyse, GéométrieDépartement d’Enseignement de Mathématiques, Bât. 425Université Paris-Sud 1191405 Orsay CEDEX

“Oh, quanta strada nei miei sandali,quanta ne avrà fatta Bartali?

Quel naso triste come una salita,quegli occhi allegri da italiano in gita.”

“Oh, how many roads under my sandals,how many did Bartali pass?

That sad nose like an ascent,those happy eyes of an Italian on an outing.”

Paolo Conte

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2

Contents

1 Weak and strong approximation 61.1 Weak approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Adelic points and strong approximation . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Torsors under groups of multiplicative type 102.1 Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Groups of multiplicative type . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Rosenlicht lemma and type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Fundamental exact sequence and elementary obstruction . . . . . . . . . . . . . . . . 132.3 Local description of torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Brauer-Manin obstruction 203.1 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 The adelic Brauer-Manin pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Introduction to descent theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 Hochschild-Serre and filtration of the Brauer group . . . . . . . . . . . . . . . 233.3.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Strong approximation for varieties containing a torsor under a torus 264.1 The case when Pic(Xk) is torsion-free . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 An example... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 ...and a counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3

Introduction

Let k be a number field, strong approximation has been widely studied for algebraic groups andtheir homogeneous spaces over k. At the same time for general non-proper varieties very little isknown.

The first results of strong approximation with Brauer-Manin obstruction appeared for the firsttime in [CTX09], where J.-L. Colliot-Thélène and F. Xu established strong approximation withBrauer-Manin obstruction for homogeneous spaces of semi-simple and simply connected algebraicgroups. Another result is the one of Y. Cao and F. Xu [CX14] on smooth toric varieties. The resultswe will present are obtained for varieties similar to toric varieties and they are achieved by Wei in[Wei14].

In his article Wei expands one of the first results of Colliot-Thélène and Sansuc’s descent theory.In the famous article [CTS87] it is proven that on a proper, smooth k-varietyX containing a k-torsorunder a torus as a dense open subset, weak approximation holds with Brauer-Manin obstruction.Wei studies strong approximation in this context, replacing the hypothesis of properness with thehypothesis that k[X]∗ = k

∗. The theorem he has proven is the following.

Theorem 0.1 (Wei). Let X be a smooth variety over k with k[X]∗ = k∗ and suppose that X

contains a dense open subset U that is a k-torsor under a torus T . For any closed subset W of Xwith codimension at least two, the commutative étale algebraic Brauer-Manin obstruction is the onlyobstruction to strong approximation off one place for X \W .

When Pic(Xk) is torsion free the commutative étale algebraic Brauer-Manin obstruction canbe replaced by algebraic Brauer-Manin obstruction. We will see at the end of the last chapter thecounterexample proposed by Wei for a variety that has a geometric Picard group with some torsion.

In the first three chapters we introduce weak and strong approximation, the Brauer-Maninobstruction and Colliot-Thélène and Sansuc’s descent theory. These topics have been studied in aWorkshop in Orsay, during the first semester of this year. Thus the first part is mainly obtainedby arranging the notes of the seminars, adding and fixing some parts. In the exposition of thesechapters we have followed the book written by Skorobogatov [Sko01].

The last chapter is an application of the theory developed in the previous chapters. We willpresent some results of the article of Wei.

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Notation

We will use the following notation:

k we will always be a field, in particular in all the chapters except the Chapter 2 and otherparts of the text where it’s stated differently it will be a number field.

We will call Ωk the set of places of k, Ω∞ the set of infinite places and we will usually denotewith ν a place of k.

We will write Γk for the absolute Galois group of k. If A is a discrete Γk-module we willdenote H i(k,A) the groups of Galois cohomology of A with respect to Γk. If X is a schemeand F is an étale (resp. fppf ) sheaf on X we will write H i

ét(X,F) (resp. H ifppf (X,F)) for

the étale (resp. fppf ) cohomology of F .

We will call Pic(X) the Picard group of X, if Z is a closed subset of X, DivZ(X) will be thegroup of divisors and Div0(X) the group of principal divisors (we will always consider divisorsover regular schemes).

5

Chapter 1

Weak and strong approximation

1.1 Weak approximation

Let k be a number field, for any ν, we can endow X(kν) with its ν-adic topology. For any finitesubset S ⊆ Ωk, we define the topological space

X(kΩk\S) :=∏

ν∈Ωk\S

X(kν),

where the topology is the product of the topologies of X(kν). We will call this topology the weaktopology. Of course there are diagonal embeddings of X(k) in X(kΩk\S). If we take X = A1

k weknow that the diagonal map

k →∏ν∈Ωk

kν

is dominant. This is a classical result of number theory and the property is usually called weakapproximation. We can wonder if this is true even for other varieties. Thus we give the followingdefinition.

Definition 1.1. Let S be a finite subset of Ωk, we say that a smooth, geometrically integralk-variety X satisfies weak approximation off S, if X(kΩk) is empty or the diagonal map

X(k)→ X(kΩk\S)

is dominant.

As A1k satisfies weak approximation, even Ank satisfies weak approximation for any n. The same

remains true for any open subset of Ank , because if U is a Zariski open subset of X, the set U(kν)is open in X(kν).

Let’s focus on the opposite problem: if U is a dense open subset of X and we have weakapproximation on U , what can we say about X? In virtue of the inverse function theorem forcomplete fields with respect to a non-trivial absolute value as it’s proven in [VA94] at the beginningof Chapter 3, we have:

Proposition 1.2. Let X be an irreducible, smooth variety defined over k, such that X(kν) is notempty. Then, for any non-empty Zariski open subset U of X, the set U(kν) is dense in X(kν) in theν-adic topology. Furthermore, any non-empty ν-adically open subset F of X(kν) is Zariski-dense inX; in particular, X(kν) is Zariski-dense in X.

As a consequence we have.

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Corollary 1.3. Let X be a smooth, geometrically integral k-variety containing an open dense subsetwhich verifies weak approximation, then X satisfies weak approximation.

Corollary 1.4. If X is a smooth, geometrically integral k-rational variety, then weak approximationholds.

Another quite elementary example of varieties satisfying weak approximation can be found inthe article of Colliot-Thélène, Sansuc and Swinnerton Dyer [CTSSD87]. On page 68, the followingtheorem is stated.

Theorem 1.5. Let k be a number field and let V ⊆ Pnk with n ≥ 6 be a pure codimension 2intersection of two quadrics over k. Assume that V is geometrically integral and not a cone. Let Xbe V smooth and assume that X(k) is not empty, then weak approximation holds for X.

1.2 Adelic points and strong approximation

If X is a variety that is not proper one can even pay attention to integral points.

Definition 1.6. Let X be a k-variety, we say that a separated scheme X of finite type over Spec(Ok)is a model of X if X ' Xη, with η the generic point of Spec(Ok).

Usually we will suppose the model to be integral. We will use the following proposition that isalmost completely proven in the book [Jah15, Chapter IV, Lemma 3.4].

Proposition 1.7. Let X be a k-variety, then there exists a model X of X. Any two models of X areisomorphic outside a finite number of places. Moreover if the variety is reduced (resp. irreducible,resp. proper) any model is reduced (resp. irreducible, resp. proper) outside a finite number of places.

For any finite place ν, we have the inclusion

X(Oν) → X(kν),

we will call any element in the image of this map, local integral points.

Remark. The injectivity of the map displayed above is a consequence of the valutative criterion ofseparatedness. If X is proper then for almost any place the map is even surjective thanks to thevalutative criterion of properness applied to X that is proper outside a finite number of places. Wealso notice that the local integral points may depend on the choice of a model of X.

Definition 1.8. We define now the set of adelic points of X away from S by

X(ASk ) := (ρν)ν ∈ X(kΩ\S)| all but finitely many ρν are integral

with the convention X(Ak) := X(A∅k).

We notice that the definition does not depend on the choice of the model because any two ofthem are isomorphic away from a finite number of places. We will not consider X(ASk ) with thetopology of subspace of X(kΩ\S), we will be more interested in the topology defined by the basis ofopen sets of the form ∏

ν∈TUν ×

∏ν 6∈T

X(Oν),

with Uν an open of X(kν) and T finite, such that Ω∞ \ (S ∩ Ω∞) ⊆ T . We will call this topologythe strong topology or the adelic topology.

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Example. If X = A1k, then X(Ak) is the set of adeles with the adelic topology; if X = Gm then

X(Ak) is the set of ideles with the idelic topology.

In analogy with weak approximation we give the following definition.

Definition 1.9. Let X be a smooth, geometrically integral k-variety, if S is a finite subset of Ωk

we will say that X satisfies strong approximation off S if X(Ak) = ∅ or

X(k)→ X(ASk )

is dominant.

Thanks to the valuative criterion of properness we can deduce that if X is proper then X(ASk ) =X(kΩk\S) as topological spaces. Thus for proper schemes weak and strong approximation areequivalent.

We also have that Ank satisfies strong approximation off one place, i.e. off S = ν, for anyν ∈ Ωk. This is a classical result of number theory, in the case when k = Q and S = ∞ this isjust the Chinese remainder theorem.

What can we say about strong approximation on open subvarieties of Ank? In general it doesn’thold off a finite set of places.

Example. Consider Gm,Q → A1Q, if we had strong approximation away from infinity, Gm(Z) would

be dense in∏`Gm(Z`), since if a point in

∏`Gm(Z`) can be approximated in the strong topology by

rational points it can be approximated by integral points. But Gm(Z) = 1,−1 and it is not densein any Gm(Z`). We can even show that strong approximation doesn’t hold off a finite place p. If itheld then as before the set Gm(Z[1/p]) would be dense in

∏`6=pGm(Z`). But Gm(Z[1/p]) = 〈−1, p〉,

so if we consider the extension Q(√−1,√p) of Q, by the Chebotarev density theorem, there exists

at least a prime ` 6= 2 that is totally split. This means that −1 and p are both squares modulo`, thus the image of Gm(Z[1/p]) in Gm(Z`) is contained in the subgroup of squares of Gm(Z`).Obviously the result can be extended to any number field k and any finite set of places S. Thanksto Dirichlet’s unit Theorem the set Gm(Ok,S) is finitely generated, let’s say by t1, . . . , tn, then wecan take k(

√t1, . . . ,

√tn) and apply the Chebotarev density theorem again.

We also have many other similar obstructions on Gm, just taking any étale cover

Gmt→tn−−−→ Gm.

This phenomenon can be generalized by the following theorem due to Minchev whose proof can befound in [Rap12, page 9].

Theorem 1.10 (Minchev 1989). Let X be an irreducible normal variety over a number field k suchthat X(k) 6= ∅. If there exists a nontrivial connected unramified covering f : Y → X defined overan algebraic closure k, then X does not satisfy strong approximation off any finite set S of placesof k.

In particular if we take any polynomial in n variables f with coefficients in k and we take theopen U in Ank that is defined by f 6= 0, we can then take as Y the closed variety in An×Gm definedby f = xmn+1 6= 0. The natural projection Y → U is unramified, thus on U we cannot have strongapproximation with respect to any finite set of places.

At the same time if we take X the complement of a closed subset in Ank of codimension at leasttwo we still have strong approximation off one place. This result has been proven independently byWei and by Cao and Xu. We will propose a slightly generalisation of this result, using a variant ofthe proof of Wei.

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Proposition 1.11. Let X be a smooth variety over the number field k and let S be a finite set ofplaces of k. Suppose that X satisfies weak approximation off S. If there exists a dense open subsetU of X with the following property:

P) For any x ∈ U(k) there exists a dense open Vx of U such that for any y ∈ Vx(k) there exists avariety Zx,y that satisfies strong approximation off S and a morphism fx,y : Zx,y → X suchthat in the fibers of fx,y with respect to x and y there are rational points.

Then X satisfies strong approximation off S.

Proof. Let X be a model of X and let T be a finite subset of Ωk \S containing Ω∞ \ (S ∩Ω∞). Forany adelic point

P = (Pν)ν∈Ωk ∈∏

ν∈T∪SX(kν)×

∏ν∈Ωk\(T∪S)

X(Oν)

we have to find a rational point of X that is as close as we want to P when ν ∈ T and integralwhen ν ∈ Ωk \ (T ∪ S).

In virtue of the inverse function theorem (Proposition 1.2), we can find local points of U thatare as close as we want to Pν for ν ∈ T . Thus by weak approximation on U off S we can chooserational points x of U that are as close as we want to Pν for ν ∈ T .

A priori the local points xν (the image of x in U(kν)) could fail to be integral when ν ∈Ωk \ (T ∪ S), let’s suppose that xν is not integral when ν ∈ T ′ ⊆ Ωk \ (T ∪ S). We take Vx as inthe property P). Thanks to the implicit function theorem, we can find local points of Vx near Pνfor ν ∈ T ′. As Vx satisfies weak approximation off S we can find a rational point y of Vx near thelocal points Pν for ν ∈ T ′, in particular we can choose y as an Oν-point of X for any ν ∈ T ′.

Now let’s take fx,y : Zx,y → X as in the property P), there exists on Zx,y an adelic pointQ = (Qν)ν∈Ωk such that fx,y(Qν) = xν when ν ∈ Ωk \ (T ′∪S) and fx,y(Qν) = yν (the image of y inU(kν)) when ν ∈ T ′. Thanks to strong approximation on Zx,y off S we can find a rational point zof Zx,y that is near Q in the strong topology of Zx,y(Ak) off S. As the morphism fx,y : Zx,y(Ak)→X(Ak) is continuous, fx,y(z) can be near to P for the places ν ∈ T and integral outside T ∪S, thuswe have the result.

Corollary 1.12. If X is a smooth k-variety1 and there exists a variety Z that satisfies strongapproximation off S and a morphism f : Z → X such that f restricted to a certain open subset Z ′

of Z is an open immersion with dense image, then X satisfies strong approximation off S.

Corollary 1.13. Let X be an open subvariety of Ank obtained by removing a closed subset W ofcodimension at least two, then X satisfies strong approximation off one place.

Proof. We take as U the space X and for any rational point x ∈ U(k) we take as Vx the openobtained removing the closed subset of X that is the union of all the lines joining x with W . Bythe hypothesis on the codimension Vx is non-empty, hence dense. For any point y ∈ Vx differentfrom x we can take as Zx,y the line ` ⊆ X that joins x and y.

Corollary 1.14. Let X be an open subvariety of a smooth quadric Q of Pnk obtained removing aclosed subset of codimension at least two, then X satisfies strong approximation.

We will see that this proposition can be used to prove that for another family of varieties strongapproximation holds off one place (Proposition 4.1).

1Here it’s not necessary to require that it satisfy weak approximation off S because it will be a consequence of theother hypotheses.

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Chapter 2

Torsors under groups of multiplicativetype

2.1 Torsors

Definition 2.1 (Torsors). If X is a scheme, G a group scheme over X, we define a left (resp. right)X-torsor under G as an fppf map Y → X, with a left (resp. right) G-action, such that there existsa covering (Ui → X) in the fppf topology that trivialises Y , in the sense that Y ×X Ui ' G×X Uicompatibly with the action of G and the projection to Ui.

We will mean left torsors if we do not specify. Observe that Y → X itself trivializes a torsorY → X as there is a natural map G ×X Y → Y ×X Y that sends (g, y) 7→ (g.y, y). We can checklocally in the fppf topology that this map is an isomorphism. If X is equal to Spec(k) the existenceof a rational point x : Spec(k)→ Y implies that G ' Y . Indeed we can check locally that the mapG→ Y that sends g to g.x is an isomorphism.

If G is commutative, quasi-projective and flat over X, there exists a bijection of pointed setsbetween the class of isomorphism of torsors over X under G and the group H1

fppf (X,G) that sendsthe class represented by G in the zero cohomological class. If moreover G is smooth, thanks to aTheorem of Grothendieck [Gro68, Théorème 11.7], H i

fppf (X,G) = H iét(X,G).

In this work we will consider only torsors over k-varieties. Let S be a group of multiplicativetype, then if f : Y → X is an X-torsor under S, we have a map θ : X(k)→ H1(k, S) which mapsx to the class [Yx] ∈ H1(k, S). We notice that θ(x) = 0 if and only if x ∈ f(Y (k)).

Definition 2.2 (Twist). Suppose we have a left torsor Y → X under a k-group G and a rightk-torsor Z under the same group, then we call the twist of Y by Z, if it exists, the quotient ofZ ×k Y by the action of G, via the map (g, z, y) 7→ (zg−1, gy) and we denote it by ZY .

Theorem 2.3. We haveX(k) =

⊔Z

Zf(ZY (k))

where the union is taken over a set of representatives of the isomorphism classes of right k-torsorsunder G.

2.1.1 Groups of multiplicative type

Let’s fix a field k. An algebraic group S is a group of multiplicative type if Sk is a subgroupscheme of Gn

m,kfor a certain n. Recall that if the characteristic of k is zero a group of multiplicative

type S over k is a commutative linear k-group which is an extension of a finite group by a torus.

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The module of characters of S is the abelian group S = Homk−groups(Sk,Gm,k), equipped with theaction of the Galois group Γk. At the same time if M is a Γk-module, we will call M the groupHomΓk(M,k

∗).

We have an equivalence of categories between groups of multiplicative type and Γk-modules offinite type. In order to understand better this equivalence we need to recall the Weil restriction.

Weil restrictionLet L/k be a finite extension of degree d. There is a functor Var /k → Var /L called extension ofscalars that sends X 7→ XL. It is also possible to construct a functor in the other direction that isjust the right adjoint. The functor is called Weil restriction or restriction of scalars, and it’s usuallydenoted RL/k(−). By construction we have

HomL(VL,W ) ∼= Homk(V,RL/k(W )).

Moreover this functor sends algebraic groups to algebraic groups and if the field is perfect tori totori.

Proposition 2.4. For any finite extension L/k and for any quasi-projective variety X over L, thefunctor RL/k(X) is representable by a quasi-projective variety over k.

Proof. You can find the proof in the book of C.Scheiderer [Sch94, Corollary 4.8.1 ].

The first example is the following:

RL/k(ANL ) ' ANdk

and this is checked by choosing a basis of L/k.If we take the open immersion Gm,L → A1

L and we apply the functor RL/k(−) we obtain a mapRL/k(Gm,L) → Adk that can be shown (Prop. 4.9 in [Sch94]) to be again an open immersion. Thuswe have

Proposition 2.5. If k is perfect the tori RL/k(Gm,L) are k-rational varieties.

We can now state the theorem.

Theorem 2.6. The association G G gives an equivalence of categories between the category ofk-groups of multiplicative type and the category of discrete Γk-modules of finite type. Moreover asequence of groups of multiplicative type is exact iff the dual of Γk-modules of characters is exact.Finally if k is perfect and L/k is a finite extension the torus RL/k(Gm) is sent to Z[Homk−alg(L, k)].

Proof. For the proof look at Proposition 1.4 Exposé X in [DG70].

We will prove here another important equivalence we will use often.

Proposition 2.7. There is an equivalence of categories that preserves cohomology, between thecategory of abelian étale sheaves over Spec(k) and the category of discrete Γk-modules.

Proof. We construct the two functors that give us the equivalence of categories. We fix a separableclosure k. For any étale abelian sheaf F and for any finite Galois subextension L of k, F(L)is a Γk-modules with the action of Γk given by the functoriality. For every pair of finite Galoissubextensions L and L′ such that L ⊆ L′ and such that the relative extension has degree n, we

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have an isomorphism α : L′ ⊗L L′ →∏gi∈Gal(L′/L) L

′ that sends `1 ⊗ `2 7→ (`1g1(`2), . . . , `1gn(`2)).Thus we have a commutative diagram

0 L L′ L′ ⊗L L′

0 L L′∏gi∈Gal(L′/L) L

′

β

γ

= = α

where β(`) = `⊗ 1− 1⊗ ` and γ(`) = (`, . . . , `)− (g1`, . . . , gn`). If we apply F we obtain

0 F(L) F(L′) F(L′ ⊗L L′)

0 F(L) F(L′)⊕

gi∈Gal(L′/L)F(L′)

F(β)

F(γ)

= = F(α)

Thus the exactness of the first row, that is the sheaf axiom for the covering L ⊆ L′, holds if andonly if the second row is exact. But the second row is exact precisely when F(L) = F(L′)Gal(L′/L).So we can associate to F the Γk-module lim−→F(L), where L runs over every finite Galois extensionin k.

For the other functor we take a discrete Γk-moduleM and we define F (A), for any étale-algebraA over k, as the set Homk−alg(A, k) with an action of Γk, induced by the natural action over k.Then we consider the presheaf that associates to any étale-algebra A over k, the group

FM (A) = HomΓk−sets(F (A),M),

with the natural transition maps. We can check that FM is a sheaf on finite Galois extension andthis is a consequence of the previous diagram. It is also easy to check that the two functors givethe equivalence of categories.

The result on the cohomology results from the isomorphism in the zero degree.

2.1.2 Rosenlicht lemma and type

We recall here another lemma we will use many times.

Lemma 2.8 (Rosenlicht). Let T be a torus, if X is a geometrically integral k-variety, every invertiblefunction of X ×k Gm is the product of an invertible function of X and a character of T .

We will see now a classical application of this lemma. Let’s start by an important definition.

Definition 2.9. For any torsor Y → X under a k-group G we will note

type(Y ) : G(k)→ Pic(Xk)

as the morphism that associates to any character χ, the image of [Y ] ∈ H1fppf (Xk, G ×k Xk) in

H1fppf (Xk,Gm,X) = Pic(Xk), via the morphism χ∗. The map type(Y ) will be the type of Y .If Pic(Xk) is of finite type and type(Y ) is an isomorphism, then we will say that Y is a universal

torsor of X.

The following exact sequence will be very important in the last chapter.

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Proposition 2.10. Let X be a smooth, geometrically integral k-variety. Then for any torsor Y → Xunder a k-torus T , we have the following exact sequence:

0→ k[X]∗ → k[Y ]∗ → Ttype(Y )−−−−−→ Pic(Xk)→ Pic(Yk)→ 0.

Proof. This fact is proven in the article Colliot-Thélène and Sansuc [CTS87, Proposition 2.1.1].

2.2 Fundamental exact sequence and elementary obstruction

We want to construct the fundamental exact sequence of Colliot-Thélène and Sansuc. Let k bea perfect field (we will keep this hypothesis during all this section). Let X be a scheme over k andlet π : X → Spec(k) be the structural map of X.

Given a Γk-module M of finite type, consider the Ext-spectral sequence:

Ep,q2 = ExtpΓk(M, (Rqπ∗)Gm)⇒ Extp+qXét(π∗M,Gm),

given by the composition of

Sh(két).

Sh(Xét) Ab

π∗ HomΓk(M,−)

HomXét (π∗M,−)

The functor π∗ sends injectives to injectives as its left adjoint is exact. The low degrees exactsequence is

0→ Ext1Γk

(M,k[X]∗)→ Ext1Xét

(π∗M,Gm)→ HomΓk(M,Pic(Xk))

∂−→ Ext2Γk

(M,k[X]∗)→ Ext2Xét

(π∗M,Gm).(2.2.1)

We want to simplify it with the following lemma.

Lemma 2.11. Let S be an X-group of multiplicative type. Then we have an isomorphism

H ifppf (X,S) = ExtiXét

(π∗S,Gm)

functorial in S and X.

Proof. We use local-to-global Ext spectral sequence that you can find in SGA4( [AGV71] ExposéV, Théorème 6.1),

Ep,q2 := Hpfppf (X, ExtqXfppf (π∗S,Gm))⇒ Extp+qXfppf

(π∗S,Gm),

with the following commutative diagram of functors:

Sh(Xfppf ).

Sh(Xfppf ) Ab

HomXfppf(π∗S,−) Γ(X,−)

HomXfppf(π∗S,−)

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The first step is to show that the spectral sequence completely degenerates, i.e. for i ≥ 1,ExtiXfppf (π∗S,Gm) = 0. As S is locally constant in the étale topology, it’s enough to show that fori ≥ 1,

ExtiXfppf (Z,Gm) = ExtiXfppf (Z/nZ,Gm) = 0.

The sheaves ExtiXfppf (Z,Gm) are zero, because they are zero in the étale site, for any X, as theycorrespond to the sheafification of the presheaves Hi(Gm) [Mil13, Chapter 1, Proposition 10.4]. Toprove that ExtiXfppf (Z/nZ,Gm) = 0 we take the exact sequence of fppf constant sheaves

0→ Z→ Z→ Z/nZ→ 0

and we apply HomXfppf (−,Gm). We obtain

Gmt→tn−−−→ Gm → Ext1

Xfppf(Z/nZ,Gm)→ 0.

As we are working in the fppf site the first map is surjective, thus Ext1Xfppf

(Z/nZ,Gm). Looking atthe other terms of the long exact sequence in cohomology we also obtain ExtiXfppf (Z/nZ,Gm) = 0

when i ≥ 2. We have proven H ifppf (X,S) = ExtiXfppf (π∗S,Gm).

Now we have to prove that

ExtiXét(π∗S,Gm) = ExtiXfppf (π∗S,Gm)

since S is locally constant for the étale topology, by the Cartan-Leray spectral sequence (for theétale coverings of X) [AGV71, Exposé V, Corollary 3.3], we can just check the equality étale-locally,so we can take S = Z and S = Z/nZ. In the first situation we can use a theorem of Grothendieck,namely Théorème 11.7, [Gro68], as Gm is smooth. For the second case we take again the exactsequence

0→ Z→ Z→ Z/nZ→ 0

and we apply once HomXfppf (−,Gm) and the second HomXfppf (−,Gm). Taking the long exactsequences in cohomology and applying the lemma of five homomorphisms we arrive to the result.

In virtue of the previous lemma and the exact sequence (2.2.1), we have proved:

Theorem 2.12 (Colliot-Thélène, Sansuc). Let k be a field, if X is a geometrically integral smoothvariety and S an X-group of multiplicative type, we have an exact sequence

0→ Ext1Γk

(S, k[X]∗)→ H1fppf (X,S)

type−−→ HomΓk(S,Pic(Xk))∂−→ Ext2

Γk(S, k[X]∗)→ H2

fppf (X,S),

functorial in S and X, called the fundamental exact sequence. The map type associates to a torsorits type.

So for any Γk-invariant morphism λ : S → Pic(Xk), the existence of a torsor of type λ is equiva-lent to ∂(λ) = 0. If Pic(Xk) is of finite type and S = Pic(Xk), we can take Id ∈ HomΓk(S,Pic(Xk)).

Definition 2.13 (Elementary obstruction). We will call the elementary obstruction of X the class∂(Id) and we will denote it as e(X).

Thanks to the fundamental exact sequence we have the following corollary.

Corollary 2.14. If Pic(Xk) is of finite type the existence of universal torsors (Definition 2.9) isequivalent to the vanishing of the elementary obstruction.

14

Proof. Let’s put S = Pic(Xk) in the fundamental exact sequence, if λ ∈ HomΓk(S,Pic(Xk)), byfunctoriality we know that ∂(λ) = λ∗(e(X)). If λ is an isomorphism then λ∗(e(X)) = 0 if and onlyif the elementary obstruction is zero.

Moreover we have another simplification of the sequence:

Corollary 2.15. If we add the hypothesis k[X]∗ = k∗, then the fundamental exact sequence becomes:

0→ H1(k, S)π∗−→ H1(X,S)

type−−→ HomΓk(S,Pic(Xk))

∂−→ H2(k, S)π∗−→ H2(X,S).

So in this case the space of torsors of a certain type is a principal homogeneous space underH1(k, S). The action of H1(k, S) is exactly the twist (Definition 2.2). In particular if k is alge-braically closed the type identifies the torsor. Still under the hypothesis k[X]∗ = k

∗, we can evendescribe the set of rational points using torsors of a given type, just rewriting the Theorem 2.3 as

X(k) =⋃

type(Y,f)=λ

f(Y (k)).

Now we want to prove two theorems about the elementary obstruction.

Theorem 2.16. Let X be a geometrically integral, smooth k-variety, then the class

−e(X) ∈ Ext2Γk

(Pic(Xk), k∗)

is represented by the 2-fold extension

0→ k∗ → k(X)∗ → Div(Xk)→ Pic(Xk)→ 0. (2.2.2)

Proof. We need a general fact of homological algebra. If π : X → Spec(k) is a k-scheme and

0→ A→ B → C → 0

is an exact sequence of étale sheaves of abelian groups over X, if the sequence

0→ π∗(A)→ π∗(B)→ π∗(C)→ R1π∗(A)→ 0 (2.2.3)

is exact then if we consider the Ext-spectral sequence

Ep,q2 = ExtpΓk(−, Rqπ∗(A))⇒ Extp+qXét(π∗(−), A)

we have:

Lemma 2.17. The transgression map (E0,12 → E2,0

2 ),

HomΓk(−, R1π∗(A))→ Ext2Γk

(−, π∗(A)),

is given by the Yoneda pairing with the opposite of the class represented by the 2-fold extension(2.2.3).

The proof of this lemma can be found in [CTS87], Lemma 1.A.4. We use the lemma with thefollowing exact sequence of étale sheaves. The exactness is proven in Milne, [Mil13] Proposition13.4.

15

Proposition 2.18. Let X be an irreducible, noetherian, regular scheme and j : η → X the inclusionof the generic point. Then we have an exact sequence of étale sheaves

0→ Gm → j∗Gm →⊕

x∈X(1)

(ix)∗Z→ 0.

We need to check that R1(π∗)(j∗Gm) is zero, but we know that it is a subsheaf of R1(πj)∗(Gm),thanks to the convergence of the Grothendieck spectral sequence for the composition of π∗ and j∗.But R1(π j)∗(Gm) is zero by Hilbert 90, so it is easy to check now that the 2-fold extension (2.2.3)becomes exactly (2.2.2).

The second important theorem of the section is the following.

Theorem 2.19. Let k be a field and X be a smooth k-variety such that k[X]∗ = k∗, we have the

following implications:

X(k) 6= ∅ ⇒(k∗→ k(X)∗ has a Γk-equivariant section.

)⇔ e(X) = 0.

As a consequence of this theorem, using the Corollary 2.14 we have:

Corollary 2.20. If Pic(Xk) is of finite type, the existence of a rational point implies the existenceof a universal torsor.

This fact is important, we will use it in the proof of the main theorem of Colliot-Thélène andSansuc descent theory. Now we will prove the theorem, we divide it in different parts.

Proposition 2.21. Let k be a field, X a smooth, geometrically integral k-variety, such that X(k) 6=∅, then the natural map

k∗→ k(X)∗

has a Γk-invariant retraction.

We need the following lemma:

Lemma 2.22. Let G be a profinite group, H a closed subgroup, B a G-module, A an H-module.Then

ExtnG(Z[G]⊗Z[H] A,B

)= ExtnH (A,BH)

where BH is the Z[H]-module obtained by restricting the action of B.

Sketch of proof. First of all we reduce to the case when G and H are finite groups. Then we choosea projective resolution of A. Since Z[G] is a free Z[H]-module, Z[G]⊗Z[H] is exact and it sendsprojectives to projectives. Thus it’s enough to check that

HomG

(Z[G]⊗Z[H] A,B

)= HomH (A,BH) ,

but this can be done similarly to the commutative case.

Now we can prove the proposition.

Proof of Proposition 2.21. Let P ∈ X(k) and consider the natural maps

k∗→ O∗Xk,P → k(X)∗

16

where O∗Xk,P is the Zariski stalk. The first map admits a section g 7→ g(P ), so it’s sufficient to finda section of the inclusion O∗Xk,P → k(X)∗. Because X is smooth, we have an exact sequence ofΓk-modules

0→ O∗Xk,P → k(X)∗ → DivP(Xk

)→ 0

whereDivP

(Xk

)=

⊕x∈Spec(OX

k,P )(1)

Zx

If this sequence splits then we get the missing section. To show this we show that

Ext1Γk

(DivP (Xk), O

∗Xk,P

)= 0.

We notice that

DivP (Xk) =∑

x∈SpecOXk,P

(1)

∑x over x

Zx =∑

x∈SpecOXk,P

(1)

Z [Γk/Hx]

where Hx is the Kernel of the transitive action of Γk on the points over x, corresponding to a certainextension Lx/k. We have

Ext1Γk

(DivP (Xk), O

∗Xk,P

)= Ext1

Γk

(∑x

Z [Γk/Hx] , O∗Xk,P

)=

=∏x

Ext1Γk

(Z [Γk/Hx] , O∗Xk,P

).

Since Z [Γk/Hx] = Z[Γk]⊗Z[Hx] Z, we can use Lemma 2.22 and we obtain that for any x,

Ext1Γk

(Z [Γk/Hx] , O∗Xk,P

)= Ext1

Hx

(Z, O∗Xk,P

).

If A := OXLx ,P and p : Spec(A)→ Spec(Lx) is the structural map,

Ext1Hx

(Z, O∗Xk,P

)= H1

ét(SpecLx, p∗Gm,A)

because the functor HomHx(Z,−) is equal to the functor M → MHx . By the Leray spectralsequence,

H1ét(SpecLx, p∗Gm,A) → H1

ét(SpecA,Gm,A).

The right group is zero by Hilbert 90 for local rings, so we are done.

In the proof of the Proposition 2.21 we have shown a useful property of the Γk-module of divisorsthat can be easily generalized as follows.

Lemma 2.23. Let X be a smooth variety over a field k, then the Γk-module of divisors on Xk,Div(Xk) is isomorphic to a certain sum ⊕

i∈IZ[Γk/Hi],

with Hi open normal subgroups (thus of finite index) of Γk and I is not necessarily finite.

17

Definition 2.24. We will call permutation module a Γk-module that contains a basis invariant (notnecessarily fixed) under the action of Γk. Thus it’s a Γk-module of the form

∑i∈I Z[Γk/Hi] with

Hi closed subgroups of Γk and I not necessarily finite.

Thus in Lemma 2.23 we have shown that Div(Xk) is a permutation module. Another fact thatwe will use many times, whose proof is the same as in the proof of Proposition 2.21 is the following.

Lemma 2.25. Let A be a local ring that is a k[Γk]-module, for any permutation module M , thenExt1

Γk(M,A∗) = 0. In particular when A = k we obtain

Ext1Γk

(M,k∗) = H1(k,M) = 0.

Let’s conclude now the proof of Theorem 2.19.

Proposition 2.26. Let k be a field and X a smooth k-variety such that k[X]∗ = k∗, then the

inclusionk∗→ k(X)∗

has a Γk-invariant retraction if and only if the 2-fold extension, (2.2.2) is zero in Ext2Γk

(Pic(Xk), k∗),

if and only if e(X) is zero.

Proof. If the map k∗ → k∗(X) has a retraction then it is a general fact that the 2-fold extension

(2.2.2) is zero in Ext2Γk

(Pic(Xk), k∗).

The other implication is not true in general for 2-fold extensions. Consider the short exactsequence

0→ k(X)∗/k∗ → Div(Xk)→ Pic(Xk)→ 0 (2.2.4)

and the long exact sequence given by the derived functor of HomΓk(−, k∗). Thanks to the Lemmas2.23 and 2.25, we know that Ext1

Γk(Div(Xk), k

∗) = 0, thus we have the injective connection map

Ext1Γk

(k(X)∗/k∗, k∗) → Ext2

Γk(Pic(Xk), k

∗)

given by the Yoneda pairing with the (2.2.4). The image of the short exact sequence

0→ k∗ → k(X)∗ → k(X)∗/k

∗ → 0 (2.2.5)

is exactly the 2-fold extension (2.2.2), thus is zero. By the injectivity we obtain that also (2.2.5) iszero in Ext1

Γk(k(X)∗/k

∗, k∗), thus the map k∗ → k(X)∗ has a retraction, as we wanted.

Thanks to the Theorem 2.16 we also know that the vanishing of the 2-fold extension (2.2.2) isequivalent to the vanishing of e(X).

2.3 Local description of torsors

If we have a certain torsor Y → X under a group of multiplicative type S we may wonder aboutthe geometry of Y . This question will be really important in the next chapters, for example afterTheorem 3.16.

If X is a smooth k-variety and k[X]∗ = k∗ there is a nice description of the restriction of Y to

certain open subsets of X.Let’s call λ the type of Y and suppose there exists an open U of X such that the composition

Sλ−→ Pic(Xk)→ Pic(Uk) is zero. Let’s call P the Kernel of the map from Pic(Xk)→ Pic(Uk), thus

we have by hypothesis a well defined map S → P that we will call λ again.

18

Then for any commutative diagram with exact rows like the following

0 R M S 0

0 k[U ]∗/k∗ Div(X\U)k

(Xk) P 0.

β

α λ

(2.3.1)

with M a permutation module of finite type we have the following result.

Theorem 2.27 (Colliot-Thélène and Sansuc). For any X-torsor Y ′ of type λ, there exist a mor-phism φ : U → R and a morphism ψ : Y ′U →M such that

Y ′U M

U R.

ψ

β

φ

is a cartesian square. Moreover the induced map φ : R → k[U ]∗ is a lifting of −α via the naturalprojection k[U ]∗ → k[U ]∗/k

∗.Finally if U is a k-torsor under a torus and all the vertical maps in (2.3.1) are isomorphisms,

φ is an isomorphism.

Proof. The proof can be found in [Sko01], Lemma 2.4.4 and Theorem 4.3.1.

In particular it is always possible to construct a diagram like the (2.3.1). We can just take N ,the fiber product of S and Div(X\U)k

(Xk) via P , then take any permutation module M with asurjective map M → N and conclude taking the Kernel.

We will often use the following corollary of the theorem.

Corollary 2.28. If X is a smooth k-variety, k[X]∗ = k∗ and if there exists a dense open subset U

of X that is a k-torsor under a torus, any universal torsor of X is a k-rational variety.

Proof. Let’s take a universal torsor f : Y → X of type λ. We notice that U is geometricallyisomorphic to Gn

m for a certain n, thus Pic(Uk) = 0. If we follow the previous construction themodule P will be Pic(Xk), we can take as M the Γk-module Div(X\U)k

(Xk) itself, because it is apermutation module (Lemma 2.23) and R becomes k[U ]∗/k

∗. As U is a k-torsor under a torus byhypothesis, we can apply the second part of the theorem, thus φ is an isomorphism. Then YU isisomorphic to M . As M is a permutation module, we know by Theorem 2.6 that YU is isomorphicto a certain product

n∏i=1

RLi/k(Gm,Li)

with Li1≤i≤n a family of finite extensions. By Proposition 2.5, YU is a k-rational variety, thusalso Y is k-rational as U is dense in X and f is an open morphism, so YU is dense in Y .

19

Chapter 3

Brauer-Manin obstruction

3.1 The Brauer group

For any field k we can construct the Brauer group with central simple algebras as explained inChapter 2 of the book [GT06]. In the book it’s even proven that the Brauer group constructed isisomorphic to H2(k, k

∗), thus it is even isomorphic to H2

ét(Spec(k),Gm). In this correspondence theclasses of quaternion algebras are in bijection with the two torsion of H2(k, k

∗). One can even define

the Brauer group of a scheme with some generalizations of central simple algebras, i.e. Azumayaalgebras. In this thesis we prefer to use the following definition.

Definition 3.1. Let X be a scheme, we will call Br(X) the group H2ét(X,Gm). If k is a field we

will call Br(k), the group Br(Spec(k)).

If X is a quasi-projective variety it is shown in the article of Gabber [Gab06] that the groupconstructed with Azumaya algebras is isomorphic to the torsion subgroup of Br(X).

Example. If a field is algebraically closed or even C1 we can show that the Brauer group is trivial,so for example Br(C) = Br(Fq) = 0. If we take R we have instead Br(R) = Z/2Z and the groupis generated by the algebra of quaternions. It can be shown that if k is a number field and ν is afinite place there is a canonical map

Br(kν)invν−−→ Q/Z

that is an isomorphism. We notice that even when ν is an infinite place we have canonical embed-dings of Br(kν) in Q/Z, we will call them invν as well.

3.1.1 Residues

An important fact of the theory of the Brauer group is that if X is a noetherian, irreducible,regular scheme then we have a canonical inclusion of Br(X) in Br(k(X)). This allows us to workonly with Brauer group of a field, that is easier to understand. For example we can use someparticular constructions as quaternion algebra and cyclic algebras.

To construct the map we take the exact sequence in Proposition 2.18 and we pass to cohomology,obtaining the exact sequence⊕

H1ét(X, (ix)∗Z)→ Br(X)

ι1−→ H2ét(X, j∗Gm).

The term⊕H1

ét(X, (ix)∗Z) is zero because every H1ét(X,Zx) is included in H1(k,Z) = 0 by the

Leray spectral sequence. This means that the map i1 is injective. Moreover we have a map

ι2 : H2ét(X, j∗Gm)→ H2

ét(η,Gm) = Br(k(X))

20

from the Leray spectral sequence that is injective because R1j∗Gm = 0. If we take ι2 ι1 this definesan injection of Br(X) in Br(k(X)). One can check that this map is the same as the map given bythe functoriality of Br with respect to the inclusion of η in X.

To understand which elements of Br(k(X)) are in Br(X) we can use residue maps. If A isa discrete valuation ring, K is the field of fractions and k the residue field. Suppose that thecharacteristic of k is zero, then if j : η → Spec(A) is the inclusion of the generic point andi : ξ → Spec(A) is the closed immersion of the closed point, the exact sequence

0→ Gm → j∗Gm → i∗Z→ 0,

gives the exact sequence

Br(A)→ H2ét(Spec(A), j∗Gm)→ H2(Spec(A), i∗Z).

Now H2(Spec(A), i∗Z) = H2(k,Z) because i∗ is exact and as Q is uniquely divisible, H1(k,Q) =H2(k,Q) = 0. So H1(k,Q/Z) is isomorphic to H2(k,Z) via the cohomology connection map. Now,as a consequence of Lang’s theorem

Proposition 3.2. Let X be an irreducible scheme of dimension 1 over a field k of characteristiczero. Then Rij∗Gm = 0 for any i ≥ 1.

Thus we have that all the Rij∗Gm = 0 when i ≥ 1, so by the Leray spectral sequence weobtain an isomorphism between H2

ét(Spec(A), j∗Gm) and Br(K). So we can construct a map ∂A :Br(K)→ H1(k,Q/Z) such that

Br(A)→ Br(K)∂A−−→ H1(k,Q/Z)

is an exact sequence.

Definition 3.3. We call the resulting map ∂A the residue map of K associated to A. If K is thefield of rational functions of an irreducible regular scheme X over a field of characteristic zero andx is a prime divisor, we will denote ∂x : Br(K)→ H1(κ(x),Q/Z) the map associated to A = OX,x.

Example. If K,A and k are as before, we want to understand the residue map

∂A : 2 Br(K)→ H1(k,Q/Z).

We notice that the image is actually inside H1(k,Z/2Z) = k∗/(k∗)2. It can be shown that if (a, b)is a quaternion algebra

∂A((a, b)) = (−1)νA(a)νA(b)[aνA(b)/bνA(a)

]where with [−] we mean the class in k∗/(k∗)2.

Now the theorem of purity of Grothendieck ([Gro68], Thm. (6.1), page 134) implies

Theorem 3.4. If X is a noetherian, irreducible, regular scheme over Q, the following sequence

0→ Br(X)→ Br(k(X))→⊕

x∈X(1)

H1(κ(x),Q/Z),

whit the last arrow induced by the residue maps ∂x, is well defined and exact.

21

3.2 The adelic Brauer-Manin pairing

In this section we define an important pairing, which will be fundamental to describe an ob-struction to the existence of rational points.

The following is a deep theorem from global class field theory.

Theorem 3.5. Let k be a number field, Ωk its set of places, then we have the following exactsequence

0→ Br(k)→⊕ν∈Ωk

Br(kν)∑

invν−−−−→ Q/Z→ 0 (3.2.1)

We will also use the following result.

Theorem 3.6. Let X be a variety over a global field k. Let A ∈ Br(X), then for some S ⊂ Ωk finite,there exists a scheme X of finite type, defined over Ok,S and a class A ∈ Br(X) with a morphism

i : X → X

identifying X with the generic fiber Xη, s.t. i∗ : Br(X)→ Br(X) sends A to A.

For the proof see Corollary 6.6.11. of [Poo11].

Definition 3.7 (Evaluation). Let X/k a variety and A ∈ Br(X). If L is a k-algebra and x ∈ X(L)then, by functoriality of Br(−), it induces a homomorphism

Br(X)→ Br(L), A 7→ A(x).

Let X/k be a smooth and geometrically integral variety over a number field k. We are interestedin the pairing

Br(X)×X(Ak)→ Q/Z

defined by the following rule:

(A, (Pν)) 7→∑ν∈Ωk

invν(A(Pν)). (3.2.2)

Where A(Pν) makes sense thanks to the previous definition and invν are the local invariant mapsappearing in the exact sequence of Theorem 3.2.1.

Lemma 3.8. The Brauer-Manin pairing is well defined, i.e. the sum of (3.2.2) is finite.

Proof. Given (Pν) ∈ X(Ak) and A ∈ Br(X) we have to show that A(Pv) = 0 for almost allν. Thanks to Theorem 3.6 we can chose a finite set of places S big enough (containing all thearchimedean places) such that Pν ∈ X(Oν) for all ν /∈ S (by the definition of the adelic ring). Thisconcludes in virtue of the following result.

Theorem 3.9. Let R be the valuation ring of a non-archimedean local field k, then Br(R) = 0.

Lemma 3.10. The Brauer-Manin pairing is trivial on Br0(X), and so it can be defined also as apairing from Br(X)/Br0(X).

Proof. This follows immediately from the exact sequence of Theorem 3.2.1 and the functoriality ofBr(−).

22

Definition 3.11. We defineX(Ak)Br(X) as the subset ofX(Ak) orthogonal to all elements of Br(X).

Lemma 3.12. The Brauer-Manin pairing is locally constant in the adelic topology.

For the proof see Corollary 8.2.11 of [Poo11].

Proposition 3.13. We have the following inclusion:

X(k) ⊆ X(Ak)Br(X) ⊆ X(Ak)

Proof. The only non trivial inclusion is the first one. But this follows from the commutativity ofthe diagram

X(k) X(Ak)

Br(k)⊕

ν∈ΩkBr(kν)

and the exact sequence of Theorem 3.2.1.

Remark. The previous two results imply that the closure of the diagonal image of X(k) via thediagonal embedding in the adelic points is contained in X(Ak)Br(X).

Remark (Functoriality). Let f : X → Y be a k-morphism of smooth geometrically integral k-varieties. Given A ∈ Br(Y ) and (Pv) an adelic point of X, we have∑

ν∈Ωk

invν(f∗A(Pν)) =∑ν∈Ωk

invv(A(f(Pν))).

It follows thatY (Ak)Br(Y ) = ∅ ⇒ X(Ak)Br(X) = ∅.

Now, if the set X(Ak)Br(X) is empty of course the variety will not have a rational point.

Definition 3.14. We will say that for a variety X the only obstruction to the Hasse principle isgiven by the Brauer-Manin obstruction if X(Ak)Br(X) 6= ∅ implies X(k) 6= ∅.

This property is weaker than the Hasse principle. An other property we will study in the lastchapter is the following.

Definition 3.15. If X is any variety and S is a finite subset of Ωk we will say the only obstructionto strong approximation off S is given by the Brauer-Manin obstruction if X(k) is dense in theimage of X(Ak)Br(X) in X(ASk ).

3.3 Introduction to descent theory

3.3.1 Hochschild-Serre and filtration of the Brauer group

We recall the Hochschild-Serre spectral sequence. If X is a scheme over k

Ep,q2 = Hp(k,Hq(Xk,Gm))⇒ Hp+q(X,Gm).

The spectral sequence is a Grothendieck spectral sequence, with respect to the composition of thetwo functors:

23

Sh(két).

Sh(Xét) Ab

Γ(Xk,−) M →MΓk

Γ(X,−)

The first functor sends injectives to injectives because it is isomorphic to π∗. In virtue of theconvergence of the spectral sequence, for any Hn there is a filtration

0 = Fn+1Hn ⊆ · · · ⊆ F 0Hn = Hn

such that Ep,q∞ ' F pHp+q/F p+1Hp+q. We notice that H2 = Br(X), we will call Br0(X) the groupF 2H2 and Br1(X) the group F 1H2. We have an exact sequence

0→ Br1(X)→ Br(X)→ E0,2∞ ,

andE0,2∞ → E0,2

2 = H2(Xk,Gm)Γk .

This implies that Br1(X) = Ker(Br(X)→ Br(Xk)). We also have the exact sequence

0→ Br0(X)→ Br1(X)→ E1,1∞ = E1,1

3 = Ker(E1,12 → E3,0

2 ).

At the end we obtain the exact sequence

0→ Br0(X)→ Br1(X)r−→ H1(k,Pic(Xk))→ H3(k,Gm).

Where r is the map defined by the spectral sequence.Notice that when k is a number field the last term is 0 by a non-trivial result of class field theory,

so this last exact sequence simplifies. The last exact sequence fits in a long exact sequence

0→ Pic(X)→ Pic(Xk)Γk → Br(k)→ Br1(X)

r−→ H1(k,Pic(Xk))→ H3(k,Gm).

For any λ ∈ HomΓk(M,Pic(Xk)) we define

Brλ(X) := r−1λ∗(H1(k,M)).

As is proven in ([Ser97], I.2.2,Cor. 2) Galois cohomology commutes with the colimits on the secondargument, thus

H1(k,Pic(Xk)) =⋃

λ: M→Pic(Xk)M of finite type

λ∗(H1(k,M)).

So we haveBr1(X) =

⋃λ: M→Pic(Xk)M of finite type

Brλ(X). (3.3.1)

3.3.2 The main theorem

Now we have all the tools to present Colliot-Thélène and Sansuc’s descent theory. This theorygeneralizes the classical descent for elliptic curves. The main goal is to show that for certain classesof k-varieties (k will always be a number field), the Brauer-Manin obstruction explains the failure ofthe Hasse principle or weak approximation. To do this we use a description of the sets X(Ak)Brλ(X)

with the help of the adelic points of torsors of type λ.

24

Theorem 3.16 (Colliot-Thélène, Sansuc, Skorobogatov, Harari). Let k be a number field, X asmooth k-variety such that k[X]∗ = k

∗. Then for any λ ∈ HomΓk(S,Pic(Xk)),

X(Ak)Brλ(X) =⋃

type(f,Y )=λ

f(Y (Ak)).

Proof. The theorem has been proven for tori by Colliot-Thélène and Sansuc in 1979 and generalizedby Skorobogatov in 1999 to groups of multiplicative type, with the additional assumption Pic(Xk) offinite type [Sko01, Theorem 6.1.1]. The Main Theorem proven in [HS10] by Harari and Skorobogatovin 2010 has as a particular case our theorem in the way we have formulated it.

If we also suppose X proper, one can shows, there are only finitely many isomorphism classesof torsors Y of any given type such that Y (Ak) 6= ∅. One can read [Sko01, Proposition 5.3.2] anduse the fundamental exact sequence of Colliot-Thélène and Sansuc.

The Theorem 3.16 is often used when Pic(Xk) is of finite type and λ is an isomorphism. In thissituation we have

X(Ak)Br1(X) =⋃

(f,Y ) universal

f(Y (Ak)),

because when λ is surjective Brλ(X) = Br1(X). In particular the algebraic Brauer-Manin obstruc-tion is empty if and only if there exists a universal torsor with an adelic point. If Pic(Xk) is not offinite type, by the equality (3.3.1) we have

X(Ak)Br1(X) =⋂

λ: M→Pic(Xk)M of finite type

⋃type(f,Y )=λ

f(Y (Ak)).

We can easily check the following corollary, recalling that the subsets X(Ak)Brλ(X) are closed inX(Ak) (Lemma 3.12).

Corollary 3.17. For X as in the theorem and for any λ ∈ HomΓk(S,Pic(Xk)), if the X-torsors oftype λ satisfy the Hasse principle, then the only obstruction to the Hasse principle for X is the onegiven by Brλ(X), i.e. X(Ak)Brλ(X) 6= ∅ ⇒ X(k) 6= ∅.Furthermore if X is proper, for any λ ∈ HomΓk(S,Pic(Xk)), if the X-torsors of type λ satisfy weakapproximation, then the only obstruction to weak approximation for X is the one given by Brλ(X),i.e. X(k) = X(Ak)Brλ(X).

As an application one can prove the following theorem. Using the local description of torsors(Section 2.3).

Theorem 3.18. Let k be a number field, X a smooth, proper k-variety that contains a k-torsor undera torus U as a dense open subset. The algebraic Brauer-Manin obstruction is the only obstructionto weak approximation.

The fact that universal torsors are k-rational gives us a good description of rational points.Since the variety we are considering is proper, thanks to Theorem 3.16 there are only finitely manyisomorphism classes of universal torsors. So the set of rational points is a finite disjoint union ofsubsets, each one parametrized by the rational points of a certain k-rational variety.

25

Chapter 4

Strong approximation for varietiescontaining a torsor under a torus

In this chapter we want to prove the main theorem (Theorem 4.6).

4.1 The case when Pic(Xk) is torsion-free

Proposition 4.1. Let X be a smooth variety over k with k[X]∗ = k∗ and Pic(Xk) = 0. Suppose

X contains an open, non-empty subset which is isomorphic to a torus T . Let W ⊆ X be a closedsubset of codimension at least two. Then X \W satisfies strong approximation off one place.

Proof. Step 1: Decoding the condition on the Picard groupFirst of all as Pic(Xk) = 0, we know that T is a permutation Γk-module. Indeed thanks toRosenlicht’s lemma we have the exact sequence

0→ T → Div(X\T )k(Xk)→ Pic(Xk) = 0.

Thus T is isomorphic to a product∏RKi/k(Gm,Ki) because Div(X\T )k

(Xk) is a permutation module(Lemma 2.23) and permutation modules correspond to those tori thanks to Theorem 2.6. We havethat

∏RKi/k(Gm,Ki) is an open of Y :=

∏RKi/k(A

1Ki

) ' ANk for a certain N .

Step 2: Checking the hypothesis of Corollary 1.12To show that strong approximation off one place holds in X\W we want to apply the Corollary 1.12.We know that on X \W is smooth by hypothesis. We know by the Corollary 1.13 that Y minusany closed subset of codimension at least two satisfies strong approximation off one place. Thusin order to verify the hypothesis of the Corollary 1.12 we have to check that the open immersionT \ (W ∩ T )→ X \W extends to a morphism f : Y \ W → X \W with W a certain closed subsetof Y of codimension at least two such that T ∩ W = T ∩W .

Step 3: Constructing the map on the algebraic closureWe want to construct first of all f : Yk \ W

′ → (X \ W )k, with W ′ a closed subset of Yk ofcodimension at least two such that Tk ∩W

′ = Tk ∩Wk and such that restricted to (T \ (W ∩ T ))kit is the natural open immersion.

We recall the isomorphism div : T → Div(X\T )k(Xk), let’s call (Di) a basis of prime divisors of

Div(X\T )k(Xk) and let’s call xi the character of T that correspond to Di. We have

Tk = Spec(k[x±11 , · · · , x±1

N ]) and Yk = Spec(k[x1, · · · , xN ]).

26

For each local ring OXk,Di we have a canonical inclusion OXk,Di → k(x1, . . . , xN ), since Xk containsTk as a dense open subset. By definition div(cxni ) = nDi for any n ∈ Z and c ∈ k∗, moreover thecxni are the only elements of k(x1, . . . , xN ) on Xk with divisor supported on Di, since k[X]∗ = k

∗.The image of the inclusion in the field of fractions is actually in the subring k[x1, . . . , xN ](xi), therational functions with valuation with respect to Di greater than or equal to 0. Thus we havenatural maps Spec(OYk,xi) → Spec(OXk,Di) that induce a map from a certain open of Yk meetingall the divisors supported outside Tk to (X \W )k that has the property we wanted.

Step 4: Descending to kNow from f we want to deduce the map f . If σ1, σ2 ∈ Γk are two automorphisms of the field k,the twisted maps fσi : Yk \ σi(W

′)→ (X \W )k coincide on Yk \ (σ1(W ′) ∪ σ2(W ′)) because theycoincide on the dense open

(T \ (W ∩ T ))k ⊆ Yk \ (σ1(W ′) ∪ σ2(W ′)).

As W ′ is defined on an finite extension of k, we can take

W :=⋃σ∈Γk

σ(W ′)

and it is again a closed subset of Yk of codimension at least two that is defined over k. Then f

descent to a morphism f : Y \ W → X \W .

Theorem 4.2 (Wei). Let X be a smooth k-variety such that k[X]∗ = k∗ and suppose that X contains

a non-empty open subset U that is a torsor under a torus T . Let’s even suppose that Pic(Xk) istorsion-free, then for any closed subset W of X, of codimension at least two, the only obstructionto strong approximation off one place for X \W is the algebraic Brauer-Manin obstruction.

Proof. Let’s fix W ⊆ X a closed subscheme of codimension at least two and let’s call X ′ := X \W .As X is smooth and k[X ′]∗ = k

∗, the same holds for X ′. Suppose that there exists at least an adelicpoint in X ′(Ak)Br1(X′), if not the result is trivial. So let’s take P ∈ X ′(Ak)Br1(X′), thanks to thedescent Theorem 3.16, we know that there exists a universal torsor f ′ : Y ′ → X ′ under S, withS ' Pic(Xk) and Q ∈ Y ′(Ak) such that P = f ′(Q).

We want to show that this torsor is actually the restriction of a universal X-torsor f : Y → X,under S. This is a consequence of the following general fact.

Proposition 4.3. Let X be a smooth k-variety such that Pic(Xk) is of finite type and such thatk[X]∗ = k

∗ and let S be a group of multiplicative type. For every open subset X ′ of X obtained byremoving a closed subset of codimension at least two, there is a canonical isomorphism

α : HomΓk

(S,Pic(Xk)

)→ HomΓk

(S,Pic(X ′

k))

that induces a bijectionIsoΓk

(S,Pic(Xk)

)! IsoΓk

(S,Pic(X ′

k)).

Moreover the restriction of X-torsors under S of type λ to X ′-torsors under S of type α(λ) is abijection.

27

Proof. The first part of the proposition is just a consequence of Pic(Xk) = Pic(X ′k). The second

part comes from the following commutative diagram.

0 H1ét(k, S) H1

fppf (X,S) HomΓk

(S,Pic(Xk)

)H2

ét(k, S)

0 H1ét(k, S) H1

fppf (X ′, S) HomΓk

(S,Pic(X ′

k))

H2ét(k, S).

type

type

Id Res α Id

The two lines are just the fundamental exact sequences of Corollary 2.12 for X and X ′ and thevertical arrows comes from the functoriality. By the lemma of five homomorphisms the map Res isan isomorphism.

Getting back to the proof of the theorem, thanks to the lemma there exists f : Y → X auniversal torsor under S such that f |f−1(X′) = f ′. As f is an open morphism, Y ′ is obtainedremoving a closed subset of codimension at least two, since f−1(W ) has codimension at least two.

Now we want to apply Proposition 4.1 with respect to Y , removing f−1(W ). The variety Yis smooth as X is smooth and f is smooth and we have that k[Y ]∗ = k

∗ and Pic(Yk) = 0 thanksto the exact sequence of Proposition 2.10. By the local description of torsors (Section 2.3) YU is atorus, thus Y contains an open subset that is a torus. Hence on Y ′ strong approximation off oneplace holds in virtue of Proposition 4.1.

Now let’s fix ν0 a place of k, then the rational points of Y ′ approximate well the point Q inY ′(Aν0

k ), thus, by continuity, the rational points of X ′ are as close as we want to P in X ′(Aν0k ).

We have shown that the rational points X ′(k) are dense in the image of X ′(Ak)Br1(X′) in X ′(Aν0)and we are done.

Remark. In the Theorem 4.2 we can actually replace the hypothesis Pic(Xk) torsion-free with theweaker one: for any universal torsor Y over X, k[Y ]∗ = k

∗ and Pic(Yk) = 0.

4.2 The general case

We will see in the next section that if Pic(Xk) is not torsion-free then Theorem 4.2 is nottrue. Luckily the defect of strong approximation off one place can be explained by the Brauer-Manin obstruction of a certain étale covering. In the paper [Sko99], Skorobogatov has proposed arefinement of the Brauer-Manin obstruction, the étale Brauer-Manin obstruction. We will give aslightly different definition.

Definition 4.4. Let k be a number field and X be a smooth, geometrically integral k-variety. Thecommutative étale Brauer-Manin set will be

X(Ak)comm,ét ,Br(X) :=⋂

f : YG−→X

⋃Z

Zf(ZY (Ak))Br(ZY ),

where the intersection is taken over all X-torsors, f : Y → X under G, with G any finitecommutative k-group and the union runs over the right k-torsors under G. We will also defineX(Ak)comm,ét ,Br1(X) just substituting Br(Y ) with Br1(Y ).

28

We haveX(k) ⊆ X(Ak)comm,ét ,Br(X) ⊆ X(Ak)Br(X).

We want to show that in our case this finer obstruction explains the defect of strong approximationoff one place.

Definition 4.5. We say that strong approximation with commutative étale algebraic Brauer-Maninobstruction holds for X off S if X(k) is dense in the image of X(Ak)comm,ét ,Br1 in X(ASk ).

Theorem 4.6 (Wei). Let X be a smooth variety over k with k[X]∗ = k∗ and suppose that X

contains a dense open subset U that is a k-torsor under a torus T . For any closed subset W of Xwith codimension at least two, the commutative étale algebraic Brauer-Manin obstruction is the onlyobstruction to strong approximation off one place for X \W .

Remark. We notice that if Pic(Xk) is torsion-free, the commutative étale algebraic Brauer-Maninobstruction is the same as the algebraic Brauer-Manin obstruction. Indeed by the Kummer exactsequence, for any n, H1

ét(Xk,Z/nZ) = H1ét(Xk, µn) = 0. Thus by the Hochschild-Serre spectral

sequenceH1ét(k,Z/nZ) = H1

ét(X,Z/nZ). This implies that for any X-torsor Y under a commutativegroup G there exists Z ∈ H1

ét(k,Z/nZ) such that ZY is a trivial X-torsor under G. In particularthis means that the Theorem 4.6 is a generalization of the Theorem 4.2.

We will end the section proving the Theorem 4.6. We will see it’s enough to construct a torsorf : Y → X under a commutative finite k-group, such that Y satisfies the hypothesis of the Theorem4.2. We will use the following lemma.

Lemma 4.7. Let X be a smooth k-variety such that k[X]∗ = k∗. Suppose there exists U , a non-

empty open subset of X, that is a k-torsor under a torus T . Let’s call S = (Pic(Xk))tors and λ theinclusion of S in Pic(Xk) and let f : Y → X be an X-torsor under S of type λ, then

i) YU is a k-torsor under a torus;

ii) Y is smooth;

iii) Pic(Yk) is torsion-free;

iv) k[Y ]∗ = k∗.

Proof. i) For the first point we will use the local description of torsors. We will construct thefollowing commutative diagram of Γk-modules with exact rows.

0 R M ′ S 0

0 T M S 0

0 T Div(X\U)k(Xk) Pic(Xk) 0.

p β

i

p

α β IdS

IdT λ

(4.2.1)

As T = k[U ]∗/k∗ by Rosenlicht’s Lemma we have the lower row exact sequence. Let M be

the preimage of S by the natural projection map p : Div(X\U)k(Xk)→ Pic(Xk). Now let M ′

29

a permutation Γk-module such that there exists a surjective map β : M ′ → M . Let’s denoteR the Kernel of p β and α the restriction of β to R with codomain T .

By the local description of torsors (Theorem 2.27) there exists a map α′ : U → R suchthat YU ' M ′ ×R U where the fiber product is respect to α′ and such that the induced mapR→ k[U ]∗ is a lifting of −α by the natural projection proj : k[U ]∗ → T .

We want to show that the natural action on M ′×RU of the fiber product M ′×R T , (the fiberproduct is with respect to α), is well defined1. We need only show that the following diagramis commutative:

U × T U

R×R R.

α′ × α α′

Where the first line is the map defining the action of T on U and the second the multiplicationonR. We can check the commutativity on k. As T andR are tori the algebras of global sectionson k are generated by the characters and the constants. Furthermore k[U ] is generated byk[U ]∗ because Uk ' Gn

m,k, so it’s enough to check the commutativity of

k[U ]∗ ⊗ T k[U ]∗

R⊗ R R,

Id⊗(− proj)

Id⊗ Id

α′ × α α′

that holds because α′ is a lifting of −α via proj.

Moreover the action we have constructed is étale-locally trivial because the same is true forthe action of T over U .

Let’s come back to the diagram (4.2.1), since β is surjective and IdSinjective, α is surjective.

Furthermore as we also have that the Kernel of β is the same as the Kernel of α, the leftupper square is cocartesian by diagram chasing. This implies that M ′ ×R T 'M , thus YU isa torsor under the torus M .

ii) Since k has characteristic zero, the map f is a smooth morphism. As X is smooth, this impliesthat Y is smooth.

iii) Let’s call j : Div(X\U)k(Xk) → Div(Y \YU )k

(Xk) the map that sends any divisor to the pull-back via the map f . As f is étale and surjective the morphism consists only in sending aprime divisor, seen as points of codimension one, to the sum of divisors defined by the pointsin its fiber. This map descends to a map q : Pic(Xk) → Pic(Yk), because it respects theprincipal divisors. Thus we have the following commutative diagram,

0 T Div(X\U)k(Xk) Pic(Xk) 0

0 M Div(Y \YU )k(Yk) Pic(Yk) 0.

i j q

1Here by natural action we mean the action such that M ′ acts on itself via the group law and T acts on U via thetorsor action.

30

We notice that the Coker(j) is torsion-free because any prime divisor of (X \ U)k is sentvia j to a sum of prime divisors of (Y \ YU )k without multiplicity. Now Coker(i) = S is atorsion module, so by the snake lemma Ker(q) ' S and Coker(q) ' Coker(j) because themap Coker(i) → Coker(j) is a map from a torsion module to a torsion-free module, hencethe zero map. This implies that Pic(Yk) is torsion-free as Pic(Xk)/Ker(q) and Coker(q) aretorsion-free.

iv) If g ∈ k[Y ]∗ we can take [g] ∈ M = k[YU ]∗/k∗. As Coker(i) is a torsion module, there exists

an element h ∈ k[U ]∗ and a positive integer n such that [gn] = i([h]). Since divY (g) = 0,divX(h) = 0 again because the pullback of divisors in this situation is injective, thus h is ink[X]∗ that is equal to k∗ by hypothesis. This means that gn ∈ k∗ that implies g ∈ k∗.

Proof of Theorem 4.6. Let’s fix W a closed subset of X of codimension at least two and let’s callX ′ := X \ W . Suppose that X ′(Ak)ét ,Br1(X′) is not empty, if not we are done. By the descentTheorem 3.16 there exist X ′-torsors of any type.

In particular consider the group of multiplicative type S, such that S = (Pic(X ′k))tors and let’s

call λ the inclusion of S in Pic(X ′k). Let’s take an adelic point P ∈ X(Ak)comm,ét ,Br1(X′), as S

is a finite commutative group, we can find a torsor f ′ : Y ′ → X ′ under S of type λ such thatP ∈ f ′(Y ′(Ak)Br1(Y ′)), thus P = f ′(Q) for a certain Q ∈ Y ′(Ak)Br1(Y ′).

By the Proposition 4.3 there exists a torsor f : Y → X of the type corresponding to λ, suchthat restricted to f−1(X ′) is isomorphic to f ′ : Y ′ → X ′. By the Lemma 4.7, Y satisfies all thehypothesis of Theorem 4.2. Thus, as Y ′ is obtained from Y removing f−1(W ), for any place ν0, theset Y ′(k) is dense in the projection of Y ′(Ak)Br1(Y ′) in Y ′(Aν0

k ). In particular we can find rationalpoints of Y ′ very close to Q, outside ν0. By the continuity of f ′ as map between adelic points, we canfind rational points of X ′ as close as we want to P in X ′(Aν0

k ). Finally we have obtained that thecommutative étale algebraic Brauer-Manin obstruction explains the defect of strong approximationoff one place.

4.3 An example...

Here we want to apply the Theorem 4.2 to a certain family of explicit diophantine equations.This example is due to Wei himself [Wei14]. Let’s start introducing some notation, let’s take Ankwith coordinates x1, . . . , xn, and K a finite extension of k of degree n. Then we can define thepolynomial NK/k(x) in the following way. We fix a basis of K over k α1, . . . , αn then NK/k(x)is the unique polynomial in k[x1, . . . , xn] such that NK/k(a1, . . . , an) = NK/k(a1α1 + · · · + anαn)for any choice of a1, . . . , an ∈ k (on the right hand side we mean the standard norm). Now we canshow the example.

Proposition 4.8. Let Ki1≤i≤m and Lj1≤n be two families of finite extensions of k. Let X bethe affine variety of AM+N

k,xi,yj, defined by the equation

m∏i=1

NKi/k(xi) = cn∏j=1

NLi/k(yj)sj

where M =∑m

i=1[Ki : k] and N =∑n

j=1[Lj : k], c ∈ k∗ and sj ≥ 1. Let’s even suppose thatgcd(s1, . . . , sn) = 1, then the smooth locus X ′ of X satisfies strong approximation with algebraicBrauer-Manin obstruction off one place.

31

Proof. We want to apply the Theorem 4.2 on X ′. We know that X ′ is smooth by construction andwe also notice that the open U ⊆ X ′ where

∏mi=1NKi/k(xi) 6= 0, is a k-torsor under the torus T

defined by the equation

m∏i=1

NKi/k(xi)n∏j=1

NLi/k(yj)sj = 1.

We need to verify two other conditions, i.e k[X ′]∗ = k∗ and Pic(X ′

k) is torsion-free, thus we can

just study X ′k. The equation defining Xk becomes

M∏k=1

zk = cN∏l=1

wrll

and the rl are just some repetition of all the sj , thus in particular gcd(r1, . . . , rN ) = 1. The firststep is to notice that X ′

kis obtained from Xk removing a closed subset of codimension at least two,

this is just an easy computation with the Jacobian criterion. Thus the Weil divisors of X ′kand Xk

are the same.Let’s call Dk,l the prime divisor of Xk (and of X ′

k) defined by zk = 0, wl = 0. We have

div(zk) =

N∑l=1

rlDk,l and div(wl) =

M∑k=1

Dk,l.

Now as k[U ]∗ is generated by the functions zk1≤k≤M and wl1≤l≤N and by k∗, then k[X ′]∗ isgenerated by some products of zk1≤k≤M and wl1≤l≤N and k∗. Let’s take f ∈ k[X ′]∗ and writeit as

f = c′M∏k=1

ztkk

N∏l=1

wull ,

with c′ ∈ k∗ and the exponents tk, ul ≥ 0. Taking the divisors we obtain

0 = div(f) =M∑k=1

N∑l=1

(tkrl + ul)Dk,l

thus for every k and l, tkrl + ul = 0. As rl 6= 0, if we make k varying, keeping l fixed, we obtaint1 = t2 = · · · = tM , so

f = c′

(M∏k=1

zk

)t1 N∏l=1

wull = c′ct1N∏l=1

wt1rl+ull = c′ct1 ∈ k∗.

Now we want to show that Pic(X ′k) is torsion-free. By definition we have to show that

Coker(k[U ]∗/k

∗ div−−→ Div(X′\U)k(X ′

k))

is torsion free. The module k[U ]∗/k∗ is isomorphic to ZM+N−1, with basis

z1, . . . , zM−1, w1, . . . , wN

and Div(X′\U)k(X ′

k) is isomorphic to ZMN with basis

D1,1, . . . , D1,N , D2,1, . . . , DM,N .

32

To compute the Cokernel we firstly quotient by the images of w1, . . . , wN , obtaining Z(M−1)N ,with basis the classes of D1,1, . . . , D1,N , D2,1, . . . , DM−1,N . Then we quotient by the images ofz1, . . . , zM−1 in Z(M−1)N . The new application is represented by the matrix:

v 0 . . . 00 v . . . 0...

.... . .

...0 0 . . . v

where we have noted

v :=

r1...rN

and 0 :=

0...0

.

As the rll are relatively prime, by the Bezout identity, we can change the target basis and the newmatrix will be an identity matrix in the first M − 1 rows and 0 elsewhere. Thus we have obtainedPic(X ′

k) = Z(M−1)(N−1). This was the last check to apply the Theorem 4.2 and finally prove the

result.

In Wei’s article it is even shown that in a very particular case of the previous example, strongapproximation off one place holds, as there is no Brauer-Manin obstruction.

Proposition 4.9. Let K and L be two finite Galois extensions of k, and let X be the smooth locusof the affine variety defined by

NL/k(x) = cNK/k(y)

with c ∈ k∗. Let T be the torus defined by NL/k(x)NK/k(y) = 1, then

Br1(X)/Br0(X) ' H2(K.L, T ).

In particular if K ∩ L = k and K and L have maximal abelian subextensions of relatively primedegrees, then X satisfies strong approximation off one place

4.4 ...and a counterexample

When Pic(Xk) has some torsion we have shown that the commutative étale algebraic Brauer-Manin obstruction explains the defect of strong approximation. Can we hope to obtain a betterresult, i.e. that standard Brauer-Manin obstruction is the only obstruction? The answer is negativeand we will explain here a counterexample of Wei.

The construction of the counterexampleLet’s take two primes p and q, with p ≡ 3 (mod 4) and q ≡ 1 (mod 8), such that(

p

q

)= 1 and

(p

q

)4

= −1,

for example p = 19 and q = 17. Let’s take f := p(qx+ y)y+ qz2 and let Z be the closed subschemeof P2

Z defined by f = 0 and X the complementary open subscheme. Let’s call Z := Z ⊗ Q andX := X⊗Q.

Proposition 4.10. The variety X defined above is smooth, it contains an open non-empty subsetthat is a torus, Q[X]∗ = Q, but the Brauer-Manin obstruction doesn’t explain the defect of strongapproximation off infinity.

33

In particular the geometric Picard group of X is not torsion-free. Let’s start to verify thehypotheses.

Lemma 4.11. The variety X is smooth, Q[X]∗ = Q, Pic(XQ) = Z/2Z and X contains a non-emptyopen subset that is a torus.

Proof. The variety X is smooth because it is an open subset of P2Q. In virtue of the exact sequence

0→ Q[X]∗/Q∗ div−−→ DivZQ(P2

Q) = Z ×2−−→ Pic(P2Q) = Z→ Pic(XQ)→ 0,

we have Q[X]∗/Q∗ = 0 and Pic(XQ) = Z/2Z. To show that X contains a torus we take U1 theprincipal open subset of P2

Q defined by y 6= 0 and we put coordinates u := x/y and v := z/y. Thenthe automorphism of U1 given by

(p(qu+ 1) + qv2, v)

sends X to the open subset defined by v 6= 0. This is isomorphic to A1Q × Gm,Q so it contains a

torus.

Now we want to show that on X strong approximation off ∞ with Brauer-Manin obstructiondoesn’t hold.

Lemma 4.12. The set of integral points of X, X(Z) is empty.

Proof. Since the only invertible elements of Z are±1 we have to show that the equation f(x, y, z) = 0has no integer solutions.

First of all consider the case f(x, y, z) = −1, if we reduce modulo p we obtain qz2 = −1 andthis is not possible since we would have(

q

p

)(z2

p

)=

(−1

p

),

but by assumption −1 is not a square modulo p and at the same time(q

p

)=

(p

q

)= 1

by the reciprocity law and by assumption.Now to show that f(x, y, z) = 1 has no integer solutions requires some more effort. Let’s consider

the variety X′ in A3Z defined by the equation f(x, y, z) = 1 and let’s note X ′ := X′ ⊗ Q. We want

to show that we have Brauer-Manin obstruction, i.e.(X ′(R)×

∏`

X′(Z`)

)Br(X′)

= ∅.

Let’s take the quaternion algebra A := (q, y) ∈ Br(k(X ′)), we want to show that A actually definesan element in Br(X ′) with the computation of residues as it’s stated in the Section 3.1.1. For anydivisor D of X ′ we have ∂D(q, y) = [qνD(y)], so if νD(y) is even we have that the residue at Dis trivial. If for a certain D the valuation νD(y) is odd, in particular it is different from 0. Asx, y and z have no poles on X ′ they are elements of the ring of integers OX′,D. If we reduce theequation p(qx+ y)y+ qz2 = 1 seen in OX′,D, modulo D, we obtain that q is a square in κ(D), thus[qνD(y)] = 1. Thus A represents an element of Br(X ′).

We want to show that there are no elements in X ′(R)×∏`X′(Z`) that are orthogonal to A. We

will compute the symbol (a, b) as explained in the book of Serre [Ser79].

34

• If P∞ ∈ X ′(R) then A(P∞) = 1 because q is positive;

• If P2 ∈ X′(Z2) then A(P2) = 1 because q ≡ 1 (mod 8);

• If P` ∈ X′(Z`) and ` is odd and(q`

)= 1 then A(P`) = 1;

• If P` ∈ X′(Z`) and ` is odd and(q`

)= −1 then if P` = (x`, y`, z`) we must have ` 6 | y`,

otherwise if we reduce modulo ` the equation p(qx` + y`)y` + qz2` = 1 we would obtain that q

is a square modulo `. This means that q and y` are two units in Z`, thus A(P`) = 1;

• If Pq ∈ X′(Zq) then if Pq = (xq, yq, zq) if we reduce modulo q the equation f(xq, yq, zq) = 1,we obtain that (

y2q

q

)4

=

(p

q

)4

= −1

by construction, thus

A(Pq) =

(yqq

)=

(y2q

q

)4

= −1.

So for any local point A(Pν) = 1 except ν = q, thus we have Brauer-Manin obstruction, thusX′(Z) = ∅ as we wanted.

At the same time we want to show that

Lemma 4.13. The set (X(R)×∏`X(Z`))Br(X) is non-empty.

Proof. Consider V ⊆ P3Z defined by f(x, y, z) = t2 and let’s call V := V⊗Q. We have a projection

π : V → P2Z that sends (x, y, z, t) to [x : y : z], ramified over Z. Let’s call Y := V \ π−1(Z) and

Y := Y⊗Q. The map induced by π on the generic fibers Y → X is an X-torsor under Z/2Z andwe will denote it with π again.

Step 1: π∗(Br(X)) ⊆ Br(V ) = Br0(V )We have Br(X)/Br0(X) ' Q∗/(Q∗)2, indeed by the Hochschild-Serre spectral sequence we have anexact sequence

Br(Q)→ Br(X)→ H1(Q,Pic(XQ)) = Q∗/(Q∗)2 → H3(Q,Gm) = 0.

At the same time we have the following commutative diagram with exact rows [CTW12, Diagram(5.7)]:

0 Br(P2Q) Br(X) H1

ét(Z,Q/Z)

0 Br(V ) Br(Y ) H1ét(Z,Q/Z).

α

π∗ ×2

As Br(X)/Br0(X) is a two-torsion group, the image via π∗ is in the Kernel of α. Finally, as it isshown in Proposition 6.9.11 of [Poo11], for smooth quadrics Br = Br0, thus Br(V ) = Br0(V ).

Step 2: Y (R)×∏`Y(Z`) is not empty.

We check place by place:

• It is trivial that Y (R) is not empty;

35

• If ` 6= p, q then (q−1(p−1 − 1) : 1 : 0 : 1) ∈ Y(Z`);

• If ` = p then (0 : 0 : 1 :√q) ∈ Y(Z`);

• If ` = q then (0 : 1 : 0 :√p) ∈ Y(Z`).

Therefore the image of Y (R)×∏

Y(Z`) in X(R)×∏`X(Z`) is orthogonal to the Brauer group

of X and it is non-empty, so we are done.

36

Acknowledgments

Firstly I would like to thank my advisor, Professor Olivier Wittenberg, for all the explanationsand the work of revision, always given in a cordial atmosphere. I also have to thank him forhis patience with my rough French and the carefulness to correct the language mistakes aswell as the mathematical ones.

I thank Professor David Harari for the time spent during the Workshop "Brauer-Manin ob-structions" that introduced us in the world of rational points.

I’m really grateful to Professor Etienne Fouvry for his help to obtain the scholarship "IDEXParis Saclay" that allowed me to come in Orsay. I want to thank him also for his interestingcourse in number theory, during my first year, and for his incredibly cheerfulness.

The life in a small town as Orsay would have been really boring without Emiliano Ambrosiand Gregorio Baldi. I have to thank them for all the time we have spent together and thethings I’ve learned from them. I also really thank Marcin Lara and all the other students ofthis year: Jokin De Rotonde, Jorge Antonio, Juanyong Wang, etc.

Last year in Orsay has also been really exciting and I have to say "Thanks!" to the peopleI met: mon voisin Mirko Mauri, Enrica Mazzon, Fabio Bernasconi, Laura Petrusi, AndreaAgostini, Yasir Kiliç, Valerio Proietti, François Delgove and Adrien Poulenard.

Many thanks to all the people of the University of Pisa, especially to "gli Invasati del ’92", theother students of "L’Aula studenti" and the students of "Scuola Normale", for the interestingdiscussions, the time spent solving problems, but even for the nights spent playing Lupus andother games and for many other things that are difficult to explain.

I cannot forget to thank all my family for the presence and the affection they have shownduring these years.

At the end I would like to thank the secretary of M1-MFA Pascale Starck and the ex-secretaryof M2-AAG, Valerie Lavigne, for all the work they have done for me these two years.

37

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