Descent theory for strong approximation for varieties containing a torsor under a torus

Transcription

1 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 varieties containing a torsor under a torus directed by Olivier Wittenberg Academic year 2015/2016

2 M2 Arithmétique, Analyse, Géométrie Département d Enseignement de Mathématiques, Bât. 425 Université Paris-Sud Orsay CEDEX

3 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 1

4 2

5 Contents 1 Weak and strong approximation Weak approximation Adelic points and strong approximation Torsors under groups of multiplicative type Torsors Groups of multiplicative type Rosenlicht lemma and type Fundamental exact sequence and elementary obstruction Local description of torsors Brauer-Manin obstruction The Brauer group Residues The adelic Brauer-Manin pairing Introduction to descent theory Hochschild-Serre and filtration of the Brauer group The main theorem Strong approximation for varieties containing a torsor under a torus The case when Pic(X k ) is torsion-free The general case An example and a counterexample

6 Introduction Let k be a number field, strong approximation has been widely studied for algebraic groups and their homogeneous spaces over k. At the same time for general non-proper varieties very little is known. The first results of strong approximation with Brauer-Manin obstruction appeared for the first time in [CTX09], where J.-L. Colliot-Thélène and F. Xu established strong approximation with Brauer-Manin obstruction for homogeneous spaces of semi-simple and simply connected algebraic groups. Another result is the one of Y. Cao and F. Xu [CX14] on smooth toric varieties. The results we 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-variety X containing a k-torsor under 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 the hypothesis 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 X with codimension at least two, the commutative étale algebraic Brauer-Manin obstruction is the only obstruction to strong approximation off one place for X \ W. When Pic(X k ) is torsion free the commutative étale algebraic Brauer-Manin obstruction can be replaced by algebraic Brauer-Manin obstruction. We will see at the end of the last chapter the counterexample 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-Manin obstruction and Colliot-Thélène and Sansuc s descent theory. These topics have been studied in a Workshop in Orsay, during the first semester of this year. Thus the first part is mainly obtained by arranging the notes of the seminars, adding and fixing some parts. In the exposition of these chapters we have followed the book written by Skorobogatov [Sko01]. The last chapter is an application of the theory developed in the previous chapters. We will present some results of the article of Wei. 4

7 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 other parts of the text where it s stated differently it will be a number field. We will call Ω k the set of places of k, Ω the set of infinite places and we will usually denote with ν a place of k. We will write Γ k for the absolute Galois group of k. If A is a discrete Γ k -module we will denote H i (k, A) the groups of Galois cohomology of A with respect to Γ k. If X is a scheme and F is an étale (resp. fppf ) sheaf on X we will write Hét i (X, F) (resp. Hi fppf (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, Div Z (X) will be the group of divisors and Div 0 (X) the group of principal divisors (we will always consider divisors over regular schemes). 5

8 Chapter 1 Weak and strong approximation 1.1 Weak approximation Let k be a number field, for any ν, we can endow X(k ν ) with its ν-adic topology. For any finite subset S Ω k, we define the topological space X(k Ωk \S) := X(k ν ), ν Ω k \S where the topology is the product of the topologies of X(k ν ). We will call this topology the weak topology. Of course there are diagonal embeddings of X(k) in X(k Ωk \S). If we take X = A 1 k we know that the diagonal map k ν Ω k k ν is dominant. This is a classical result of number theory and the property is usually called weak approximation. We can wonder if this is true even for other varieties. Thus we give the following definition. Definition 1.1. Let S be a finite subset of Ω k, we say that a smooth, geometrically integral k-variety X satisfies weak approximation off S, if X(k Ωk ) is empty or the diagonal map is dominant. X(k) X(k Ωk \S) As A 1 k satisfies weak approximation, even An k satisfies weak approximation for any n. The same remains true for any open subset of A n k, because if U is a Zariski open subset of X, the set U(k ν) is open in X(k ν ). Let s focus on the opposite problem: if U is a dense open subset of X and we have weak approximation on U, what can we say about X? In virtue of the inverse function theorem for complete fields with respect to a non-trivial absolute value as it s proven in [VA94] at the beginning of Chapter 3, we have: Proposition 1.2. Let X be an irreducible, smooth variety defined over k, such that X(k ν ) is not empty. Then, for any non-empty Zariski open subset U of X, the set U(k ν ) is dense in X(k ν ) in the ν-adic topology. Furthermore, any non-empty ν-adically open subset F of X(k ν ) is Zariski-dense in X; in particular, X(k ν ) is Zariski-dense in X. As a consequence we have. 6

9 Corollary 1.3. Let X be a smooth, geometrically integral k-variety containing an open dense subset which verifies weak approximation, then X satisfies weak approximation. Corollary 1.4. If X is a smooth, geometrically integral k-rational variety, then weak approximation holds. Another quite elementary example of varieties satisfying weak approximation can be found in the article of Colliot-Thélène, Sansuc and Swinnerton Dyer [CTSSD87]. On page 68, the following theorem is stated. Theorem 1.5. Let k be a number field and let V P n k with n 6 be a pure codimension 2 intersection of two quadrics over k. Assume that V is geometrically integral and not a cone. Let X be 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(O k ) is a model of X if X X η, with η the generic point of Spec(O k ). Usually we will suppose the model to be integral. We will use the following proposition that is almost 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 are isomorphic 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(O ν ) X(k ν ), 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 of separatedness. If X is proper then for almost any place the map is even surjective thanks to the valutative criterion of properness applied to X that is proper outside a finite number of places. We also 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(A S k ) := {(ρ ν) ν X(k Ω\S ) all but finitely many ρ ν are integral} with the convention X(A k ) := X(A k ). We notice that the definition does not depend on the choice of the model because any two of them are isomorphic away from a finite number of places. We will not consider X(A S k ) with the topology of subspace of X(k Ω\S ), we will be more interested in the topology defined by the basis of open sets of the form X(O ν ), ν T U ν ν T with U ν an open of X(k ν ) and T finite, such that Ω \ (S Ω ) T. We will call this topology the strong topology or the adelic topology. 7

10 Example. If X = A 1 k, then X(A k) is the set of adeles with the adelic topology; if X = G m then X(A k ) 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 Ω k we will say that X satisfies strong approximation off S if X(A k ) = or is dominant. X(k) X(A S k ) Thanks to the valuative criterion of properness we can deduce that if X is proper then X(A S k ) = X(k Ωk \S) as topological spaces. Thus for proper schemes weak and strong approximation are equivalent. We also have that A n k satisfies strong approximation off one place, i.e. off S = {ν}, for any ν Ω k. This is a classical result of number theory, in the case when k = Q and S = { } this is just the Chinese remainder theorem. What can we say about strong approximation on open subvarieties of A n k? In general it doesn t hold off a finite set of places. Example. Consider G m,q A 1 Q, if we had strong approximation away from infinity, G m(z) would be dense in l G m(z l ), since if a point in l G m(z l ) can be approximated in the strong topology by rational points it can be approximated by integral points. But G m (Z) = {1, 1} and it is not dense in any G m (Z l ). We can even show that strong approximation doesn t hold off a finite place p. If it held then as before the set G m (Z[1/p]) would be dense in l p G m(z l ). But G m (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 l 2 that is totally split. This means that 1 and p are both squares modulo l, thus the image of G m (Z[1/p]) in G m (Z l ) is contained in the subgroup of squares of G m (Z l ). Obviously the result can be extended to any number field k and any finite set of places S. Thanks to Dirichlet s unit Theorem the set G m (O k,s ) is finitely generated, let s say by t 1,..., t n, then we can take k( t 1,..., t n ) and apply the Chebotarev density theorem again. We also have many other similar obstructions on G m, just taking any étale cover G m t t n G m. This phenomenon can be generalized by the following theorem due to Minchev whose proof can be found in [Rap12, page 9]. Theorem 1.10 (Minchev 1989). Let X be an irreducible normal variety over a number field k such that X(k). If there exists a nontrivial connected unramified covering f : Y X defined over an algebraic closure k, then X does not satisfy strong approximation off any finite set S of places of k. In particular if we take any polynomial in n variables f with coefficients in k and we take the open U in A n k that is defined by f 0, we can then take as Y the closed variety in An G m defined by f = x m n+1 0. The natural projection Y U is unramified, thus on U we cannot have strong approximation with respect to any finite set of places. At the same time if we take X the complement of a closed subset in A n k of codimension at least two we still have strong approximation off one place. This result has been proven independently by Wei and by Cao and Xu. We will propose a slightly generalisation of this result, using a variant of the proof of Wei. 8

11 Proposition Let X be a smooth variety over the number field k and let S be a finite set of places of k. Suppose that X satisfies weak approximation off S. If there exists a dense open subset U of X with the following property: P) For any x U(k) there exists a dense open V x of U such that for any y V x (k) there exists a variety Z x,y that satisfies strong approximation off S and a morphism f x,y : Z x,y X such that in the fibers of f x,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 Ω k \ S containing Ω \ (S Ω ). For any adelic point P = (P ν ) ν Ωk X(k ν ) X(O ν ) ν T S ν Ω k \(T S) we have to find a rational point of X that is as close as we want to P when ν T and integral when ν Ω k \ (T S). In virtue of the inverse function theorem (Proposition 1.2), we can find local points of U that are as close as we want to P ν for ν T. Thus by weak approximation on U off S we can choose rational points x of U that are as close as we want to P ν for ν T. A priori the local points x ν (the image of x in U(k ν )) could fail to be integral when ν Ω k \ (T S), let s suppose that x ν is not integral when ν T Ω k \ (T S). We take V x as in the property P). Thanks to the implicit function theorem, we can find local points of V x near P ν for ν T. As V x satisfies weak approximation off S we can find a rational point y of V x near the local points P ν for ν T, in particular we can choose y as an O ν -point of X for any ν T. Now let s take f x,y : Z x,y X as in the property P), there exists on Z x,y an adelic point Q = (Q ν ) ν Ωk such that f x,y (Q ν ) = x ν when ν Ω k \ (T S) and f x,y (Q ν ) = y ν (the image of y in U(k ν )) when ν T. Thanks to strong approximation on Z x,y off S we can find a rational point z of Z x,y that is near Q in the strong topology of Z x,y (A k ) off S. As the morphism f x,y : Z x,y (A k ) X(A k ) is continuous, f x,y (z) can be near to P for the places ν T and integral outside T S, thus we have the result. Corollary If X is a smooth k-variety 1 and there exists a variety Z that satisfies strong approximation 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 Let X be an open subvariety of A n k obtained by removing a closed subset W of codimension 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 V x the open obtained removing the closed subset of X that is the union of all the lines joining x with W. By the hypothesis on the codimension V x is non-empty, hence dense. For any point y V x different from x we can take as Z x,y the line l X that joins x and y. Corollary Let X be an open subvariety of a smooth quadric Q of P n k obtained removing a closed 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 strong approximation holds off one place (Proposition 4.1). 1 Here it s not necessary to require that it satisfy weak approximation off S because it will be a consequence of the other hypotheses. 9

12 Chapter 2 Torsors under groups of multiplicative type 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 exists a covering (U i X) in the fppf topology that trivialises Y, in the sense that Y X U i G X U i compatibly with the action of G and the projection to U i. We will mean left torsors if we do not specify. Observe that Y X itself trivializes a torsor Y X as there is a natural map G X Y Y X Y that sends (g, y) (g.y, y). We can check locally in the fppf topology that this map is an isomorphism. If X is equal to Spec(k) the existence of a rational point x : Spec(k) Y implies that G Y. Indeed we can check locally that the map G 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 sets between the class of isomorphism of torsors over X under G and the group Hfppf 1 (X, G) that sends the class represented by G in the zero cohomological class. If moreover G is smooth, thanks to a Theorem of Grothendieck [Gro68, Théorème 11.7], Hfppf i (X, G) = Hi ét (X, G). In this work we will consider only torsors over k-varieties. Let S be a group of multiplicative type, then if f : Y X is an X-torsor under S, we have a map θ : X(k) H 1 (k, S) which maps x to the class [Y x ] H 1 (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 right k-torsor Z under the same group, then we call the twist of Y by Z, if it exists, the quotient of Z k Y by the action of G, via the map (g, z, y) (zg 1, gy) and we denote it by Z Y. Theorem 2.3. We have X(k) = Z Zf( Z Y (k)) where the union is taken over a set of representatives of the isomorphism classes of right k-torsors under G Groups of multiplicative type Let s fix a field k. An algebraic group S is a group of multiplicative type if S k is a subgroup scheme of G n for a certain n. Recall that if the characteristic of k is zero a group of multiplicative m,k type S over k is a commutative linear k-group which is an extension of a finite group by a torus. 10

13 The module of characters of S is the abelian group Ŝ = Hom k groups (S k, G m,k ), equipped with the action of the Galois group Γ k. At the same time if M is a Γ k -module, we will call M the group Hom Γk (M, k ). We have an equivalence of categories between groups of multiplicative type and Γ k -modules of finite type. In order to understand better this equivalence we need to recall the Weil restriction. Weil restriction Let L/k be a finite extension of degree d. There is a functor Var /k Var /L called extension of scalars that sends X X L. It is also possible to construct a functor in the other direction that is just the right adjoint. The functor is called Weil restriction or restriction of scalars, and it s usually denoted R L/k ( ). By construction we have Hom L (V L, W ) = Hom k (V, R L/k (W )). Moreover this functor sends algebraic groups to algebraic groups and if the field is perfect tori to tori. Proposition 2.4. For any finite extension L/k and for any quasi-projective variety X over L, the functor R L/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 ]. The first example is the following: R L/k (A N L ) A Nd k and this is checked by choosing a basis of L/k. If we take the open immersion G m,l A 1 L and we apply the functor R L/k( ) we obtain a map R L/k (G m,l ) A d k that can be shown (Prop. 4.9 in [Sch94]) to be again an open immersion. Thus we have Proposition 2.5. If k is perfect the tori R L/k (G m,l ) are k-rational varieties. We can now state the theorem. Theorem 2.6. The association G Ĝ gives an equivalence of categories between the category of k-groups of multiplicative type and the category of discrete Γ k -modules of finite type. Moreover a sequence of groups of multiplicative type is exact iff the dual of Γ k -modules of characters is exact. Finally if k is perfect and L/k is a finite extension the torus R L/k (G m ) is sent to Z[Hom k 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 the category of abelian étale sheaves over Spec(k) and the category of discrete Γ k -modules. Proof. We construct the two functors that give us the equivalence of categories. We fix a separable closure k. For any étale abelian sheaf F and for any finite Galois subextension L of k, F(L) is a Γ k -modules with the action of Γ k given by the functoriality. For every pair of finite Galois subextensions L and L such that L L and such that the relative extension has degree n, we 11

14 have an isomorphism α : L L L g i Gal(L /L) L that sends l 1 l 2 (l 1 g 1 (l 2 ),..., l 1 g n (l 2 )). Thus we have a commutative diagram 0 L L L L L = = α 0 L L γ β g i Gal(L /L) L where β(l) = l 1 1 l and γ(l) = (l,..., l) (g 1 l,..., g n l). If we apply F we obtain 0 F(L) F(L ) F(L L L ) 0 F(L) F(L ) F(β) = = F(α) F(γ) g i Gal(L /L) F(L ) Thus the exactness of the first row, that is the sheaf axiom for the covering L L, holds if and only 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 Γ k -module lim F(L), where L runs over every finite Galois extension in k. For the other functor we take a discrete Γ k -module M and we define F (A), for any étale-algebra A over k, as the set Hom k alg (A, k) with an action of Γ k, induced by the natural action over k. Then we consider the presheaf that associates to any étale-algebra A over k, the group F M (A) = Hom Γk sets(f (A), M), with the natural transition maps. We can check that F M is a sheaf on finite Galois extension and this is a consequence of the previous diagram. It is also easy to check that the two functors give the equivalence of categories. The result on the cohomology results from the isomorphism in the zero degree 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 invertible function of X k G m 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 ) : Ĝ(k) Pic(X k ) as the morphism that associates to any character χ, the image of [Y ] H 1 fppf (X k, G k X k ) in H 1 fppf (X k, G m,x) = Pic(X k ), via the morphism χ. The map type(y ) will be the type of Y. If Pic(X k ) 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. 12

15 Proposition Let X be a smooth, geometrically integral k-variety. Then for any torsor Y X under a k-torus T, we have the following exact sequence: 0 k[x] k[y ] T type(y ) Pic(X k ) Pic(Y k ) 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 be a perfect field (we will keep this hypothesis during all this section). Let X be a scheme over k and let π : X Spec(k) be the structural map of X. Given a Γ k -module M of finite type, consider the Ext-spectral sequence: given by the composition of E p,q 2 = Ext p Γ k (M, (R q π )G m ) Ext p+q X ét (π M, G m ), Sh(X ét ) Hom Xét (π M, ) Ab π HomΓk (M, ) Sh(k ét ). The functor π sends injectives to injectives as its left adjoint is exact. sequence is The low degrees exact 0 Ext 1 Γ k (M, k[x] ) Ext 1 X ét (π M, G m ) Hom Γk (M, Pic(X k )) Ext 2 Γ k (M, k[x] ) Ext 2 X ét (π M, G m ). (2.2.1) We want to simplify it with the following lemma. Lemma Let S be an X-group of multiplicative type. Then we have an isomorphism functorial in S and X. H i fppf (X, S) = Exti X ét (π Ŝ, G m ) Proof. We use local-to-global Ext spectral sequence that you can find in SGA4( [AGV71] Exposé V, Théorème 6.1), E p,q 2 := H p fppf (X, Extq X fppf (π Ŝ, G m )) Ext p+q X fppf (π Ŝ, G m ), with the following commutative diagram of functors: Sh(X fppf ) Hom Xfppf (π Ŝ, ) Ab Hom Xfppf (π Ŝ, ) Γ(X, ) Sh(X fppf ). 13

16 The first step is to show that the spectral sequence completely degenerates, i.e. for i 1, Ext i X fppf (π Ŝ, G m ) = 0. As S is locally constant in the étale topology, it s enough to show that for i 1, Ext i X fppf (Z, G m ) = Ext i X fppf (Z/nZ, G m ) = 0. The sheaves Ext i X fppf (Z, G m ) are zero, because they are zero in the étale site, for any X, as they correspond to the sheafification of the presheaves H i (G m ) [Mil13, Chapter 1, Proposition 10.4]. To prove that Ext i X fppf (Z/nZ, G m ) = 0 we take the exact sequence of fppf constant sheaves and we apply Hom Xfppf (, G m ). We obtain G m 0 Z Z Z/nZ 0 t t n G m Ext 1 X fppf (Z/nZ, G m ) 0. As we are working in the fppf site the first map is surjective, thus Ext 1 X fppf (Z/nZ, G m ). Looking at the other terms of the long exact sequence in cohomology we also obtain Ext i X fppf (Z/nZ, G m ) = 0 when i 2. We have proven H i fppf (X, S) = Exti X fppf (π Ŝ, G m ). Now we have to prove that Ext i X ét (π Ŝ, G m ) = Ext i X fppf (π Ŝ, G m ) since Ŝ is locally constant for the étale topology, by the Cartan-Leray spectral sequence (for the étale coverings of X) [AGV71, Exposé V, Corollary 3.3], we can just check the equality étale-locally, so we can take Ŝ = Z and Ŝ = Z/nZ. In the first situation we can use a theorem of Grothendieck, namely Théorème 11.7, [Gro68], as G m is smooth. For the second case we take again the exact sequence 0 Z Z Z/nZ 0 and we apply once Hom Xfppf (, G m ) and the second Hom Xfppf (, G m ). Taking the long exact sequences 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 smooth variety and S an X-group of multiplicative type, we have an exact sequence 0 Ext 1 Γ (Ŝ, k[x] k ) Hfppf 1 type (X, S) Hom (Ŝ, Pic(X Γk k )) Ext 2 Γ (Ŝ, k[x] k ) Hfppf 2 (X, S), functorial in S and X, called the fundamental exact sequence. The map type associates to a torsor its type. So for any Γ k -invariant morphism λ : Ŝ Pic(X k ), the existence of a torsor of type λ is equivalent to (λ) = 0. If Pic(X k ) is of finite type and Ŝ = Pic(X k ), we can take Id Hom Γ (Ŝ, Pic(X k k )). 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 If Pic(X k ) is of finite type the existence of universal torsors (Definition 2.9) is equivalent to the vanishing of the elementary obstruction. 14

17 Proof. Let s put Ŝ = Pic(X k ) in the fundamental exact sequence, if λ Hom Γ (Ŝ, Pic(X k k )), by functoriality we know that (λ) = λ (e(x)). If λ is an isomorphism then λ (e(x)) = 0 if and only if the elementary obstruction is zero. Moreover we have another simplification of the sequence: Corollary If we add the hypothesis k[x] = k, then the fundamental exact sequence becomes: 0 H 1 (k, S) π H 1 (X, S) type Hom (Ŝ, Pic(X Γk k )) H 2 (k, S) π H 2 (X, S). So in this case the space of torsors of a certain type is a principal homogeneous space under H 1 (k, S). The action of H 1 (k, S) is exactly the twist (Definition 2.2). In particular if k is algebraically closed the type identifies the torsor. Still under the hypothesis k[x] = k, we can even describe the set of rational points using torsors of a given type, just rewriting the Theorem 2.3 as X(k) = f(y (k)). type(y,f)=λ Now we want to prove two theorems about the elementary obstruction. Theorem Let X be a geometrically integral, smooth k-variety, then the class is represented by the 2-fold extension e(x) Ext 2 Γ k (Pic(X k ), k ) 0 k k(x) Div(X k ) Pic(X k ) 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) R 1 π (A) 0 (2.2.3) is exact then if we consider the Ext-spectral sequence we have: E p,q 2 = Ext p Γ k (, R q π (A)) Ext p+q X ét (π ( ), A) Lemma The transgression map (E 0,1 2 E 2,0 2 ), Hom Γk (, R 1 π (A)) Ext 2 Γ k (, 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 the following exact sequence of étale sheaves. The exactness is proven in Milne, [Mil13] Proposition

18 Proposition Let X be an irreducible, noetherian, regular scheme and j : η X the inclusion of the generic point. Then we have an exact sequence of étale sheaves 0 G m j G m x X (1) (i x ) Z 0. We need to check that R 1 (π )(j G m ) is zero, but we know that it is a subsheaf of R 1 (π j) (G m ), thanks to the convergence of the Grothendieck spectral sequence for the composition of π and j. But R 1 (π j) (G m ) 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 Let k be a field and X be a smooth k-variety such that k[x] = k, we have the following implications: ( ) X(k) k k(x) has a Γ k -equivariant section. e(x) = 0. As a consequence of this theorem, using the Corollary 2.14 we have: Corollary If Pic(X k ) is of finite type, the existence of a rational point implies the existence of a universal torsor. This fact is important, we will use it in the proof of the main theorem of Colliot-Thélène and Sansuc descent theory. Now we will prove the theorem, we divide it in different parts. Proposition Let k be a field, X a smooth, geometrically integral k-variety, such that X(k), then the natural map k k(x) has a Γ k -invariant retraction. We need the following lemma: Lemma Let G be a profinite group, H a closed subgroup, B a G-module, A an H-module. Then Ext n G ( Z[G] Z[H] A, B ) = Ext n H (A, B H ) where B H 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 choose a projective resolution of A. Since Z[G] is a free Z[H]-module, Z[G] Z[H] is exact and it sends projectives to projectives. Thus it s enough to check that Hom G ( Z[G] Z[H] A, B ) = Hom H (A, B H ), but this can be done similarly to the commutative case. Now we can prove the proposition. Proof of Proposition Let P X(k) and consider the natural maps k O X k,p k(x) 16

19 where OX k,p is the Zariski stalk. The first map admits a section g g(p ), so it s sufficient to find a section of the inclusion OX k,p k(x). Because X is smooth, we have an exact sequence of Γ k -modules where 0 O X k,p k(x) Div P ( Xk ) 0 Div P ( Xk ) = x Spec(O Xk,P ) (1) Z x If this sequence splits then we get the missing section. To show this we show that ) Ext 1 Γ k (Div P (X k ), OX k,p = 0. We notice that Div P (X k ) = Z x = x Spec O Xk,P (1) x over x x Spec O Xk,P (1) Z [Γ k /H x ] where H x is the Kernel of the transitive action of Γ k on the points over x, corresponding to a certain extension L x /k. We have ( ) ) Ext 1 Γ k (Div P (X k ), OX k,p = Ext 1 Γ k Z [Γ k /H x ], OX k,p = x = ) Ext 1 Γ k (Z [Γ k /H x ], OX k,p. x Since Z [Γ k /H x ] = Z[Γ k ] Z[Hx] Z, we can use Lemma 2.22 and we obtain that for any x, Ext 1 Γ k (Z [Γ k /H x ], O X k,p ) ( ) = Ext 1 H x Z, OX k,p. If A := O XLx,P and p : Spec(A) Spec(L x ) is the structural map, ) Ext 1 H x (Z, OX k,p = Hét(Spec 1 L x, p G m,a ) because the functor Hom Hx (Z, ) is equal to the functor M M Hx. sequence, Hét(Spec 1 L x, p G m,a ) Hét(Spec 1 A, G m,a ). The right group is zero by Hilbert 90 for local rings, so we are done. By the Leray spectral In the proof of the Proposition 2.21 we have shown a useful property of the Γ k -module of divisors that can be easily generalized as follows. Lemma Let X be a smooth variety over a field k, then the Γ k -module of divisors on X k, Div(X k ) is isomorphic to a certain sum Z[Γ k /H i ], i I with H i open normal subgroups (thus of finite index) of Γ k and I is not necessarily finite. 17

20 Definition We will call permutation module a Γ k -module that contains a basis invariant (not necessarily fixed) under the action of Γ k. Thus it s a Γ k -module of the form i I Z[Γ k/h i ] with H i closed subgroups of Γ k and I not necessarily finite. Thus in Lemma 2.23 we have shown that Div(X k ) is a permutation module. Another fact that we will use many times, whose proof is the same as in the proof of Proposition 2.21 is the following. Lemma Let A be a local ring that is a k[γ k ]-module, for any permutation module M, then Ext 1 Γ k (M, A ) = 0. In particular when A = k we obtain Ext 1 Γ k (M, k ) = H 1 (k, M) = 0. Let s conclude now the proof of Theorem Proposition Let k be a field and X a smooth k-variety such that k[x] = k, then the inclusion k k(x) has a Γ k -invariant retraction if and only if the 2-fold extension, (2.2.2) is zero in Ext 2 Γ k (Pic(X k ), 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 Ext 2 Γ k (Pic(X k ), k ). The other implication is not true in general for 2-fold extensions. Consider the short exact sequence 0 k(x) /k Div(X k ) Pic(X k ) 0 (2.2.4) and the long exact sequence given by the derived functor of Hom Γk (, k ). Thanks to the Lemmas 2.23 and 2.25, we know that Ext 1 Γ k (Div(X k ), k ) = 0, thus we have the injective connection map Ext 1 Γ k (k(x) /k, k ) Ext 2 Γ k (Pic(X k ), 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) is zero in Ext 1 Γ k (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) is equivalent 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 about the geometry of Y. This question will be really important in the next chapters, for example after Theorem 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 Ŝ λ Pic(X k ) Pic(U k ) is zero. Let s call P the Kernel of the map from Pic(X k ) Pic(U k ), thus we have by hypothesis a well defined map Ŝ P that we will call λ again. 18

21 Then for any commutative diagram with exact rows like the following β 0 R M Ŝ 0 α λ 0 k[u] /k Div (X\U)k (X k ) P 0. (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 morphism φ : U R and a morphism ψ : Y U M such that Y U ψ M φ U R. is a cartesian square. Moreover the induced map φ : R k[u] is a lifting of α via the natural projection 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 and Theorem In particular it is always possible to construct a diagram like the (2.3.1). We can just take N, the fiber product of Ŝ and Div (X\U) k (X k ) via P, then take any permutation module M with a surjective map M N and conclude taking the Kernel. We will often use the following corollary of the theorem. Corollary 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 geometrically isomorphic to G n m for a certain n, thus Pic(U k ) = 0. If we follow the previous construction the module P will be Pic(X k ), we can take as M the Γ k -module Div (X\U)k (X k ) itself, because it is a permutation module (Lemma 2.23) and R becomes k[u] /k. As U is a k-torsor under a torus by hypothesis, we can apply the second part of the theorem, thus φ is an isomorphism. Then Y U is isomorphic to M. As M is a permutation module, we know by Theorem 2.6 that Y U is isomorphic to a certain product n R Li /k(g m,li ) i=1 with {L i } 1 i n a family of finite extensions. By Proposition 2.5, Y U is a k-rational variety, thus also Y is k-rational as U is dense in X and f is an open morphism, so Y U is dense in Y. β 19

22 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 in Chapter 2 of the book [GT06]. In the book it s even proven that the Brauer group constructed is isomorphic to H 2 (k, k ), thus it is even isomorphic to H 2 ét (Spec(k), G m). In this correspondence the classes of quaternion algebras are in bijection with the two torsion of H 2 (k, k ). One can even define the Brauer group of a scheme with some generalizations of central simple algebras, i.e. Azumaya algebras. 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 H 2 ét (X, G m). 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 group constructed with Azumaya algebras is isomorphic to the torsion subgroup of Br(X). Example. If a field is algebraically closed or even C 1 we can show that the Brauer group is trivial, so for example Br(C) = Br(F q ) = 0. If we take R we have instead Br(R) = Z/2Z and the group is generated by the algebra of quaternions. It can be shown that if k is a number field and ν is a finite place there is a canonical map Br(k ν ) invν Q/Z that is an isomorphism. We notice that even when ν is an infinite place we have canonical embeddings of Br(k ν ) in Q/Z, we will call them inv ν as well 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 work only with Brauer group of a field, that is easier to understand. For example we can use some particular 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 H 1 ét (X, (i x ) Z) Br(X) ι 1 Hét(X, 2 j G m ). The term H 1 ét (X, (i x) Z) is zero because every H 1 ét (X, Z x) is included in H 1 (k, Z) = 0 by the Leray spectral sequence. This means that the map i 1 is injective. Moreover we have a map ι 2 : H 2 ét(x, j G m ) H 2 ét(η, G m ) = Br(k(X)) 20

23 from the Leray spectral sequence that is injective because R 1 j G m = 0. If we take ι 2 ι 1 this defines an injection of Br(X) in Br(k(X)). One can check that this map is the same as the map given by the 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 is a discrete valuation ring, K is the field of fractions and k the residue field. Suppose that the characteristic of k is zero, then if j : η Spec(A) is the inclusion of the generic point and i : ξ Spec(A) is the closed immersion of the closed point, the exact sequence gives the exact sequence 0 G m j G m i Z 0, Br(A) H 2 ét(spec(a), j G m ) H 2 (Spec(A), i Z). Now H 2 (Spec(A), i Z) = H 2 (k, Z) because i is exact and as Q is uniquely divisible, H 1 (k, Q) = H 2 (k, Q) = 0. So H 1 (k, Q/Z) is isomorphic to H 2 (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 characteristic zero. Then R i j G m = 0 for any i 1. Thus we have that all the R i j G m = 0 when i 1, so by the Leray spectral sequence we obtain an isomorphism between H 2 ét (Spec(A), j G m ) and Br(K). So we can construct a map A : Br(K) H 1 (k, Q/Z) such that is an exact sequence. Br(A) Br(K) A H 1 (k, Q/Z) Definition 3.3. We call the resulting map A the residue map of K associated to A. If K is the field of rational functions of an irreducible regular scheme X over a field of characteristic zero and x is a prime divisor, we will denote x : Br(K) H 1 (κ(x), Q/Z) the map associated to A = O X,x. Example. If K, A and k are as before, we want to understand the residue map A : 2 Br(K) H 1 (k, Q/Z). We notice that the image is actually inside H 1 (k, Z/2Z) = k /(k ) 2. It can be shown that if (a, b) is a quaternion algebra A ((a, b)) = ( 1) ν A(a)ν A (b) [ a ν A(b) /b ν A(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) H 1 (κ(x), Q/Z), whit the last arrow induced by the residue maps x, is well defined and exact. 21

24 3.2 The adelic Brauer-Manin pairing In this section we define an important pairing, which will be fundamental to describe an obstruction 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, Ω k its set of places, then we have the following exact sequence 0 Br(k) invν Br(k ν ) Q/Z 0 (3.2.1) ν Ω k 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 Ω k finite, there exists a scheme X of finite type, defined over O k,s and a class A Br(X) with a morphism i : X X identifying X with the generic fiber X η, s.t. i : Br(X) Br(X) sends A to A. For the proof see Corollary 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 A(x). Let X/k be a smooth and geometrically integral variety over a number field k. We are interested in the pairing Br(X) X(A k ) Q/Z defined by the following rule: (A, (P ν )) ν Ω k inv ν (A(P ν )). (3.2.2) Where A(P ν ) makes sense thanks to the previous definition and inv ν are the local invariant maps appearing in the exact sequence of Theorem Lemma 3.8. The Brauer-Manin pairing is well defined, i.e. the sum of (3.2.2) is finite. Proof. Given (P ν ) X(A k ) and A Br(X) we have to show that A(P v ) = 0 for almost all ν. Thanks to Theorem 3.6 we can chose a finite set of places S big enough (containing all the archimedean places) such that P ν X(O ν ) for all ν / S (by the definition of the adelic ring). This concludes 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 The Brauer-Manin pairing is trivial on Br 0 (X), and so it can be defined also as a pairing from Br(X)/ Br 0 (X). Proof. This follows immediately from the exact sequence of Theorem and the functoriality of Br( ). 22

25 Definition We define X(A k ) Br(X) as the subset of X(A k ) orthogonal to all elements of Br(X). Lemma The Brauer-Manin pairing is locally constant in the adelic topology. For the proof see Corollary of [Poo11]. Proposition We have the following inclusion: X(k) X(A k ) Br(X) X(A k ) Proof. The only non trivial inclusion is the first one. But this follows from the commutativity of the diagram X(k) X(A k ) Br(k) and the exact sequence of Theorem ν Ω k Br(k ν ) Remark. The previous two results imply that the closure of the diagonal image of X(k) via the diagonal embedding in the adelic points is contained in X(A k ) Br(X). Remark (Functoriality). Let f : X Y be a k-morphism of smooth geometrically integral k- varieties. Given A Br(Y ) and (P v ) an adelic point of X, we have It follows that ν Ω k inv ν (f A(P ν )) = ν Ω k inv v (A(f(P ν ))). Y (A k ) Br(Y ) = X(A k ) Br(X) =. Now, if the set X(A k ) Br(X) is empty of course the variety will not have a rational point. Definition We will say that for a variety X the only obstruction to the Hasse principle is given by the Brauer-Manin obstruction if X(A k ) Br(X) implies X(k). This property is weaker than the Hasse principle. An other property we will study in the last chapter is the following. Definition If X is any variety and S is a finite subset of Ω k we will say the only obstruction to strong approximation off S is given by the Brauer-Manin obstruction if X(k) is dense in the image of X(A k ) Br(X) in X(A S k ). 3.3 Introduction to descent theory Hochschild-Serre and filtration of the Brauer group We recall the Hochschild-Serre spectral sequence. If X is a scheme over k E p,q 2 = H p (k, H q (X k, G m )) H p+q (X, G m ). The spectral sequence is a Grothendieck spectral sequence, with respect to the composition of the two functors: 23

26 Sh(X ét ) Γ(X, ) Ab Γ(X k, ) Sh(k ét ). M M Γ k The first functor sends injectives to injectives because it is isomorphic to π. convergence of the spectral sequence, for any H n there is a filtration In virtue of the 0 = F n+1 H n F 0 H n = H n such that E p,q F p H p+q /F p+1 H p+q. We notice that H 2 = Br(X), we will call Br 0 (X) the group F 2 H 2 and Br 1 (X) the group F 1 H 2. We have an exact sequence and 0 Br 1 (X) Br(X) E 0,2, E 0,2 E 0,2 2 = H 2 (X k, G m ) Γ k. This implies that Br 1 (X) = Ker(Br(X) Br(X k )). We also have the exact sequence 0 Br 0 (X) Br 1 (X) E 1,1 = E 1,1 3 = Ker(E 1,1 2 E 3,0 2 ). At the end we obtain the exact sequence 0 Br 0 (X) Br 1 (X) r H 1 (k, Pic(X k )) H 3 (k, G m ). 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(X k ) Γ k Br(k) Br 1 (X) r H 1 (k, Pic(X k )) H 3 (k, G m ). For any λ Hom Γk (M, Pic(X k )) we define Br λ (X) := r 1 λ (H 1 (k, M)). As is proven in ([Ser97], I.2.2,Cor. 2) Galois cohomology commutes with the colimits on the second argument, thus H 1 (k, Pic(X k )) = λ (H 1 (k, M)). So we have The main theorem Br 1 (X) = λ: M Pic(X k ) M of finite type λ: M Pic(X k ) M of finite type Br λ (X). (3.3.1) Now we have all the tools to present Colliot-Thélène and Sansuc s descent theory. This theory generalizes the classical descent for elliptic curves. The main goal is to show that for certain classes of k-varieties (k will always be a number field), the Brauer-Manin obstruction explains the failure of the Hasse principle or weak approximation. To do this we use a description of the sets X(A k ) Br λ(x) with the help of the adelic points of torsors of type λ. 24

27 Theorem 3.16 (Colliot-Thélène, Sansuc, Skorobogatov, Harari). Let k be a number field, X a smooth k-variety such that k[x] = k. Then for any λ Hom Γk (Ŝ, Pic(X k )), X(A k ) Br λ(x) = type(f,y )=λ f(y (A k )). Proof. The theorem has been proven for tori by Colliot-Thélène and Sansuc in 1979 and generalized by Skorobogatov in 1999 to groups of multiplicative type, with the additional assumption Pic(X k ) of finite type [Sko01, Theorem 6.1.1]. The Main Theorem proven in [HS10] by Harari and Skorobogatov in 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 classes of torsors Y of any given type such that Y (A k ). One can read [Sko01, Proposition 5.3.2] and use the fundamental exact sequence of Colliot-Thélène and Sansuc. The Theorem 3.16 is often used when Pic(X k ) is of finite type and λ is an isomorphism. In this situation we have X(A k ) Br 1(X) = (f,y ) universal f(y (A k )), because when λ is surjective Br λ (X) = Br 1 (X). In particular the algebraic Brauer-Manin obstruction is empty if and only if there exists a universal torsor with an adelic point. If Pic(X k ) is not of finite type, by the equality (3.3.1) we have X(A k ) Br1(X) = f(y (A k )). λ: M Pic(X k ) type(f,y )=λ M of finite type We can easily check the following corollary, recalling that the subsets X(A k ) Br λ(x) are closed in X(A k ) (Lemma 3.12). Corollary For X as in the theorem and for any λ Hom (Ŝ, Pic(X Γk k )), if the X-torsors of type λ satisfy the Hasse principle, then the only obstruction to the Hasse principle for X is the one given by Br λ (X), i.e. X(A k ) Br λ(x) X(k). Furthermore if X is proper, for any λ Hom (Ŝ, Pic(X Γk k )), if the X-torsors of type λ satisfy weak approximation, then the only obstruction to weak approximation for X is the one given by Br λ (X), i.e. X(k) = X(A k ) Brλ(X). As an application one can prove the following theorem. Using the local description of torsors (Section 2.3). Theorem Let k be a number field, X a smooth, proper k-variety that contains a k-torsor under a torus U as a dense open subset. The algebraic Brauer-Manin obstruction is the only obstruction to 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 many isomorphism classes of universal torsors. So the set of rational points is a finite disjoint union of subsets, each one parametrized by the rational points of a certain k-rational variety. 25