Pathologies of the Brauer–Manin obstruction
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1 Introduction
It had long been suspected that varieties X over a number field k without kpoints but with a nonempty Brauer–Manin set \(X({\mathbf {A}}_k)^{{\mathrm {Br}}}\) are fairly common. The first examples were found in [27] and then in [2]. An earlier, conditional example is given in [21]. One should also expect that there are many varieties X without kpoints for which the étale Brauer–Manin set \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\subset X({\mathbf {A}}_k)^{{\mathrm {Br}}}\) is nonempty (we refer to [19] or [29] for the definition of these subsets of the space \(X({\mathbf {A}}_{k})\) of adèles of X). Different methods to construct such varieties have been found recently. In [19] Poonen constructs a threefold X with a surjective morphism to a curve C that has exactly one kpoint P and the fibre \(X_P\) has points everywhere locally but not globally. In Poonen’s example \(X_P\) is a smooth Châtelet surface. The trick with a curve with just one rational point was also used in [11] where the fibres of \(X\rightarrow C\) are curves of high genus and \(X_P\) is a singular curve which geometrically is a union of projective lines. In retrospect one could note that the examples in [27] and [2] are families of genus 1 curves parameterised by elliptic curves of Mordell–Weil rank 0.

a conic bundle surface \(X\rightarrow E\) over a real quadratic field k, where E is an elliptic curve such that \(E(k)=\{0\}\), see Sect. 5.2;

a threefold over an arbitrary real number field \(k\subset {\mathbb {R}}\), which is a family \(X \rightarrow C\) of 2dimensional quadrics parameterised by a curve C with exactly one kpoint (one can choose C to be an elliptic curve), see Sect. 3.1;

a threefold over an arbitrary number field k, which is a family \(X \rightarrow C\) of geometrically rational surfaces parameterised by a curve C with exactly one kpoint, the fibre above which is singular, see Sect. 3.2.
The examples we construct in this paper show that this is no longer the case for conic or quadric bundles over curves of genus at least 1.
In a nutshell, the idea is this. Let k be a number field. Following Poonen we use a base variety B such that \(B(k)=\{P\}\). By a continuous deformation of the adèle attached to P at an archimedean component we see that B(k) is not dense in \(B({\mathbf {A}}_k)^{{\mathrm {Br}}}\). Density may also fail due to places of k that need not be archimedean. Suppose B contains an irreducible singular conic S so that \(P=S_{{\mathrm {sing}}}\). If a place v of k splits in the quadratic extension given by the discriminant of the binary quadratic form that defines S, then \(B\times _k k_v\) contains two copies of \({\mathbb {P}}^1_{k_v}\) meeting at P. Since \({\mathrm {Br}}({\mathbb {P}}^1_{k_v})={\mathrm {Br}}(k_v)\), we can modify the adèle of P at v while staying inside \(B({\mathbf {A}}_k)^{{\mathrm {Br}}}\). However, the kpoint P cannot be moved in B, so B(k) is not dense in \(B({\mathbf {A}}_k)^{{\mathrm {Br}}}\).
Next, one constructs a surjective morphism \(X\rightarrow B\) for which the fibre \(X_P\) has local points in all but one or two completions of k, and ensures that X has \(k_v\)points for missing places v such the resulting adelic point of X projects to \(B({\mathbf {A}}_k)^{{\mathrm {Br}}}\). Now, if the natural map \({\mathrm {Br}}(B)\rightarrow {\mathrm {Br}}(X)\) is surjective we have found an adelic point in \(X({\mathbf {A}}_k)^{{\mathrm {Br}}}\). But since \(X(k)\subset X_P\) we have \(X(k)=\emptyset \). With more work one can find examples such that \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\) is nonempty, too.
The paper is organised as follows. After some preparations in Sect. 2 we realise the aforementioned programme for threefolds in Sect. 3. Making it work for surfaces requires rather more effort. For this purpose in Sect. 4 we establish some Bashmakovstyle properties of elliptic curves with a large Galois image on torsion points. These properties are used in the proof of our main result in the case of surfaces in Sect. 5. Some general observations on the Brauer–Manin set are collected in Sect. 6.
The authors are grateful to Nathan Jones and Chris Wuthrich for their helpful advice on Serre curves. We used sage [30] in our calculations with elliptic curves. This work started in April 2013 when the authors were guests of the Hausdorff Institut für Mathematik (Bonn) during the special programme “Arithmetic and geometry”.
2 Brauer groups and torsors on quadric bundles
For the convenience of the reader we recall the following well known lemma.
Lemma 2.1
Let k be a field of characteristic zero. Let X be a smooth projective quadric over k of dimension at least 1. Then the natural map \({\mathrm {Br}}(k)\rightarrow {\mathrm {Br}}(X)\) is surjective. If \({\mathrm {dim}}(X)\ge 3\), then this map is an isomorphism.
Proof
In this paper a quadric bundle is a surjective flat morphism \(f : X \rightarrow B\) of smooth, projective, geometrically integral varieties over a field k, the generic fibre of which is a smooth quadric of dimension at least 1, and all geometric fibres are reduced.
We denote by k(B) the function field of B, and by \(X_{k(B)}\) the generic fibre of \(f : X \rightarrow B\). If \({\mathrm {dim}}(X_{k(B)})=1\), then \(f : X \rightarrow B\) is called a conic bundle.
The following proposition is essentially well known, at least when \(B={\mathbb {P}}^1_k\), see [26, Cor. 3.2], [6, Thm. 2.2.1, Thm. 2.3.1], [3, Prop. 2.1].
Proposition 2.2

\(\mathrm{(i)}\) \({\mathrm {dim}}(X_{k(B)})=1\) and there is a point \(P\in B\) of codimension 1 such that for each point \(Q\not =P\) of codimension 1 in B the fibre \(X_Q\) contains a geometrically integral component of multiplicity 1;

\(\mathrm{(ii)}\) \({\mathrm {dim}}(X_{k(B)})=2\) and for each point \(Q\in B\) of codimension 1 the fibre \(X_Q\) contains a geometrically integral component of multiplicity 1;

\(\mathrm{(iii)}\) \({\mathrm {dim}}(X_{k(B)})\ge 3\).
Proof
(i) Let \(\gamma \in {\mathrm {Br}}(k(B))\) be the class of the conic \(X_{k(B)}\). Since \(\gamma \) is in the kernel of the natural map \(f^*:{\mathrm {Br}}(k(B)) \rightarrow {\mathrm {Br}}(X_{k(B)})\), the assumption of (i) implies that the residue \({\mathrm {res}}_Q(\gamma )\in H^1(k(Q),{\mathbb {Q}}/{\mathbb {Z}})\) is zero if \(Q\not =P\). Take any \(\alpha \in {\mathrm {Br}}(X)\). By Lemma 2.1 the map \(f^*:{\mathrm {Br}}(k(B)) \rightarrow {\mathrm {Br}}(X_{k(B)})\) is surjective, so the image of \(\alpha \) in \({\mathrm {Br}}(X_{k(B)})\) comes from some \(\beta \in {\mathrm {Br}}(k(B))\). Again, by the assumption of (i) we have \({\mathrm {res}}_Q(\beta )=0\) if \(Q\not =P\). Moreover, \({\mathrm {res}}_P(\beta )=0\) or \({\mathrm {res}}_P(\beta )={\mathrm {res}}_P(\gamma )\). By the purity theorem for the Brauer group [10, III, Thm. 6.1, p. 134] we conclude that \(\beta \in {\mathrm {Br}}(B)\) or \(\beta \gamma \in {\mathrm {Br}}(B)\). Since \(f^*(\beta )=f^*(\beta \gamma )=\alpha \) in \({\mathrm {Br}}(X_{k(B)})\), and the natural map \({\mathrm {Br}}(X)\rightarrow {\mathrm {Br}}(X_{k(B)})\) is injective, we have proved (i).
The proof of (ii) uses Lemma 2.1 and the arguments from the proof of (i).
In case (iii) it is well known that each fibre of f at a point of B of codimension 1 contains a geometrically integral component of multiplicity 1. Then (iii) follows from the last statement of Lemma 2.1. \(\square \)
Proposition 2.3
Let \(f : X \rightarrow B\) be a quadric bundle over a field k of characteristic zero. Then any torsor \(X' \rightarrow X\) of a finite kgroup scheme G is the inverse image under f of a torsor \(B' \rightarrow B\) of G.
Proof
By our definition of quadric bundles, the morphism f is flat and all its geometric fibres are connected and reduced. The generic geometric fibre of f is simply connected. By [25, X, Cor. 2.4] this implies that each geometric fibre of such a fibration is simply connected. The result then follows from [25, IX, Cor. 6.8]. \(\square \)
3 Threefolds
3.1 Example based on real deformation
Let k be a number field with a real place. We fix a real place v, so we can think of k as a subfield of \(k_v={\mathbb {R}}\).
Let C be a smooth, projective, geometrically integral curve over k such that C(k) consists of just one point, \(C(k)=\{P\}\). By [20] such a curve exists for any number field k, and by [18, Thm. 1.1] we can take C to be an elliptic curve over k. Let \(\Pi \subset C({\mathbb {R}})\) be an open interval containing P. Let \(f:C \rightarrow {\mathbb {P}}^1_k\) be a surjective morphism that is unramified at P. Choose a coordinate function t on \({\mathbb {A}}^1_k={\mathbb {P}}^1_k{\setminus } f(P)\) such that f is unramified above \(t=0\). We have \(f(P)=\infty \). Take any \(a>0\) in k such that a is an interior point of the interval \(f(\Pi )\) and f is unramified above \(t=a\).
Let w be a finite place of k. There exists a quadratic form \(Q(x_0,x_1,x_2)\) of rank 3 that represents zero in all completions of k other than \(k_v\) and \(k_w\), but not in \(k_v\) or \(k_w\). We can assume that Q is positive definite over \(k_v={\mathbb {R}}\). Choose \(n \in k\) with \(n>0\) in \(k_{v}\) and \(nQ(1,0,0)\in k_w^{*2}\). Let \(Y_1\subset {\mathbb {P}}^3_k\times {\mathbb {A}}^1_k\) be given by \(Q(x_0,x_1,x_2)+nt(ta)x_3^2=0\), and let \(Y_2\subset {\mathbb {P}}^3_k\times {\mathbb {A}}^1_k\) be given by \(Q(X_0,X_1,X_2)+n(1aT)X_3^2=0\). We glue \(Y_1\) and \(Y_2\) by identifying \(T=t^{1}\), \(X_3=tx_3\), and \(X_i=x_i\) for \(i=0,1,2\). This produces a quadric bundle \(Y\rightarrow {\mathbb {P}}^1_k\) with exactly two degenerate fibres (over \(t=a\) and \(t=0\)), each given by the quadratic form \(Q(x_0,x_1,x_2)\) of rank 3. Define \(X=Y\times _{{\mathbb {P}}^1_k} C\). This is a quadric bundle \(X\rightarrow C\) with geometrically integral fibres.
Proposition 3.1
In the above notation we have \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\not =\emptyset \) and \(X(k)=\emptyset \).
Proof
Since \(C(k)=\{P\}\) we have \(X(k)\subset X_P\). The fibre \(X_P\) is the smooth quadric \(Q(x_0,x_1,x_2)+nx_3^2=0\). This quadratic form is positive definite thus \(X_P\) has no points in \(k_v={\mathbb {R}}\) and so \(X(k)=\emptyset \). By assumption \(X_P\) has local points in all completions of k other than \(k_v\) and \(k_w\). The condition \(nQ(1,0,0)\in k_w^{*2}\) implies that \(X_P\) contains \(k_w\)points, so \(X_P\) has local points in all completions of k but one. Choose \(N_u\in X_P(k_u)\) for each place \(u\not =v\). Consider a small real \(\varepsilon >0\) such that \(a\varepsilon \in f(\Pi )\) and \(\varepsilon <a\). Let \(M\in \Pi \) be such that \(f(M)=a\varepsilon \). Then the smooth real fibre \(X_M\) is given by an indefinite quadratic form and so \(X_M(k_v)\not =\emptyset \). Choose any \(N_v \in X_M(k_v)\). We now have an adelic point \((N_u)\), where we allow \(u=v\).
We claim that \((N_u)\in X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\).
Let G be a finite kgroup scheme. Proposition 2.3 implies that any torsor \(X'/X\) of G comes from a torsor \(C'/C\) of G, in the sense that \(X\times _C C'\rightarrow X\) and \(X'\rightarrow X\) are isomorphic as Xtorsors with the structure group G. Let \(\sigma \in Z^1(k,G)\) be a 1cocycle defining the ktorsor which is the fibre of \(C' \rightarrow C\) at P. Twisting \(X'/X\) and \(C'/C\) by \(\sigma \) and replacing the group G by the twisted group \(G^{\sigma }\) and changing notation, we can assume that \(C'\) contains a kpoint \(P'\) that maps to P in C. The irreducible component \(C''\) of \(C'\) that contains \(P'\) is a geometrically integral curve over k. Let \(X'' \subset X'\) denote the inverse image of \(C''\) in \(X'\). The fibres of the morphism \(X\rightarrow C\) are geometrically integral, hence such are also the fibres of \(X' \rightarrow C'\) and \(X'' \rightarrow C''\). Thus \(X''\) is a geometrically integral variety over k.
There are natural isomorphisms \(X''_{P' } \cong X'_{P'} \cong X_P\), so we can define \(N'_u\in X''(k_u)\) as the point that maps to \(N_u\in X(k_u)\) for each \(u\not =v\). The map \(C'' \rightarrow C\) is finite and étale. The image of \(C''({\mathbb {R}})\) in \(C({\mathbb {R}})\) is thus closed and open. The image of the connected component of \(P' \in C''({\mathbb {R}})\) is the whole connected component of \(P \in C({\mathbb {R}})\), hence contains \(\Pi \). The inverse image of the interval \(\Pi \) in \(C''({\mathbb {R}})\) is a disjoint union of intervals, one of which contains \(P'\) and maps bijectively onto \(\Pi \). Let us call this interval \(\Pi '\). Let \(M'\) be the unique point of \(\Pi '\) over M. Let \(N'_v\in X''_{M'}({\mathbb {R}})\) be the point that maps to \(N_v\in X_M({\mathbb {R}})\). Thus the adelic point \((N'_u)\in X''({\mathbf {A}}_k)\subset X'({\mathbf {A}}_k)\) projects to the adelic point \((N_u)\in X({\mathbf {A}}_k)\).
By the definition of the étale Brauer–Manin obstruction, to prove that \((N_u)\) is contained in \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\) it suffices to show that \((N'_u)\) is orthogonal to \({\mathrm {Br}}(X')\). For this it is enough to show that \((N'_u)\) is orthogonal to \({\mathrm {Br}}(X'')\). By Proposition 2.2 (ii) applied to \(X'' \rightarrow C''\) we know that the natural map \({\mathrm {Br}}(C'') \rightarrow {\mathrm {Br}}(X'')\) is surjective. Thus it is enough to show that the adèle on \(C''\) such that its uadic component is \(P'\) when \(u\not =v\) and and its vcomponent is \(M'\), is orthogonal to \({\mathrm {Br}}(C'')\). The real point \(M'\) is pathconnected to \(P'\), so this adèle is in the connected component of the diagonal image of the kpoint \(P'\) in \(C''({\mathbf {A}}_k)\). But the latter adèle is certainly in \(C''({\mathbf {A}}_k)^{\mathrm {Br}}\), and the proposition follows. \(\square \)
Remark 3.2
 (1)
Our method gives simple examples of threefolds with points everywhere locally but not globally and no Brauer–Manin obstruction. An even simpler proof is available in the case of fibrations into quadrics of dimension at least 3 over a curve.
 (2)
By a theorem of Wittenberg [32, Thm. 1.3] the variety X has a 0cycle of degree 1 over k, that is, there exist field extensions \(k_1,\ldots ,k_r\) of k whose degrees have no common factor such that \(X(k_i)\not =\emptyset \) for \(i=1,\ldots , r\). Although [32, Thm. 1.3] requires the finiteness of the Shafarevich–Tate group of the Jacobian of C, Wittenberg pointed out that in the proof of his theorem this assumption is only used to ensure the existence of a suitable 0cycle of degree 1 on C. In our case such a 0cycle is directly provided by the kpoint P, so the assumption on the Shafarevich–Tate group is not needed. For more details see Remark 5.7 below.
3.2 Examples based on deformation along a rational curve defined over a completion of k
Lemma 3.3

B contains a curve S isomorphic to an irreducible singular projective conic;

the singular point of S is the unique kpoint of B;

there is a surjective morphism \(\pi :B\rightarrow {\mathbb {P}}^1_k\) with smooth and geometrically integral generic fibre such that \(\pi (S)={\mathbb {P}}^1_k\).
Proof
According to [20] there is a smooth, projective and geometrically integral curve C over k with exactly one kpoint, \(C(k)=\{O\}\). Moreover, by [18, Thm 1.1] there is an elliptic curve over k with this property. Let \(f:Z\rightarrow C\) be any conic bundle such that the fibre \(Z_O\) over \(O\in C(k)\) is an irreducible singular conic. The singular point of \(S=Z_O\) is then the unique kpoint of Z.
There is a closed embedding \(Z\subset {\mathbb {P}}^n_k\) for some \(n\ge 1\). By the Bertini theorem [12, II.8.18, III.7.9], there exists a hyperplane \(H_1\subset {\mathbb {P}}^n_k\) such that \(Z\cap H_1\) is a smooth and geometrically integral curve. This implies that S is not a subset of \(H_1\).
Let d be the degree of Z in \({\mathbb {P}}^n_k\). We can find a hyperplane \(H_2\subset {\mathbb {P}}^n_k\) such that \(Z\cap H_1\cap H_2\) is a set of d distinct \(\bar{k}\)points not in S. It follows that no geometric irreducible component of S is contained in a hyperplane passing through \(H_1\cap H_2\).
Let \(\tilde{{\mathbb {P}}}^n_k\) be the blowingup of \({\mathbb {P}}^n_k\) at \(H_1\cap H_2\simeq {\mathbb {P}}^{n2}_k\). The projection from \(H_1\cap H_2\) defines a morphism \(\tilde{{\mathbb {P}}}^n_k \rightarrow {\mathbb {P}}^1_k\). Let B be the Zariski closure of \(Z{\setminus } (Z\cap H_1\cap H_2)\) in \(\tilde{{\mathbb {P}}}^n_k\), so that B is the blowingup of Z in d distinct points. Thus B is a smooth, projective, geometrically integral surface with a unique kpoint, equipped with a surjective morphism \(\pi :B\rightarrow {\mathbb {P}}^1_k\) with smooth and geometrically integral generic fibre. Moreover, S is contained in B, and \(\pi (S)={\mathbb {P}}^1_k\). \(\square \)
Let P be the unique kpoint of B, and let \(Q=\pi (P)\in {\mathbb {P}}_k^1(k)\). Let K be the quadratic extension of k over which the components of S are defined. If w is a place of k that splits in K, then the \(k_w\)variety \(S\times _k k_w\) is the union of two projective lines meeting at P. Let \(L_w\subset B\times _k k_w\) be one of these rational curves. Since \(\pi (L_w)={\mathbb {P}}^1_{k_w}\), there is a point \(N_w\in L_w(k_w)\) such that \(\pi (N_w)\not =Q\).
Proposition 3.4

there exists a closed point \(R\in {\mathbb {P}}^1_k\), \(R \ne Q\), such that the restriction \(Y{\setminus } Y_R \rightarrow {\mathbb {P}}^1_k{\setminus } R\) is a smooth morphism, and the fibre of \(\pi :B\rightarrow {\mathbb {P}}^1_k\) at R is smooth;
 the fibre \(Y_Q\) is a smooth conic that has \(k_v\)points for all completions of k except \(w_1\) and \(w_2\), in particular \(Y_Q(k)=\emptyset \);$$\begin{aligned} Y_{\pi (N_w)}(k_{w})\not =\emptyset \quad \hbox { for }w=w_1, w_2. \end{aligned}$$
Proof
Let \(p:X\rightarrow B\) be the natural projection. Since \(B(k)=\{P\}\), \(\pi (P)=Q\) and \(Y_Q(k)=\emptyset \), we see that \(X(k)=\emptyset \).
The fibre \(X_P\) is naturally isomorphic to \(Y_Q\). For a place v such that \(v\not =w_1\), \(v\not =w_2\) choose \(M_v\in X_P(k_v)\). For \(w=w_1,w_2\), choose a \(k_{w}\)point \(M_{w}\) in the \(k_{w}\)fibre \(X_{N_{w}}\) (which is isomorphic to \(Y_{\pi (N_{w})}\)). We claim that \((M_v)\in X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\).
One easily checks that the projection map \(X \rightarrow B\) is a quadric bundle as defined in this paper: both X and B are smooth and projective over k, the generic fibre is a geometrically integral conic, the morphism \(X \rightarrow B\) is flat and all its geometric fibres are reduced. Let G be a finite kgroup scheme. By Proposition 2.3 every torsor \(X'\rightarrow X\) of G is the pullback of a torsor \(B'\rightarrow B\) of G. After a twist by a ktorsor of G, as detailed in the proof of Proposition 3.1, we may assume that \(B'\) has a kpoint \(P'\) over P. Let \(B''\) be the irreducible component of \(B'\) that contains \(P'\). Then \(B''\) is geometrically integral. Let \(X''\) be the inverse image of \(B''\) under the map \(X'\rightarrow B'\). The kvariety \(X''\) is geometrically integral, and \(X''\rightarrow B''\) is a conic bundle.
For all \(v\not =w_1,w_2\) the fibre \(X''_{P'}\) contains a \(k_v\)point \(M'_v\) that maps to \(M_v\) in \(X_P\). Since \(B''\rightarrow B\) is finite and étale, for \(w=w_1,w_2\) the inverse image of \(L_{w}\) in \(B''\) contains a rational \(k_{w}\)curve \(L'_{w}\) through \(P'\) that maps isomorphically to \(L_{w}\). Let \(N'_{w}\) be the \(k_{w}\)point of \(L'_{w}\) that maps to \(N_{w}\). Then there is a \(k_{w}\)point \(M'_{w}\) in \(X''_{N'_{w}}\) that maps to \(M_{w}\in X_{N_{w}}(k_{w})\). To prove our claim it is enough to show that the adelic point \((M'_v)\) in \(X''\) is orthogonal to \({\mathrm {Br}}(X'')\).
Let \(\bar{x}\in R\) be a \(\bar{k}\)point. Since the singular loci of the morphisms \(\overline{Y}\rightarrow {\mathbb {P}}^1_{\bar{k}}\) and \(\pi :\overline{B}\rightarrow {\mathbb {P}}^1_{\bar{k}}\) are disjoint, the fibre \(\overline{B}_{\bar{x}}=\pi ^{1}(\bar{x})\) is a smooth integral curve over \(\bar{k}\). The fundamental group of \({\mathbb {P}}^1_{\bar{k}}\) is trivial. Therefore, by [25, Chap. X, Cor. 1.4], the map \(\pi _1(\overline{B}_{\bar{x}})\rightarrow \pi _1(\overline{B})\) is surjective. Since \(\overline{B}''\) is integral, this implies that \(B''\times _B B_{\bar{x}}\) is also integral. As a consequence, the inverse image of R in \(B''\) is a geometrically integral k(R)curve D. Hence D is integral.
Since the restriction of the conic bundle \(X''\rightarrow B''\) to the complement \(B''\setminus D\) of the integral curve D is a smooth morphism, this conic bundle satisfies the condition of Proposition 2.2 (i). It follows that the induced map \({\mathrm {Br}}(B'') \rightarrow {\mathrm {Br}}(X'')\) is surjective. Thus it remains to prove that the image of \((M'_v)\) in \(B''\) is orthogonal to \({\mathrm {Br}}(B'')\).
This image is the adelic point such that for all \(v\not =w_1,w_2\) the \(k_v\)component is \(P'\). For \(w=w_1, w_2\) the \(k_w\)component is \(N'_w\). But \(L'_w\) is a projective line over \(k_w\) containing both \(P'\) and \(N'_w\). The natural map \({\mathrm {Br}}(k_w)\rightarrow {\mathrm {Br}}(L'_w)\) is an isomorphism. Thus when pairing the image of \((M'_v)\) in \(B''\) with \({\mathrm {Br}}(B'')\) we may replace \((M'_v)\) by the diagonal adèle \((P')\), which by the reciprocity law is orthogonal to \({\mathrm {Br}}(B'')\). \(\square \)
Remark 3.5
Consider a surface B with a morphism \(\pi :B\rightarrow {\mathbb {P}}^1_k\) as in Lemma 3.3. Let \(Y \rightarrow {\mathbb {P}}^1_k\) be a quadric bundle all fibres of which are of dimension \(d\ge 2\) and contain a geometrically integral component of multiplicity one (which is automatic if \(d \ge 3\)). Assume that the singular loci of \(B\rightarrow {\mathbb {P}}^1_k\) and \(Y\rightarrow {\mathbb {P}}^1_k\) do not intersect. Let \(X=Y \times _{{\mathbb {P}}^1_k}B\). Suppose that the fibre \(Y_Q\), where \(Q=\pi (P)\in {\mathbb {P}}^1(k)\), is a smooth quadric such that \(Y_Q(k_v)\not =\emptyset \) for all \(v\not =w\) but \(Y_Q(k_w)=\emptyset \) (such quadrics exist in dimension 2 and higher, but not in dimension 1). Assume that \(L_w\) has a nonempty intersection with the image of \(X(k_w) \rightarrow B(k_w)\). In view of Proposition 2.2 (ii), (iii), an argument similar but shorter than the one above shows that \(X(k)=\emptyset \) and \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\not =\emptyset \). In this example \({\mathrm {dim}}(X)=2+d\ge 4\).
Example 3.6
Let \(i:X_P\rightarrow X_O\) be the closed embedding. Let \(\nu :Y_K\rightarrow X_O\) be the normalisation morphism and let \(C=\nu ^{1}(X_P)\). Then \(C=X_P\times _{\mathbb {Q}}K\) is the intersection of the quadric \(Q_K=Q\times _{\mathbb {Q}}K\) given by the second equation in (1) with the plane \(u=0\). The morphism \(\nu :C\rightarrow X_P\) is the natural projection \(X_P\times _{\mathbb {Q}}K\rightarrow X_P\).
To conclude, our example resembles that of Poonen [19] in that the fibres of \(X\rightarrow E\) are birationally equivalent to intersections of two quadrics in \({\mathbb {P}}^4\). However, in our case the fibre above the unique \({\mathbb {Q}}\)point is geometrically simply connected and satisfies \({\mathrm {Br}}(X_O)={\mathrm {Br}}({\mathbb {Q}})\) and \(X_O({\mathbf {A}}_{\mathbb {Q}})\not =\emptyset \), so we see that the étale Brauer–Manin set of X is nonempty without studying X any further. Note that it is essential that the fibres of \(X\rightarrow E\) have dimension at least 2. Indeed, each everywhere locally solvable geometrically connected and simply connected curve over a number field k has a kpoint [11, Remark 2.2].
4 Elliptic curves with a large Galois image
Let k be a field of characteristic zero, with an algebraic closure \(\bar{k}\). For a field K such that \(k\subset K\subset \bar{k}\) we denote by \(\Gamma _K\) the Galois group \({\mathrm {Gal}}(\bar{k}/K)\). Let \(k^{{\mathrm {cyc}}}\subset \bar{k}\) be the cyclotomic extension of k, i.e. the abelian extension of k obtained by adjoining all roots of unity.
Let E be an elliptic curve over k, and let \(\rho :\Gamma _k\rightarrow {\mathrm {GL}}_2(\hat{{\mathbb {Z}}})\) be the Galois representation in the torsion subgroup of E. The group \({\mathrm {SL}}_2({\mathbb {Z}}/2)\) is isomorphic to the symmetric group \(S_3\). Let us define \({\mathrm {SL}}_2^+(\hat{{\mathbb {Z}}})\) as the kernel of the composition of the reduction modulo 2 map \({\mathrm {SL}}_2(\hat{{\mathbb {Z}}})\rightarrow {\mathrm {GL}}_2({\mathbb {Z}}/2)\) with the unique nontrivial homomorphism \(\varepsilon :{\mathrm {GL}}_2({\mathbb {Z}}/2)\rightarrow \{\pm 1\}\). (The finite group \({\mathrm {GL}}_2({\mathbb {Z}}/2)\) can be identified with the symmetric group \(S_3\), and then \(\varepsilon \) is the signature character.)
In this section we prove the following theorem using methods that go back to Bashmakov [1, Ch. 5].
Theorem 4.1
Let E be an elliptic curve over a field k of characteristic zero such that \({\mathrm {SL}}_2^+(\hat{{\mathbb {Z}}})\) is a subgroup of \(\rho (\Gamma _k)\subset {\mathrm {GL}}_2(\hat{{\mathbb {Z}}})\). Let K be a field such that \(k\subset K\subset k^{{\mathrm {cyc}}}\), and let \(\varphi :E'\rightarrow E\times _kK\) be an isogeny of elliptic curves over K. Then for any point \(P\in E(K)\) that cannot be written as \(P=mQ\) with \(m>1\) and \(Q\in E(K)\), the scheme \(\varphi ^{1}(P)\) is integral.
In the assumption of the theorem, Lemma 4.7 (b) below shows that the isogeny \(\varphi :E'\rightarrow E\) can be identified with the multiplication by n map \(E\rightarrow E\) for some integer n. The theorem then follows from Proposition 4.8.
Serre [24, Prop. 22] proved that for any elliptic curve E over \({\mathbb {Q}}\) the image \(\rho (\Gamma _{\mathbb {Q}})\) is contained in a certain subgroup \(H_\Delta \) which only depends on the discriminant \(\Delta \) of E. The group \(H_\Delta \) has index 2 in \({\mathrm {GL}}_2(\hat{{\mathbb {Z}}})\) and contains \({\mathrm {SL}}_2^+(\hat{{\mathbb {Z}}})\). N. Jones [14] showed that almost all elliptic curves over \({\mathbb {Q}}\) are Serre curves which means that \(\rho (\Gamma _{\mathbb {Q}})=H_\Delta \). Theorem 4.1 thus applies to almost all elliptic curves E over \({\mathbb {Q}}\).
4.1 Group cohomology
Proposition 4.2
Let n be a positive integer, and let \(G\subset {\mathrm {GL}}_{2}({\mathbb {Z}}/n)\) be a subgroup containing \({\mathrm {SL}}_{2}^+({\mathbb {Z}}/n)\). If n is odd, then \(H^1(G, ({\mathbb {Z}}/n)^2)=0\). For \(n=2^rm\), where m is odd and \(r\ge 1\), the abelian group \(H^1(G, ({\mathbb {Z}}/n)^2)\) is annihilated by \(2^{r1}\).
The proof of Proposition 4.2 is based on a few lemmas.
Lemma 4.3
For any integer \(n>1\) we have \(H^0({\mathrm {SL}}_{2}^+({\mathbb {Z}}/n), ({\mathbb {Z}}/n)^2)=0\).
Proof
The group \({\mathrm {SL}}_{2}^+({\mathbb {Z}}/n)\) contains the transformation \((x,y) \mapsto (x+y,x)\). \(\square \)
Lemma 4.4
Proof
Lemma 4.5
For an odd prime p we have \(H^1({\mathrm {SL}}_{2}({\mathbb {Z}}/p^r), ({\mathbb {Z}}/p^r)^2)=0\) for any positive integer r.
Proof
Lemma 4.6
For a positive integer r the group \(H^1({\mathrm {SL}}_2^+({\mathbb {Z}}/2^r), ({\mathbb {Z}}/2^r)^2)\) is annihilated by \(2^{r1}\).
Proof
When \(r=1\) the group \({\mathrm {SL}}_2^+({\mathbb {Z}}/2^r)\) has order 3, so the claim is obvious.
Proof of Proposition 4.2
4.2 Isogenies of elliptic curves
The multiplication by n map on an elliptic curve E is denoted by \([n]:E\rightarrow E\). Let \(\rho _n:\Gamma _k\rightarrow {\mathrm {GL}}_2({\mathbb {Z}}/n)\) be the Galois representation in the ntorsion subgroup E[n].
Lemma 4.7

(i) Let M be a finite \(\Gamma _k\)submodule of \(E(\bar{k})\). Then \(M=E[m]\) for an integer \(m\not =0\). In particular E(k) is torsionfree.

(ii) Let \(\varphi :C\rightarrow E\) be an isogeny of elliptic curves. Then there is an integer \(n>0\) and an isomorphism of elliptic curves \(\psi :E\tilde{\longrightarrow } C\) such that \(\varphi \circ \psi =[n]\).
Proof

(i) Let n be the smallest positive integer such that \([n]M=0\). We claim that \(M=E[n]\). The \(\Gamma _k\)module M is the direct sum of its \(\ell \)primary components \(M\{\ell \}\). If \(\ell ^r\) is the highest power of the prime number \(\ell \) that divides n, then \(\ell ^r\) is the smallest positive integer that annihilates \(M\{\ell \}\). The tautological \({\mathrm {SL}}_2^+({\mathbb {Z}}/\ell )\)module \(({\mathbb {Z}}/\ell )^2\) is simple, hence the \(\Gamma _k\)module \(E[\ell ]\) is simple by assumption. But \(\ell ^{r1}M\{\ell \}\not =0\), so we have \(\ell ^{r1}M\{\ell \}=E[\ell ]\). The abelian group \(E[\ell ^r]\) is isomorphic to \(({\mathbb {Z}}/\ell ^r{\mathbb {Z}})^2\), so any subgroup that is mapped by multiplication by \(\ell ^{r1}\) onto \((\ell ^{r1}{\mathbb {Z}}/\ell ^r{\mathbb {Z}})^2\) is equal to the whole group. Thus \(M\{\ell \}=E[\ell ^r]\), and hence \(M=E[n]\).

(ii) Passing to the dual isogeny of \(\varphi :C\rightarrow E\) we see that it is enough to prove that every isogeny \(\alpha :E\rightarrow E'\) is \([m]:E\rightarrow E\) for some integer m. This follows from (i) by taking \(M={\mathrm {Ker}}(\alpha )\). \(\square \)
Proposition 4.8

(i) \({\mathrm {SL}}_2^+(\hat{{\mathbb {Z}}})\subset \rho (\Gamma _K)\), hence \({\mathrm {SL}}_2^+({\mathbb {Z}}/n)\subset \rho _n(\Gamma _K)\) for any positive integer n.

(ii) For any point \(P\in E(K)\) that cannot be written as \(P=mQ\) for \(m>1\) with \(Q\in E(K)\), and for any integer \(n>0\), the scheme \([n]^{1}(P)\) is integral.
Proof

(i) The composition \(\det \rho :\Gamma _k\rightarrow {\mathrm {GL}}_{2}(\hat{{\mathbb {Z}}})\rightarrow \hat{{\mathbb {Z}}}^*\) is the cyclotomic character, so \(\Gamma _{k^{{\mathrm {cyc}}}}\) is the subgroup of \(\Gamma _k\) given by the condition \(\det \rho (x)=1\). In view of the natural surjections \({\mathrm {SL}}_2(\hat{{\mathbb {Z}}})\rightarrow {\mathrm {SL}}_{2}({\mathbb {Z}}/n)\), this proves (i).
 (ii) Let \(K_n=K(E[n])\). In (i) we proved that \(G=\Gamma _K/\Gamma _{K_n}\) is a subgroup of \({\mathrm {GL}}_2({\mathbb {Z}}/n)\) containing \({\mathrm {SL}}_2^+({\mathbb {Z}}/n)\). The scheme \([n]^{1}(P)\) is a Ktorsor of E[n] of the class \(\kappa (P)\in H^1(K,E[n])\), where \(\kappa :E(K)/n\hookrightarrow H^1(K,E[n])\) is the Kummer map. There is an inflationrestriction exact sequence$$\begin{aligned} 0\rightarrow H^1\left( G,E[n]\right) \rightarrow H^1\left( K,E[n]\right) \rightarrow H^1 \left( K_n,E[n]\right) ^G. \end{aligned}$$(2)
Write \(n=2^r s\), where \(r\ge 0\) and s is odd. We first deal with the case \(r=0\), and then proceed by induction in r. Since P is not divisible in E(K) by assumption, and E(K) is torsionfree, the order of \(\kappa (P) \in H^1(K,E[n])\) is n.
If n is odd, by Proposition 4.2 the exact sequence (2) gives rise to an embedding of \(H^1(K,E[n])\) into \({\mathrm {Hom}}(\Gamma _{K_n},E[n])\), so that \(\varphi \) has order n. It follows that \(\varphi (\Gamma _{K_n})=E[n]\) in this case. This implies that \([n]^{1}(P)\times _K K_n\) is irreducible, hence \([n]^{1}(P)\) is irreducible.
Now suppose that \(n=2n'\) and the scheme \([n']^{1}(P)\) is irreducible. The multiplication by 2 map defines a surjective morphism \([n]^{1}(P)\rightarrow [n']^{1}(P)\) which is a torsor of E[2]. We know that \(\Gamma _K\) acts transitively on \([n']^{1}(P)(\overline{K})\) and we want to show that \(\Gamma _K\) acts transitively on \([n]^{1}(P)(\overline{K})\). For this we must show that each \(\overline{K}\)fibre of \([n]^{1}(P)\rightarrow [n']^{1}(P)\) is contained in one \(\Gamma _K\)orbit. Recall that the \(\Gamma _{K_n}\)set \([n]^{1}(P)(\overline{K})\) is identified with E[n] so that \(g\in \Gamma _{K_n}\) acts as the translation by \(\varphi (g)\). The \(\overline{K}\)fibres of \([n]^{1}(P)\rightarrow [n']^{1}(P)\) are the E[2]orbits in E[n]. Therefore, it is enough to show that \(\varphi (\Gamma _{K_n})\) contains E[2]. As the order of \(\kappa (P)\) is n, by Proposition 4.2 the exact sequence (2) shows that the order of \(\varphi \) is divisible by 2s. Thus \(\varphi (\Gamma _{K_n})\) contains E[2s] and hence contains E[2]. This finishes the proof. \(\square \)
5 Surfaces
5.1 An elliptic curve
For an elliptic curve E over a field k of characteristic zero we denote by \(E^c\) the quadratic twist of E by \(c\in k^*\).
Lemma 5.1
Proof
Let \(\sigma \) be the nonzero element of \({\mathrm {Gal}}(K/k)\). We have \(E(k)\simeq E(K)^\sigma \). The choice of a square root of d in K defines an isomorphism \(E^d\times _k K\simeq E\times _k K\). This gives an identification \(E^d(k)\simeq \{x\in E(K)\sigma (x)=x\}\). Sending \(x\in E(K)\) to \(x+\sigma (x)\) defines a homomorphism \(E(K)\rightarrow E(k)\) with kernel \(E^d(k)\). \(\square \)
Proposition 5.2

a real quadratic field \(k={\mathbb {Q}}(\sqrt{c})\), where c is a squarefree positive integer not congruent to 1 modulo 8;

a totally real biquadratic field \(K={\mathbb {Q}}(\sqrt{c},\sqrt{d})\), where d is a squarefree positive integer;

an elliptic curve E over \({\mathbb {Q}}\) of discriminant \(\Delta <0\), such that \({\mathrm {SL}}_2^+(\hat{{\mathbb {Z}}})\subset \rho (\Gamma _k)\), \(E(k)=\{0\}\), and E(K) is torsionfree of positive rank.
Proof

\(\rho _\ell (\Gamma _{\mathbb {Q}})={\mathrm {GL}}_2({\mathbb {Z}}/\ell )\) for all primes \(\ell \) (this is checked using sage),

\(\rho _8(\Gamma _{\mathbb {Q}})={\mathrm {GL}}_2({\mathbb {Z}}/8)\) (this is checked using [7]), and

\(\rho _9(\Gamma _{\mathbb {Q}})={\mathrm {GL}}_2({\mathbb {Z}}/9)\) (this is checked using [8]).
Remark 5.3
For c and \(k={\mathbb {Q}}(\sqrt{c})\) as above, the conic \(x^2+y^2+z^2=0\) has points in all nonArchimedean completions of k, but no points in the two real completions of k. Indeed, this conic has \({\mathbb {Q}}_p\)points for all odd primes p. Since 2 is ramified or inert in k, this conic also has points in the completion of k at the unique prime over 2.
5.2 A conic bundle over the elliptic curve
Let \(P\in E(K)\) be a point not divisible in E(K). Let \(\sigma \in {\mathrm {Gal}}(K/k)\) be the generator. Since \(P+\sigma (P)\in E(k)=0\) we obtain that \(\pi (P)\) is a point of \({\mathbb {A}}^1_k(k)=k\), say \(\pi (P)=a \in k\). The Kpoint P gives rise to a solution of \(y^2=r(a)\) in K, hence \(r(a) \in k\) is totally positive. We have \(r(a)\not =0\) since E(K) is torsionfree.
Proposition 5.4
(Albert) Let F be a field, \(char(F)\ne 2\), and let \(\alpha ,\beta ,\gamma ,\delta \in F^*\). The tensor product of quaternion algebras \((\alpha ,\beta ) \otimes (\gamma ,\delta )\) is a division algebra if and only if the diagonal quadratic form \(\langle \alpha ,\beta ,\alpha \beta ,\gamma ,\delta ,\gamma \delta \rangle \) is anisotropic. If it is isotropic, then \((\alpha ,\beta ) \otimes (\gamma ,\delta )\) is similar to a quaternion algebra over F.
For the proof see [15, § 16.A]. The quadratic form \(\langle \alpha ,\beta ,\alpha \beta ,\gamma ,\delta ,\gamma \delta \rangle \) is called an Albert form associated to \((\alpha ,\beta ) \otimes (\gamma ,\delta )\).
Lemma 5.5
Let k, K, a, r(t) be as above. If \(b \in k\) is such that r(b) is totally negative, then the algebra A over the field \(k({\mathbb {P}}_k^1)=k(x)\) is similar to a quaternion algebra.
Proof
The associated Albert form contains the subform \(\Phi =\langle r(b),1,1,1\rangle \). By Remark 5.3 the form \(\langle 1,1,1\rangle \) is isotropic over all finite completions of k. Since r(b) is totally negative, \(\Phi \) is isotropic over both real completions of k. By the Hasse–Minkowski theorem the quadratic form \(\Phi \) is isotropic over k. An application of Proposition 5.4 concludes the proof. \(\square \)
We are now ready to state one of the main results of this paper. (For an explanation why we do not consider the case \(k={\mathbb {Q}}\) see Proposition 6.5 below.)
Theorem 5.6

(i) the fibres of \(f :X\rightarrow E\) are conics;

(ii) there exists a closed point \(P\in E\) such that the field k(P) is a totally real biquadratic extension of \({\mathbb {Q}}\) and the restriction \(X{\setminus } f^{1}(P)\rightarrow E{\setminus } P\) is a smooth morphism;

(iii) \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\not =\emptyset \) and \(X(k)=\emptyset \).
Proof
We keep the above notation: \(k={\mathbb {Q}}(\sqrt{10})\), E is the curve of Proposition 5.2 given by its short Weierstraß equation \(y^2=r(x)\), viewed as a curve over k, and \(K=k(P)={\mathbb {Q}}(\sqrt{10},\sqrt{2})\). We fix \(b \in k\) such that r(b) is totally negative, and define a central simple algebra A by (3). By Lemma 5.5, A is similar to a quaternion algebra B over \(k({\mathbb {P}}_k^1)\). Let \(f :X \rightarrow E\) be a relatively minimal conic bundle such that the generic fibre \(X_{k(E)}\) is the conic over k(E) defined by the quaternion algebra \(B\otimes _{k({\mathbb {P}}^1)}k(E)\). (We call a conic bundle \(X\rightarrow E\) relatively minimal if for every conic bundle \(Y\rightarrow E\) any birational morphism \(X\rightarrow Y\) compatible with the projections to E, is an isomorphism. See [16, Thm. 1.6] for the well known description of the fibres of \(X\rightarrow E\).) The closed point \(P=\pi ^{1}(a)\simeq {\mathrm {Spec}}\,K\) is a solution to \(y^2=r(a)\). The residue of \(B\otimes _{k({\mathbb {P}}^1)}k(E)\) at P is the class of r(b) in \(K^{\times }/K^{\times 2}\), which is nontrivial since r(b) is totally negative. We have \(\pi ^{1}(b)={\mathrm {Spec}}\,k(\sqrt{r(b)})\), and the residue at this closed point is the class of r(b) in \(k(\sqrt{r(b)})^*/k(\sqrt{r(b)})^{*2}\) which is trivial. Thus \(f :X \rightarrow E\) is smooth away from P. This gives (i) and (ii).
If we go over to one of the two real completions \(k_v\) of k, the point P breaks up into two real points \(P_{1}\) and \(P_{2}\). The residue at each of these points is an element of \({\mathbb {R}}^*/{\mathbb {R}}^{*2}\) given by the image of r(b), and that element is totally negative. Thus the algebra A is ramified at both \(P_1\) and \(P_2\). The fibre of \(X \rightarrow E\) above each point \(P_{i} \in E(k_{v})\) is thus a singular conic, hence has a \(k_{v}\)point. In particular, \(X(k_v) \ne \emptyset \) for each of the real completions of k. The condition \(\Delta <0\) implies that \(E(k_v)\) is connected, thus any point in the image of \(X(k_v) \rightarrow E(k_v)\) is path connected to the point 0, which is the unique point of E(k). (In fact, the image of \(X({\mathbb {R}})\) in \(E({\mathbb {R}})\) is the closed interval between \(P_{1}\) and \(P_{2} \) which does not contain 0.) Let \(M_v\) be any point of \(X(k_v)\), and let \(I_v\subset E(k_v)\simeq S^1\) be a real interval linking \(f(M_v)\) and 0.
The value of A at \(\infty \in {\mathbb {P}}^1_{k} \), hence also at \(0 \in E(k)\), is \((1,1)\), hence the fibre \(X_0=f^{1}(0)\) is the conic \(x^2+y^2+z^2=0\). By Remark 5.3, \(X_0\) has points in all completions of k except the two real completions, hence \(X_0(k)=\emptyset \). Since \(E(k)=\{0\}\) it follows that \(X({\mathbf {A}}_k)\not =\emptyset \), but \(X(k)=\emptyset \). For each finite place v of k choose \(M_v\in X_0(k_v)\).
We now prove that \((M_v)\in X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\). By Proposition 2.3 every torsor \(X'\rightarrow X\) of a finite kgroup scheme G is the pullback \(X'=X\times _E E\rightarrow X\) of a torsor \(E'\rightarrow E\) of G. By twisting \(E'\) and \(X'\) with a ktorsor of G (and replacing G with the corresponding inner form) we can assume that \(E'\) has a kpoint \(0'\) over \(0\in E(k)\). The connected component C of \(E'\) containing this kpoint is a smooth, projective, geometrically integral curve. The kmorphism \(\varphi :C\rightarrow E\) is finite and étale, hence C has genus 1. Choosing \(0'\) for the origin of the group law on C we make \(\varphi :C\rightarrow E\) into an isogeny of elliptic curves and write \(0=0'\).
Let \(Y=X\times _E C\). Then Y is a smooth, projective, geometrically integral surface over k which is an irreducible component of \(X'\). The morphism \(g :Y\rightarrow C\) is a conic bundle. By Proposition 5.2 (3) we have \({\mathrm {SL}}_2^+(\hat{{\mathbb {Z}}})\subset \rho (\Gamma _k)\). Since the point \(P \in E(K)\) is not divisible, we can apply Theorem 4.1 to the elliptic curve E and the isogeny \(\varphi :C\rightarrow E \). It follows that the inverse image \(Q=\varphi ^{1}(P)\) of the closed point P of the kcurve E is a closed point of the kcurve C. We see that \(g:Y\rightarrow C\) is a conic bundle such that \(Y{\setminus } Y_{Q}\rightarrow C{\setminus } Q\) is a smooth morphism. By Proposition 2.2 (i), the induced map \(g^*:{\mathrm {Br}}(C)\rightarrow {\mathrm {Br}}(Y)\) is surjective.
For each finite place v of k let \(M'_v\) be the \(k_v\)point in the fibre \(Y_0\) over \(0\in C(k)\) that projects to \(M_v\in X_0(k_v)\). Now let v be a real place of k. By Lemma 4.7 (ii) the isogeny \(\varphi :C\rightarrow E\) is identified with \([n]:E\rightarrow E\) for some n. Since \(\Delta <0\), the induced map \(C(k_{v})\simeq E(k_v)\rightarrow E(k_{v})\) is a surjective étale map \(S^1 \rightarrow S^1\). Since \(I_v\) is contractible, we see that \(\varphi ^{1}(I_v)\) is a disjoint union of copies of \(I_v\) exactly one of which contains 0. Let us call this interval \(I'_v\). One of its ends is 0 and the other end is a point \(R\in C(k_v)\) such that \(\varphi (R)=f(M_v)\). Hence the real fibre \(Y_R\) is naturally isomorphic to the fibre of \(f :X\rightarrow E\) that contains \(M_v\). Let \(M'_v\) be the point in \(Y_R({\mathbb {R}})\) that projects to \(M_v\). Since \({\mathrm {Br}}(Y)=g^*({\mathrm {Br}}(C))\), we see that \((M'_v)\in Y({\mathbf {A}}_k)^{\mathrm {Br}}\). But Y is an irreducible component of \(X'\) hence \((M'_v)\in X'({\mathbf {A}}_k)^{\mathrm {Br}}\). This is a lifting of the adelic point \((M_v)\), and therefore \((M_v)\in X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\). \(\square \)
Remark 5.7
In the proof above the adèle \((f(M_v))\in E({\mathbf {A}}_k)\) and the kpoint \(0 \in E(k)\) have the following property: for each place v the class of the 0cycle \(f(M_v)0\) is infinitely divisible in \({\mathrm {Pic}}(E\times _k k_v)\), in fact it is even zero if v is nonarchimedean. Methods initiated by one of the authors and developed by Frossard [9, Thm. 0.3], by van Hamel, and by Wittenberg [32, Thm. 1.3], then show that there exists a 0cycle of degree 1 on X. In these various papers, finiteness of the Shafarevich–Tate group of the Jacobian of the base curve is assumed in order to appeal to the Cassels–Tate dual exact sequence, which guarantees the existence of a 0cycle of degree 1 on the curve satisfying a divisibility property analogous to the one above. In our case we can take this 0cycle of degree 1 to be the kpoint 0, so there is no need to assume the finiteness of the Shafarevich–Tate group of E.
6 Remarks on the Brauer–Manin set
6.1 Birational invariance
Recall that \({\mathrm {Br}}_{0}(X)\) denotes the image of the map \({\mathrm {Br}}(k) \rightarrow {\mathrm {Br}}(X)\).
Proposition 6.1

(i) If \(X({\mathbf {A}}_k)^{\mathrm {Br}}\not =\emptyset \), then \(Y({\mathbf {A}}_{k})^{\mathrm {Br}}\not =\emptyset \).

(ii) If \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\not =\emptyset \), then \(Y({\mathbf {A}}_{k})^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\not =\emptyset \).

(iii) Assume, in addition, that \({\mathrm {Br}}(X)/{\mathrm {Br}}_0(X)\) is finite. Then the density of X(k) in \(X({\mathbf {A}}_k)^{\mathrm {Br}}\) implies the density of Y(k) in \(Y({\mathbf {A}}_k)^{\mathrm {Br}}\).
Proof

(i) By Hironaka’s theorem, there exist a smooth, projective variety Z over k and birational morphisms \(f : Z \rightarrow X\) and \(g: Z \rightarrow Y\). Let v be a place of k. Since Z is proper, \(f(Z(k_{v}))\) is closed in \(X(k_{v})\). Let us show that \(X(k_{v})=f(Z(k_{v}))\), for which it is enough to show that \(f(Z(k_{v}))\) is dense in \(X(k_{v})\). There exists a nonempty Zariski open set \(U\subset X\) such that f induces an isomorphism \(f^{1}(U)\tilde{\longrightarrow } U\). Then \(U(k_{v})\subset f(Z(k_{v}))\), but \(U(k_{v})\) is dense in \(X(k_{v})\) by the implicit function theorem.
Thus for any \((M_{v}) \in X({\mathbf {A}}_k)\) there exists \((N_{v}) \in Z({\mathbf {A}}_k)\) such that \((f(M_v))=(N_v)\). The birational morphism f induces an isomorphism \(f^* : {\mathrm {Br}}(X) \tilde{\longrightarrow } {\mathrm {Br}}(Z)\). From the projection formula we conclude that if \((M_{v})\in X({\mathbf {A}}_{k})^{{\mathrm {Br}}}\), then \((N_{v})\in Z({\mathbf {A}}_{k})^{{\mathrm {Br}}}\). By the covariant functoriality of the Brauer–Manin set we have \(g\left( Z({\mathbf {A}}_{k})^{{\mathrm {Br}}}\right) \subset Y({\mathbf {A}}_{k})^{{\mathrm {Br}}}\).

(ii) Let G be a finite kgroup scheme. By the birational equivalence of the fundamental group [25, X, Cor. 3.4] there is a natural bijection between Xtorsors and Ytorsors of G in which a torsor \(X'/X\) corresponds to \(Y'/Y\) if \(X'\times _X Z=Y'\times _Y Z\). (This bijection respects the twisting by a ktorsor of G.) Let us denote this Ztorsor of G by \(Z'\). The natural morphism \(Z'\rightarrow X'\) is a componentwise birational morphism of smooth and projective varieties, so it induces an isomorphism \({\mathrm {Br}}(X') \tilde{\longrightarrow } {\mathrm {Br}}(Z')\).
Consider any \((M_v) \in X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\). Let \((N_v)\in Z({\mathbf {A}}_k)\) be a lifting of \((M_v)\) as in part (i). By the definition of \(X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\), for any torsor \(X'/X\) the adèle \((M_v)\) is the image of some \((M'_v) \in X'({\mathbf {A}}_k)^{\mathrm {Br}}\) (after twisting \(X'/X\) by a ktorsor of G and replacing G by the corresponding inner form). If \(N'_v\in Z'(k_v)\) is the point \(M'_v\times _{M_v}N_v\) for each place v, then \((N'_v)\in Z'({\mathbf {A}}_k)^{\mathrm {Br}}\) by the projection formula. This implies that \((N_v)\in Z({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\). By the covariant functoriality of the étale Brauer–Manin set we have$$\begin{aligned} g(Z({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}})\subset Y({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}. \end{aligned}$$ 
(iii) Let \(U \subset X\) and \(V \subset Y\) be nonempty Zariski open sets such that there is an isomorphism \(h : V \tilde{\longrightarrow } U\). Under the assumption of (ii), the group \({\mathrm {Br}}(Y)/{\mathrm {Br}}_0(Y) \simeq {\mathrm {Br}}(X)/{\mathrm {Br}}_0(X)\) is finite. Let \(\beta _{1}, \dots , \beta _{n} \in {\mathrm {Br}}(Y)\) be coset representatives. There exists a finite set \(\Sigma \) of places of k such that each \(\beta _{i}\) vanishes on \(Y(k_{v})\) for each \(v\notin \Sigma \). Thus every adèle \((M_v)\in Y({\mathbf {A}}_k)^{\mathrm {Br}}\) can be approximated by an adèle \((N_v)\in Y({\mathbf {A}}_k)^{\mathrm {Br}}\) such that \(N_v\in V(k_v)\) for all completions \(k_v\) of k. Let \(M_{v}=h(N_{v})\). Since h induces an isomorphism \({\mathrm {Br}}(Y)\simeq {\mathrm {Br}}(X)\), the adèle \((M_{v})\) is in \(X({\mathbf {A}}_k)^{\mathrm {Br}}\). If X(k) is dense in \(X({\mathbf {A}}_k)^{\mathrm {Br}}\), we can approximate \((M_{v})\) by a kpoint M in U. Then \(N=h^{1}(M) \in V(k)\) is close to \((N_v)\in Y({\mathbf {A}}_k)^{\mathrm {Br}}\). \(\square \)
Remark 6.2
 (1)
We do not know if the analogue of Proposition 6.1 (iii) holds for the étale Brauer–Manin set.
 (2)
We do not know if Proposition 6.1 (iii) still holds when \({\mathrm {Br}}(X)/{\mathrm {Br}}_0(X)\) is infinite, but we can make the following observation. Recall that \(X({\mathbf {A}}_{k})_{\bullet }\) denotes the quotient of \(X({\mathbf {A}}_{k})\) by the relation which identifies two points in the same connected component. For smooth, projective, geometrically integral varieties X and Y that are birationally equivalent, if one does not assume the finiteness of \({\mathrm {Br}}(X)/{\mathrm {Br}}_0(X)\), then Y(k) can be dense in \(Y({\mathbf {A}}_k)^{\mathrm {Br}}_\bullet \) without X(k) being dense in \(X({\mathbf {A}}_k)^{\mathrm {Br}}_\bullet \). Indeed, let \(Y=A\) be an abelian surface over \({\mathbb {Q}}\) such that \(A({\mathbb {Q}})= \{0\}\) and the Shafarevich–Tate group of A is finite. Let Z be the blowingup of A at the \({\mathbb {Q}}\)point 0, and let X be the blowingup of Z at some \({\mathbb {Q}}\)point (all \({\mathbb {Q}}\)points of Z are contained in the exceptional divisor). The surface X contains two copies of \({\mathbb {P}}^1_{{\mathbb {Q}}}\) meeting at a \({\mathbb {Q}}\)point P; let us call them E and F. The finiteness of the Shafarevich–Tate group of A implies [28, Prop. 6.2.4] that \(A({\mathbf {A}}_{\mathbb {Q}})^{\mathrm {Br}}\) is the connected component of 0 in \(A({\mathbf {A}}_{\mathbb {Q}})\), which is isomorphic to the real connected component of 0 in \(A({\mathbb {R}})\). Choose a \({\mathbb {Q}}\)point \(M\not =P\) in E, and a \({\mathbb {Q}}\)point \(N\not =P\) in F. Let q be a prime. Consider the adelic point \((M_p)\) of X, where \(M_p=M\) for all \(p\not =q\) (including \(p=\infty \)) and \(M_q=N\). Since the morphism \(f:X\rightarrow A\) induces an isomorphism \(f^*:{\mathrm {Br}}(A)\tilde{\longrightarrow }{\mathrm {Br}}(X)\) and \(f(M_p)=0\) for all p, we see that \((M_p) \in X({\mathbf {A}}_{\mathbb {Q}})^{\mathrm {Br}}\). However, the connected component of \((M_p)\) in \(X({\mathbf {A}}_{\mathbb {Q}})\) contains no \({\mathbb {Q}}\)points. Indeed, a \({\mathbb {Q}}\)point of X is either in E or in F. In the first case it cannot approximate \( M_q\) in the qadic topology, and in the second case it cannot approximate \( M_p\) in the padic topology where p is any prime different from q. It is easy to see that \((M_p) \in X({\mathbf {A}}_{\mathbb {Q}})^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\), so the previous discussion applies to the set of connected components of the étale Brauer–Manin set as well.
6.2 Cases where the Brauer–Manin obstruction suffices
The following result shows that for the conclusion of Theorem 5.6 to hold, the conic bundle \(f : X \rightarrow E\) must contain degenerate fibres.
Proposition 6.3
Let E be an elliptic curve over a number field k with a finite Shafarevich–Tate group. Let \(f : X \rightarrow E\) be a Severi–Brauer scheme over E. Then \(X({\mathbf {A}}_{k})^{{\mathrm {Br}}} \ne \emptyset \) implies \(X(k) \ne \emptyset \). Moreover, X(k) is dense in \(X({\mathbf {A}}_{k})_{\bullet }^{{\mathrm {Br}}}\).
Proof
Remark 6.4
The same argument works more generally for a projective morphism \(f: X \rightarrow E\) such that each fibre contains a geometrically integral component of multiplicity 1, provided that the smooth kfibres satisfy the Hasse principle. For the last statement to hold, the smooth kfibres also need to satisfy weak approximation.
The following proposition is a complement to Theorem 5.6 which explains why a similar counterexample cannot be constructed over \({\mathbb {Q}}\).
Proposition 6.5
Let E be an elliptic curve over a number field k such that both E(k) and the Shafarevich–Tate group of E are finite. Let \(f : X \rightarrow E\) be a conic bundle. Suppose that there exists a real place \(v_0\) of k such that for each real place \(v\not =v_0\) no singular fibre of \(f : X \rightarrow E\) is over a \(k_v\)point of E. Then \(X({\mathbf {A}}_{k})^{{\mathrm {Br}}}\ne \emptyset \) implies \(X(k)\ne \emptyset \).
Proof
If a kfibre of f is not smooth, then this fibre contains a kpoint. We may thus assume that the fibres above E(k) are smooth. Let \((M_{v})\in X({\mathbf {A}}_{k})^{{\mathrm {Br}}}\). Then \((f(M_{v}))\in E({\mathbf {A}}_{k})^{{\mathrm {Br}}}\). Set \(N_{v}=f(M_{v})\) for each place v. The finiteness of the Shafarevich–Tate group of E implies the exactness of the Cassels–Tate dual sequence (4). Hence there exists \(N \in E(k)\) such that \(N=N_{v} \) for each finite place v and such that N lies in the same connected component as \(N_{v}\) for v archimedean. The fibre \(X_{N}\) is a smooth conic with points in all finite completions of k. For an archimedean place \(v\not =v_0\), the map \(X(k_{v} ) \rightarrow E(k_{v})\) sends each connected component of \(X(k_{v})\) onto a connected component of \(E(k_{v})\). Since N and \(N_{v}\) are in the same connected component of \(E(k_{v})\), this implies \(X_{N}(k_{v}) \ne \emptyset \). Thus the conic \(X_{N}\) has points in all completions of k except possibly \(k_{v_0}\). By the reciprocity law it has points in all completions of k and hence in k. \(\square \)
Remark 6.6
If E(k) is finite we cannot in general expect X(k) to be dense in \(X({\mathbf {A}}_k)_\bullet ^{\mathrm {Br}}\) or even in \(X({\mathbf {A}}_k)_\bullet ^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\). Indeed, if the fibre \(X_N\) over some kpoint N of E is an irreducible singular conic, then the singular point P of \(X_N\) is the unique kpoint of \(X_N\). Let \((M_v)\) be any adèle in \(X_N({\mathbf {A}}_k)\). Note that if v splits in the quadratic extension of k over which the components of \(X_N\) are defined, then \(X_N\times _k k_v\) is a union of two projective lines meeting at P. Using the fact that \({\mathrm {Br}}({\mathbb {P}}^1_{k_v})={\mathrm {Br}}(k_v)\) we see that \((M_v)\in X({\mathbf {A}}_k)^{\mathrm {Br}}\). Furthermore, we have \((M_v)\in X({\mathbf {A}}_k)^{\acute{{\mathrm {e}}}{\mathrm {t}},{\mathrm {Br}}}\), cf. [11, Remark 2.4]. On the other hand, \((M_v)\) is not in the closure of X(k) in \(X({\mathbf {A}}_k)_\bullet \) provided \(M_v\not =P\) for at least one finite place v of k.
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