Hasse principle for pencils of curves of genus one

whose Jacobians have rational 2-division points

J.-L. Colliot-Thélène, A. N. Skorobogatov and Sir Peter Swinnerton-Dyer

Introduction

Let k be a number field, X a smooth, proper, connected surface over k equipped with a map π : X → P_{k}^{1} whose generic fibre over the function field K = k(P^{1}) is a curve of genus one X_{K}, and whose special fibres are all reduced. Let E_{K} be the Jacobian of X_{K}. Assume that the class of X_{K} in the Weil-Châtelet group H^{1}(K, E_{K}) is of order two but is not divisible by 2. Suppose that all 2-division points of E_{K} are defined over K, and that the only possible bad reduction of E_{K} at closed points of P_{k}^{1} is multiplicative with two components. Let

Br_{vert}(X) = Br(X) ∩ π^{∗}(Br(k(P^{1})) ⊂ Br(X) ⊂ Br(k(X))

be the vertical subgroup of the Brauer group. Under a condition slightly stronger than the hypothesis that the 2-primary torsion of Br(X) lies entirely in Br_{vert}(X), and at the expense of assuming two difficult conjectures, we prove that the Brauer-Manin obstruction attached to Br_{vert}(X) is the only obstruction to the Hasse principle for the existence of rational points on X. As a matter of fact, when there is no such obstruction, we prove that k-rational points are Zariski-dense on X.

The two conjectures are : finiteness of the Tate-Shafarevich groups of elliptic curves, and Schinzel’s hypothesis (H), which is a well-known and wild generalization of the conjecture on twin primes and of Bouniakowsky’s conjecture.

In a series of earlier papers (see the survey [CT97]), the Hasse principle has been studied for varieties equipped with a (linear) pencil of algebraic varieties. Under Schinzel’s hypothesis, general results were obtained for the total space of a pencil of Severi-Brauer varieties (e.g. surfaces which admit a conic bundle structure over the projective line) ([CT/SwD94]), and more generally for the total space of a pencil of varieties, when these varieties satisfy the Hasse principle and there is no multiple fibre in the pencil ([CT/Sk/SwD97b]).

Building upon Schinzel’s hypothesis, and also on the conjectural finiteness of the Tate-Shafarevich group of elliptic curves, Swinnerton-Dyer in [SwD95] gave sufficient conditions for the existence of a rational point on some surfaces which are defined over the rationals and are equipped with a pencil of curves of genus one.

A number of the computations in [SwD95] are rather mysterious, and some of them seem to rely on a series of lucky coincidences. Because [SwD95] represents the first, albeit conjectural, approach to the study of rational points on elliptic surfaces without a section, it is important to understand the scope of the method. The present paper represents the result of our joint efforts in this direction.

Although one is still a long way from the ideal results, we have succeeded in this enterprise in various ways :

(i) Condition (D) of [SwD95] is generalized. It is interpreted in terms of an algebraicogeometric version of a Selmer group attached to the Jacobian of X_{K}.

(ii) This condition is also interpreted in terms of the Brauer group of X.

(iii) On the number-theoretic side, we compute the 2-Selmer group as the kernel of a certain symmetric bilinear form. This replaces the case-by-case discussion in section 3 of [SwD95].

As a consequence, our main result holds over an arbitrary number field, and it holds for a much bigger class of elliptic surfaces.

In order to state our main theorem precisely, we need some preparation.

Let k be a field, char(k) ≠ 2. We start with a family of elliptic curves E/P_{k}^{1}, over the projective line P_{k}^{1}, given by the equation

Y^{2} = (X − c_{1}(t))(X − c_{2}(t))(X − c_{3}(t))

in affine space A_{k}^{2} × A_{k}^{1} (with coordinates (X, Y; t)), where the c_{i}(t) ∈ k[t] are polynomials. For {i, j, k} a cyclic permutation of {1, 2, 3}, we write p_{i}(t) = c_{j}(t) − c_{k}(t) and

R(t) = p_{1}(t)p_{2}(t)p_{3}(t) = (c_{2}(t) − c_{3}(t))(c_{3}(t) − c_{1}(t))(c_{1}(t) − c_{2}(t)).

We assume that R(t) is square free and not constant. We shall assume that the p_{i}(t) have the same even degree. These conditions ensure that the above elliptic curve E_{K} has good reduction at the point at infinity of the projective line and that the bad fibres have multiplicative reduction of type I_{2} (i.e. with two components). Conversely, using Tate’s algorithm [Ta72], one may check that any elliptic curve over K whose 2-torsion is rational over K = k(P^{1}) and which has bad reduction only of type I_{2} may be given by an equation as above with the same conditions on R(t) and on the degrees of the polynomials p_{i}.

We let T ⊂ A_{k}^{1} be the set of closed points defined by R(t) = 0. This is the set of bad reduction points for E/P_{k}^{1}. We let U ⊂ A_{k}^{1} be the open set defined by R(t) ≠ 0 and k[U]^{∗} = H^{0}(U, O_{U}^{∗}) be the group of rational functions invertible on U. Thus E_{U}/U is an abelian scheme over U. The group of sections E(U) of E_{U} over U coincides with the group E(k(P_{k}^{1})) = E(k(t)) of k(t)-rational points of the generic fibre.

Given an integer n > 0 and an abelian group A, we let _{n}A = {x ∈ A, nx = 0}. The exact sequence of U-group schemes

0 → _{2}E_{U} → E_{U} ⟶^{x↦2x} E_{U} → 0

gives rise to the Kummer map ∂ : E(U) → H^{1}(U,_{2}E_{U}) in étale cohomology, which defines an embedding E(U)/2 ↪ H^{1}(U,_{2}E_{U}) (sometimes also denoted ∂).

The generic fibre of E/P_{k}^{1} is an elliptic curve over the field k(P^{1}) all 2-torsion points of which are defined over k(P^{1}). Thus the finite U-group scheme _{2}E_{U} is isomorphic to the constant group scheme (Z/2)^{2} over U. The group H^{1}(U,_{2}E_{U}) may therefore be identified with the group of triples m = (m_{1}, m_{2}, m_{3}) ∈ (k[U]^{∗}/k[U]^{∗2})^{3} such that the product m_{1}m_{2}m_{3} = 1 ∈ k[U]^{∗}/k[U]^{∗2}. Given an element m ∈ H^{1}(U,_{2}E_{U}), we may represent it by a triple of square-free polynomials (m_{1}(t), m_{2}(t), m_{3}(t)) whose product is a square and divides R(t)^{2}. In the product P_{k}^{3} ×A_{k}^{1}, with coordinates (X_{0}, X_{1}, X_{2}, X_{3}; t), the system of equations

m_{i}(t)X_{i}^{2} − m_{j}(t)X_{j}^{2} = (c_{j}(t) − c_{i}(t))X_{0}^{2} i, j ∈ {1, 2, 3}

then defines a surface equipped with a pencil of curves of genus one, pencil whose fibres are smooth over the open set U of P_{k}^{1} (informally speaking, the above system is a one-parameter family of 2-descents). One knows that any such fibration may be extended to an essentially unique relatively minimal model π : X → P_{k}^{1} with X = X(m) smooth over k, projective and geometrically connected (for surfaces over a field of characteristic zero, this is a very classical fact, which has been extended to more general situations, see [Sha66] and [BLR90] p. 21 for further references). The singular fibres X_{M} = π^{−1}(M) of π lie over T and possibly the point at infinity of P_{k}^{1}.

We shall need to impose some conditions on the fibration X(m)/P_{k}^{1}, i.e. on m. Let 𝔖 ⊂ H^{1}(U,_{2}E_{U}) be the subgroup consisting of triples m = (m_{1}, m_{2}, m_{3}) ∈ (k[U]^{∗}/k[U]^{∗2})^{3}, with product equal to 1, such that for each i and each irreducible p(t) dividing p_{i}, the p-valuation of m_{i} is even. Once m is represented by a triple m = (m_{1}, m_{2}, m_{3}) of square-free polynomials with product a square dividing R(t)^{2}, the condition that the class of m lies in 𝔖 reads : for each i the polynomials m_{i} and p_{i} are coprime. We also introduce the subgroup 𝔖_{0} ⊂ 𝔖 consisting of classes m = (m_{1}, m_{2}, m_{3}) ∈ 𝔖 such that moreover the valuation at infinity of each m_{i}(t) is even. Once m = (m_{1}, m_{2}, m_{3}) ∈ (k[U]^{∗}/k[U]^{∗2})^{3} is represented by a triple m = (m_{1}, m_{2}, m_{3}) of square-free polynomials with product a square dividing R(t)^{2}, it lies in 𝔖_{0} if and only if for each i, the polynomials m_{i} and p_{i} are coprime, and moreover the degree of any m_{i} is even. An easy valuation computation shows that ∂(E(U)) ⊂ H^{1}(U,_{2}E_{U}) lies in 𝔖_{0}.

The condition that m ∈ H^{1}(U,_{2}E_{U}) belongs to 𝔖 implies (and is essentially equivalent to) the following geometric properties of the relatively minimal model X/P_{k}^{1}. The set of points M ∈ A_{k}^{1} whose fibre X_{M} is singular coincides with the set T ⊂ A_{k}^{1} of points of bad reduction for the restriction of E/P_{k}^{1} over A_{k}^{1}. For any such M ∈ T, with residue field k_{M}, the fibre X_{M} is reduced (as a divisor on X) and there exists a constant or quadratic extension K_{M} of k_{M} such that X_{M} ×_{k_{M}} K_{M} is the union of two smooth conics defined over K_{M} and meeting transversally ; over a separable closure of K_{M}, the intersection of these two conics consists of two points. If moreover m lies in 𝔖_{0}, then the fibre of X(m)/P_{k}^{1} over the point at infinity of P_{k}^{1} is smooth.

Given m ∈ 𝔖 and M ∈ T as above, let δ_{M}(m) ∈ k_{M}^{∗}/k_{M}^{∗2} be the class of the quadratic extension K_{M}/k_{M}. We have thus defined a map

δ : 𝔖 → ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2}

where T denotes the set of points of bad reduction for E/P_{k}^{1}. One checks that the map δ is a homomorphism and that its kernel is finite and contains ∂(E(U)) (lemma 2.2.1 below).

Before stating the main Theorem, let us say a few words on the Brauer group and the Brauer-Manin obstruction. For a more detailed discussion of the Brauer-Manin obstruction, see [CT/SwD94, section 3] and [CT97]. The Brauer group Br(X) is the étale cohomology group H_{ét}^{2}(X, G_{m}). Given π : X → P_{k}^{1} as above, the vertical subgroup Br_{vert}(X) of the Brauer group of X with respect to π has been defined in the beginning. In the situation above (m ∈ 𝔖_{0}), all geometric fibres of π are reduced. This implies that the quotient Br_{vert}(X)/Br(k) is finite (see [Sk96, Cor. 4.5]).

Let now k be a number field. Let Ω be the set of places of k. For v ∈ Ω, let k_{v} be the completion of k at v. For any proper k-variety X, let X(𝔸_{k}) = ∏_{v∈Ω} X(k_{v}) be the space of adèles of X, equipped with its natural topology, which is none but the product topology. For any element A ∈ Br(X), the map which sends the adèle {P_{v}} to ∑_{v∈Ω} inv_{v}(A(P_{v})) ∈ Q/Z is a continuous function θ_{A} : X(𝔸_{k}) → Q/Z with finite image. Given B ⊂ Br(X), we let X(𝔸_{k})^{B} ⊂ X(𝔸_{k}) denote the closed subset which is the intersection of the kernels θ_{A}^{−1}(0) for A ∈ B. The diagonal inclusion X(k) ⊂ X(𝔸_{k})^{B} of the set of k-rational points of X in the set of adèles of X shows that X(𝔸_{k})^{B} ≠ ∅ is a necessary condition for the existence of a k-rational point on X. For π : X → P_{k}^{1} as above, the finiteness of Br_{vert}(X)/Br(k) implies that X(𝔸_{k})^{Br_{vert}(X)} is open in X(𝔸_{k}).

We are now in a position to state our main result.

Theorem Let k be a number field. Let E/P_{k}^{1} be an elliptic surface as above. Let m ∈ H^{1}(U,_{2}E) and let X = X(m) be the associated surface, equipped with the relatively minimal k-morphism π : X → P_{k}^{1}. Assume :

(1) The class m belongs to 𝔖_{0}.

(2) The class m does not belong to ∂(_{2}E(U)), the image of the 2-torsion subgroup _{2}E(U) of E(U) under the Kummer map.

(3) The kernel of the composite map

𝔖_{0} →^{δ} ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) >

is generated by m and by ∂(_{2}E(U)). (This will later be referred to as condition (D).) (4) The Tate-Shafarevich group of each elliptic curve occurring as a fibre of E/P_{k}^{1} is finite.

(5) Schinzel’s hypothesis (H) holds.

Let R ⊂ P^{1}(k) be the set of k-points x ∈ P^{1}(k) such that the fibre X_{x} = π^{−1}(x) is smooth and has infinitely many k-points. Let R_{E} ⊂ P^{1}(k) be the set of k-points x ∈ P^{1}(k) such that the fibre E_{x} of E/P_{k}^{1} is an elliptic curve over k of rank (= dim_{Q}(E(k) ⊗ Q)) at least equal to one. We have R ⊂ R_{E}. Then :

(a) The closure of R in P^{1}(𝔸_{k}) coincides with π(X(𝔸_{k})^{Br_{vert}(X)}).

(b) Assume X(𝔸_{k})^{Br_{vert}(X)} ≠ ∅, i.e. assume that there is no vertical Brauer-Manin obstruction to the existence of a rational point (this is certainly the case if X(k) ≠ ∅). Then for any nonempty finite set S ⊂ Ω, the closure of R in ∏_{v∈S} P^{1}(k_{v}) contains a non-empty open set. The same therefore holds for R_{E}. In particular R and R_{E} are Zariski-dense in P_{k}^{1}, the set X(k) is Zariski-dense in X and the set E(k) is Zariski-dense in the surface E.

(c) Assume X(𝔸_{k})^{Br_{vert}(X)} ≠ ∅. Then there exists a finite set S_{0} ⊂ Ω such that for any finite set S ⊂ Ω with S∩S_{0} = ∅, the closure of R under the diagonal embedding P^{1}(k) → ∏_{v∈S} P^{1}(k_{v}) coincides with ∏_{v∈S} π(X(k_{v})).

Let us comment on the various hypotheses in the Theorem.

As mentioned above, hypothesis (1) ensures in particular that all fibres of π : X → P_{k}^{1} are reduced. Some condition of that kind is required, if we hope to conclude that infinitely many kfibres of π have rational points (see Cor. 2.2, Cor. 2.4, Prop. 4.1, Prop. 4.2 of [CT/Sk/SwD97a]). Hypothesis (4) (finiteness of Tate-Shafarevich groups) is precisely condition (A) of [SwD95], and Hypothesis (5) (Schinzel’s hypothesis) is essentially condition (B) of [SwD95]. If (2) is not satisfied, then the surface X/P_{k}^{1} is k-birational to E/P_{k}^{1} over P_{k}^{1} and under the assumption (3), the rank of E(k(t)) is zero. In this case X(k) is not empty, but we do not know whether X(k) is Zariski-dense in X. Hypothesis (3) (condition (D)) is a generalization of condition (D) in [SwD95] (p. 290), which is of a kind which had not appeared in previous papers on the Hasse principle. The algebraic significance of this condition in [SwD95] was unclear. In section 4, we show that (D) is closely connected with the hypothesis that the 2-primary part of the Brauer group of X lies in the vertical subgroup Br_{vert}(X). This fits in well with the obvious remark that the necessary condition X(𝔸_{k})^{Br_{vert}(X)} ≠ ∅ for the existence of a rational point is a priori weaker than the condition X(𝔸_{k})^{Br(X)} ≠ ∅. For the pencils of algebraic varieties under consideration in [CT/SwD94], the whole Brauer group is vertical, but it is not the case any more here.

The last two assumptions in the Theorem are of a very different kind to the first three ; and one way of looking at the Theorem is to regard it as providing strong support for the following conjecture :

Conjecture Let k be a number field and let E, m, X be as in the Theorem. Assume that hypotheses (1), (2) and (3) of the Theorem hold. Then so do the conclusions of the Theorem. In particular, if there exists a k-rational point on X, then rational points are Zariski dense on X.

The most surprising feature of this conjecture is that the assumptions are all effectively computable, and that condition (3) (= condition (D)), as well as the additional condition Br_{vert}(X)/Br(k) = 0, tend to be satisfied by general members of the families of surfaces to which this conjecture applies, thereby predicting a Hasse principle in many cases. In particular, there are large families of K3 surfaces to which this conjecture applies – for example the 13parameter family of surfaces given by an equation of the form

q_{0}(x_{0}, ..., x_{3})q_{1}(x_{0}, ..., x_{3}) = q_{2}(x_{0}, ..., x_{3})q_{3}(x_{0}, ..., x_{3}),

where the q_{i} are homogeneous diagonal quadratic forms in x_{0}, ..., x_{3}. Such examples lead us to ask whether there are more general families of K3 surfaces for which there is an effectively computable sufficient condition for the Hasse principle to hold. Until we embarked on the current research, we would not have supposed that this was a sensible question to ask.

We hope that, for the pencils of curves of genus 1 with which we are concerned in this paper (and perhaps for more general ones also), further progress will enable us to forget about hypothesis (3) (i.e. condition (D)) and establish that the closure of R in P^{1}(𝔸_{k}) coincides with π(X(𝔸_{k})^{Br(X)}). Still granting (4) and (5), this would in particular imply that X(k) ≠ ∅ as soon as X(𝔸_{k})^{Br(X)} ≠ ∅, i.e. the (total) Brauer-Manin obstruction to the Hasse principle would be the only obstruction to the existence of a rational point. But this would not be an unmixed gain, for it would replace a condition which is too strong but which is effectively computable by one which, although natural as well as necessary and sufficient, is at most semi-computable.

Here is the idea of the proof, which is given in section 2 after some preliminary work in the first section. Start with an adèle {P_{v}} ∈ X(𝔸_{k})^{Br_{vert}(X)}. Let S_{1} be a finite set of places of k. Using Schinzel’s Hypothesis (H), find x ∈ U(k) ⊂ A^{1}(k), close to each π(P_{v}) for v ∈ S_{1} and such that

(a) the curve X_{x} = π^{−1}(x) (the fibre at x) has points in all the completions of k;

(b) the 2-Selmer group of the elliptic curve E_{x} has order exactly 8 (generators being m(x) and ∂(_{2}E_{x})), so that the 2-torsion subgroup of the Tate-Shafarevich group of E_{x} has order at most 2.

Assume that the Tate-Shafarevich group �(E_{x}) of E_{x} is finite. The Cassels-Tate pairing, which is alternating (Cassels, see [Milne86], Remark 6.12, p. 100), is then non-degenerate (Cassels [Cass62] ; [Milne86] thm. 6.13, p. 101), hence the order of the 2-primary subgroup of �(E_{x}) is a square. Thus the 2-torsion of the Tate-Shafarevich group of E_{x} is trivial, whence X_{x}(k) ≠ ∅.

The proof of (a) follows a general pattern (see [CT/SwD94] and [CT/Sk/SwD97b]). It ‘only’ uses assumption (5) (Schinzel’s hypothesis) and the fact that m lies in 𝔖_{0} (assumption (1)), which guarantees that the reducible fibres of X/P_{k}^{1} are not too bad. However, to get (a) together with (b), one also needs to use assumptions (2) and (3) in the theorem.

Section 1 contains results on the computation of the 2-Selmer group of an elliptic curve over a number field k, when its 2-torsion is entirely defined over k. These results are crucial for the proof in section 2, but they seem to be of independent interest. They provide a general explanation for the computations made in section 3 of [SwD95], including the symmetry statement in Lemma 3 of that paper. This Lemma is now replaced by our proposition 1.2.3, which enables one to compute the 2-Selmer group as the kernel of a certain symmetric bilinear form. Some class field theory is used to cover the case of arbitrary number fields (in [SwD95], attention was restricted to the rational field Q).

Section 2 contains the proof of the Theorem. The next sections are independent. Section 3 discusses the geometrical types of surfaces to which the Theorem applies. It also gives a readily computable version of condition (D). In Section 4, which uses more sophisticated tools than in the previous sections, such as the the geometry of Néron minimal models, we provide a general context for some of the previous computations. In particular the groups 𝔖 and 𝔖_{0} turn out to be geometric versions of the 2-Selmer group. We define a generalized condition (D) and relate it with the Grothendieck Brauer group. The special case of interest in this paper is described in subsection 4.7.

1. Arithmetical prerequisites

1.1. A statement from global class field theory, and some linear algebra

Let k be a number field, Ω the set of its places and L ⊂ Ω a finite subset. Consider the vector space ⊕_{v∈L}k_{v}^{∗}/k_{v}^{∗2} over the finite field F_{2}. We equip this space with the bilinear form ( , ) = ⊕_{v∈L}( , )_{v}, where ( , )_{v} is the bilinear pairing (the local symbol)

H^{1}(k_{v}, Z/2) × H^{1}(k_{v}, Z/2) → H^{2}(k_{v}, Z/2) ↪ Z/2

(where the right arrow is an isomorphism provided v is not a complex place). It is a standard fact of local class field theory that the pairing ( , )_{v}, and hence the pairing ( , ), is non-degenerate. This pairing is skew-symmetric : for any a, b ∈ k_{v}^{∗}, we have (a, b)_{v} + (b, a)_{v} = 0. Let O_{L} be the ring of L-integers of k, and let A_{L} be the image of O_{L}^{∗}/O_{L}^{∗2} in ⊕_{v∈L}k_{v}^{∗}/k_{v}^{∗2}.

Proposition 1.1.1 Let L be a finite set of places containing all the archimedean places, and all the places over 2. Then the dimension of the F_{2}-vector space O_{L}^{∗}/O_{L}^{∗2} is equal to half the dimension of the F_{2}-vector space ⊕_{v∈L}k_{v}^{∗}/k_{v}^{∗2}. If moreover L contains a set of generators of the ideal class group of k, then the natural map O_{L}^{∗}/O_{L}^{∗2} → ⊕_{v∈L}k_{v}^{∗}/k_{v}^{∗2} is injective and its image A_{L} is a maximal isotropic subspace of ⊕_{v∈L}k_{v}^{∗}/k_{v}^{∗2}.

Proof Let us compare F_{2}-dimensions. By Dirichlet’s theorem, dim(O_{L}^{∗}/O_{L}^{∗2}) = #L. On the other hand dim(⊕_{v∈L}k_{v}^{∗}/k_{v}^{∗2}) = 2#L (note that #(k_{v}^{∗}/k_{v}^{∗2}) = 4/|2|_{v} and use the product formula). This proves the first assertion. The fact that A_{L} is isotropic follows from the global reciprocity law. Assume Pic(O_{L}) = 0. Lemma 9.2 of [ANT67], VII, together with our assumptions, implies that the map O_{L}^{∗}/O_{L}^{∗2} → ⊕_{v∈L}k_{v}^{∗}/k_{v}^{∗2} is injective (see [ANT67], Exercise 2.16). By dimension count, we conclude that A_{L} is maximal isotropic.

Remarks

a) One could also prove the proposition by using Poitou-Tate duality : in [Milne86], combine theorem 4.10 p. 70, proposition 2.9 p. 209, the duality statement on top of p. 70, and finally the injection of the Brauer group of a number field into the direct sum of the Brauer groups of its completions.

b) One may prove a similar statement with 2 replaced by an arbitrary prime p, provided k contains the p-th roots of unity.

Let S_{0} denote a fixed family of places of k satisfying all the conditions of the proposition. Clearly, for any finite set of places L containing S_{0}, the statement of the proposition holds.

For any place v of k, we let V_{v} = H^{1}(k_{v}, (Z/2)^{2}) = (k_{v}^{∗}/k_{v}^{∗2})^{2}. We equip this F_{2}-vector space with the bilinear form :

e_{v}((a, b), (c, d)) = (a, d)_{v} − (c, b)_{v} ∈ Z/2

(this is also equal to (a, d)_{v} +(b, c)_{v}.) This bilinear form is clearly alternating : e_{v}((a, b), (a, b)) = 0, and it is non-degenerate since e_{v} is non-degenerate.

For any finite set L of places containing S_{0}, we let V_{L} = ⊕_{v∈L}V_{v}. This space comes equipped with the symplectic structure e_{L} = ⊕_{v∈L}e_{v} given by the orthogonal sum of the structures on the individual V_{v}.

Because L contains S_{0}, we have Pic(O_{L}) = 0. Hence the diagonal map

H^{1}(O_{L}, (Z/2)^{2}) → ⨁_{v∈L} H^{1}(k_{v}, (Z/2)^{2})

may be read

(O_{L}^{∗}/O_{L}^{∗2})^{2} → ⨁_{v∈L}(k_{v}^{∗}/k_{v}^{∗2})^{2}

and it is injective by the previous proposition. By the global reciprocity law, the image U_{L} of this map is isotropic with respect to the symplectic pairing e = ⊕_{v∈L}e_{v} defined above. By dimension count (using proposition 1.1.1), we then see that it is maximal isotropic.

Note that for any L, the natural map H^{1}(O_{L}, (Z/2)^{2}) → H^{1}(k, (Z/2)^{2}) is injective. For L containing S_{0}, we shall identify (O_{L}^{∗}/O_{L}^{∗2})^{2} = H^{1}(O_{L}, (Z/2)^{2}) with the corresponding subgroup of (k^{∗}/k^{∗2})^{2} = H^{1}(k, (Z/2)^{2}), and also with U_{L}. The elements of U_{L} may thus be identified with the elements of (k^{∗}/k^{∗2})^{2} = H^{1}(k, (Z/2)^{2}) whose image in (k_{v}^{∗}/k_{v}^{∗2})^{2} = H^{1}(k_{v}, (Z/2)^{2}) belongs to (O_{v}^{∗}/O_{v}^{∗2})^{2} = H^{1}(O_{v}, (Z/2)^{2}) for each v ∉ L.

Proposition 1.1.2 Let S_{0} be as above, and let L be a finite family of places containing S_{0}. There exist subspaces K_{v} ⊂ V_{v}, v ∈ L , such that :

(a) for each place v ∈ L, K_{v} is a maximal isotropic subspace of V_{v};

(b) V_{L} = U_{L} ⊕ K_{L}, where K_{L} := ⊕_{v∈L}K_{v}.

For v ∉ S_{0}, one may choose K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} ⊂ V_{v}.

Proof The space U_{L} ⊂ V_{L} is maximal isotropic. Thus except for the last sentence, the statement follows from the general lemma :

Lemma 1.1.3 Let k be a field and V_{i}, i = 1, ⋯, n, be vector spaces each equipped with a non-degenerate alternating bilinear form. Let V = ⊕_{i}V_{i} be equipped with the direct sum of these alternating forms. Let U ⊂ V be a maximal isotropic subspace. Then there exist maximal isotropic subspaces K_{i} ⊂ V_{i} such that U ⊕ (⊕_{i}K_{i}) = V.

Proof Let 2d be the dimension of V. We have d = dim(U). For j = 0, 1, …, d we shall construct isotropic subspaces K^{j} = ⊕_{i}K_{i}^{j} with K_{i}^{j} ⊂ V_{i} such that dim(K^{j}) = j and U ∩K^{j} = 0. We shall set U^{j} := U ∩ (K^{j})^{⊥}.

From U ∩ K^{j} = 0 and U = U^{⊥} we deduce V = U + (K^{j})^{⊥}, hence

2d = dim(U) + dim((K^{j})^{⊥}) − dim(U ∩ (K^{j})^{⊥}) = d + (2d − j) − dim(U^{j}).

Thus dim(U^{j}) = d − j.

For j = 0, one simply takes K^{0} = 0. Here is how to pass from j < d to j + 1. The composite map

U^{j} → (K^{j})^{⊥} → (K^{j})^{⊥}/K^{j},

where the first arrow is the inclusion U^{j} := U ∩ (K^{j})^{⊥} → (K^{j})^{⊥}, is injective, since U ∩ K^{j} = 0. The dimension of U^{j} is d − j, that of (K^{j})^{⊥}/K^{j} is 2d − 2j, as one easily checks. The space (K^{j})^{⊥}/K^{j} is equipped with an induced non-degenerate alternating form, and via the above embedding U^{j} is an isotropic hence (dimension count) maximal isotropic subspace. One may find a class in (K^{j})^{⊥}/K^{j} which does not vanish on U^{j}. We may choose it as the image of v ∈ (K_{i}^{j})^{⊥} ⊂ V_{i} for a suitable i. One then defines K^{j+1} = K^{j} + k < v >. This is isotropic in V. We clearly have dim(K^{j+1}) = j + 1. We have U ∩ K^{j+1} ⊂ U ∩ (K^{j+1})^{⊥} = U^{j} ∩ (K^{j+1})^{⊥} hence U ∩ K^{j+1} = U^{j} ∩ K^{j+1}. However U^{j} and K^{j} are orthogonal and v does not vanish on U^{j}. Thus U ∩ K^{j+1} = 0. For j = d, the equation dim(K^{d}) = d ensures that K^{d} is maximal isotropic. The way in which it is built guarantees that it is a sum K^{d} = ⊕_{i}K_{i}^{d} of isotropic, hence maximal isotropic, subspaces K_{i}^{d} ⊂ V_{i}.

We apply the lemma to L = S_{0} and the maximal isotropic subspace U_{S_{0}} ⊂ V_{S_{0}}, thus producing maximal isotropic subspaces K_{v} ⊂ V_{v} for each v ∈ S_{0}, with the property V_{S_{0}} = U_{S_{0}} ⊕ K_{S_{0}}, where K_{S_{0}} := ⊕_{v∈S_{0}}K_{v}. This proves the proposition for L = S_{0}.

For arbitrary L, take the just found K_{v}’s for v ∈ S_{0}, and let K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} for v ∈ L, v ∉ S_{0}. Each K_{v} ⊂ V_{v} is maximal isotropic. By dimension count, it is enough to show U_{L} ∩K_{L} = 0. But any element of U_{L} = (O_{L}^{∗}/O_{L}^{∗2})^{2} which belongs to U_{L}∩K_{L} actually belongs to U_{S_{0}} = (O_{S_{0}}^{∗}/O_{S_{0}}^{∗2})^{2} (both groups being viewed as subgroups of (k^{∗}/k^{∗2})^{2}), since K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} for v ∉ S_{0}. Thus U_{L} ∩ K_{L} ⊂ U_{S_{0}} ∩ K_{S_{0}} = 0.

In the sequel, when applying proposition 1.1.2, we shall always choose K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} ⊂ V_{v} for v ∉ S_{0}.

1.2. The 2-Selmer group of an elliptic curve (with rational 2-torsion) as the kernel of a symmetric bilinear form

Let E be an elliptic curve over a number field k, let _{2}E be the k-subgroup of points of order 2. Assume all 2-division points are defined over k. Fix an isomorphism _{2}E ≃ (Z/2)^{2} of k-group schemes. The curve E may be given by an equation Y^{2} = (X − c_{1})(X − c_{2})(X − c_{3}), c_{i} ∈ k. We shall fix the isomorphism above in such a way that the point (X, Y) = (c_{1}, 0), resp. (X, Y) = (c_{2}, 0), corresponds to (1, 0) ∈ (Z/2)^{2}, resp. (0, 1) ∈ (Z/2)^{2}. There is a corresponding isomorphism H^{1}(k,_{2}E) ≃ (k^{∗}/k^{∗2})^{2}.

Consider the following diagram, deduced from the Kummer sequence :

0 → E(k)/2 → H^{1}(k,_{2}E) → _{2}H^{1}(k, E) → 0
↓ r_{v} ↓ ρ_{v} ↓
0 → E(k_{v})/2 ⟶^{∂_{v}} H^{1}(k_{v},_{2}E) ⟶^{i_{v}}  _{2}H^{1}(k_{v}, E) → 0

where the vertical arrows (e.g. r_{v} and ρ_{v}) are the restrictions to k_{v}. If P ∈ E(k_{v}) has coordinates (X, Y) and is not a point of order 2, then the Kummer map ∂_{v} : E(k_{v}) → H^{1}(k_{v},_{2}E) sends P to ∂_{v}(P) = (X − c_{1}, X − c_{2}).

For any place v of k, we let V_{v} = H^{1}(k_{v},_{2}E) = H^{1}(k_{v}, (Z/2)^{2}) = (k_{v}^{∗}/k_{v}^{∗2})^{2}. The Weil pairing and cup-product define a non-degenerate alternating pairing V_{v} × V_{v} → Z/2, i.e. a symplectic space structure e_{v} on V_{v} which coincides with the pairing described in subsection 1.1. For any finite set L of places containing S_{0}, we let V_{L} = ⊕_{v∈L}V_{v}. This space comes equipped with the symplectic structure e_{L} = ⊕_{v∈L}e_{v} given by the orthogonal sum of the structures on the individual V_{v}.

For each place v ∈ Ω, let W_{v} = W_{v}(E) = ∂_{v}(E(k_{v})/2) ⊂ V_{v}. According to Tate (see [Milne86], p. 56), W_{v}(E) is a maximal isotropic subspace of the symplectic space V_{v}. If v ∉ S is a place of good reduction for E, then W_{v} = H^{1}(O_{v},_{2}E) ⊂ V_{v}, in other words W_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} ⊂ (k_{v}^{∗}/k_{v}^{∗2})^{2}. For any finite set L of places containing S_{0}, let W_{L} = W_{L}(E) = ⊕_{v∈L}W_{v}(E). This is clearly a maximal isotropic subspace of V_{L}.

Recall that the 2-torsion subgroup of the Tate-Shafarevich group of E is given by :

2�(E) := {x ∈ _{2}H^{1}(k, E) | ρ_{v}(x) = 0 for all places v}.

As for the 2-Selmer group S(E, 2), it is defined by

S(E, 2) := {x ∈ H^{1}(k,_{2}E) | i_{v}r_{v}(x) = 0 for all places v}.

There is the well-known exact sequence

0 → E(k)/2 → S(E, 2) → _{2}�(E) → 0.

Proposition 1.2.1 Let S_{0} be as in subsection 1.1, and let L be a finite family of places containing S_{0} and the places of bad reduction for the elliptic curve E. Let V_{L} and U_{L} ⊂ V_{L} be as in subsection 1.1, and W_{L} = W_{L}(E) ⊂ V_{L} as above. The 2-Selmer group S(E, 2) is isomorphic to each of the following groups :

(1) The intersection U_{L} ∩ W_{L} ⊂ V_{L}.

(2) The left kernel of the pairing e_{L} : U_{L} × W_{L} → Z/2 induced by the symplectic pairing e_{L} : V_{L} × V_{L} → Z/2.

(3) The right kernel of the pairing e_{L} : U_{L} × W_{L} → Z/2 induced by the symplectic pairing e_{L} : V_{L} × V_{L} → Z/2.

Proof We have to compare two subgroups of H^{1}(k,_{2}E). Since

im(∂_{v}) = W_{v}(E) = H^{1}(O_{v},_{2}E) ⊂ V_{v}

for v ∉ L, the class α ∈ H^{1}(k,_{2}E) belongs to U_{L} if and only if for each v ∉ L, the restriction α_{v} ∈ H^{1}(k_{v},_{2}E) belongs to the image of ∂_{v}. The condition α_{v} ∈ W_{v} for v ∈ S exactly says that α_{v} belongs to the image of ∂_{v}. This proves (1). As for (2) and (3), they follow from (1) and from the fact that U_{L} and W_{L} are maximal isotropic subspaces, hence each equal to its own orthogonal.

Given any finite set of places L containing S_{0}, but not necessarily containing the primes of bad reduction for E, let us define several important objects, whose definition depends on the K_{v}’s and the W_{v}’s.

We define t_{L} : V_{L} → U_{L} to be the projector along K_{L}.

We set U′_{L} = U′_{L}(E) = U_{L} ∩ (W_{L}(E) + K_{L}) ⊂ V_{L}.

We set W′_{L} = W′_{L}(E) = W_{L}(E)/(W_{L}(E) ∩ K_{L}) = ⨁_{v∈L}(W_{v}(E)/W_{v}(E) ∩ K_{v}).

Note that for S_{0} ⊂ L ⊂ L_{1}, the natural inclusion U_{L} ⊂ U_{L_{1}} induces the equality U′_{L} = U′_{L_{1}} ∩ U_{L}.

Lemma 1.2.2 Let x, y ∈ V_{L}. Then e_{L}(x, y) = e_{L}(t_{L}(x), y) + e_{L}(x, t_{L}(y)).

Proof The homomorphism t_{L} (resp. 1−t_{L}) is a projector to the isotropic subspace U_{L} (resp. K_{L}). Hence e_{L}(t_{L}(x), t_{L}(y)) = e_{L}(x − t_{L}(x), y − t_{L}(y)) = 0, which implies the lemma.

Proposition 1.2.3 Assume that L contains S_{0}.

(1) The projection map t_{L} : V_{L} → U_{L}, when restricted to W_{L} ⊂ V_{L}, induces an isomorphism

τ_{L} : W′_{L} ⟶^{≃} U′_{L}.

(2) The map τ_{L}^{−1} associates to u = ∑_{v∈L}(α_{v} + β_{v}) ∈ U′_{L} (with α_{v} ∈ W_{v} and β_{v} ∈ K_{v}) the class of ∑_{v∈L} α_{v} ∈ W_{L}/(W_{L} ∩ K_{L}) = W′_{L}.

(3) The pairing e_{L} : U_{L} × W_{L} → Z/2 induces a pairing e′_{L} : U′_{L} × W′_{L} → Z/2. Assume that L also contains all the places of bad reduction for the elliptic curve E.

(4) The left and right kernel of the pairing e′_{L} are both isomorphic to S(E, 2).

(5) The pairing U′_{L} × U′_{L} → Z/2 given by (x, y) → e′_{L}(x, τ_{L}^{−1}(y)) is symmetric and its kernel is isomorphic to S(E, 2).

(6) The pairing W′_{L} × W′_{L} → Z/2 given by (x, y) → e′_{L}(τ_{L}(x), y) is symmetric and its kernel is isomorphic to S(E, 2).

Proof Statements (1) and (2) follow from the definitions. We have

U′_{L} = U_{L} ∩ (W_{L} + K_{L}) = U_{L} ∩ (W_{L}^{⊥} + K_{L}^{⊥}) = U_{L} ∩ (W_{L} ∩ K_{L})^{⊥}.

Here the first equality follows from the definitions, and the second one follows from W_{L} = W_{L}^{⊥} and K_{L} = K_{L}^{⊥}. This implies (3).

The subgroup S(E, 2) ⊂ U_{L} is equal to U_{L} ∩W_{L}, hence is contained in U′_{L} = U_{L} ∩(W_{L} +K_{L}). From (3) and proposition 1.2.1 (2) we conclude that the left kernel of e′_{L} is isomorphic to S(E, 2). By dimension count, this also holds for the right kernel, hence (4).

Given x, y ∈ U′_{L}, there exist α, β ∈ W_{L} such that t_{L}(α) = x and t_{L}(β) = y. We have e_{L}(α, β) = 0. From the lemma we conclude e_{L}(x, β)+e_{L}(α, y) = 0, hence e_{L}(x, β)−e_{L}(y, α) = 0, hence finally e′_{L}(x, τ_{L}^{−1}(y)) − e′_{L}(y, τ_{L}^{−1}(x)) = 0. This proves the symmetry in (5). The proof of the symmetry in (6) is also a straightforward application of the lemma and of the fact that W_{L} is isotropic. That the kernel of either symmetric pairing is isomorphic to S(E, 2) then follows from (4).

Describing U′_{L}(E) ⊂ U_{L}

From the definition U′_{L}(E) = U_{L} ∩ (W_{L}(E) + K_{L}) we have that U′_{L} as a subspace of U_{L} = (O_{L}^{∗}/O_{L}^{∗2})^{2} is given by local conditions, namely, for each v ∈ L the image of the corresponding global element ξ ∈ (O_{L}^{∗}/O_{L}^{∗2})^{2} in (k_{v}^{∗}/k_{v}^{∗2})^{2} must belong to W_{v} + K_{v}, where

W_{v} = W_{v}(E) = ∂_{v}(E(k_{v})/2) ⊂ H^{1}(k_{v},_{2}E) = (k_{v}^{∗}/k_{v}^{∗2})^{2}.

For some places v ∈ L this condition on ξ ∈ (O_{L}^{∗}/O_{L}^{∗2})^{2} can be made explicit. For v ∉ S_{0}, letting κ_{v} be the residue field at v we have a residue map :

δ_{v} : H^{1}(k_{v}, (Z/2)^{2}) → H^{0}(κ_{v}, (Z/2)^{2})

simply given by valuation modulo 2 on each summand :

δ_{v} : (k_{v}^{∗}/k_{v}^{∗2})^{2} → (Z/2)^{2}.

If v is not in S_{0} and v is a place of good reduction of E, then K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} = H^{1}(O_{v},_{2}E) = W_{v}(E) ; hence the condition at such a v simply says that the image of ξ under the composite map H^{1}(O_{S}, (Z/2)^{2}) → H^{1}(k_{v}, (Z/2)^{2}) → H^{0}(κ_{v}, (Z/2)^{2}) is zero.

In general, as long as v does not belong to S_{0}, since K_{v} = H^{1}(O_{v}, (Z/2)^{2}) the condition that ξ belongs to W_{v} + K_{v} is equivalent to the condition that δ_{v}(ξ) ∈ H^{0}(κ_{v}, (Z/2)^{2}) belongs to δ_{v}(W_{v}) ⊂ H^{0}(κ_{v}, (Z/2)^{2}).

In this paper, we shall be interested in the following concrete case of bad reduction. We consider an elliptic curve given by an affine equation y^{2} = (x − c_{1})(x − c_{2})(x − c_{3}). We are concerned with a (finite) place v ∉ S_{0} such that all c_{i} are in O_{v} and that (c_{1}−c_{2})(c_{2}−c_{3})(c_{3}−c_{1}) has valuation one. If c_{1} − c_{2} is the uniformizing parameter at v (hence the other two are units), then δ_{v}(W_{v}) = {(0, 0), (1, 1)} ⊂ (Z/2)^{2}. If c_{2} − c_{3} is the uniformizing parameter, then δ_{v}(W_{v}) = {(0, 0), (0, 1))} ⊂ (Z/2)^{2}. If c_{3} − c_{1} is the uniformizing parameter, then δ_{v}(W_{v}) = {(0, 0), (1, 0))} ⊂ (Z/2)^{2}.

In the particular case under consideration, one can also phrase this by saying that the valuations of an element of U′_{L}(E) must belong to the subgroup of (Z/2)^{2} consisting of the image of _{2}E(k) under the composite map

2E(k) → E(k_{v})/2 → H^{1}(k_{v},_{2}E) → H^{0}(κ_{v}, (Z/2)^{2}) = (Z/2)^{2}.

2. Proof of the Theorem

2.1 Fibres with points everywhere locally

Let k be a number field. We are given a smooth projective surface E/k equipped with a fibration p : E → P_{k}^{1}, the generic fibre of which is given by the affine equation

Y^{2} = (X − c_{1}(t))(X − c_{2}(t))(X − c_{3}(t)).

We assume that the product R(t) = (c_{1}(t) − c_{2}(t))(c_{2}(t) − c_{3}(t))(c_{3}(t) − c_{1}(t)) is squarefree and non-constant, and that the various polynomials p_{i}(t) = c_{j}(t) − c_{k}(t) all have the same even degree.

We are given a triple (m_{1}(t), m_{2}(t), m_{3}(t)) of non-zero squarefree elements of k[t], whose product is a square and divides R(t)^{2}. We shall represent each m_{i}(t) as the product of a constant ρ_{i} and polynomials of the type R_{M}(t), where R_{M}(t) runs through the monic irreducible polynomials dividing R(t).

By assumptions (1) and (2) of the Theorem, we have :

The class m of (m_{1}(t), m_{2}(t), m_{3}(t)) lies in 𝔖_{0} ⊂ H^{1}(U,_{2}E) and it does not belong to ∂(_{2}E(U)) ⊂ H^{1}(U,_{2}E).

The surface given in affine coordinates by the system of equations

x − c_{i}(t) = m_{i}(t)y_{i}^{2}, i = 1, 2, 3

in affine space with coordinates x, t, y_{1}, y_{2}, y_{3}, together with its projection on the t-axis, may be extended to a unique smooth proper k-variety X, equipped with a relatively minimal fibration π : X → P_{k}^{1} (see [Sha66]). The singular fibres of this fibration are over the closed points M ∈ A_{k}^{1} which correspond to the irreducible factors of R(t). We let T be the set of such points. Over the quadratic (or trivial) extension K_{M} = k_{M}(\sqrt{δ_{M}(m)}) of k_{M}, the fibre X_{M} at M ∈ T breaks up into a union of two smooth conics. Over an algebraic closure of k_{M}, this fibre consists of two smooth conics intersecting transversally in two points.

We let Br_{vert}(X) ⊂ Br(X) be the vertical Brauer group with respect to the map π, as defined in the introduction. Since the geometric fibres are reduced, the quotient Br_{vert}(X)/Br(k) is finite ([Sk96], Cor. 4.5). We start with an adèle {P_{v}} ∈ X(𝔸_{k}) = ∏_{v∈Ω} X(k_{v}). We assume that for all A ∈ Br_{vert}(X), we have

∑_{v∈Ω} inv_{v}(A(P_{v})) = 0 ∈ Q/Z,

in other words we assume that {P_{v}} is in the kernel of all reciprocity maps θ_{A} : X(𝔸_{k}) → Q/Z associated to elements A ∈ Br_{vert}(X) (see [CT97]).

Let S_{1} be a finite set of places of k containing all the archimedean places ; for v ∈ S_{1}, let U_{v} ⊂ P^{1}(k_{v}) be an open neighbourhood of π(P_{v}). Under the five assumptions of the Theorem, we want to show that we may find a k-point x ∈ P^{1}(k), lying in U_{v} for v ∈ S_{1}, and such that moreover the fibre X_{x} = π^{−1}(x) is smooth and has infinitely many rational points. This will prove statement (a) in the Theorem. Statements (b) and (c) will then follow easily.

A standard procedure, which uses the finiteness of Br_{vert}(X)/Br(k), and is described in full detail in [CT/Sk/SwD97b] (proof of Theorem 1.1) allows us to choose the rational point at infinity and the parameter t on A_{k}^{1} = Spec(k[t]) = P_{k}^{1} \ ∞ so that :

(a) For each place v ∈ Ω, the point P_{v} lies on a smooth fibre of π and does not lie on the fibre at infinity (i.e. π(P_{v}) lies in U(k_{v}), for U ⊂ A_{k}^{1} as in the introduction).

(b) For each place v ∈ S_{1}, the projection X(k_{v}) → P^{1}(k_{v}) admits an analytic section σ_{v} : U_{v} → X(k_{v}) over U_{v}.

(c) For each real place v, the neighbourhood U_{v} coincides with the open set t_{v} > 0, and for v complex, U_{v} contains the point at infinity.

The mentioned procedure involves a projective transformation of the projective line P_{k}^{1} sending a k-point m ∈ P^{1}(k) to the point at infinity. Because the polynomials p_{i}(t) = c_{j}(t)−c_{k}(t) all have the same even degree, after suitable and obvious birational transformations we find that the restriction to A_{k}^{1} of E/P_{k}^{1} in the new system of coordinates still reads as it did, namely

Y^{2} = (X − c_{1}(t))(X − c_{2}(t))(X − c_{3}(t)),

with the same degree and coprimality conditions on the p_{i} = c_{j} − c_{k}. One easily checks that the new 𝔖 and the new 𝔖_{0} are the obvious transforms of the old ones, that the new m lies in the new 𝔖_{0} and that condition (2) in the main Theorem (Introduction) is preserved (the other ones are obviously preserved). From now on, we assume that conditions (a),(b),(c) above are fulfilled.

Given (the new) m and its associated model X = X(m)/P_{k}^{1}, we shall define a finite set S_{0} of ‘bad’ places of k. As in proposition 1.1.1, we want it to contain all the archimedean places and all the places above 2, and we want the class group of O_{S_{0}} to be trivial. We also want it to contain the set S_{1} given above. Given v ∉ S_{0}, we want all polynomials c_{i}(t) and m_{i}(t) to be O_{v}-integral, we want the constants ρ_{i} to be O_{v}-units, and we want R(t) to define a closed subscheme of Spec(O_{S_{0}}[t]) which is finite and étale over O_{S_{0}}, the ring of S_{0}-integers of k. Given any monic irreducible factor R_{M}(t) of R(t), we have the closed subscheme M̃ = Spec(O_{S_{0}}[t]/R_{M}(t)) of A_{O_{S_{0}}}^{1} , which is finite and étale over Spec(O_{S_{0}}). For each M ∈ T, we want the (trivial or) quadratic extension K_{M}/k_{M} (see above) to be unramified over the ring of S_{0}-integers of k_{M}. For much later use, we want S_{0} to contain all the places above a prime of Q less than or equal to the product of the degree [k : Q] and the degree of the polynomial R(t). We also want S_{0} to be big enough so that the fibration π : X → P_{k}^{1} extends to a fibration X /P_{O_{S_{0}}}^{1} with the following property. The scheme Spec(O_{S_{0}}[t]/R(t)) = ∪_{M∈T} M̃ is the set of points of P_{O_{S_{0}}}^{1} whose fibre is singular. Over any geometric point of this subscheme, the fibre consists of two smooth conics intersecting transversally in two points. (That this can be achieved may be seen directly on projective equations, as in the introduction.)

Given a finite set of primes S containing S_{0}, we shall be interested in the affine line A_{O_{S}}^{1} over O_{S} and in its open set U_{S} = Spec(O_{S}[t][1/R]). The group of units of O_{S}[t][1/R] is the direct product of O_{S}^{∗} and the free group generated by the polynomials P_{M} for M ∈ T, as one easily checks. Also, one easily checks that the Picard group of O_{S}[t][1/R] vanishes (this uses the vanishing of the Picard group of O_{S}).

A point x ∈ P^{1}(k) = P^{1}(O_{S_{0}}) (resp. x ∈ P^{1}(k_{v}) = P^{1}(O_{v}) for v ∉ S_{0}) defines a section x̃ : Spec(O_{S_{0}}) → P_{O_{S_{0}}}^{1} (resp. a local section x̃ : Spec(O_{v}) → P_{O_{v}}^{1}). We shall call such a point transversal if the corresponding section (resp. local section) x̃ defined over Spec(O_{S_{0}}) (resp. Spec(O_{v})) transversally intersects with the finite étale O_{S_{0}}-scheme (resp O_{v}-scheme) which is the union of all the (restrictions over O_{S_{0}} of the) M̃’s (M ∈ T) and the section at infinity of P_{O_{S_{0}}}^{1} (resp. which is the pull-back to O_{v} of this étale scheme). In less geometric terms, this simply means that for each M ∈ T, the element R_{M}(x) ∈ k^{∗} (resp. R_{M}(x) ∈ k_{v}^{∗}) is either a unit or a uniformizing parameter at each v ∉ S_{0} (resp. at v), and that if v(x) < 0, then v(x) = −1.

Lemma 2.1.1 Let S_{0} be as above. Let v ∉ S_{0} and x ∈ A^{1}(k_{v}) be a transversal point.

(a) If x̃ does not intersect any of the M̃’s, then X_{x}(k_{v}) ≠ ∅;

(b) if x̃ intersects M̃ at some place w of k_{M} (w unramified over v, of degree one over v), then X_{x}(k_{v}) ≠ ∅ if and only if w splits in K_{M}.

Proof Let X_{x}/O_{v} be the O_{v}-scheme which is the inverse image of X /P_{O}^{1} under x̃.

(a) The reduction mod v of the fibre X_{x}/O_{S} is a smooth curve of genus one over a finite field, hence has a rational point by F. K. Schmidt’s theorem (or Lang’s theorem). Such a point can be lifted to a point over O_{v}, hence over k_{v}, by Hensel’s lemma.

(b) Since X_{x}/O_{S} is proper we have X_{x}(k_{v}) = X_{x}(O_{v}). Thus a point of X_{x}(k_{v}) gives a local section over Spec(O_{v}) which intersects just one component of the closed fibre ; the closed fibre is therefore split. Thus the unramified extension K_{M}/k_{M} splits over w. Conversely, if this is so, the two conics which constitute the closed fibre of X_{x}/O_{S} at v are individually defined over F_{w} = F_{v}. Any conic contains at least 3 points, thus we can always find a smooth rational point in such a fibre. Again, this point can be lifted to a point over k_{v} by Hensel’s lemma.

Given M ∈ T, we may write R_{M}(t) = N_{k_{M} /k}(t − a_{M}) with a_{M} ∈ k_{M}. Let us define

α_{M}(t) = Cores_{k_{M} /k}((K_{M}/k_{M}, t − a_{M})) ∈ Br(k(P_{k}^{1})).

One may check that the inverse image of such a classe on X is unramified on the complement in X of the fibre at infinity of X/P_{k}^{1}, but this will be used only in remark 2.5.4 (v).

Lemma 2.1.2 Let π : X → P_{k}^{1} and S_{0} be as above, and let {P_{v}} ∈ ∏_{v∈Ω} X(k_{v}) be an adelic point of X such that ∑_{v∈Ω} inv_{v}(A(P_{v})) = 0 for all A ∈ Br_{vert}(X). There exists a finite set of places S of k, containing S_{0}, and there exist points Q_{v} ∈ X_{U}(k_{v}) for v ∈ S, with Q_{v} = P_{v} for v ∈ S_{0} (hence for v ∈ S_{1}), such that

∑_{v∈S} inv_{v}(α_{M}(π(Q_{v}))) = 0

for all M ∈ T.

Proof This is just a special case of Harari’s ‘formal lemma’ ([Ha94], Cor. 2.6.1 ; [CT/SwD94], Theorem 3.2.1).

From now on and until the end of section 2, the set S of places will be fixed and taken as in lemma 2.1.2 above. So will be x_{v} = π(Q_{v}) (with smooth fibres E_{x_{v}} and X_{x_{v}}) for v ∈ S.

Proposition 2.1.3 Let π : X → P_{k}^{1}, the set of places S and the Q_{v} ∈ X(k_{v}), v ∈ S be as in the conclusion of the previous lemma. For each point M ∈ T ∪ ∞, let L_{M} be a finite set of places of k such that L_{M} ∩ S = ∅, L_{M} ∩ L_{N} = ∅ for M ≠ N. Let x ∈ U(k) ⊂ P^{1}(k) = P^{1}(O_{S}).

Assume :

(1) For each place v ∈ L_{M}, there is an associated place w of k_{M} (of degree one over k) which splits in the quadratic extension K_{M} = k_{M}(\sqrt{δ_{M}(m)})/k_{M} (i.e. δ_{M}(m) is a square in k_{Mw}) ;

(2) each place v ∈ L_{∞} splits in each quadratic extension k(\sqrt{N_{k_{M} /k}(δ_{M}(m))})/k (i.e. N_{k_{M} /k}(δ_{M}(m)) is a square in k_{v} for each v ∈ L_{∞}) ;

(3) x is transversal over O_{S};

(4) for any M ∈ T, the (transversal) intersection of x and M̃, when viewed on M̃, consists of the places w ∈ L_{M} and just one place w_{M} of k_{M}, of degree one over a place v_{M} of k;

(5) the transversal intersection of x and the section at infinity of P_{O_{S}}^{1} consists exactly of the places v ∈ L_{∞} (hence for any such v, we have v(x) = −1) ;

(6) x is very close to x_{v} = π(Q_{v}) ∈ U(k_{v}) for v ∈ S and lies in U_{v} for v ∈ S_{1}.

Let L = S ∪ (∪_{M∈T} L_{M}) and L(x) = L ∪ (∪_{M∈T} {v_{M}}).

Then one has :

(i) X_{x}(k_{v}) ≠ ∅ for all places v of k;

(ii) the ‘evaluation map’ ev_{x} : (O_{L∪L_{∞}}[U]^{∗}/O_{L∪L_{∞}}[U]^{∗2})^{2} → (O_{L(x)∪L_{∞}}^{∗}/O_{L(x)∪L_{∞}}^{∗2})^{2} is an isomorphism ;

(iii) the images of the points of order 2 on E_{x} under the Kummer map ∂ and the class of m(x) in (k^{∗}/k^{∗2})^{2} lie in the subgroup U_{L(x)} = (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2} and are independent.

Proof (i) We proceed as in [CT/SwD94] and [CT/Sk/SwD97b]. We have

∑_{v∈S} inv_{v}(α_{M}(π(Q_{v}))) = 0

for all M ∈ T. For x ∈ k close enough to each x_{v} for v ∈ S (condition (6)), we therefore have

∑_{v∈S} inv_{v}(α_{M}(x)) = 0;

hence

∑_{v∉S} inv_{v}(α_{M}(x)) = 0

by the global reciprocity law. Now inv_{v}(α_{M}(x)) = inv_{v}(Cores_{k_{M} /k}((K_{M}/k_{M}, x−a_{M}))) vanishes if R_{M}(x) = N_{k_{M} /k}(x − a_{M}) is a unit at v ∉ S. Thus for v ∉ S we have inv_{v}(α_{M}(x)) = 0 except perhaps if v ∈ L_{∞} ∪ L_{M} ∪ {v_{M}}. For v ∈ L_{M}, condition (1) guarantees that v splits in the unramified extension K_{M}/k_{M}, hence

inv_{v}(α_{M}(x)) = inv_{v}(Cores_{k_{M} /k}((K_{M}/k_{M}, x − a_{M}))) = 0.

For v ∈ L_{∞}, using (5) and the projection formula, we have

inv_{v}(α_{M}(x)) = inv_{v}(Cores_{k_{M} /k}((K_{M}/k_{M}, x − a_{M}))) = inv_{v}(N_{k_{M} /k}(δ_{M}(m)), 1/π_{v})

where π_{v} denotes a uniformizer for k_{v}, and this last quantity is zero since N_{k_{M} /k}(δ_{M}(m)) is a square in k_{v} (hypothesis (2)). We now conclude

inv_{v_{M}} (α_{M}(x)) = inv_{v_{M}} (Cores_{k_{M} /k}((K_{M}/k_{M}, x − a_{M}))) = 0.

In view of the transversality condition at v_{M} ((3) and (4)), this last vanishing implies that K_{M}/k_{M} splits at v_{M}.

Now for any place v ∉ S, either x does not intersect any of the M̃, whence X_{x}(k_{v}) ≠ ∅ by lemma 2.1.1 (a), or x meets one M̃ transversally at v (viewed as a place of k or of k_{M}) and this place of k_{M} splits in K_{M}, whence X_{x}(k_{v}) ≠ ∅ by lemma 2.1.1 (b). If x is close enough to x_{v} for v ∈ S (condition (6) – already used above), we have X_{x}(k_{v}) ≠ ∅ by the implicit function theorem. This completes the proof of (i).

(ii) Evaluating ‘functions’ from O_{L∪L_{∞}}[U]^{∗}/O_{L∪L_{∞}}[U]^{∗2} at x gives the identity map on O_{L∪L_{∞}}^{∗}/O_{L∪L_{∞}}^{∗2}. The group O_{L∪L_{∞}}[U]^{∗} is generated by O_{L∪L_{∞}}^{∗} and the polynomials R_{M}(t) for M ∈ T.

Consider the diagram

(2.1.1)

0 → (O_{L∪L_{∞}}^{∗}/O_{L∪L_{∞}}^{∗2})^{2} → (O_{L∪L_{∞}}[U]^{∗}/O_{L∪L_{∞}}[U]^{∗2})^{2} → ⊕_{M∈T} (Z/2)^{2} → 0
↓ =  ↓ ev_{x}  ↓ =
0 → (O_{L∪L_{∞}}^{∗}/O_{L∪L_{∞}}^{∗2})^{2} → (O_{L(x)∪L_{∞}}^{∗}/O_{L(x)∪L_{∞}}^{∗2})^{2} → ⊕_{M∈T} (Z/2)^{2} → 0

where the top right hand side arrow is induced by the valuation maps k(P^{1})^{∗} → Z at the points M ∈ T, and where the bottom right hand side arrow is induced by the valuation maps k^{∗} → Z at the places v_{M} of k. Using property (4), we find that this diagram is commutative. Hence the middle vertical map is an isomorphism, as claimed in (ii).

(iii) Let us first consider the Kummer map ∂ : E(U) → (k[U]^{∗}/(k[U]^{∗2})^{2}. The point (X, Y) = (c_{1}, 0) is mapped to the class e_{1} = ((c_{1}−c_{2})(c_{1}−c_{3}), c_{1}−c_{2})) and the point (X, Y) = (c_{2}, 0) to the class e_{2} = (c_{2}−c_{1}, (c_{2}−c_{1})(c_{2}−c_{3})). As for the point (X, Y) = (c_{3}, 0), it is sent to (c_{3}−c_{1}, c_{3}−c_{2}), which is the product of the two previous classes. By assumption, R(t) = (c_{2}−c_{3})(c_{3}−c_{1})(c_{1}−c_{2}) is a non-constant squarefree polynomial and the various c_{i} − c_{j} all have the same even degree, which is therefore positive. Comparing the irreducible factors of the coordinates of e_{1} and e_{2} shows that these classes are independent in (k[U]^{∗}/(k[U]^{∗2})^{2} (here and later, independence refers to independence in the F_{2}-vector spaces under consideration). As for m = (m_{1}, m_{2}), it lies in 𝔖_{0} and by assumption does not belong to the subgroup of (k[U]^{∗}/(k[U]^{∗2})^{2} spanned by the two previous classes. The classes m, e_{1}, e_{2} all lie in (O_{S_{0}}[U]^{∗}/O_{S}[U]^{∗2})^{2}. By (ii), evaluating these classes at x gives rise to independent classes in (O_{L(x)∪L_{∞}}^{∗}/O_{L(x)∪L_{∞}}^{∗2})^{2}. To complete the proof of (iii), it only remains to observe that since any c_{i} − c_{j} has even degree and since m belongs to 𝔖_{0}, the degrees (modulo 2) of the coordinates of m, e_{1}, e_{2} are all even, hence for any v ∈ L_{∞}, condition (5) implies that the v-adic valuation of the coordinates of ev_{x}(m), ev_{x}(e_{1}), ev_{x}(e_{2}) is even.

2.2 The group 𝔖 and the map δ

We want to give a precise description of the group 𝔖 defined in the introduction. In this subsection, k is any field of characteristic not 2. We have the split exact sequence

0 → (k^{∗}/k^{∗2})^{2} → (k[U]^{∗}/k[U]^{∗2})^{2} → ⊕_{M∈T} (Z/2)^{2} → 0

where the right hand side map is given by taking valuation at points in T, one splitting being given by the polynomials R_{M}.

Recall that the product R(t) = (c_{2} − c_{3})(c_{3} − c_{1})(c_{1} − c_{2}) is squarefree and differs from ∏_{M∈T} R_{M} by a constant. Hence each R_{M} divides a well-defined c_{i} − c_{j}. For any M ∈ T, let us define (β_{M}, γ_{M}) ∈ (Z/2)^{2} in the following explicit fashion. If R_{M} divides c_{1} − c_{2}, let (β_{M}, γ_{M}) = (1, 1). If R_{M} divides c_{2} − c_{3}, let (β_{M}, γ_{M}) = (0, 1). If R_{M} divides c_{3} − c_{1}, let (β_{M}, γ_{M}) = (1, 0). Note that since none of the c_{i} − c_{j} is constant, any non-zero element of (Z/2)^{2} is equal to (β_{M}, γ_{M}) for some M.

Comparing this definition with the definition of 𝔖 in the introduction, we find that the group 𝔖 ⊂ (k[U]^{∗}/k[U]^{∗2})^{2} is the inverse image of the subgroup

⊕_{M∈T} (Z/2)(β_{M}, γ_{M}) ⊂ ⊕_{M∈T} (Z/2 × Z/2)

under the right hand side map in the above diagram. In other words, the group 𝔖 ⊂ (k[U]^{∗}/k[U]^{∗2})^{2} is isomorphic to the direct sum of (k^{∗}/k^{∗2})^{2} and the free Z/2-module with generators (R_{M}^{β_{M}} , R_{M}^{γ_{M}}) for M running through T. Since the polynomials R_{M} all belong to O_{S_{0}}[U]^{∗}, in the description we have just given, we may replace k everywhere by any ring between O_{S_{0}} and k.

We shall later need to consider the map η : (k[U]^{∗}/k[U]^{∗2})^{2} → (Z/2)^{2} given by the valuation at infinity (degree map). Indeed, the subgroup 𝔖_{0} ⊂ 𝔖 is precisely the kernel of the composite map 𝔖 ⊂ (k[U]^{∗}/k[U]^{∗2})^{2} ⟶^{η} (Z/2)^{2}, which we shall also denote η. This map sends the element (R_{M}^{β_{M}} , R_{M}^{γ_{M}}) to (β_{M}.deg(R_{M}), γ_{M}.deg(R_{M})) ∈ (Z/2)^{2}.

We now discuss the map δ. Let b(t) = (b_{1}(t), b_{2}(t)) ∈ 𝔖 ⊂ (k[U]^{∗}/k[U]^{∗2})^{2}. Let M ∈ T. Let i ∈ {1, 2, 3} be the unique value such that R_{M} divides c_{j} − c_{k}, where {i, j, k} = {1, 2, 3}. Let Q_{j} be a point of order exactly 2 of E(U) = E(k(t)) other than the point with coordinate (X, Y) = (c_{i}, 0). Consider the image ε_{j} = (ε_{j,1}, ε_{j,2}) ∈ H^{1}(U, (Z/2)^{2}) of Q_{j} under the Kummer map ∂ : E(U)/2 → H^{1}(U, (Z/2)^{2}). We can then form the cup-product

< b, ε_{j} >= (b_{1}, ε_{j,2}) − (ε_{j,1}, b_{2}) ∈ H^{2}(U, Z/2) ⟶^{≃} _{2}Br(U).

Lemma 2.2.1 Let δ_{M} : 𝔖 → k_{M}^{∗}/k_{M}^{∗2} be the set-theoretic map defined in the introduction.

(i) For any b ∈ 𝔖 as above, δ_{M}(b) coincides with the image of < b, ε_{j} >∈ _{2}Br(U) under the residue map _{2}Br(U) → H^{1}(k_{M}, Z/2) = k_{M}^{∗}/k_{M}^{∗2}. In particular, δ_{M} is a homomorphism.

(ii) The image ∂(E(U)) of E(U) under the Kummer map lies in the kernel of δ_{M}.

(iii) For (c, d) ∈ (k^{∗}/k^{∗2})^{2} ⊂ 𝔖, we have δ_{M}(c, d) = c^{γ_{M}} .d^{β_{M}} ∈ k_{M}^{∗}/k_{M}^{∗2}.

(iv) The kernel of the diagonal map δ : 𝔖 → ⊕_{M∈T} k_{M}^{∗}/k_{M}^{∗2} is a finite group.

(v) The kernel of the composite map

𝔖 →^{δ} ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) >

is a finite group.

Proof (i) This is checked by a straightforward but tedious computation. Let us assume that the polynomial R_{M} divides p_{1} = c_{2} − c_{3}. We may assume that b_{1}, b_{2} are in k[t] and are squarefree and that b_{3} is b_{1}.b_{2} divided by its square factors. Since b ∈ 𝔖, we know that b_{1} is prime to p_{1}, hence in particular b_{1}(M) ≠ 0 ∈ k(M). The image of the 2-torsion point Q_{2} (X = c_{2}, Y = 0) under the Kummer map is ε_{2} = (ε_{2,1}, ε_{2,2}) = (c_{2} − c_{1}, (c_{2} − c_{1})(c_{2} − c_{3})) ∈ (k[U]^{∗}/k[U]^{∗2})^{2}. Hence

< b, ε_{2} >= (b_{1}, (c_{2} − c_{1})(c_{2} − c_{3})) − (c_{2} − c_{1}, b_{2}) ∈ H^{2}(U, Z/2) = _{2}Br(U).

Now

(b_{1}, (c_{2}−c_{1})(c_{2}−c_{3}))−(c_{2}−c_{1}, b_{2}) = (b_{1}, (c_{2}−c_{1})(c_{2}−c_{3}))+(b_{2}, c_{2}−c_{1}) = (b_{1}, c_{2}−c_{3})+(b_{3}, c_{2}−c_{1}).

Since the R_{M} valuation of b_{1} and of c_{2} − c_{1} is 0, and that of c_{2} − c_{3} is one, we find that the residue of < b, ε_{2} > at M is

(2.2.1)  b_{1}(M).(c_{2}(M) − c_{1}(M))^{v_{M} (b_{3})} ∈ k_{M}^{∗}/k_{M}^{∗2},

where v_{M} denotes the valuation at M. We leave it to the reader to write down the equations of the surface X(b) above a neighbourhood of M and check that this class coincides with δ_{M}(m) as defined in the introduction : one has to distinguish two cases, depending on whether the R_{M}-valuation of each b_{i} is even (hence zero), or not.

(ii) This is proved by a case-by-case discussion involving valuation arguments and formula (2.2.1), in the case where R_{M} divides p_{1}, and is checked similarly in the other cases.

(iii) The statement follows from formula (2.2.1) in the case where R_{M} divides p_{1}, and is checked similarly in the other cases.

(iv) Since 𝔖 is spanned by the finitely many elements (R_{M}^{β_{M}} , R_{M}^{γ_{M}}) and by (k^{∗}/k^{∗2})^{2}, it is enough to check that the kernel of the induced map δ : (k^{∗}/k^{∗2})^{2} → ⊕_{M∈T} k_{M}^{∗}/k_{M}^{∗2} is finite. There exists a point M_{1} ∈ T with (β_{M_{1}}, γ_{M_{1}}) = (1, 0) and a point M_{2} ∈ T with (β_{M_{2}}, γ_{M_{2}}) = (0, 1). It is enough to show that the kernel of the induced map k^{∗}/k^{∗2} ⊕k^{∗}/k^{∗2} → k_{M_{1}}^{∗}/k_{M_{1}}^{∗2} ⊕k_{M_{2}}^{∗}/k_{M_{2}}^{∗2} is finite. By (ii), this map simply reads (c, d) ↦ (c, d), and we are reduced to the simple algebraic fact : if K/k is a finite separable extension of a field k of characteristic not 2, the kernel of the map k^{∗}/k^{∗2} → K^{∗}/K^{∗2} is finite (reduce to the case where K/k is Galois with group G, and use the restriction inflation sequence, which shows that the kernel is the finite group H^{1}(G, Z/2) = Hom(G, Z/2)).

(v) This immediately follows from (iv).

Remark To any b = (b_{1}, b_{2}) ∈ (k(t)^{∗}/k(t)^{∗2})^{2} one may associate the element

ξ = (X − c_{1}(t), b_{2}(t)) + (X − c_{2}(t), b_{1}(t))

in the (2-torsion subgroup of the) Brauer group of the generic fibre of the regular minimal model E/P_{k}^{1}. The assumption b ∈ 𝔖 guarantees that all residues of ξ at generic points of components of closed fibres over A_{k}^{1} are constant. Given M ∈ T, the fibre E_{M} has two components, the neutral component C_{M} and the ‘second’ one C′_{M}. The residue δ_{C′_{M}} ∈ H^{1}(k(C′_{M}), Z/2) at the second component comes from H^{1}(k_{M}, Z/2), and one checks that it comes from δ_{M}(b) ∈ k_{M}^{∗}/k_{M}^{∗2} = H^{1}(k_{M}, Z/2) as defined in the introduction. Suppose R_{M} divides p_{i}. Then any 2-torsion point Q_{j}, j ≠ i, defines a section of E/P_{k}^{1} which hits the second component (whereas 0 and the point (X, Y) = (c_{i}, 0) hit the neutral component). The class < b, ε_{j} > is none other than the inverse image under the section Q_{j} of the class ξ, and the residue of this inverse image at M is just the inverse image under Q_{j} of the (constant) residue δ_{M}(b), hence equal to δ_{M}(b).

2.3 Independence on x : the space

Given x ∈ k as in proposition 2.1.3, let us consider the curve E_{x}. It has good reduction at all places outside L(x) (it has indeed good reduction at places in L_{∞}). In subsection 1.1 we defined U_{L(x)} = (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2} and in subsection 1.2 we considered the subspace U′_{L(x)} ⊂ U_{L(x)} attached to the curve E_{x}. By proposition 1.2.3, this is the space with which one computes the Selmer group. It is defined by local conditions, namely, it consists of elements in (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2} whose image under the map

(O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2} → (k_{v}^{∗}/k_{v}^{∗2})^{2}

belongs to W_{v}(E_{x}) + K_{v}, this for all v ∈ L(x). In particular, statement (iii) in proposition 2.1.3 may be completed as follows : for x as in that proposition, the images of the points of order 2 on E_{x} under the Kummer map ∂ and the class of m(x) in (k^{∗}/k^{∗2})^{2} lie in the subgroup U′_{L(x)} and are independent.

The spaces K_{v} do not depend on x. By the approximation condition (6) in proposition 2.1.3, for v ∈ S the elliptic curve E_{x} is very close to E_{x_{v}} in the v-adic topology ; hence, with the notation of the first section, the subspaces W_{v}(x) = W_{v}(E_{x}) and W_{v}(x_{v}) = W_{v}(E_{x_{v}}) of V_{v} are identical. Hence the conditions defining U′_{L(x)} at the places of S do not depend on x.

Let us revert to the considerations at the end of subsection 1.2. They imply the following : given R_{M}(t) a monic irreducible divisor of R(t), and (β_{M}, γ_{M}) ≠ 0 ∈ (Z/2)^{2} as in the beginning of subsection 2.2, for any x ∈ U(k_{v}) which intersects M̃ transversally above the place v of k, the subspace W_{v}(x) + K_{v} ⊂ V_{v} consists of the elements of (k_{v}^{∗}/k_{v}^{∗2})^{2} whose image under the valuation map lies in {(0, 0), (β_{M}, γ_{M})} ⊂ (Z/2)^{2}. Thus for a prime v ∈ L(x) \ S, the condition that an element of (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2} lies in W_{v}(x)+K_{v} only depends on which L_{M} ∪v_{M} the place v belongs to, it does not depend on x. Note that for v ∈ L(x) \ S, the quotient (W_{v} + K_{v})/K_{v} ≃ W_{v}/W_{v} ∩ K_{v} is isomorphic to Z/2.

Lemma 2.3.1 Let L and x be as in proposition 2.1.3.

a) The inverse image N_{0} = U′_{S} in (O_{S}^{∗}/O_{S}^{∗2})^{2} ⊂ V_{L} of U′_{L(x)} ⊂ V_{L(x)} (under the diagonal map (O_{S}^{∗}/O_{S}^{∗2})^{2} → (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2}) depends neither on L nor on x.

b) The inverse image U′_{L} of U′_{L(x)} ⊂ V_{L(x)} in U_{L} = (O_{L}^{∗}/O_{L}^{∗2})^{2} ⊂ V_{L} (under the diagonal map (O_{L}^{∗}/O_{L}^{∗2})^{2} → (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2}) only depends on L = S ∪ (∪_{M∈T} L_{M}), it does not depend on x.

Proof For the subgroup (O_{S}^{∗}/O_{S}^{∗2})^{2} ⊂ (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2}, the condition at a place v ∉ S is automatically satisfied, since K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} for such a v, and we have seen that the conditions at the places in S are independent of L and x.

For the subgroup (O_{L}^{∗}/O_{L}^{∗2})^{2} ⊂ (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2}, the condition at a place v_{M} (M ∈ T) is automatically satisfied, since K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} for any v ∉ S, and we have seen that the conditions at other places are independent of x.

The proof to follow will involve the construction of a ‘subdiagram’ of the diagram appearing in the proof of 2.1.3. Basically, we want to look at the inverse image of U′_{L(x)} ⊂ U_{L(x)∪L_{∞}} under ev_{x}. Before doing this, we shall fix L_{∞}.

Consider the finite subgroup 𝔖_{S} of 𝔖 spanned by U_{S} and the elements (R_{M}(t)^{β_{M}} , R_{M}(t)^{γ_{M}}) for M running through T. This subgroup contains m. For each M ∈ T, we have the finite subgroup δ_{M}(𝔖_{S}) ⊂ k_{M}^{∗}/k_{M}^{∗2}, which defines a multiquadratic extension F_{M}/k_{M}. Let F/k be the composite of the extensions F_{M}/k. It is a consequence of Tchebotarev’s theorem that there exist infinitely many principal ideals (π_{v}) in the ring of integers O of k such that the (chosen) generator π_{v} is a square (as a matter of fact, as close as one wishes to 1) in each completion k_{w} for w ∈ S, and such that moreover the place v splits in each quadratic extension k(\sqrt{N_{k_{M} /k}(δ_{M}(m))})/k, M ∈ T. Choose v_{∞} ∉ S a place which satisfies these properties (required in proposition 2.1.3), and which moreover is unramified in the extension F/k. Let μ = π_{v_{∞}} ∈ O be a generator of the corresponding principal ideal. For each w ∈ S, μ is a square in k_{w}. For the rest of section 2, we fix the set L_{∞} = {v_{∞}}, and we let x_{∞} = x_{v_{∞}} = 1/μ ∈ k_{v_{∞}}.

Remark In the special case where all the R_{M}’s are of even degree, or more generally under an assumption slightly stronger than (D) (see remark 2.5.4 (iv)), one may take L_{∞} = ∅ and μ = 1 in the proof to follow.

Lemma 2.3.2 Let X and S be as above. For each M ∈ T, there exists a uniquely defined (a_{M}, b_{M}) ∈ U_{S} = (O_{S}^{∗}/O_{S}^{∗2})^{2} such that for any L_{∞} = {v_{∞}} and μ as above, any L and x as in proposition 2.1.3,

(a_{M}(μ^{deg(R_{M})}R_{M}(x))^{β_{M}} , b_{M}(μ^{deg(R_{M})}R_{M}(x))^{γ_{M}}) ∈ U_{L(x)∪L_{∞}}

is in the subspace of U′_{L(x)} ⊂ U_{L(x)∪L_{∞}} defined by the additional conditions that the local projection at v ∈ S is in K_{v} for all v ∈ S.

Proof Let us write ζ = (a_{M}(μ^{deg(R_{M})}R_{M}(x))^{β_{M}} , b_{M}(μ^{deg(R_{M})}R_{M}(x))^{γ_{M}}) ∈ U_{L(x)∪L_{∞}}. For any (a_{M}, b_{M}) ∈ U_{S}, and x as in proposition 2.1.3, for any valuation v ∉ L(x), the coordinates of ζ have even valuation : this is clear for v ∉ L_{∞}, and for v ∈ L_{∞}, it follows from the transversality conditions (3) and (5), together with the definition of μ. Thus ζ ∈ U_{L(x)}. For any (a_{M}, b_{M}) ∈ U_{S}, the local condition ζ ∈ W_{v}(E_{x}) + K_{v} is satisfied at places v ∈ L_{M_{1}} ∪ {v_{M_{1}}} for M_{1} ≠ M, since at such places the coordinates have even valuation and K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} for any v ∉ S. For any (a_{M}, b_{M}) ∈ U_{S}, the local condition is also satisfied at places in L_{M} ∪ {v_{M}} (see comment before lemma 2.3.1). Since x is v-adically very close to x_{v} for v ∈ S (condition (6) in proposition 2.1.3), the class of R_{M}(x) in k_{v}^{∗}/k_{v}^{∗2} for such v does not depend on x. Since μ is a square in each k_{v} for v ∈ S, this implies that the projection of ((μ^{deg(R_{M})}R_{M}(x))^{β_{M}} , (μ^{deg(R_{M})}R_{M}(x))^{γ_{M}}) to V_{S} is independent of x, L and L_{∞}. Since U_{S} ⊕ K_{S} = V_{S}, by modifying ((μ^{deg(R_{M})}R_{M}(x))^{β_{M}} , (μ^{deg(R_{M})}R_{M}(x))^{γ_{M}}) by a well-defined (a_{M}, b_{M}) ∈ U_{S}, we can achieve that the projection of

ζ = (a_{M}(μ^{deg(R_{M})}R_{M}(x))^{β_{M}} , b_{M}(μ^{deg(R_{M})}R_{M}(x))^{γ_{M}})

to V_{v} belong to K_{v} for all v ∈ S.

Let A_{M} = (a_{M}(μ^{deg(R_{M})}R_{M}(t))^{β_{M}} , b_{M}(μ^{deg(R_{M})}R_{M}(t))^{γ_{M}}) ∈ (O_{S∪L_{∞}}[U]^{∗}/O_{S∪L_{∞}}[U]^{∗2})^{2}. As we have seen above, these are elements of 𝔖 ⊂ (k[U]^{∗}/k[U]^{∗2})^{2}.

Notation 2.3.3 Let A ⊂ (O_{S∪L_{∞}}[U]^{∗}/O_{S∪L_{∞}}[U]^{∗2})^{2} ⊂ (k[U]^{∗}/k[U]^{∗2})^{2} be the subspace spanned by the A_{M} for M ∈ T. Let N = A ⊕ U′_{L} ⊂ (O_{L∪L_{∞}}[U]^{∗}/O_{L∪L_{∞}}[U]^{∗2})^{2} be the subspace generated by A and U′_{L} = U′_{L(x)} ∩ (O_{L}^{∗}/O_{L}^{∗2})^{2}.

The definition of N depends on the choice of the family of subspaces K_{v} (fixed once and for all) as well as on the points Q_{v} ∈ X(k_{v}), v ∈ S (which determine the spaces W_{v}(x) = W_{v}(Q_{v}) for v ∈ S), as well as on L and L_{∞}. But once these choices have been made, lemma 2.3.1 shows that N does not depend on the particular point x chosen as in proposition 2.1.3.

If we fix L_{∞} and μ as above, but still let the L_{M}’s and v_{M}’s vary, there is a subspace of (k[U]^{∗}/k[U]^{∗2})^{2} which is contained in any N, namely the subspace N_{0} ⊕ A ⊂ (O_{S∪L_{∞}}[U]^{∗}/O_{S∪L_{∞}}[U]^{∗2})^{2}, where N_{0} = U′_{S} is as above, and may be defined as N_{0} = N ∩ (O_{S}^{∗}/O_{S}^{∗2})^{2} ⊂ (O_{L∪L_{∞}}[U]^{∗}/O_{L∪L_{∞}}[U]^{∗2})^{2}. Both N_{0} and A are independent of L, and they clearly do not intersect.

As we observed in the proof of part (ii) of proposition 2.1.3, the evaluation map ev_{x} : (O_{L∪L_{∞}}[U]^{∗}/O_{L∪L_{∞}}[U]^{∗2})^{2} → (O_{L(x)∪L_{∞}}^{∗}/O_{L(x)∪L_{∞}}^{∗2})^{2} is an isomorphism as soon as properties (3) and (4) hold.

Proposition 2.3.4 Let L, L_{∞} and x be as in proposition 2.1.3, and let N be as above.

(a) The evaluation map ev_{x} from (O_{L∪L_{∞}}[U]^{∗}/O_{L∪L_{∞}}[U]^{∗2})^{2} to (O_{L(x)∪L_{∞}}^{∗}/O_{L(x)∪L_{∞}}^{∗2})^{2} induces an isomorphism between N and U′_{L(x)}.

(b) Denote by φ_{x} the composition of this isomorphism with the isomorphism τ_{L(x)}^{−1} : U′_{L(x)} ⟶^{≃} W′_{L(x)} = ⊕_{v∈L(x)}(W_{v}/W_{v} ∩ K_{v}). There is a direct sum decomposition :

N = N_{0} ⊕ A ⊕ φ_{x}^{−1}(⊕_{v∈L\S}(W_{v}/W_{v} ∩ K_{v})).

Proof That the isomorphism ev_{x} sends N into U′_{L(x)} follows from lemma 2.3.2. The image of U′_{L(x)} under the bottom horizontal map

(O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2} → ⊕_{M∈T} (Z/2)^{2}

in diagram (2.1.1) (map induced by the valuations v_{M} of k) has dimension equal to the order of T. Indeed, for each M, the image of the corresponding projection U′_{L(x)} → (Z/2)^{2} falls into {(0, 0), (β_{M}, γ_{M})}. On the other hand, the image of

A_{M}(x) = (a_{M}(μ^{deg(R_{M})}R_{M}(x))^{β_{M}} , b_{M}(μ^{deg(R_{M})}R_{M}(x))^{γ_{M}})

has projection (β_{M}, γ_{M}) at M and (0, 0) at the points of T \ M. Inspection of diagram (2.1.1) completes the proof of (a).

As for (b), note that φ_{x}(N_{0}) = φ_{x}(U′_{S}) = τ_{L(x)}^{−1}(U′_{S}) = ⊕_{v∈S}(W_{v}/W_{v} ∩ K_{v}) ⊂ W′_{L(x)}, and that, by the choice of μ and A_{M} (see lemma 2.3.2), the image of A_{M} ∈ N under φ_{x} has zero component at all places in S, as well as at places in L_{M_{1}} ∪ v_{M_{1}} for M_{1} ≠ M, whereas it has component 1 ∈ Z/2 = W_{v}/W_{v} ∩ K_{v} for v ∈ L_{M} ∪ v_{M}.

2.4 Independence on x : the pairing

In the previous subsection, given any L , L_{∞} and any x as in proposition 2.1.3, and N as in definition 2.3.3, we have obtained an isomorphism

ev_{x} : N ⟶^{≃} U′_{L(x)}.

According to proposition 1.2.3, there is an isomorphism τ_{L(x)} : W′_{L(x)} ⟶^{≃} U′_{L(x)} and the e_{L(x)}-pairing on V_{L(x)} induces a pairing e′_{L(x)} : U′_{L(x)} × W′_{L(x)} → Z/2 and then a symmetric pairing U′_{L(x)} × U′_{L(x)} → Z/2 defined by (x, y) ↦ e′_{L(x)}(x, τ_{L(x)}^{−1}(y)).

Notation 2.4.1 Let e_{N} : N × N → Z/2 be the symmetric pairing given by the composite map

N × N ⟶^{(ev_{x},ev_{x})} U′_{L(x)} × U′_{L(x)} ⟶^{(id,τ_{L(x)}^{−1})} U′_{L(x)} × W′_{L(x)} ⟶^{e′_{L(x)}} Z/2

Note that the space N depends on L and L_{∞} and that the pairing e_{N} should be denoted e_{N,L,L_{∞},x}, since it depends on all these data. The proof of the following important proposition will occupy the rest of this subsection.

Proposition 2.4.2 Let L, L_{∞} = {v_{∞}} and x be as above, and let N be as above. Then provided x is close enough to x_{∞} = 1/μ at v_{∞}, the restriction to the subspace N_{0} ⊕ A ⊂ (O_{S}[U]^{∗}/O_{S}[U]^{∗2})^{2} of the pairing e_{N} on N ⊂ (k[U]^{∗}/k[U]^{∗2})^{2} depends neither on L nor on x.

Proof As we have seen in subsection 2.2, the space N_{0} ⊕ A depends neither on x nor on L. Under the evaluation map ev_{x}, the space N_{0} = U′_{S} ⊂ (O_{S}^{∗}/O_{S}^{∗2})^{2} is mapped identically to itself.

Let W′_{S} → W′_{L(x)} be the map induced by the map W_{S} → W_{L(x)} sending the family {α_{v}}_{v∈S} to {α_{v}^{1}}_{v∈L(x)} with α_{v}^{1} = α_{v} for v ∈ S and α_{v}^{1} = 0 for v ∈ L(x) \ S. Then the diagram

U′_{L(x)} ⟶^{τ_{L(x)}^{−1}} W′_{L(x)}
↑  ↑
U′_{S}  ⟶^{τ_{S}^{−1}} W′_{S}

commutes, where the left vertical map is the natural inclusion of subgroups of (k^{∗}/k^{∗2})^{2}. One now easily checks that the following diagram is commutative

N × N ⟶^{(ev_{x},ev_{x})} U′_{L(x)} × U′_{L(x)} ⟶^{(id,τ_{L(x)}^{−1})} U′_{L(x)} × W′_{L(x)} ⟶^{e′_{L(x)}} Z/2
↑  ↑  ↑  ↑  ↑  ↑  ↑
N_{0} × N_{0} ⟶^{(ev_{x},ev_{x})} U′_{S} × U′_{S}  ⟶^{(id,τ_{S}^{−1})} U′_{S} × W′_{S}  ⟶^{e′_{S}} Z/2

From this diagram it follows that the restriction of the pairing e_{N} : N ×N → Z/2 to N_{0} ×N_{0} depends neither on x nor on L.

We now discuss the induced pairing on A × N_{0}. Given A_{M} ∈ A and ρ ∈ N_{0} ⊂ (O_{S}^{∗}/O_{S}^{∗2})^{2}, we have

e_{N}(A_{M}, ρ) = e′_{L(x)}(A_{M}(x), τ_{L(x)}^{−1}(ρ)).

From ρ ∈ N_{0} = U′_{S} we conclude as above (commutativity of the last but one diagram) that τ_{L(x)}^{−1}(ρ) ∈ W′_{L(x)} = ⊕_{v∈L(x)}W_{v}/(W_{v} ∩ K_{v}) is of the shape ∑_{v∈S} β_{v} (all the factors in L(x) \ S are zero). The β_{v}’s clearly do not depend on x. Now e_{N}(A_{M}, ρ) = ∑_{v∈S} e′_{v}(A_{M}(x), β_{v}) only depends on the image of x in ⊕_{v∈S}k_{v}, hence by condition (6) of proposition 2.1.3 is independent of x and L. (Note also that since μ is a square in each k_{v} for v ∈ S, the pairing essentially does not depend on L_{∞} = {v_{∞}}.) Thus the restriction of e_{N} to A × N_{0} does not depend on x or L. By symmetry of the pairing e_{N}, we conclude that the restriction of e_{N} to N_{0} × A also does not depend on x or L.

It remains to study the pairing e_{N} on A × A. In other terms, we want to discuss e_{N}(A_{P} , A_{Q}) for P and Q in T. Recall that

A_{M}(t) = (a_{M}(μ^{deg(R_{M})}R_{M}(t))^{β_{M}} , b_{M}(μ^{deg(R_{M})}R_{M}(t))^{γ_{M}})

with a_{M} and b_{M} in O_{S}^{∗}/O_{S}^{∗2}. We have e_{N}(A_{P} , A_{Q}) = e′_{L(x)}(A_{P} (x), τ_{L(x)}^{−1}(A_{Q}(x))). Let us write

A_{Q}(x) = ∑_{v∈L(x)}(α_{v} + β_{v}) ∈ ⨁_{v∈L(x)} V_{v}

with α_{v} ∈ W_{v} and β_{v} ∈ K_{v}. For v ∉ S ∪ L_{Q} ∪ v_{Q}, we may and will choose α_{v} = 0. We have

τ_{L(x)}^{−1}(A_{Q}(x)) = ∑_{v∈L(x)} α_{v} ∈ ⨁_{v∈L(x)} W_{v}/(W_{v} ∩ K_{v}),

with α_{v} = 0 for v ∉ S ∪ L_{Q} ∪ v_{Q}, hence :

e_{N}(A_{P} , A_{Q}) = ∑_{v∈L(x)} e_{v}(A_{P} (x), α_{v}) = ∑_{v∈S} e_{v}(A_{P} (x), α_{v}) + ∑_{v∈L_{Q}∪v_{Q}} e_{v}(A_{P} (x), α_{v}).

In view of property (6) of proposition 2.1.3, the sum ∑_{v∈S} e_{v}(A_{P} (x), α_{v}) depends neither on L nor on x (and it essentially does not depend on L_{∞} = {v_{∞}}, since μ is a square in each k_{v} for v ∈ S). Let us study ∑_{v∈L_{Q}∪v_{Q}} e_{v}(A_{P} (x), α_{v}).

Assume first P ≠ Q. Since L_{Q}∪{v_{Q}} does not meet L_{P} ∪{v_{P} }∪L_{∞}, and μ is a unit away from L_{∞}, the coordinates of A_{P} (x) are units at each v ∈ L_{Q}∪v_{Q}, hence e_{v}(A_{P} (x), β_{v}) = 0 at each such v (recall β_{v} ∈ K_{v} and K_{v} = (O_{v}^{∗}/O_{v}^{∗2})^{2} for v ∉ S) hence e_{v}(A_{P} (x), α_{v}) = e_{v}(A_{P} (x), α_{v} + β_{v}) = e_{v}(A_{P} (x), A_{Q}(x)) at each such v. We thus find

∑_{v∈L_{Q}∪v_{Q}} e_{v}(A_{P} (x), α_{v}) = ∑_{v∈L_{Q}∪v_{Q}} e_{v}(A_{P} (x), A_{Q}(x)).

All components of A_{P} (x) and A_{Q}(x) have even valuation at any v_{∞}, hence for any such place we have e_{v}(A_{P} (x), A_{Q}(x)) = 0. The sum above is thus equal to

∑_{v∈L_{Q}∪v_{Q}∪L_{∞}} e_{v}(A_{P} (x), A_{Q}(x)).

We claim that this sum does not depend on L or x, provided that x satisfies the proximity condition (6) in proposition 2.1.3 and also satisfies a suitable proximity condition at v_{∞}. Given the definition of the A_{M}’s, and the computation of the pairing e_{v}, this reduces to the following, which will be applied to S ∪ L_{∞} rather than S. Suppose we have two irreducible polynomials f(t) and g(t) in O_{S}[t] such that the product f(t)g(t) defines a closed subscheme of A_{O_{S}}^{1} which is finite and étale over O_{S}, and suppose that x ∈ P^{1}(O_{S}) is transversal to this closed subscheme. Let k_{g} = k[t]/g(t) = k(a_{g}), with g(t) = uN_{k_{g}/k}(t − a_{g}), u ∈ O_{S}^{∗}. Note that f(a_{g}) is a unit at all places of k_{g} not above a place of S. Then

∑_{v∉S, v(g(x))>0} inv_{v}(f(x), g(x)) = ∑_{v∉S, v(g(x))>0} inv_{v}(f(x), N_{k_{g}/k}(x − a_{g}))

since f(x) is a unit at the places v where v(g(x)) > 0. This last sum is equal to

∑_{v∉S, v(g(x))>0} inv_{v}(Cores_{k_{g}/k}(f(x), x − a_{g}))

by the projection formula, and in turn this sum may be written

∑_{v∉S, v(g(x))>0} inv_{v}(Cores_{k_{g}/k}(f(a_{g}), x − a_{g}))

since f(a_{g}) is a unit away from S. This also coincides with

∑_{v∉S} inv_{v}(Cores_{k_{g}/k}(f(a_{g}), x − a_{g})) = − ∑_{v∈S} inv_{v}(Cores_{k_{g}/k}(f(a_{g}), x − a_{g}))

where the last equality follows from the reciprocity law in global class field theory. Now for x close enough to given values in k_{v} for v ∈ S, this last term is constant. (For similar computations, see [SwD94], Lemma 2, [SwD95] and [CT/SwD94], Lemmas 7.2.1 and 7.2.4.)

Let us now assume P = Q. To fix ideas, let us assume that R_{P} (t) divides p_{3}(t) = c_{1}(t)−c_{2}(t). Then (β_{P} , γ_{P}) = (1, 1), hence A_{P} (t) = (a_{P} R_{P} (t), b_{P} R_{P} (t)). The point of order 2 on E_{x}(k_{v}) given by X = c_{1}(x) has image ((c_{1}(x) − c_{2}(x))(c_{1}(x) − c_{3}(x)), c_{1}(x) − c_{2}(x)) ∈ (k_{v}^{∗}/k_{v}^{∗2})^{2} under the Kummer map. For any v ∈ L_{P} ∪ v_{P}, it follows that one may write A_{P} (x) ∈ V_{v} as the sum of α_{v} = ((c_{1}(x) − c_{2}(x))(c_{1}(x) − c_{3}(x)), c_{1}(x) − c_{2}(x)) ∈ W_{v} and an element in K_{v}. Thus

∑_{v∈L_{P} ∪v_{P}} e_{v}(A_{P} (x), α_{v}) = ∑_{v∈L_{P} ∪v_{P}} e_{v}(A_{P} (x), ((c_{1}(x) − c_{2}(x))(c_{1}(x) − c_{3}(x)), c_{1}(x) − c_{2}(x))).

Let B(t) = (C(t), D(t)) be the quotient of ((c_{1}(t) − c_{2}(t))(c_{1}(t) − c_{3}(t)), c_{1}(t) − c_{2}(t)) by A_{P} (t) ; the polynomials C(t) and D(t) are now coprime with R_{P} (t), with coefficients in O_{S∪L_{∞}}, and leading coefficients in O_{S∪L_{∞}}^{∗}. We have

e_{v}(A_{P} (x), A_{P} (x)) = 0

for any place v (using the skew-symmetry of e_{v}). Thus

∑_{v∈L_{P} ∪v_{P}} e_{v}(A_{P} (x), α_{v}) = ∑_{v∈L_{P} ∪v_{P}} e_{v}(A_{P} (x), B(x)).

Now the same argument as for the case P ≠ Q enables us to conclude that provided x satisfies the proximity condition (6) in proposition 2.1.3 and also satisfies a suitable proximity condition at v_{∞}, this sum is independent both of x and L, hence so is also e_{N}(A_{P} , A_{P}).

2.5 Completion of the proof of the Theorem

In this subsection, we shall complete the proof of the Theorem, as stated in the introduction.

Proposition 2.5.1 Let m = (m_{1}(t), m_{2}(t), m_{3}(t)) ∈ 𝔖_{0} be as in the beginning of this section, and let X/P_{k}^{1} be the associated pencil of curves of genus one. Assume that m ∈ H^{1}(k(t), (Z/2)^{2}) is not the image of a point of order 2 under the Kummer map. Assume :

(D) The kernel of the composite map

𝔖_{0} → ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) >

is generated by m and the image of the 2-torsion subgroup _{2}E(U) of E(U) under the Kummer map.

Let L_{∞} = {v_{∞}} be as above. Then there exist finite sets L_{M} of places of k, for M ∈ T, satisfying properties (1) and (2) of proposition 2.1.3 and points x_{v} ∈ O_{v} for v ∈ ∪_{M∈T} L_{M} such that for any x ∈ U(k) satisfying properties (3) to (6) of that same proposition, close enough to x_{∞} = 1/μ at v_{∞} and close enough to each x_{v} for v ∈ ∪_{M∈T} L_{M}, the 2-Selmer group S(E_{x}, 2) has order 8 and is generated by m(x) and the images of the 2-torsion points of E_{x}(k).

Proof First of all m(x) and the image of the 2-torsion points of E_{x}(k) under the Kummer map are in the 2-Selmer group of E_{x}. This is clear for the 2-torsion points. As for m(x), this is the content of (i) in proposition 2.1.3 : X_{x} has points everywhere locally. By proposition 2.1.3 (iii), these three classes define independent elements in the F_{2}-vector space U′_{L(x)}, which contains the 2-Selmer group (proposition 1.2.3).

Under the isomorphism ev_{x} : N ⟶^{≃} U′_{L(x)} these classes correspond to elements in the group N_{0} ⊕ A. Indeed, from the definition of the (β_{M}, γ_{M}) one checks that there exists a suitable product of the A_{M}(x) such that the quotient of m(x) by this product, which a priori lies in U′_{L(x)} ⊂ (O_{L(x)}^{∗}/O_{L(x)}^{∗2})^{2}, actually lies in (O_{S}^{∗}/O_{S}^{∗2})^{2} hence lies in U′_{S}. The inverse image of m(x) under ev_{x} thus lies in N_{0} ⊕ A. The argument is similar for the points of order 2. Let B_{0} ⊂ N_{0} ⊕ A ⊂ (O_{S∪L_{∞}}[U]^{∗}/O_{S∪L_{∞}}[U]^{∗2})^{2} be the subgroup they generate, which is clearly independent of L. Since the corresponding elements in U′_{L(x)} are in the 2-Selmer group, the group B_{0} is in the kernel of the pairing e_{N} on N × N (see proposition 1.2.3). Let B ⊂ N_{0} ⊕ A be the kernel of the restriction of e_{N} to N_{0} ⊕A ⊂ N; this kernel is independent of x and L (proposition 2.4.2). Let B_{1} ⊂ B (independent of x and L) be such that B = B_{0} ⊕ B_{1}. Let B_{2} ⊂ N_{0} ⊕ A be a supplementary space for B ⊂ N_{0} ⊕ A. Finally, let B_{3} = φ_{x}^{−1}(⊕_{v∈L\S}(W_{v}/W_{v} ∩ K_{v})) be as in proposition 2.3.4. This is a supplementary space for N_{0} ⊕ A ⊂ N. We thus have N = B_{0} ⊕ B_{1} ⊕ B_{2} ⊕ B_{3}.

Lemma 2.5.2 Let S_{1} be a finite set of places containing S. Assume that b(t) = (b_{1}(t), b_{2}(t)) ∈ (O_{S_{1}}[U]^{∗}/O_{S_{1}}[U]^{∗2})^{2} lies in 𝔖 but is not in the kernel of the map

𝔖 → ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) > .

Then there exist infinitely many places v ∉ S_{1} such that

(a) there exists a point M ∈ T and a place v^{M} of k_{M} which is of degree one over k and which splits in K_{M} ;

(b) for any point x_{v} ∈ A^{1}(O_{v}) which meets M̃ ⊂ A_{O_{S}}^{1} at the place v^{M} and is transversal to it there, there exists an element σ_{v} ∈ W_{v}(x_{v}) such that e_{v}(ev_{x_{v}}(b), σ_{v}) ≠ 0.

Proof By assumption, there exists M ∈ T such that K_{M}(\sqrt{δ_{M}(b)})/K_{M} is a non-trivial extension. A suitable version of Tchebotarev’s theorem (see [Ha94], Prop. 2.2.1) ensures that there exist infinitely many places v of k such that v extends to a place w of K_{M} of degree one over v, the place w itself being inert in the extension K_{M}(\sqrt{δ_{M}(b)})/K_{M}. Let v be any such place. Let v^{M} be the trace of the corresponding w on k_{M}. Because of the degree one condition, this place splits in K_{M}, as required in (a).

Let U = U_{S_{1}} = Spec(O_{S_{1}}[t][1/R]). The residue map H^{2}(U, Z/2) → H^{1}(k_{M}, Z/2) induces a residue map H^{2}(U, Z/2) → H^{1}(M̃, Z/2) ⊂ H^{1}(k_{M}, Z/2).

Given a point x_{v} ∈ U(k_{v}) which comes from a point x_{v} ∈ A^{1}(O_{v}) and which meets M̃ and is transversal to it in v^{M}, there is a commutative diagram :

H^{2}(k_{v}, Z/2) ⟶^{inv_{v}} H^{1}(F_{v}, Z/2)
↑ x_{v}^{∗}  ↑
H^{2}(U, Z/2) ⟶^{ResM̃} H^{1}(M̃, Z/2)

Here the horizontal maps are residue maps (the top one giving the invariant of local class field theory, which is an isomorphism), the left vertical map is restriction to x_{v} ∈ U(k_{v}) ⊂ k_{v}, the right vertical arrow is restriction to the closed point Spec(F_{v}) of M̃ defined by v^{M}.

Let i ∈ {1, 2, 3} be such that R_{M}(t) divides p_{i}(t). Let Q_{j} and ε_{j} be as in lemma 2.2.1. Note that the classes b and ε_{j} actually belong to H^{1}(U, (Z/2)^{2}). Let us now compute the image of < b, ε_{j} >∈ H^{2}(U, Z/2) under the two routes in the above commutative diagram. Applying lemma 2.2.1 and using the fact that the place v^{M} of k_{M} is inert in k_{M}(\sqrt{δ_{M}(b)}), we find that the image is non-zero. Since < b, ε_{j} > goes to e_{v}(ev_{x_{v}}(b), ε_{j}(x_{v})) under the left vertical map in the diagram, we see :

e_{v}(ev_{x_{v}}(b), ε_{j}(x_{v})) ≠ 0.

In other words, for each x_{v} satisfying the above transversality condition we have found an element σ_{v} = ε_{j}(x_{v}) ∈ W_{v}(x_{v}) ⊂ V_{v} (as a matter of fact the image of an element of order 2 in E_{x_{v}}(k_{v})) such that

e_{v}(ev_{x_{v}}(b), σ_{v}) ≠ 0.

We now resume the proof of proposition 2.5.1. The finite set B_{1} \ {1} lies in 𝔖 and does not meet B_{0}. Condition (D) ensures that the hypothesis of lemma 2.5.2 (with S_{1} any set of places containing S) is satisfied by the elements of B_{1} \ {1} which lie in 𝔖_{0}.

Any element ξ ∈ B_{1} which does not lie in 𝔖_{0} is the product of an element of U′_{S} and a certain product of elements

A_{M} = (a_{M}(μ^{deg(R_{M})}R_{M}(t))^{β_{M}} , b_{M}(μ^{deg(R_{M})}R_{M}(t))^{γ_{M}}) ∈ (O_{S∪L_{∞}}[U]^{∗}/O_{S∪L_{∞}}[U]^{∗2})^{2},

with the property that η(ξ) = (η_{1}(ξ), η_{2}(ξ)) ≠ 0 ∈ (Z/2)^{2}. In the group 𝔖, we may rewrite this product as ξ = (μ^{η_{1}(ξ)}, μ^{η_{2}(ξ)}).ξ_{1}, where ξ_{1} belongs to the finite subgroup of 𝔖_{S} ⊂ 𝔖 spanned by U_{S} and the elements (R_{M}(t)^{β_{M}} , R_{M}(t)^{γ_{M}}). Let us consider the set M of elements ξ ∈ 𝔖_{S} in this finite subgroup with the property that η(ξ) ≠ 0. For any point M ∈ T, we have

δ_{M}(ξ) = δ_{M}((μ^{η_{1}(ξ)}, μ^{η_{2}(ξ)})).δ_{M}(ξ_{1}) = μ^{γ_{M} .η_{1}(ξ)+β_{M} .η_{2}(ξ)}.δ_{M}(ξ_{1}).

For any element ξ ∈ M, there exist at least one closed point M ∈ T and a pair (β_{M}, γ_{M}) such that γ_{M}.η_{1}(ξ) + β_{M}.η_{2}(ξ) = 1 ∈ Z/2. Now by the very choice of L_{∞} = {v_{∞}} before lemma 2.3.2, for such a point M the element δ_{M}(ξ) = μ.δ_{M}(ξ_{1}) ∈ k_{M}^{∗}/k_{M}^{∗2} differs both from 1 and from δ_{M}(m).

The hypothesis of lemma 2.5.2 is therefore satisfied by all elements in B_{1} \ {1}. Applying lemma 2.5.2 to each element of B_{1} \ {1} in turn, we produce a finite set of distinct places of k, away from S, indexed by B_{1} \ {1}, such that conditions (a) and (b) of the lemma hold for each b ∈ B_{1} \ {1} and its associated place. For each such place v, fix x_{v} ∈ O_{v} as in part (b) of the lemma (such points exist), together with some associated σ_{v} ∈ W_{v}(x_{v}). Let 1_{v} ∈ ⊕_{v∈S_{1}}Z/2 be the element with all coordinates zero except the one at v. The bilinear pairing

B_{1} × (⊕_{v∈S_{1}}Z/2) → Z/2

sending (b, 1_{v}) to e_{v}(ev_{x_{v}}(b), σ_{v}) is non-degenerate on the left hand side. By linear algebra, we may find a subset S_{2} ⊂ S_{1} of order equal to the rank of B_{1}, such that the induced pairing

B_{1} × (⊕_{v∈S_{2}}Z/2) → Z/2

is non-degenerate on both sides.

We now take L to be S ∪ S_{2}, together with the decomposition L \ S = (∪_{M∈T} L_{M}) coming from the above proof. Note that the order of L \ S is precisely equal to the rank of B_{1}. For any x ∈ O_{S} close enough to each x_{v} for v ∈ L, the corresponding pairing

B_{1} × (⊕_{v∈S_{2}}Z/2) → Z/2

with x in place of x_{v} is still non-degenerate. Since B_{3} = φ_{x}^{−1}(⊕_{v∈L\S}(W_{v}/W_{v} ∩ K_{v})), we find that the pairing B_{1} × B_{3} → Z/2 induced by e_{N} is non-degenerate.

If we now apply lemma 2.5.3 below to V = B_{1} ⊕ B_{2}, V′ = B_{3}, V_{0} = B_{1} and Ψ the restriction of e_{N} to B_{1} ⊕ B_{2} ⊕ B_{3}, we find that this restriction is non-degenerate. Recall that B_{0} was orthogonal to all of N. We thus find that the left kernel of the pairing e_{N}, which by proposition 1.2.3 is the 2-Selmer group, is precisely B_{0}, is of rank 3, and is spanned by the points of order 2 and m(x).

The following lemma replaces the matrix game of [SwD95].

Lemma 2.5.3 Let Ψ be a symmetric bilinear form on a vector space V ⊕ V′ over a field F. Let V_{0} ⊂ V be the kernel of the restriction of Ψ to a bilinear form on V. Suppose that the pairing between V_{0} and V′ given by Ψ is non-degenerate (that is, defines an isomorphism between V′ and the dual V_{0}^{∗} of V_{0}). Then Ψ is non-degenerate.

Proof Let V_{1} ⊂ V be the left kernel of the pairing between V and V′ given by Ψ, then V = V_{0} ⊕V_{1}. We have V ⊕V′ = V_{1} ⊥ (V_{0} ⊕V′) where ⊥ denotes the orthogonal direct sum. The restriction of Ψ to a bilinear form on V_{1} is non-degenerate. The same is true for the restriction of Ψ to a bilinear form on V_{0} ⊕ V′, because Ψ identifies V′ and V_{0}^{∗}, and the restriction of Ψ to V_{0} is zero. Hence our lemma is proved.

The final step

Having chosen L = ∪_{M∈T} L_{M} as in proposition 2.5.1, for each M ∈ T and each v ∈ L_{M}, we choose a point x_{v} ∈ A^{1}(O_{v}) such that x_{v} meets M̃ in v and is transversal there. For a finite place v ∈ S, x_{v} was defined in proposition 2.1.3. For v = v_{∞}, we have defined x_{v} = 1/μ (see proposition 2.4.2).

We now apply Schinzel’s hypothesis (H), in its (H_{1}) version (due to Serre, see [CT/SwD94], p. 71), to the family of irreducible polynomials R_{M} and to the family of points x_{v} ∈ k_{v}, for v ∈ S_{finite} ∪L∪{v_{∞}} (recall that all ‘small’ primes had been included in S_{0} ⊂ S at the beginning of subsection 2.1). This produces an x ∈ k which satisfies all assumptions of proposition 2.1.3, in particular such that x ∈ U_{v} for v ∈ S_{1}. According to proposition 2.5.1, for any such x the 2-Selmer group S(E_{x}, 2) has order 8 and is generated by m(x) and the 2-torsion points of E_{x}(k). Thus the 2-torsion of the Tate-Shafarevich group of E_{x} has order at most 2. Assuming that the Tate-Shafarevich group is finite, we know (Cassels [Cass62] ; [Milne86], thm. 6.13, p. 101) that the order of its 2-torsion must be a square. Hence _{2}�(k, E_{x}) = 0, and the element m(x) of the 2-Selmer group must come from E_{x}(k)/2, hence its image in H^{1}(k, E_{x}) is trivial : the principal homogeneous space X_{x} has a k-point. Note that the 2-Selmer group S(E_{x}, 2) is of order 8 and coincides with the image of E_{x}(k)/2 under the Kummer map. Thus each E_{x} is an elliptic curve over k of rank exactly one. Since the principal homogeneous space X_{x} under E_{x} has a k-point, it is trivial, i.e. X_{x} is k-isomorphic to E_{x}, and X_{x}(k) is infinite.

This completes the proof of statement (a) in the main Theorem. Since E_{x}(k) is infinite if X_{x}(k) is infinite, the proof also shows that under assumptions (1) to (4) and X(𝔸_{k})^{Br_{vert}(X)} ≠ ∅, the set of k-points x with E_{x}(k) of rank one is Zariski-dense in P_{k}^{1}, hence E(k) is Zariski-dense in the surface E. Since the group Br_{vert}(X)/Br(k) is finite, the results on X stated in (b) and (c) of the Theorem are direct consequences of (a) (see [CT/Sk/SwD97b], proof of Theorem 1.1).

Remarks 2.5.4

(i) Under the hypotheses of the theorem, the quotient E(U)/(2E(U) +_{2}E(U)) is either zero or is isomorphic to Z/2, the generator having the same image as m in the quotient of 𝔖_{0} by the image of _{2}E(U). The rank of E(k(t)) is zero in the first case. In the second case, the rank of E(k(t)) is one, and X = X(m) and E are birational over P_{k}^{1}. In the second case, the main Theorem is therefore essentially obvious. In the first case however, although the generic fibre of E/P_{k}^{1} has rank zero, under the assumptions of the theorem and X(𝔸_{k})^{Br_{vert}(X)} ≠ ∅ (e.g. X(k) ≠ ∅) we find that the set of k-points x such that the specialized curve E_{x} has rank at least one is Zariski-dense in P_{k}^{1}.

(ii) Under the hypotheses of the theorem, when X(k) ≠ ∅, we conclude that the rational points on both surfaces E and X are Zariski dense. This is worth pointing out, since some of our examples (see section 3) include K3-surfaces, for which little was known in such respect.

(iii) Suppose we start with E/P_{k}^{1} as in the Introduction. There is an associated map δ : 𝔖_{0} → ⊕_{M∈T} k_{M}^{∗}/k_{M}^{∗2}. We know (lemma 2.2.1) that the kernel Ker(δ) is a finite abelian group of exponent 2. This group is of order at least 4, since it contains the image of _{2}E(U) under the Kummer map. One may wonder whether one may find m ∈ 𝔖_{0} such that the hypotheses of the main theorem are fulfilled. A necessary condition is that the order of Ker(δ) be exactly 8. Indeed, if the order is 4, then Hypothesis (1) in the main Theorem cannot be fulfilled, and if the order is at least 16, then (2) cannot be fulfilled.

(iv) Instead of hypothesis (D), let us make the stronger assumption :

(D_{0}) The kernel of the composite map

𝔖 → ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) >

is generated by m and by the image of the 2-torsion points of E(U) under the Kummer map.

Then, as the reader will check, we can take L_{∞} = ∅ in the above proof. We have the exact sequence

0 → 𝔖_{0} → 𝔖 ⟶^{η} (Z/2)^{2}

with η as in the beginning of subsection 2.2. Thus when all R_{M}’s are of even degree, the two conditions (D) and (D_{0}) agree, and there is no need for L_{∞}.

(v) Under (D_{0}), the x’s produced in the proof are integral away from the set S. One can actually go further. We shall here ignore the difficulties produced by the real places, assume that k is totally imaginary and ignore approximation at the complex places. Then there is no need to make the projective transformation mentioned at the beginning of subsection 2.1. As pointed out after lemma 2.1.1, the image of each α_{M}(t) under the map Br(k(t)) → Br(k(X)) lies in the subgroup Br(X_{aff}), where X_{aff} denotes the inverse image of A_{k}^{1} under X → P_{k}^{1}. We can produce a finite set S_{0} as in lemma 2.1.2, with the additional condition that the α_{M} lie in the Brauer group of an integral model of Y over O_{S_{0}}. Assume that there is no Brauer-Manin obstruction to the existence of an integral point (in an obvious sense) on Y. Then we have the same conclusion as in lemma 2.1.2, with the additional property that each x_{v} lies in O_{v} for v ∉ S_{0}. Under (D_{0}), we ultimately produce an element x ∈ k integral away from S_{0} and such that X_{x}(k) ≠ ∅.

(vi) It is quite likely that the theorem goes through if we replace (D) by the assumption : (D_{1}) The kernel of the composite map

𝔖_{0} →^{δ} ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) >

is generated by m and by the image of E(U) under the Kummer map, and m does not belong to the image of the whole group E(U) under the Kummer map.

3. Surfaces to which the theorem applies

3.1 Equations

The surfaces to which the theorem applies can be given by more symmetrical looking systems of equations. Let k be a field, char(k) ≠ 2. Given a_{i}(t) and b_{i}(t) in k[t], consider the surface W ⊂ P_{k}^{3} ×_{k} A_{k}^{1} defined by the system of equations

∑_{i=0}^{3} a_{i}(t)x_{i}^{2} = 0, ∑_{i=0}^{3} b_{i}(t)x_{i}^{2} = 0.

Here A_{k}^{1} = Spec(k[t]), the a_{i} and b_{i} belong to k[t], and (x_{0}, x_{1}, x_{2}, x_{3}) are homogeneous coordinates for P_{k}^{3}.

We shall make the following ‘generic’ assumptions :

(i) deg a_{i} − deg b_{i} does not depend on i, deg a_{i} − deg a_{j} is even for all i, j, and for i ≠ j the difference Δ_{ij}(t) = a_{i}(t)b_{j}(t) − a_{j}(t)b_{i}(t) has degree equal to the degree of a_{i}b_{j} (equal to the degree of a_{j}b_{i} by the previous assumptions).

(ii) The product R(t) = ∏_{i<j} Δ_{ij}(t) is separable.

There then exists a model π : X → P_{k}^{1} of W/A_{k}^{1} such that X/k is smooth and proper, the fibre at infinity is smooth (this is guaranteed by condition (i)), the singular fibres of π correspond to the zeros of ∏_{i<j} Δ_{ij}(t), and each geometric singular fibre is the union of two smooth conics in P^{3} which are not coplanar and which meet in two points.

The number of geometric singular fibres is given by

∑_{i<j} deg(Δ_{ij}) = (3/2) ∑_{i}(deg(a_{i}) + deg(b_{i})),

which is always divisible by 6.

Note that we have identically

Δ_{01}(t)Δ_{23}(t) + Δ_{02}(t)Δ_{31}(t) + Δ_{03}(t)Δ_{12}(t) = 0.

The system of equations above is equivalent (over R(t) ≠ 0) to the system

Δ_{01}(t)x_{0}^{2} + Δ_{21}(t)x_{2}^{2} + Δ_{31}(t)x_{3}^{2} = 0
Δ_{02}(t)x_{0}^{2} + Δ_{12}(t)x_{1}^{2} + Δ_{32}(t)x_{3}^{2} = 0
Δ_{03}(t)x_{0}^{2} + Δ_{13}(t)x_{1}^{2} + Δ_{23}(t)x_{2}^{2} = 0,

where any one of the three equations may be deduced from the other two. The jacobian of this curve may be written in the standard form

Y^{2} = (X − c_{1}(t))(X − c_{2}(t))(X − c_{3}(t)),

with

c_{1} − c_{2} = Δ_{12}Δ_{30}, c_{2} − c_{3} = Δ_{23}Δ_{10}, c_{3} − c_{1} = Δ_{31}Δ_{20}.

The relatively minimal fibration X/P_{k}^{1} associated to the surface we started with, in terms of the introduction of this paper, corresponds to the element m(t) = (m_{1}(t), m_{2}(t), m_{3}(t)) with

m_{1} = −Δ_{12}Δ_{13}, m_{2} = −Δ_{21}Δ_{23}, m_{3} = −Δ_{31}Δ_{32} .

The hypotheses on the polynomials a_{i} and b_{i} ensure that all the hypotheses of the introduction are satisfied ; in particular m lies in 𝔖_{0}.

Conversely, provided the coordinates are general enough, one may go from the equations given in the introduction to a system of two simultaneously diagonal quadratic equations as the one we started with here. The details will be left to the reader.

Deciding whether condition (3) (= condition (D)) in the Theorem holds reduces to the following explicit computation. Let T be the set of monic irreducible divisors of R(t). For each monic irreducible factor R_{M}(t) of R(t), with residue field k_{M} = k[t]/R_{M}(t), one computes δ_{M}(m) ∈ k_{M}^{∗}/k_{M}^{∗2} according to the recipe given in the introduction. If e.g. R_{M} divides Δ_{01}, then δ_{M}(m) is given by the class of −Δ_{21}Δ_{31} modulo R_{M}, as is clear from the system of equations above. As for 𝔖_{0} ⊂ (k(t)^{∗}/k(t)^{∗2})^{3}, it consists of classes (a ∏_{M∈T} R_{M}^{r_{M}} , b ∏_{M∈T} R_{M}^{s_{M}} , c ∏_{M∈T} R_{M}^{t_{M}}) with a, b, c ∈ k^{∗}/k^{∗2} such that abc = 1, and with r_{M}, s_{M}, t_{M} each equal to 0 or 1, satisfying the conditions : r_{M} + s_{M} + t_{M} is even ; if R_{M} divides c_{2} − c_{3} = Δ_{23}Δ_{10}, then r_{M} = 0, if R_{M} divides c_{3} − c_{1} = Δ_{31}Δ_{20}, then s_{M} = 0, if R_{M} divides c_{1} − c_{2} = Δ_{12}Δ_{30}, then t_{M} = 0 ; finally, each of ∑_{M∈T} r_{M}deg(R_{M}), ∑_{M∈T} s_{M}deg(R_{M}), ∑_{M∈T} t_{M}deg(R_{M}) is even. One computes the map δ on 𝔖_{0} as we did above for δ(m), or by using the recipe given in the proof of lemma 2.2.1. The composite map

𝔖_{0} →^{δ} ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) >

is now completely explicit. One knows that its kernel is finite (lemma 2.2.1). Condition (D) holds if and only if this kernel is of rank 3, generated by the classes of m and of the elements

((c_{1} − c_{2})(c_{1} − c_{3}), (c_{1} − c_{2}), (c_{1} − c_{3})) = (Δ_{12}Δ_{30}Δ_{13}Δ_{20}, Δ_{12}Δ_{30}, Δ_{13}Δ_{20})

and

(c_{2} − c_{1}, (c_{2} − c_{1})(c_{2} − c_{3}), (c_{2} − c_{3})) = (−Δ_{12}Δ_{30}, −Δ_{12}Δ_{30}Δ_{23}Δ_{10}, Δ_{23}Δ_{10}).

3.2 The Egloffstein case

In [SwD95], one starts with a smooth complete intersection of two quadrics W ⊂ P_{k}^{4} given by a system of equations

∑_{i=0}^{4} a_{i}x_{i}^{2} = 0, ∑_{i=0}^{4} b_{i}x_{i}^{2} = 0,

where all a_{i}, b_{i} belong to k, and a_{i}b_{j} − a_{j}b_{i} ≠ 0 for i ≠ j. This is a Del Pezzo surface of degree 4. One may transform this surface into a pencil of curves of genus one by considering the pencil x_{4} = tx_{0}. The new system now reads

∑_{i=1}^{3} a_{i}x_{i}^{2} + (a_{0} + t^{2}a_{4})x_{0}^{2} = 0,

∑_{i=1}^{3} b_{i}x_{i}^{2} + (b_{0} + t^{2}b_{4})x_{0}^{2} = 0

in A_{k}^{1} × P_{k}^{3}, which one may extend to a pencil of curves of genus one π : X → P_{k}^{1} over the projective line. One readily checks that the number of singular geometric fibres is 6.

Proposition 3.2.1 Let π : X → P_{k}^{1} be as above.

(1) If none of the −d_{i0}d_{i4} is a square in k, then the vertical Brauer group of X (with respect to π) is reduced to the image of Br(k).

(2) If moreover the three classes −d_{12}d_{13}, −d_{21}d_{23}, ∏_{i≠4} d_{4i} are independent in k^{∗}/k^{∗2}, then condition (D) holds for π.

(3) Assume that k is a number field. Assume that the various indepence hypotheses above are fulfilled. Assume Schinzel’s hypothesis, and assume that the Tate-Shafarevich group of elliptic curves over k is finite. Then the Hasse principle holds for the intersection of two quadrics

∑_{i=0}^{4} a_{i}x_{i}^{2} = 0, ∑_{i=0}^{4} b_{i}x_{i}^{2} = 0.

Proof In the above notation we have Δ_{ij}(t) = d_{ij} if i and j are in {1, 2, 3}, and by assumption Δ_{i0}(t) = d_{i4}t^{2} + d_{i0} is irreducible for i ∈ {1, 2, 3}. The closed points M ∈ P_{K}^{1} such that X_{M} is singular are given by Δ_{i0}(t) = 0 for i ∈ {1, 2, 3}. Let us call these points M_{1}, M_{2}, M_{3}. Their residue fields are k_{i} = k(\sqrt{β_{i}}) where β_{i} is the class of −d_{i4}d_{i0} in k^{∗}/k^{∗2}. The class m = (m_{1}, m_{2}) ∈ (k(t)^{∗}/k(t)^{∗2})^{2} is given by the constants (α_{1}, α_{2}) ∈ (k^{∗}/k^{∗2})^{2}, where α_{i} = −d_{ij}d_{ir} if {i, j, r} = {1, 2, 3}.

Let A ∈ Br_{vert}(X) ⊂ Br(k(X)). Write A = π^{∗}(a) with a ∈ Br(k(t)). The class a can have non-trivial residues only at the points M_{i} (i ∈ {1, 2, 3}). Moreover, at the point M_{i}, its residue is either trivial or equal to the image of α_{i}. The quaternion algebra (α_{i}, Δ_{i0}(t)) has residue α_{i} ∈ k_{i}^{∗}/k_{i}^{∗2} = H^{1}(k_{i}, Z/2) at M_{i} and has residue zero at all other closed points of P_{k}^{1}. By the Faddeev exact sequence, the element a belongs to the subgroup of Br(k(t)) spanned by Br(k) and the quaternion algebras (α_{i}, Δ_{i0}(t)). But a glance at the equations defining X shows that each of these quaternion algebras becomes trivial over k(X). This proves (1).

Since each Δ_{i0}(t) is irreducible and of even degree, the group 𝔖 ⊂ ((k(t)^{∗}/k(t)^{∗2}))^{2} coincides with the group 𝔖_{0}. One easily checks that 𝔖_{0} coincides with the subgroup spanned by (k^{∗}/k^{∗2})^{2}, the image ∂(_{2}E(k(t))) of the points of order 2 under the Kummer map and the class (Δ_{30}(t), Δ_{30}(t)).

Using the explicit description of the map

δ : 𝔖_{0} → ⊕_{i=1}^{3}k_{i}^{∗}/k_{i}^{∗2}

given by lemma 2.2.1, one obtains the following results. For (c_{1}, c_{2}) ∈ (k^{∗}/k^{∗2})^{2}, letting c_{3} = c_{1}c_{2} ∈ k^{∗}/k^{∗2}. Then δ_{M_{i}}((c_{1}, c_{2})) = c_{i} ∈ k_{i}^{∗}/k_{i}^{∗2}. In particular, δ_{M_{i}}(m) = α_{i} ∈ k_{i}^{∗}/k_{i}^{∗2}.

Since δ vanishes on the image of E(k(t)) under the Kummer map, to describe δ it only remains to compute δ((Δ_{30}(t), Δ_{30}(t))). One finds :

δ_{M_{1}}(Δ_{30}(t), Δ_{30}(t)) = −d_{13}d_{04}d_{14} ∈ k_{1}^{∗}/k_{1}^{∗2},
δ_{M_{2}}(Δ_{30}(t), Δ_{30}(t)) = −d_{23}d_{04}d_{24} ∈ k_{2}^{∗}/k_{2}^{∗2},
δ_{M_{3}}(Δ_{30}(t), Δ_{30}(t)) = −d_{13}d_{32}d_{04}d_{34} ∈ k_{3}^{∗}/k_{3}^{∗2}.

Let ρ_{1} = −d_{13}d_{04}d_{14} ∈ k^{∗}/k^{∗2}, ρ_{2} = −d_{23}d_{04}d_{24} ∈ k^{∗}/k^{∗2} and ρ_{3} = −d_{13}d_{32}d_{04}d_{34} ∈ k_{3}^{∗}/k_{3}^{∗2} ∈ k^{∗}/k^{∗2}. Let ρ = ρ_{1}.ρ_{2}.ρ_{3} = ∏_{j≠4} d_{4j} ∈ k^{∗}/k^{∗2}.

Let ε ∈ {0, 1}, and c, d ∈ k^{∗}/k^{∗2}. If the class ((Δ_{30}(t), Δ_{30}(t)))^{ε}.(c, d) is in the kernel of the composite map

𝔖_{0} →^{δ} ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2} → ⨁_{M∈T} k_{M}^{∗}/ < k_{M}^{∗2}, δ_{M}(m) >,

then for i ∈ {1, 2, 3} there exist ε_{i} ∈ {0, 1} such that ρ_{1}^{ε}.c = α_{1}^{ε_{1}}, ρ_{2}^{ε}.d = α_{2}^{ε_{2}}, ρ_{3}^{ε}.c.d = α_{3}^{ε_{3}} = (α_{1}.α_{2})^{ε_{3}}. Elimination yields α_{1}^{ε_{1}−ε_{3}}.α_{2}^{ε_{2}−ε_{3}}.ρ^{ε} = 1. The assumptions in 1) now imply ε = 0 and ε_{1} = ε_{2} = ε_{3}, which shows that hypothesis (D) is fulfilled, and completes the proof of (2).

Statement (3) now follows from the main Theorem.

Remarks 3.2.2

(1) Let k be a number field. For a_{0}, …, a_{4}, b_{0}, …, b_{4} in a Hilbert subset of affine space A^{10}(k), all hypotheses above are fulfilled. Indeed, K = k(A_{0}, …, A_{4}, B_{0}, …, B_{4}) be the function field of A_{k}^{10} and let D_{ij} = A_{i}B_{j} − A_{j}B_{i}. Looking at divisors, it is an easy matter to check that the elements −D_{12}D_{13}, D_{21}D_{23}, ∏_{i≠4} D_{4i} are independent in K^{∗}/K^{∗2}. Similarly, if {i, j, r} is a permutation of {1, 2, 3}, then for each i = 1, 2, 3, the classes −D_{i0}D_{i4}, −D_{ij}D_{ir} are independent in K^{∗}/K^{∗2}. The assertion now follows from Hilbert’s irreducibility theorem. Thus for ‘general’ a_{0}, …, a_{4}, b_{0}, …, b_{4}, in the number field k, provided the two mentioned conjectures hold, the Hasse principle holds for the corresponding intersection of two quadrics.

(2) Conditions (1) and (2) in the previous proposition are a priori weaker than the original condition (D) (p. 290 of [SwD95]).

(3) Note that there are 10 different obvious ways in which one may transform the original surface W into a pencil of curves of genus one, namely one may set x_{i} = tx_{j} for any pair (i, j), i ≠ j. This leads to 10 versions of condition (D). We have examples where none of these 10 conditions holds, and where nevertheless the quotient Br(W)/Br(k) = 0. Standard conjectures on rational surfaces suggest that the Hasse principle holds for such a W. However the main theorems of [SwD95] and of the present paper are not enough to prove this. This also indicates that condition (D) is not a necessary condition for the Hasse principle to hold.

3.3 Geometry

Let us consider the surface X/P^{1} over the algebraic closure of k. Let T be the set of points of P^{1} with singular fibres, and let r be its order. The topological Euler-Poincaré characteristic e(X) of X is given by the formula ([BPV84], Prop. (11.4) p. 97) :

e(X) = e(X_{η})e(P^{1}) + ∑_{M∈T}(e(X_{M}) − e(X_{η})),

where X_{η} denotes the geometric generic fibre, a curve of genus one, and X_{M} denotes the fibre at M. We have e(P^{1}) = 2, e(X_{η}) = 0, and e(X_{M}) = 2. Thus e(X) = 2r.

(Note : the formula in [BPV84], p. 162 also covers the case of multiple fibres. This is useful when describing ‘wild’ cases, as in [CT/Sk/SwD97a].)

Let χ(X) = χ(O_{X}) denote the Euler-Poincaré characteristic of the structural sheaf of a proper variety X. Since p : X → P^{1} is a smooth surface fibred into curves of genus one, and has no multiple fibre, the canonical bundle K_{X} is of the shape K_{X} = p^{∗}(L) with L a line bundle of degree χ(X) − 2χ(P^{1}) = χ(X) − 2 on P^{1} ([BPV84], Corollary (12.3) p. 162). In particular, c_{1}^{2}(X) = (K_{X}.K_{X}) = 0. From the formula c_{1}^{2}(X) + e(X) = 12χ(X), and from the above result, we deduce

r = 6χ(X).

Assume χ(X) = 1, i.e. r = 6. We have K_{X} = O_{X}(−X_{P}), where X_{P} denote a fibre. Thus 2K_{X} has no section and in particular H^{0}(X, K_{X}) = 0. The vector space H^{2}(X, O_{X}) is dual to H^{0}(X, K_{X}). Hence the geometric genus p_{g} = dim(H^{2}(X, O_{X})) vanishes. From χ(X) = 1 we conclude H^{1}(X, O_{X}) = 0. Castelnuovo’s criterion now ensures that the surface is rational.

In terms of simultaneously diagonal equations as above, the possibilities for the degrees of the a_{i}’s and b_{i}’s are ((0,0,0,0) ;(1,1,1,1)) which gives a surface obviously birational to a quadric, or ((0,0,0,2) ;(0,0,0,2)) which is indeed a generalisation of the case considered in [SwD95] and discussed in subsection 3.2 above.

Assume χ(X) = 2, i.e. r = 12. Then K_{X} = O_{X}. Thus dim(H^{2}(X, O_{X})) = 1, hence dim(H^{1}(X, O_{X})) = 0. In this case, X is a K3-surface.

In terms of simultaneously diagonal equations as above, the possibilities for the degrees are : ((0,0,0,4) ;(0,0,0,4)), ((0,0,2,2) ;(0,0,2,2)), ((1,1,1,1) ;(1,1,1,1)), ((0,0,0,2) ;(1,1,1,3)), ((0,0,0,0) ;(2,2,2,2)).

Let us discuss the case ((1, 1, 1, 1); (1, 1, 1, 1)) some more. Let us write

a_{i}(t) = α_{i0}t + α_{i1}, b_{i}(t) = α_{i2}t + α_{i3}.

In the product P_{k}^{1}×P_{k}^{3} with bihomogeneous coordinates (λ, μ; x_{0}, x_{1}, x_{2}, x_{3}), the original surface X is given by the system of equations

∑_{i=0}^{3}(α_{i0}λ + α_{i1}μ)x_{i}^{2} = 0, ∑_{i=0}^{3}(α_{i2}λ + α_{i3}μ)x_{i}^{2} = 0,

projection to P^{1} yielding the original fibration π : X → P^{1}. Let x = (x_{0}, x_{1}, x_{2}, x_{3}). For j = 0, 1, 2, 3, let

A_{j}(x) = α_{0j}x_{0}^{2} + α_{1j}x_{1}^{2} + α_{2j}x_{2}^{2} + α_{3j}x_{3}^{2}.

Then we have

λA_{0}(x) + μA_{1}(x) = λA_{2}(x) + μA_{3}(x) = 0

and the surface X is isomorphic to the quartic surface Z ⊂ P_{k}^{3} given by the homogeneous equation

A_{0}(x)A_{3}(x) − A_{1}(x)A_{2}(x) = 0.

This is also isomorphic to the surface X′ given in the product P_{k}^{1} × P_{k}^{3}, with bihomogeneous coordinates (σ, τ; x_{0}, x_{1}, x_{2}, x_{3}), by the equations :

σA_{0}(x) + τA_{2}(x) = σA_{1}(x) + τA_{3}(x) = 0.

Thus on Z there is a second fibration into curves of genus one, given by the projection X′ → P^{1}.

However, there is a natural correspondence between the singular fibres of the two pencils. Geometrically, this is given as follows. Let C_{1} ⊂ X and C_{2} ⊂ X be the two smooth conics which are the components of a singular fibre of the first pencil. Viewed in Z ⊂ P^{3}, these curves are still conics. Let Π_{i} ⊂ P^{3} be the plane containing C_{i}, and let C′_{i} be the intersection of Π_{i} and Z residual to C_{i}. This is a plane conic. An easy calculation with intersection numbers shows that C′_{i} does not meet the general fibre of the second pencil. So it is a component of some singular fibre of that pencil, and it is now obvious that this singular fibre is C′_{1} ∪ C′_{2}. Algebraically, (λ, μ) ∈ k^{2} \ (0, 0) corresponds to a singular fibre if and only if there exists (σ, τ) ∈ k^{2} \ (0, 0) such that

σ(λA_{0}(x) + μA_{1}(x)) + τ(λA_{2}(x) + μA_{3}(x))

is a quadratic form of rank 2 in x = (x_{0}, x_{1}, x_{2}, x_{3}) ; and this condition has an obvious symmetry.

4. Condition (D), Néron models and the Brauer group

In this section we establish a connection between Condition (D) and the structure of the Brauer group of the surface X in the main Theorem. We also give a general definition of the group 𝔖 as ‘the geometric Selmer group’, as well as an intrinsic interpretation of the maps δ_{M}.

In this whole section, we let C be an excellent integral Dedekind scheme (= noetherian, normal, of dimension one). We assume that the residue fields at closed points are perfect. (For the purposes of the present paper, the reader may assume C to be a smooth integral curve over a field of characteristic zero, or a localisation of such a scheme.) We let K be the fraction field of C. For a closed point M ∈ C, we denote by O_{M} the local ring at M, by k_{M} its residue field, by O_{M}^{h}, resp. O_{M}^{sh}, the henselisation, resp. the strict henselisation of O_{M}, and we denote by K_{M}^{h}, resp. K_{M}^{sh}, the fraction fields of these rings. We denote by i_{M} : Spec(k_{M}) → C the natural closed immersion of a closed point M into C, and by j_{K} : Spec(K) → C the natural map of the generic point of C into C.

Given an elliptic curve J_{K}/K, there exists a unique proper, flat, relatively minimal model J^{c}/C with J^{c} integral and regular ([Sha66], [BLR90]). The open set of smooth points for the projection J^{c}/C is the Néron minimal model J/C of J_{K}/K. A key rôle in the discussion to follow will be played by the étale cohomology group H^{1}(C, J).

4.1 Geometric incarnation of H^{1}(C, J) : unramified torsors

The following lemma is well known.

Lemma 4.1.1 Let X be a regular integral scheme, proper and flat over C, such that the generic fibre of the structural morphism π : X → C is a curve of genus 1. Let K be the function field of C, and let J_{K} be the jacobian of the generic fibre X_{K} of π. Assume that the residual characteristics are coprime with the multiplicities of the irreducible components of π. Then the following conditions are equivalent :

(1) there are no multiple fibres (that is the g.c.d. of the multiplicities of the irreducible components is 1) ;

(2) for every closed point M ∈ C, the fibre X_{M} contains an irreducible component of multiplicity 1 ;

(3) for every closed point M ∈ C the class [X_{K}] ∈ H^{1}(K, J_{K}) goes to zero under the restriction to K_{M}^{sh}.

Proof The question is local for the étale topology on C. One thus reduces to the case where C is the spectrum of a strictly henselian discrete valuation ring. Condition (3) is then equivalent to X_{K}(K) ≠ ∅, itself equivalent to the existence of a section for π : X → C. This section meets the special fibre in a component of multiplicity one. Thus (3) implies (2) which trivially implies (1). If (1) is satisfied, then X_{K} possesses a zero-cycle of degree one ([BLR90], Corollary 9.1.9), hence a K-point since it is a curve of genus one. Thus (3) holds.

These conditions depend only on the generic fibre X_{K}. Condition (3) is the definition of an unramified torsor ([BLR90], p.158). Let J → C be the Néron model of J_{K}. By [BLR90], 6.5.3, under this assumption there exists a local Néron model Z_{O_{M}} of Z_{K} = X_{K} (that is, smooth, separated, of finite type, and satisfying the Néron universal mapping property) which can be built by descent from J_{K_{M}^{sh}}. Moreover, Z_{O_{M}} is a torsor under J_{O_{M}} ([BLR90], loc.cit.). These local models glue together into a (unique) global Néron model Z/C ([BLR90], 1.4.1) which is a torsor under J/C. To Z one can canonically attach the proper regular relatively minimal model X, also denoted by Z^{c}. This model is unique [Sha66]. From this model X one can then recover Z as the maximal subscheme of X on which X → C is smooth.

Thus the unramified torsors are classified by the group

H^{1}(C, J) ⟶^{≃} H_{nr}^{1}(K, J) := [Ker : H^{1}(K, J_{K}) ⟶ ∏_{M∈C} H^{1}(K_{M}^{sh}, J_{K})]

(here and below, the notation M ∈ C is shorthand for M running through the closed points of C). That the map H^{1}(C, J) → H^{1}(K, J_{K}) is injective is a consequence of the Néron property. Thus the group H^{1}(C, J) is a kind of ‘geometric’ Tate-Shafarevich group for J (cf. [Gr68], III, section 4).

4.2 Geometric Selmer groups

Let J/C and J_{K}/K be as above. Let n > 0 be an integer invertible on C. From the Kummer sequence

0 → _{n}J_{K} → J_{K} ⟶^{n} J_{K} → 0

we deduce the exact sequence

0 → J(K)/n → H^{1}(K,_{n}J) → _{n}H^{1}(K, J) → 0.

For each closed point M ∈ C, we have a similar exact sequence

0 → J(K_{M}^{sh})/n → H^{1}(K_{M}^{sh},_{n}J) → _{n}H^{1}(K_{M}^{sh}, J) → 0.

Definition 4.2.1 The ‘geometric’ n-Selmer group of J/C, denoted 𝔖(J, n), is the group of elements α ∈ H^{1}(K,_{n}J) such that for each closed point M ∈ C, the image of α in H^{1}(K_{M}^{sh},_{n}J) lies in the image of J(K_{M}^{sh})/n.

For each M ∈ C, let us introduce the group F_{M} of connected components of the (special fibre of the) Néron model of J_{O_{M}^{sh}}. This is a finite module under the Galois group Gal(k_{M}^{sep}/k_{M}) and is zero for almost all M ∈ C. We have the exact sequence of Gal(k_{M}^{sep}/k_{M})-modules

0 → J^{0}(O_{M}^{sh}) → J(O_{M}^{sh}) → F_{M} → 0,

where J^{0} denotes the connected component of identity in the Néron model. The group J^{0}(O_{M}^{sh}) is divisible by n ([BLR90], subsection 7.3) hence we have J(O_{M}^{sh})/n ⟶^{≃} F_{M}/n. On the other hand, the basic property of the Néron minimal model implies J(O_{M}^{sh})/n ⟶^{≃} J(K_{M}^{sh})/n. We thus have J(K_{M}^{sh})/n ≃ F_{M}/n.

Proposition 4.2.2 (a) The geometric n-Selmer group 𝔖(J, n) ⊂ H^{1}(K,_{n}J) is the inverse image in H^{1}(K,_{n}J) of _{n}H^{1}(C, J) ⊂ _{n}H^{1}(K, J) : it fits into the exact sequence

0 → J(K)/n → 𝔖(J, n) → _{n}H^{1}(C, J) → 0.

(b) There is a natural exact sequence

0 → H^{1}(C,_{n}J) → 𝔖(J, n) → ⊕_{M∈C}F_{M}/nF_{M}.

(c) Assume Pic(C) = 0 and assume that _{n}J_{K}/K is isomorphic to the constant group scheme (Z/n)_{K}^{2}. Then the previous sequence may be rewritten as :

0 → (k[C]^{∗}/k[C]^{∗2})^{2} → 𝔖(J, n) → ⊕_{M∈C}F_{M}/nF_{M} → 0.

Proof We claim that there exists a commutative diagram

0  0
↓  ↓
H^{1}(C,_{n}J)  _{n}H^{1}(C, J)
↓  ↓
0 ⟶  J_{K}(K)/n  ⟶  H^{1}(K,_{n}J_{K})  ⟶  _{n}H^{1}(K, J_{K})  ⟶ 0
↓  ↓  ↓
0 ⟶ ⨁_{M∈C}J_{K}(K_{M}^{sh})/n ⟶ ⨁_{M∈C}H^{1}(K_{M}^{sh},_{n}J_{K_{M}^{sh}}) ⟶ ⨁_{M∈C}_{n}H^{1}(K_{M}^{sh}, J_{K_{M}^{sh}}) ⟶ 0

with exact rows and columns. The horizontal exact sequence has already been discussed, and so has been the bottom one. A class in H^{1}(K,_{n}J) comes from a class in H^{1}(U,_{n}J_{U}) for a nonempty open set U ⊂ C. Now H^{1}(O_{M}^{sh},_{n}J) = 0 for any M, for trivial reasons, hence the diagonal map does send the middle exact sequence to the bottom exact sequence (we may write sums rather than products).

The right vertical exact sequence comes from the isomorphism

H^{1}(C, J) ⟶^{≃} Ker : [H^{1}(K, J_{K}) ⟶ ∏_{M∈C} H^{1}(K_{M}^{sh}, J_{K})]

mentioned in subsection 4.1. This much of the diagram is enough to give the exact sequence in (a), which, it should be noted, is a ‘geometric’ analogue of the classical sequence (over number fields) involving the n-Selmer group and the n-torsion of the Tate-Shafarevich group.

Given any Galois-module F over K, we have the general exact sequence :

0 → H^{1}(C, j_{K∗}F) → H^{1}(K, F) → ∏_{M∈C} H^{1}(K_{M}^{sh}, F).

Applying this to the module _{n}J_{K}, we get the exact sequence

0 → H^{1}(C, j_{K∗}(_{n}J_{K})) → H^{1}(K,_{n}J_{K}) → ∏_{M∈C} H^{1}(K_{M}^{sh},_{n}J_{K}),

where for reasons as above the right hand side map lands in the direct sum. By the Néron property we have the isomorphism of étale sheaves J ⟶^{≃} j_{K∗}J_{K} over C, hence also the isomorphism _{n}J ⟶^{≃} j_{K∗}(_{n}J_{K}). We thus find that the middle vertical sequence in the diagram is exact. As discussed before the proposition, we have J(K_{M}^{sh})/n ≃ F_{M}/n for all M ∈ C. A chase through the diagram now gives the exact sequence in (b).

If _{n}J_{K}/K is isomorphic to the constant group scheme (Z/n)_{K}^{2} over K, then

nJ ⟶^{≃} j_{K∗}(_{n}J_{K}) ≃ j_{K∗}((Z/n)_{K}^{2}) = (Z/n)_{C}^{2}.

Thus H^{1}(C,_{n}J) = H^{1}(C, (Z/n)_{C}^{2}). That _{n}J_{K}/K is isomorphic to the constant group scheme (Z/n)_{K}^{2} over K also implies, via the Weil pairing, that the Galois module of n-th roots of unity μ_{n,K} is trivial, i.e. isomorphic to (Z/n)_{K}. If moreover Pic(C) = 0, then the Kummer sequence now yields (k[C]^{∗}/k[C]^{∗n})^{2} ≃ H^{1}(C,_{n}J). In order to prove (c), it only remains to show the surjectivity of the map 𝔖(J, n) → ⊕_{M∈C}F_{M}/nF_{M}. But this follows from the surjectivity of the map H^{1}(K,_{n}J_{K}) → ⊕_{M∈C}H^{1}(K_{M}^{sh},_{n}J_{K_{M}^{sh}}), which translates as the valuation map (K^{∗}/K^{∗n})^{2} → ⊕_{M∈C}(Z/n)^{2}, map which is surjective since Pic(C) = 0.

4.3 Divisors on the special fibres of the Néron model and the map δ

In this subsection, we collect results from Raynaud [Ray70], Artin [Art86], Bosch, Lütkebohmert and Raynaud ([BLR90], subsections 9.5 and 9.6).

Let C be an integral Dedekind scheme whose residue fields are perfect. Let K be the fraction field of K. Let π : X → C be a relatively minimal, proper and flat morphism, with X integral and regular. Assume that the generic fibre is a smooth, geometrically integral curve of genus one over the fraction field K of R, that for each M ∈ C the multiplicities of the components of the geometric special fibre at M are prime to the characteristic of k, and that the greatest common divisor of these multiplicities is one. Let J_{K} = \underline{Pic}_{X_{K}/K}^{0} be the jacobian of the smooth projective curve X_{K} = X ×_{C} Spec(K). The Néron minimal model J/C exists and is of finite type over C. Because of the hypotheses on the multiplicities of the special fibres, to X one associates a torsor Z over C under J, which is just the smooth locus of π (subsection 4.1).

Let \underline{Pic}_{X/C} = R^{1}π_{∗}G_{m} be the Picard functor. This is the fppf sheaf on C associated to the presheaf U → Pic(X ×_{C} U). Taking degrees on the generic fibre defines a short exact sequence of fppf sheaves on C

0 → P_{X} → \underline{Pic}_{X/C} → Z_{C} → 0

defining the open and closed fppf subsheaf P_{X}. Restriction of this sequence to the (small) étale site of C still yields an exact sequence

(4.3.1)  0 → P_{X} → Pic_{X/C} → Z_{C} → 0.

Surjectivity of Pic_{X/C} → Z_{C} follows from the assumptions made on the special fibres.

Let E_{X} ⊂ P_{X} be the schematic closure in P_{X} of the unit section of \underline{Pic}_{X_{K}/K}^{0}. The quotient P_{X}/E_{X} (the biggest separated quotient of P_{X}) is representable by the abelian group scheme J/C. The connected component \underline{Pic}_{X/C}^{0} is separated and coincides with the identity component J^{0} of J.

Let us assume for a moment that C is the spectrum C = Spec(R) of a strictly henselian discrete valuation ring R. Let D be the free, finitely generated, abelian group of divisors supported on the special fibre X_{s}. The subgroup of principal divisors is a free summand, of rank one, generated by the special fibre X_{s} of X/C (as defined by the divisor of the uniformizing parameter of R). Let Δ := D/ZX_{s}. Because of the assumption we made on the multiplicities, this is a finitely generated torsionfree abelian group : there is an exact sequence

0 → Z ⟶^{{n_{Y} }} ⊕_{Y ⊂X_{s}}Z → Δ → 0,

where Y runs through the irreducible components of X_{s}, and n_{Y} is the multiplicity of Y. There is a natural isomorphism Δ ≃ E_{X}(C), which composed with the map E_{X}(C) ⊂ Pic(X) is the natural map associating to a divisor its class. Intersection of divisors defines a surjective map Pic(X) → D^{o} = Hom(D, Z), hence a surjective map P_{X}(C) → Δ^{o} = Hom(Δ, Z). This latter map induces a homomorphism α : Δ → Δ^{o}. One has Coker(α) ≃ J(R)/J^{0}(R). This latter group is none other than the (finite) group of connected components of (the special fibre of) J, which we have denoted F above.

Let us now go back to the general case of an integral Dedekind scheme C. For each M ∈ C and each irreducible component Y ⊂ X_{M}, let n_{Y} be the multiplicity of Y and let k_{Y} be the algebraic closure of k_{M} in the function field k(Y) of Y. Let β_{Y} : Spec(k_{Y}) → Spec(k_{M}) be the natural projection. We then have the exact sequence of sheaves on C :

(4.3.2)  0 → i_{M∗}Z ⟶^{{n_{Y} }} ⊕_{Y ⊂X_{M}} β_{Y ∗}Z → i_{M∗}Δ_{M} → 0.

The various facts recalled above give us several commutative diagrams of exact sequences of sheaves for the étale topology on C.

The first diagram is

(4.3.3)

0 →  P_{X}  →  Pic_{X/C}  →  Z  → 0
↓  ↓  ↓
0 → ⊕_{M∈C}i_{M∗}Δ_{M}^{o} → ⊕_{M∈C}i_{M∗}D_{M}^{o} → ⊕_{M∈C}i_{M∗}Z → 0

The next diagram is

(4.3.4)

0  0
↓  ↓
J^{0} = J^{0}
↓  ↓
0 ⟶ ⊕_{M∈C}i_{M∗}Δ_{M} ⟶^{cl}  P_{X}  ⟶  J  ⟶ 0
||  ↓  ↓
0 ⟶ ⊕_{M∈C}i_{M∗}Δ_{M} ⟶^{α} ⊕_{M∈C}i_{M∗}(Δ_{M}^{o}) ⟶ ⊕_{M∈C}i_{M∗}F_{M} ⟶ 0
↓  ↓
0  0

where cl denotes the map which sends a divisor to its class in the Picard group and the sheaf F_{M} is the Galois module (over k_{M}) defined by the components of the geometric special fibre at M (see subsection 4.2). This diagram is none other than (1.21) in [Art86], where attention is restricted to the case where X_{K}(K) ≠ ∅.

The last diagram is

(4.3.5)

0  0
↓  ↓
0 → ⊕_{M∈C}i_{M∗}Δ_{M} → P_{X}  →  J  → 0
||  ↓  ↓
0 → ⊕_{M∈C}i_{M∗}Δ_{M} → Pic_{X/C} → j_{K∗}Pic_{X_{K}/K} → 0
↓  ↓
Z_{C} = Z_{C}
↓  ↓
0  0

Surjectivity of Pic_{X/C} → j_{K∗}Pic_{X_{K}/K} follows from the regularity of X.

Observe that the restrictions of the two right hand side vertical exact sequences to the generic point of C coincide with the exact sequence of étale sheaves on K :

0 → Pic_{X_{K}/K}^{0} → Pic_{X_{K}/K} → Z_{K} → 0,

i.e.

0 → J_{K} → Pic_{X_{K}/K} → Z → 0,

The boundary map Z → H^{1}(K, J_{K}) maps 1 to the class of the principal homogeneous space X_{K} = Z_{K} under J_{K}. Since the map H^{1}(C, J) → H^{1}(K, J_{K}) is injective by the Néron property we conclude that the image of 1 under the boundary map Z → H^{1}(C, J) in the cohomology sequence associated to the right hand side exact sequence is the class [Z]. From diagram (4.3.5) we see that this boundary map factorizes as Z ⟶^{d_{X}} H^{1}(C, P_{X}) → H^{1}(C, J), where the map d_{X} : Z → H^{1}(C, P_{X}) is the boundary map in the cohomology sequence associated to the middle vertical exact sequence.

For each M ∈ C, there is a map δ_{M} : H^{1}(C, J) → H^{1}(k_{M}, F_{M}) coming from diagram (4.3.4) and there is a boundary map d_{M} : H^{1}(C, J) → H^{2}(k_{M}, Δ_{M}) associated to the exact sequence

(4.3.6)  0 → ⊕_{M∈C}i_{M∗}Δ_{M} ⟶^{cl} P_{X} → J → 0.

For the sake of simplicity, let us assume that C is local, with closed point M. Diagram (4.3.4) gives rise to the commutative diagram of exact sequences :

H^{1}(C, P_{X}) → H^{1}(C, J) → H^{2}(k_{M}, Δ_{M})
↓  ↓ δ_{M}  ||
H^{1}(k_{M}, Δ_{M}^{o}) → H^{1}(k_{M}, F_{M}) → H^{2}(k_{M}, Δ_{M})

In diagram (4.3.3), D_{M}^{o} is a permutation module over the absolute Galois group of k_{M}, hence H^{1}(k_{M}, D_{M}^{o}) = 0. Cohomology of (4.3.3) therefore yields a commutative diagram

Z ⟶^{d_{M}} H^{1}(C, P_{X})
||  ↓
Z ⟶ H^{1}(k_{M}, Δ_{M}^{o})

where the lower map is surjective. Let us now revert to the case of an arbitrary C. Let us now revert to the case of an arbitrary C. We have proved :

Proposition 4.3.1 For each closed point M ∈ C, the coboundary map d_{M} : H^{1}(C, J) → H^{2}(k_{M}, Δ_{M}) coming from (4.3.6) factorizes as :

H^{1}(C, J) → H^{1}(C, J)/Z[Z] ⟶^{δ_{M}} H^{1}(k_{M}, F_{M})/Zδ_{M}([Z]) ↪ H^{2}(k_{M}, Δ_{M}),

the last map being an injection.

This enables us to give the following definition :

Definition 4.3.2 Let D(X) be the subgroup of H^{1}(C, J)/Z[Z] defined by either of the equivalent definitions

D(X) := Ker[H^{1}(C, J)/Z[Z] ⟶^{⊕_{M∈C}δ_{M}} ⨁_{M∈C} H^{1}(k_{M}, F_{M})/Zδ_{M}([Z])],

where the maps δ_{M} are induced by the maps J → i_{M∗}F_{M}, or

D(X) := Ker[H^{1}(C, J)/Z[Z] ⟶^{⊕_{M∈C}d_{M}} ⨁_{M∈C} H^{2}(k_{M}, Δ_{M})],

where d_{M} comes from (4.3.6). (We have abused notation and denoted by δ_{M} and d_{M} maps induced by maps bearing these names.)

The first definition uses the Néron model J/C and the class [Z] ∈ H^{1}(C, J). In the second definition, the maps d_{M} are induced by the coboundary maps d_{M} : H^{1}(C, J) → H^{2}(k_{M}, Δ_{M}) associated to (4.3.6) ; this exact sequence involves the geometry of X.

4.4 Cohomology of G_{m}

In this subsection we develop the (hyper)cohomological machinery necessary to establish theorem 4.5.1 below and its various corollaries. Although theorem 4.5.1 could be stated without reference to derived categories, the proof of the stated commutativities in easier to give from this point of view. We refer the reader to [BBD82] and the literature cited there for the necessary background, including the definition and properties of the truncation functors τ_{≤n}.

Let π : X → C be as in the beginning of subsection 4.3. Note that since π is proper and has geometrically connected fibres one has G_{m,C} →^{≃} π_{∗}G_{m,X}, and G_{m,K} →^{≃} π_{∗}G_{m,X_{K}}. Let j_{X_{K}} : X_{K} → X be the map induced by j_{K} : Spec(K) → C.

Lemma 4.4.1 Let X/C be as above.

(a) We have R^{n}j_{K∗}G_{m,K} = 0 for all n ≥ 1 ;

(b) we have R^{1}j_{X_{K}∗}G_{m,X_{K}} = 0 ; (c) if the local rings of C are Japanese, we have R^{n}π_{∗}G_{m,X} = 0 for n ≥ 2.

Proof (a) Recall that we assumed all residue fields of C to be perfect. The result now follows from [Gr68], III, Thm. 1.1 : for n = 1 this is a consequence of Hilbert’s theorem 90, and for n ≥ 2 the fraction field of an excellent henselian discrete valuation ring with algebraically closed residue field is a C_{1} field (Lang’s theorem) – when C is a smooth curve over a field of characteristic zero, the result follows from Tsen’s theorem – hence has cohomological dimension 1.

(b) Let P be a point of X. Then on Spec(O_{X,P}^{sh}) the sheaf R^{1}j_{∗}G_{m,X_{K}} is computed as P ic(O_{X,P}^{sh} ×_{C} K). There is a natural restriction map

P ic(O_{X,P}^{sh}) → P ic(O_{X,P}^{sh} ×_{C} K).

Since X is regular, this map is surjective. It is clear that P ic(O_{X,P}^{sh}) = 0. Thus our statement is proved.

(c) This is Artin’s theorem ([Gr68], III, Cor. 3.2, p. 98). Here we are using the fact that the fibres have dimension one.

Given a sheaf F on C, we denote by F^{•} its image in the derived category, under the map which sends it to the class of the complex consisting of F in degree zero and 0 elsewhere.

Theorem 4.4.2 Let X/C be as above. In the derived category D_{+}(C_{ét}) of bounded below complexes of sheaves for the étale topology on C, there is a commutative diagram of distinguished triangles (arrows will be defined in the proof) :

G_{m,C}^{•}  →  τ_{≤1}(Rπ_{∗})(G_{m,X}^{•})  →  Pic_{X/C}^{•}[−1]
↓  ↓  ↓
j_{K∗}G_{m,K}^{•}  →  τ_{≤1}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•})  → j_{K∗}Pic_{X_{K}/K}^{•}[−1]
↓  ↓  ↓
⊕_{M∈C}i_{M∗}Z_{k(M)}^{•} → ⊕_{M∈C} ⊕_{Y ⊂X_{M}} τ_{≤1}(Rπ_{∗})(i_{Y ∗}Z_{k(Y)}^{•}) → ⊕_{M∈C}i_{M∗}Δ_{M}^{•}

Proof Given a sheaf F on X and the complex (Rπ_{∗})(F^{•}), we have a distinguished triangle :

(π_{∗}F)^{•} → τ_{≤1}(Rπ_{∗})(F^{•}) → (R^{1}π_{∗}F)^{•}[−1]

and this associated distinguished triangle is functorial in F.

Let us apply this first to the sheaf G_{m,X}. From G_{m,C} →^{≃} π_{∗}(G_{m,X}) and R^{1}π_{∗}(G_{m,X}) = Pic_{X/C}, we obtain the top distinguished triangle in the diagram.

Let us now apply this to the sheaf j_{X_{K}∗}(G_{m,X_{K}}). The two maps π∘j_{X_{K}} and j_{K} ∘π_{K} coincide, and define a map ρ : X_{K} → C. We have π_{∗}(j_{X_{K}∗}(G_{m,X_{K}})) = ρ_{∗}(G_{m,X_{K}}) = j_{K∗}(π_{∗}(G_{m,X_{K}})) = j_{K∗}(G_{m,K}). This identifies the left term of the middle horizontal distinguished triangle.

Let us show (R^{1}π_{∗})(j_{X_{K}∗}G_{m,X_{K}}) →^{≃} j_{K∗}P ic_{X_{K}/K}. The spectral sequence of composite functors Rρ_{∗} = Rπ_{∗} ∘ Rj_{X_{K}∗} yields the exact sequence of sheaves

0 → R^{1}π_{∗}(j_{X_{K}∗}G_{m,X_{K}}) → R^{1}ρ_{∗}(G_{m,X_{K}}) → π_{∗}(R^{1}j_{X_{K}∗}(G_{m,X_{K}})).

By Lemma 4.4.1 (b), we have R^{1}j_{X_{K}∗}(G_{m,X_{K}}) = 0, hence R^{1}π_{∗}(j_{X_{K}∗}G_{m,X_{K}}) →^{≃} R^{1}ρ_{∗}(G_{m,X_{K}}). The spectral sequence of composite functors Rρ_{∗} = Rj_{K∗} ∘ Rπ_{∗} gives rise to the exact sequence

0 → R^{1}j_{K∗}(G_{m,K}) → R^{1}ρ_{∗}(G_{m,X_{K}}) → j_{K∗}(R^{1}π_{∗}(G_{m,X_{K}})) → R^{2}j_{K∗}(G_{m,K}),

where we have used G_{m,K} →^{≃} π_{∗}G_{m,X_{K}}. By Lemma 4.4.1 (a), we have R^{1}j_{K∗}(G_{m,K}) = 0 and R^{2}j_{K∗}(G_{m,K}) = 0. We thus have a natural isomorphism R^{1}ρ_{∗}(G_{m,X_{K}}) →^{≃} j_{K∗}(R^{1}π_{∗}(G_{m,X_{K}})), which composed with R^{1}π_{∗}(j_{∗}G_{m,X_{K}}) →^{≃} R^{1}ρ_{∗}(G_{m,X_{K}}) yields the announced isomorphism R^{1}π_{∗}(j_{X_{K}∗}G_{m,X_{K}}) →^{≃} j_{K∗}P ic_{X_{K}/K} and completes the construction of the middle distinguished triangle in the diagram. It is an easy matter to check that the vertical maps induced by functoriality in F of the construction above are the obvious ones.

Since C is regular we have an exact sequence of étale sheaves over C :

1 → G_{m,C} → j_{K∗}G_{m,K} → ⊕_{M∈C}i_{M∗}Z_{k(M)} → 0.

Since X is also regular, there is a similar exact sequence of étale sheaves over X :

1 → G_{m,X} → j_{X_{K}∗}G_{m,X_{K}} → ⊕_{M∈C} ⊕_{Y ⊂X_{M}} i_{Y ∗}Z_{k(Y)} → 0,

where i_{Y} : Spec(k(Y)) → X is the inclusion of the generic point of Y.

The second exact sequence gives rise to the following distinguished triangle in the derived category of étale sheaves over C :

Rπ_{∗}(G_{m,X}^{•}) → Rπ_{∗}(j_{X_{K}∗}G_{m,X_{K}} ^{•}) → ⊕_{M∈C} ⊕_{Y ⊂X_{M}} Rπ_{∗}(i_{Y ∗}Z_{k(Y)}^{•}).

By Artin’s theorem (lemma 4.4.1 (c)) we have R^{2}π_{∗}(G_{m,X}) = 0. This implies that if we apply τ_{≤1} to the previous distinguished triangle, we still get a distinguished triangle. By the universal property of truncation, we find that there is a map of distinguished triangles :

G_{m,C}^{•}  →  τ_{≤1}(Rπ_{∗})(G_{m,X}^{•})
↓  ↓
j_{K∗}G_{m,K}^{•} → τ_{≤1}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•})
↓  ↓
⊕_{M∈C}i_{M∗}Z_{k(M)}^{•} → ⊕_{M∈C} ⊕_{Y ⊂X_{M}} τ_{≤1}(Rπ_{∗})(i_{Y ∗}Z_{k(Y)}^{•})

Let us analyze the map i_{M∗}Z_{k(M)}^{•} → τ_{≤1}(Rπ_{∗})(i_{Y ∗}Z_{k(Y)}^{•}). Let σ_{Y} = π ∘i_{Y} : Spec(k(Y)) → C. This map also factorizes as σ_{Y} = i_{M} ∘ β_{Y} ∘ α_{Y}, with α_{Y} : Spec(k(Y)) → Spec(k_{Y}) and β_{Y} = Spec(k_{Y}) → Spec(k_{M}) the obvious maps. Since β_{Y} and i_{M} are finite maps, we have R^{n}β_{Y ∗} = 0 and R^{n}i_{M∗} = 0 for n ≥ 1. By definition, k_{Y} is algebraically closed in k(Y). Let k_{Y} ⊂ F_{Y} be the embedding of k_{Y} into a separable closure, and let F (Y) = k(Y) ⊗_{k_{Y}} F_{Y}, which is a field. We have H^{0}(F (Y), Z) = Z and H^{1}(F (Y), Z) = 0. Hence the natural map Z_{k_{Y}} → α_{Y ∗}Z_{k(Y)} is an isomorphism, and R^{1}α_{Y ∗}Z_{k(Y)} = 0. The spectral sequences of composite functors now easily yield : R^{1}π_{∗}(i_{Y ∗}Z_{k(Y)}) = 0. On the other hand, π_{∗}(i_{Y ∗}Z_{k(Y)}) = σ_{Y ∗}(Z_{k(Y)}) = i_{M∗}β_{Y ∗}Z_{k_{Y}}. We thus have a natural quasi-isomorphism

(i_{M∗}β_{Y ∗}Z_{k_{Y}})^{•} →^{≃} τ_{≤1}(Rπ_{∗})(i_{Y ∗}Z_{k(Y)}^{•})

hence the bottom map in the above diagram may be rewritten as

⊕_{M∈C}(i_{M∗}Z_{k_{M}})^{•} → ⊕_{M∈C} ⊕_{Y ⊂X_{M}} (i_{M∗}β_{Y ∗}Z_{k_{Y}})^{•}.

It is an easy matter to check that this map is induced by the maps n_{Y} : Z_{k_{M}} → β_{Y ∗}Z_{k_{Y}}. Indeed, for any irreducible component Y ⊂ X_{M} with multiplicity n_{Y}, a valuation computation shows that the following diagram of sheaves is commutative :

j_{K∗}(G_{m,K}) ⟶ j_{X_{K}∗}(G_{m,X_{K}})
↓  ↓
i_{M∗}Z_{k_{M}}  ⟶^{n_{Y}} i_{Y ∗}(Z_{k(Y)})

We have now identified all terms of the above diagram of distinguished triangles. As any such diagram, it may be completed into a diagram ‘of the nine’ (see [BBD82] Prop. 1.1.11). This diagram of the nine is the one announced in the theorem, except that we have not yet identified the term in the bottom right hand side. This we can do in two ways :

Using the exact sequences

0 → i_{M∗}Z ⟶^{{n_{Y} }} ⊕_{Y ⊂X_{M}} β_{Y ∗}Z → i_{M∗}Δ_{M} → 0,

we find that this term is ⊕_{M∈C}i_{M∗}(Δ_{M})^{•}.

Or we use the exact sequence of sheaves :

0 → ⊕_{M∈C}i_{M∗}Δ_{M} → Pic_{X/C} → j_{K∗}Pic_{X_{K}/K} → 0

which gives us the distinguished triangle

Pic_{X/C}^{•}[−1] → (j_{K∗}Pic_{X_{K}/K})^{•}[−1] → ⊕_{M∈C}(i_{M∗}Δ_{M})^{•}.

This completes the proof.

Let us now identify various hypercohomology groups of the complexes appearing in theorem 4.4.2.

Proposition 4.4.3 Let assumptions be as in theorem 4.4.2.

(a) For n ≥ 0, we have H^{n}(C, G_{m,C}^{•}) = H^{n}(C, G_{m,C}).

(b) For n ≥ 0, we have H^{n}(C, τ_{≤1}(Rπ_{∗})(G_{m,X}^{•})) = H^{n}(X, G_{m,X}).

(c) For n ≥ 0, we have H^{n}(C, j_{K∗}G_{m,K}^{•}) = H^{n}(C, j_{K∗}G_{m,K}) = H^{n}(K, G_{m,K}).

(d) We have H^{2}(C, Pic_{X/C}^{•}[−1]) = H^{1}(C, Pic_{X/C}) = H^{1}(C, P_{X})/d(1), where the map d is induced by the cohomology of sequence (4.3.1).

(e) We have H^{2}(C, (j_{K∗}Pic_{X_{K}/K})^{•}[−1]) = H^{1}(C, j_{K∗}Pic_{X_{K}/K}) = H^{1}(C, J)/[Z].

Proof Quite generally, for F a sheaf on C, we have H^{i}(C, F^{•}) = H^{i}(C, F). Statement (a) is thus clear. Because of Artin’s theorem (Lemma 4.4.1 (c)), the natural map τ_{≤1}(Rπ_{∗})(G_{m,X}^{•}) → (Rπ_{∗})(G_{m,X}^{•}) is a quasi-isomorphism. For any integer n, we therefore have

H^{n}(C, τ_{≤1}(Rπ_{∗})(G_{m,X}^{•})) →^{≃} H^{n}(C, (Rπ_{∗})(G_{m,X}^{•})) = H^{n}(X, G_{m,X}),

which is claim (b). By Lemma 4.4.1 (a), we have R^{n}j_{K∗}(G_{m,K}) = 0 for n ≥ 1. Statement (c) follows. Statement (d) is obvious since H^{1}(C, Z_{C}) = 0 (since C is regular, hence in particular geometrically unibranch). As for statement (e), the exact sequence of sheaves

0 → J → j_{K∗}Pic_{X_{K}/K} → Z_{C} → 0

from diagram (4.3.5) gives rise to an exact sequence

Z → H^{1}(C, J) → H^{1}(C, j_{K∗}Pic_{X_{K}/K}) → 0

(the last zero for the same reason as above). It only remains to use the fact that the image of 1 ∈ Z is none other than the class of the homogeneous space Z (comments after (4.3.5)).

Theorem 4.4.4 The following groups are canonically isomorphic :

(a) the subgroup Br^{gnr}(X_{K}) of the Brauer group Br(X_{K}) consisting of geometrically unramified elements, that is, elements of Br(X_{K}) whose image in Br(X_{K_{M}^{sh}}) lies in the subgroup Br(X_{O_{M}^{sh}}), for any closed point M ∈ C;

(b) the subgroup of the Brauer group Br(X_{K}) consisting of elements of Br(X_{K}) which vanish in Br(X_{K_{M}^{sh}}) for any closed point M ∈ C;

(c) the hypercohomology group H^{2}(C, τ_{≤1}(Rπ_{∗})j_{∗}G_{m,X_{K}}).

Asssume that C is a smooth geometrically integral curve over a field k of characteristic zero. Then Br^{gnr}(X_{K}) also coincides with :

(d) the group of elements A ∈ Br(X_{K}) such that for each irreducible component Y ⊂ X_{M} of a fibre X_{M} above a closed point M, the residue res_{Y} (A) ∈ H^{1}(k(Y), Q/Z) is ‘constant’, i.e. comes from H^{1}(k_{Y} , Q/Z) ;

(e) the group of elements A ∈ Br(X_{K}) = Br(X_{k(C)}) whose image in Br(X_{k^{s}(C)}) is unramified, i.e. comes from Br(X ×_{k} k^{s}).

Proof Since the relative dimension is one, Br(X_{O_{M}^{sh}}) = 0 by Artin’s theorem. The equivalence of (a) and (b) immediately follows.

By Lemma 4.4.1 (c), we have R^{1}j_{X_{K}∗}G_{m,X_{K}} = 0. Hence the spectral sequence

E_{2}^{pq} = H^{p}(X, R^{q}j_{X_{K}∗}G_{m,X_{K}}) ⟹ H^{n}(X_{K}, G_{m,X_{K}})

yields the short exact sequence

0 → H^{2}(X, j_{X_{K}∗}(G_{m,X_{K}})) → H^{2}(X_{K}, G_{m,X_{K}}) → H^{0}(X, R^{2}j_{X_{K}∗}(G_{m,X_{K}}))

and we have a compatible exact sequence for each map X_{O_{M}^{sh}} → X_{K_{M}^{sh}}. Thus if A ∈ H^{2}(X_{K}, G_{m,X_{K}}) lies in Br^{gnr}(X_{K}), then its image b ∈ H^{0}(X, R^{2}j_{X_{K}∗}(G_{m,X_{K}})) restricts to zero in each H^{0}(X_{O_{M}^{sh}}, R^{2}j_{X_{K{sh \atop M}∗}}(G_{m,X_{K{sh \atop M}}})). We conclude that each closed point of X admits an étale neighbourhood over which the section b vanishes, hence b = 0. The exact sequence above now implies that Br^{gnr}(X_{K}) actually lies in H^{2}(X, j_{X_{K}∗}(G_{m,X_{K}})) ⊂ H^{2}(X_{K}, G_{m,X_{K}}), more precisely it coincides with the subgroup of elements of H^{2}(X, j_{X_{K}∗}(G_{m,X_{K}})) which restrict to zero in each H^{2}(X_{O_{M}^{sh}}, j_{X_{K{sh \atop M}} ∗}(G_{m,X_{K{sh \atop M}}})).

We have the distinguished triangle

τ_{≤1}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•}) → (Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•}) → τ_{≥2}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•}).

By the definition of truncation, we have H^{n}(C, τ_{≥2}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•})) = 0 for n ≤ 1 and H^{2}(C, τ_{≥2}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•})) = H^{0}(C, R^{2}π_{∗}(j_{X_{K}∗}G_{m,X_{K}})). Hence the hypercohomology exact sequence associated to the above triangle gives in particular the exact sequence :

0 → H^{2}(C, τ_{≤1}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•})) → H^{2}(X, j_{X_{K}∗}G_{m,X_{K}}) → H^{0}(C, R^{2}π_{∗}(j_{X_{K}∗}G_{m,X_{K}})),

and we have compatible exact sequences for each X_{O_{M}^{sh}}/O_{M}^{sh} in place of X/C. Now a class in H^{2}(X, j_{X_{K}∗}G_{m,X_{K}}) which restricts to zero in each H^{2}(X_{O_{M}^{sh}}, j_{X_{K{sh \atop M}} ∗}(G_{m,X_{K{sh \atop M}}})) has for image

a section of the sheaf R^{2}π_{∗}(j_{X_{K}∗}G_{m,X_{K}}) with trivial value in a suitable neighbourhood of any closed point of C, hence this image is zero. We conclude that Br^{gnr}(X_{K}) is identified with a subgroup of

H^{2}(C, τ_{≤1}Rπ_{∗}(j_{X_{K}∗}G_{m,X_{K}} ^{•})) ⊂ H^{2}(X, j_{X_{K}∗}G_{m,X_{K}}) ⊂ H^{2}(X_{K}, G_{m}).

On the other hand, we may consider a similar and compatible distinguished triangle as the one above over O_{M}^{sh}. Since a strictly henselian local ring has sheaf cohomology only in degree zero, we clearly have H^{2}(O_{M}^{sh}, τ_{≤1}Rπ_{∗}(j_{X_{K{sh \atop M}} ∗}G_{m,X_{K{sh \atop M}}})^{•}) = 0 as we would have with any complex concentrated in degrees at most one. Thus the image of the composite map

H^{2}(C, τ_{≤1}(Rπ_{∗})(j_{X_{K}∗}G_{m,X_{K}} ^{•}) → H^{2}(X, j_{X_{K}∗}G_{m,X_{K}}) → H^{2}(X_{K}, G_{m})

vanishes in each H^{2}(X_{K_{M}^{sh}}, G_{m}). This completes the proof that the group in (c) is the same as the group in (a) and (b).

Let A be a discrete valuation ring with field of fractions L and perfect residue field κ. Let j_{L} : Spec(L) → Spec(A) (resp. i_{κ}) be the inclusion of the generic (resp. special) point. By lemma 4.1.1 we have R^{n}j_{L∗}G_{m,L} = 0 for all n ≥ 1. From the exact sequence of sheaves

1 → G_{m,A} → j_{L∗}G_{m,L} → i_{κ∗}Z_{κ} → 0

we now deduce the exact sequence

0 → Br(A) → Br(L) ⟶^{res} H^{1}(κ, Q/Z).

Let X be a two-dimensional, regular, integral scheme whose residue class fields are of characteristic zero. By classical results on the purity of the Brauer group ([Gr68], II, Prop. 2.3) an element α in the Brauer group of the function field of X belongs to Br(X) if and only if for any closed irreducible subvariety Y ⊂ X of codimension one with function field κ(Y), we have res_{Y} (α) = 0 ∈ H^{1}(κ(Y), Q/Z).

For any given closed point M ∈ C we have a natural commutative diagram :

X ← X ×_{k} k^{s} ←  X_{O_{M}^{sh}}
↓  ↓  ↓
C ← C ×_{k} k^{s} ← Spec(O_{M}^{sh})

Since the horizontal maps are limits of étale maps, for Y ⊂ X_{M} a component of the fibre X_{M}, the upper part of the diagram gives rise to the following commutative diagram :

Br(X_{K})  →  Br(X_{K⊗_{k}k^{s}})  ⟶  Br(X_{K_{M}^{sh}})
↓ res  ↓ res  ↓ res
H^{1}(k(Y), Q/Z) → H^{1}(k(Y) ⊗_{k} k^{s}, Q/Z) ⟶^{≃} H^{1}(k(Y) ⊗_{k} k^{s}, Q/Z)

The kernel of the map H^{1}(k(Y), Q/Z) → H^{1}(k(Y)⊗_{k} k^{s}, Q/Z) is precisely H^{1}(k_{Y} , Q/Z), where k_{Y} is the separable closure of k in k(Y). Combining these remarks, we conclude that Br^{gnr}(X_{K}) coincides with the groups in (d) and (e).

When the field k is of positive characteristic p, the identification of Br^{gnr}(X_{K}) with the groups in (d) and (e) still holds, provided one restricts to the prime-to-p torsion.

4.5 The Brauer group of X, the group D(X), and the geometric Selmer group

Theorem 4.5.1 Let X/C be as in the beginning of subsection 4.3, Br^{gnr}(X_{K}) as in theorem 4.4.4 and D(X) as in definition 4.3.2.

a) There is a commutative diagram

0  0  0
↓  ↓  ↓
Br(C)  ⟶  Br(X)  ⟶  D(X)
↓  ↓  ↓
Br(K)  ⟶  Br^{gnr}(X_{K})  ⟶  H^{1}(C, J)/[Z]
↓ res  ↓  ↓ {d_{M}}
⨁_{M∈C} H^{1}(k_{M}, Q/Z) ⟶ ⨁_{M∈C} ⨁_{Y ⊂X_{M}} H^{1}(k_{Y} , Q/Z) ⟶ ⨁_{M∈C} H^{2}(k_{M}, Δ_{M})

with exact columns and exact middle and bottom rows, the map Br(X) → D(X) being induced from the rest of the diagram.

b) If k is a number field and C is a dense open set of P_{k}^{1}, then the middle and bottom rows of this diagram are right exact.

Proof Applying hypercohomology to the diagram of theorem 4.4.2 and using the identifications of hypercohomology groups in Proposition 4.4.3 gives a). Note that it gives more, in particular the horizontal middle exact sequence extends to the right, the next term being H^{3}(K, G_{m}), and the bottom exact sequence also extends to the right, the next term being ⊕_{M∈C}H^{3}(k_{M}, Z) ≃ ⊕_{M∈C}H^{2}(k_{M}, Q/Z).

If F is a number field, then H^{2}(F, Q/Z) = 0 ([Schn79]). Thus if k is a number field and C an open of P_{k}^{1} the bottom sequence is right exact. Under the same assumptions, H^{3}(k(P_{k}^{1}), G_{m}) = 0 : this is an easy consequence (cf. [Ha94], p. 241) of the previous vanishing result for all finite extensions F/k, together with the vanishing of H^{3}(k, G_{m}) ([ANT67], p. 199). This completes the proof of b).

Note that if C is a curve over a field of characteristic zero, the map Br^{gnr}(X_{K}) → H^{1}(k_{Y} , Q/Z) in the diagram above is induced by the residue map at the generic point of Y.

Let us define the vertical Brauer group of X (with respect to π) as :

Br_{vert}(X) := Br(X) ∩ π^{∗}(Br(K)) ⊂ Br(X_{K}),

and let the ‘horizontal’ (or rather non-vertical) Brauer group be :

Br_{hor}(X) := Br(X)/Br_{vert}(X).

Theorem 4.5.2 Let π : X → C be as above, and let Br(X) → D(X) be the map in the above diagram.

a) This map induces an embedding Br_{hor}(X) ↪ D(X).

b) Assume that k is a number field and C = A_{k}^{1}. Then the map Br_{hor}(X) ↪ D(X) is an isomorphism. c) Assume that k is a number field and C = P_{k}^{1}. There is an exact sequence

0 → Br_{hor}(X) → D(X) → H^{1}(k, Q/Z)/[∑_{M} t_{M}(H^{1}(k_{M}, Δ_{M}))].

Here t_{M} : H^{1}(k_{M}, Δ_{M}) → H^{1}(k, Q/Z) is the composite map H^{1}(k_{M}, Δ_{M}) → H^{2}(k_{M}, Z) ≃ H^{1}(k_{M}, Q/Z) → H^{1}(k, Q/Z), the first map coming from the exact sequence 0 → Z → D_{M} → Δ_{M} → 0, the last map being the norm map from k_{M} to k.

d) In the situation of c), let A_{k}^{1} ⊂ P_{k}^{1} be the complement of a k-point. Then Br_{hor}(X) injects into D(X ×_{P_{k}^{1}} A_{k}^{1}).

Proof This follows from the previous theorem and a chase through the big diagram in that theorem, together with the following facts. Firstly, the bottom exact sequence extends on the left to a map H^{1}(k_{M}, Δ_{M}) → H^{2}(k_{M}, Z) = H^{1}(k_{M}, Q/Z). This is clear from the preliminaries. Secondly, we have the following well-known results, which go back to D. K. Faddeev. Let C = A_{k}^{1} = Spec(k[t]). Then the left vertical exact sequence in this diagram extends to an exact sequence :

0 → Br(k) → Br(k(t)) → ⨁_{M∈A_{k}^{1}} H^{1}(k_{M}, Q/Z) → 0.

Let C = P_{k}^{1}. Then the left vertical exact sequence in this diagram extends to an exact sequence :

0 → Br(k) → Br(k(t)) → ⨁_{M∈P_{k}^{1}} H^{1}(k_{M}, Q/Z) → H^{1}(k, Q/Z) → 0,

where maps H^{1}(k_{M}, Q/Z) → H^{1}(k, Q/Z) are norm maps from k_{M} to k.

Let us revert to the geometric Selmer groups discussed in subsection 4.2. Let n be the order of [Z] ∈ H^{1}(C, J), which is the order of [X_{K}] ∈ H^{1}(K, J_{K}). Assume n > 0, i.e. X_{K}(K) = ∅. Let m ∈ 𝔖(J, n) be a lift of [Z] under the surjective map 𝔖(J, n) → _{n}H^{1}(C, J). From proposition 4.2.2 a) we get an isomorphism

𝔖(J, n)/(Zm + J(K)/n) ⟶^{≃} (_{n}H^{1}(C, J))/Z[Z]

which we may compose with the inclusions

(_{n}H^{1}(C, J))/Z[Z] ⊂ _{n}(H^{1}(C, J)/Z[Z]) ⊂ H^{1}(C, J)/Z[Z]

and the map

H^{1}(C, J)/Z[Z] ⟶^{⊕_{M∈C}δ_{M}} ⨁_{M∈C} H^{1}(k_{M}, F_{M})/Zδ_{M}([Z])

whose kernel is by definition D(X). Let D^{n}(X) be the kernel of this composite map :

D^{n}(X) := Ker[𝔖(J, n)/(Zm + J(K)/n) ⟶ ⨁_{M∈C} H^{1}(k_{M}, F_{M})/Zδ_{M}([Z])]

(the notation D^{n}(X) instead of D^{n}(m) is legitimate, since the group 𝔖(J, n)/(Zm + J(K)/n), hence the above kernel, only depend on the image of m in _{n}H^{1}(K, J), which is the class of Z_{K} = X_{K}).

Corollary 4.5.3 Let notation be as above.

a) Assume that n = l is a prime number and that [Z] is not divisible by l in H^{1}(C, J). Then the group _{l}Br_{hor}(X) injects into D^{l}(X). In particular, if D^{l}(X) = 0, then _{l}Br_{hor}(X) = 0, and the whole l-primary part of the Brauer group of X is vertical : Br(X)_{vert}{l} = Br(X){l}.

b) Assume that k is a number field and C = A_{k}^{1}. If _{n}Br_{hor}(X) = 0, i.e. if Br(X)_{vert}{l} = Br(X){l} for each prime l dividing n, then we have D^{n}(X) = 0.

Proof That D^{n}(X) injects into _{n}D(X) is clear from the definitions. If n = l is prime, one immediately checks that this injection is an isomorphism if [Z] is not divisible by l in H^{1}(C, J). By the previous theorem, _{n}Br_{hor}(X) injects into _{n}D(X), and under the hypothesis of b), this map is an isomorphism. Since Br(X) is a torsion group, Br(X)/Br_{vert}(X) is l-torsion free if and only if the inclusion Br_{vert}(X) ⊂ Br(X) induces an isomorphism on l-primary subgroups.

4.6 The Brauer group of the Jacobian fibration

We shall now discuss the special case where X_{K}(K) ≠ ∅. In this subsection, we thus assume X = J^{c} and Z = J.

Let Br_{0}(J) ⊂ Br(J) be the subgroup consisting of elements which vanish when restricted to the zero section of J/C, and let similarly Br_{0}(J_{K}) ⊂ Br(J_{K}) consist of elements vanishing at 0 ∈ J_{K}(K). The following proposition gives an algebraic incarnation of the elements in H^{1}(C, J) in terms of the Brauer group of J_{K}.

Proposition 4.6.1 The following groups are naturally isomorphic :

a) the group H^{1}(C, J) of C-torsors under J;

b) the group H_{nr}^{1}(K, J_{K}) of unramified K-torsors under J_{K};

c) the subgroup of Br_{0}(J_{K}) consisting of elements α such that for every closed point M ∈ C the image of α in Br_{0}(J_{K_{M}^{sh}}) is zero.

Asssume that C is a smooth geometrically integral curve over a field k of characteristic zero. Then these groups also coincide with

d) the subgroup of Br_{0}(J_{K}) consisting of elements whose residues at components of fibres of J^{c}/C are constant ;

e) the subgroup Br_{0}^{gnr}(J_{K}^{c}) ⊂ Br_{0}(J_{K}^{c}) = Br_{0}(J_{K}) consisting of elements whose image in Br(J_{K⊗_{k}k^{s}}^{c}) is unramified, i.e. lies in Br(J^{c} ×_{k} k^{s}).

The exact sequence

0 → Z → D_{M} → Δ_{M} → 0

is split, and one may identify Δ_{M} with the permutation module on components which do not meet the zero section σ_{0} ⊂ J. The commutative square of theorem 4.5.1

Br^{gnr}(J_{K})  ⟶ H^{1}(C, J)
↓  ↓ d_{M}
⨁_{Y ⊂X_{M}} H^{1}(k_{Y} , Q/Z) ⟶ H^{2}(k_{M}, Δ_{M})

gives here rise to the commutative square

Br_{0}^{gnr}(J_{K})  ⟶^{≃} H^{1}(C, J)
↓  ↓ d_{M}
⨁_{Y ⊂X_{M} ,Y ∩σ_{0}=∅} H^{1}(k_{Y} , Q/Z) ⟶^{≃} H^{2}(k_{M}, Δ_{M})

We thus have two ways of viewing the map H^{1}(C, J) → ⨁_{Y ⊂X_{M} ,Y ∩σ_{0}=∅} H^{1}(k_{Y} , Q/Z) : from

the geometric point of view, it comes from the composition

H^{1}(C, J) → H^{1}(k_{M}, F_{M}) → H^{2}(k_{M}, Δ_{M}),

where F_{M} is the (geometric) group of connected components of the special fibre at M. From the algebraic point of view, where H^{1}(C, J) is identified with a subgroup of the Brauer group, it corresponds to residue maps at the generic points of components of the special fibre of J^{c}.

The middle and bottom exact sequences in the main diagram of theorem 4.5.1 are here left and right exact. They are in fact split in a compatible way, the splitting being given by the zero section. From the mentioned diagram one deduces

Br(J^{c}) ≃ Br(C) ⊕ D(J^{c}).

4.7 The case under study in this paper : glossary

For the family of elliptic curves E/P_{k}^{1} under consideration in the introduction, let J_{K} = E_{K} denote the generic fibre over K = k(P^{1}), let J^{c}/C be the regular minimal model over C = P_{k}^{1}, and let J/C be the Néron minimal model.

One easily checks that the singular fibres J_{M}^{c} of J^{c}/C are reduced and consist of two components, which are two smooth curves of genus zero meeting transversally in two points (over the algebraic closure of k_{M}). The special fibres J_{M} of J are thus extensions of F_{M} = Z/2 (with a clearly trivial Galois action) by a one-dimensional k_{M}-torus. In particular, we have H^{1}(k_{M}, F_{M}) = k_{M}^{∗}/k_{M}^{∗2}. The group _{2}J/C is equal to the constant group scheme (Z/2)_{C}^{2}. The map (K^{∗}/K^{∗2})^{2} = H^{1}(K,_{2}J_{K}) → _{2}H^{1}(K, J_{K}) associates to the pair (m_{1}, m_{2}) the class of the curve X(m) given by the equations

m_{1}X_{1}^{2} + c_{1}X_{0}^{2} = m_{2}X_{2}^{2} + c_{2}X_{0}^{2} = m_{1}m_{2}X_{3}^{2} + c_{3}X_{0}^{2}

in P_{K}^{3}, which is a principal homogeneous space under J_{K}.

Easy valuation computations show that the group 𝔖 ⊂ (K^{∗}/K^{∗2})^{2}, resp. 𝔖_{0} ⊂ (K^{∗}/K^{∗2})^{2}, of the introduction coincides with the group 𝔖(J/A_{k}^{1}, 2) ⊂ H^{1}(K,_{2}J_{K}), resp. 𝔖(J/P_{k}^{1}, 2) ⊂ H^{1}(K,_{2}J_{K}). If m lies in 𝔖_{0}, then X(m) extends to a torsor Z over P_{k}^{1} under J.

Next, one checks that for any point M of bad reduction for J_{K}/K (hence M ∈ A_{k}^{1}), the composite map

𝔖 = 𝔖(J/A_{k}^{1}, 2) → _{2}H^{1}(A_{k}^{1}, J) ⟶^{δ_{M}} H^{1}(k_{M}, F_{M}) = k_{M}^{∗}/k_{M}^{∗2}

coincides with the map δ_{M} defined in the introduction.

Let us now look at the statement of the main Theorem. We start from m ∈ 𝔖_{0} ⊂ 𝔖. We assume that m is not in the image of _{2}J_{K}(K)/2. If m lies in the image of J_{K}(K) under the Kummer map, then the rank of J_{K}(K) is at least one, and J_{K} ≃ X(m)_{K}. In this case, rational points are clearly Zariski dense on X.

Let E/P_{k}^{1}, m, X = X(m) be as in the main Theorem. Assume that m ∈ 𝔖_{0} does not lie in the image of J_{K}(K)/2. Let us consider the following statements.

(i) Condition (3) (=(D)) of the main Theorem holds. (ii) D^{2}(X/P_{k}^{1}) = 0 . (iii) _{2}D(X/P_{k}^{1}) = 0 . (iv) The 2-primary subgroups Br_{vert}(X){2} and Br(X){2} coincide. Statement (i) holds if and only if (ii) holds and the rank of J(K) is zero. Statement (iii)

implies (ii) and the converse holds as soon as the class of Z is not divisible by 2 in H^{1}(P_{k}^{1}, J). Statement (iii) implies (iv).

Most of this is immediate from subsection 4.5. Let us simply note that the image of J_{K}(K)/2 in 𝔖_{0} ⊂ 𝔖 lies in the kernel of the map 𝔖 → ⨁_{M∈T} k_{M}^{∗}/k_{M}^{∗2}. If (D) holds, then the inclusion 2J_{K}(K) ⊂ J_{K}(K) induces an isomorphism _{2}J_{K}(K) ≃ J_{K}(K)/2 (see (iii) in the proof of proposition 2.1.3), and the rank of J_{K}(K) is zero. Note that the class [Z] is not divisible by 2 if δ_{M}(m) ≠ 0 for some M ∈ T.

The situation over the affine line is somewhat simpler.

Let E/P_{k}^{1}, m, X = X(m) be as in the main theorem. Assume that m ∈ 𝔖_{0} does not lie in the image of J_{K}(K)/2. Let X_{aff} := X ×_{P_{k}^{1}} A_{k}^{1}.

Let us consider the following statements.

(i) Condition (D_{0}) in Remark 2.5.4 (iv) holds. (ii) D^{2}(X_{aff}/A_{k}^{1}) = 0 . (iii) _{2}D(X_{aff}/A_{k}^{1}) = 0 . (iv) The 2-primary subgroups Br_{vert}(X_{aff}){2} and Br(X_{aff}){2} coincide. Statement (i) holds if and only if (ii) holds and the rank of J(K) is zero. Statement (iii)

implies (ii) and the converse holds as soon as the class of Z_{aff} is not divisible by 2 in H^{1}(A_{k}^{1}, J). Statement (iii) is equivalent to (iv).

Acknowledgements A. N. Skorobogatov acknowledges support from Centre National de la Recherche Scientifique,

Ecole Normale Supérieure (Paris) and the Russian Foundation for Basic Research (project 96-01-01378). He also thanks l’Institut de Mathématiques de Luminy and l’Université de Paris-Sud for their hospitality.

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J.-L. Colliot-Thélène, C.N.R.S., URA D0752, Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay France e-mail : colliot@math.u-psud.fr

A. N. Skorobogatov,

Institute for Problems of Information Transmission,  Max-Planck-Institut für Mathematik
Russian Academy of Sciences,  Gottfried-Claren-Straße 26
19, Bolshoi Karetnyi  D-53225 Bonn
Moscow 101447  Deutschland

Russia e-mail : skoro@mpim-bonn.mpg.de

Sir Peter Swinnerton-Dyer, Isaac Newton Institute, 20 Clarkson Road, Cambridge CB3 0EH United Kingdom e-mail : hpfs100@newton.cam.ac.uk