Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point (Close variation on a paper of Bender and Swinnerton-Dyer)

J.-L. Colliot-Thélène

Let k be a number field and X/k a smooth projective surface equipped with a morphism f : X → P_{k}^{1} the generic fibre of which is a curve of genus one. Assume that each geometric fibre of f contains a multiplicity one component. If the set X(k) of rational points of X is not empty, one may ask whether it is then Zariski-dense in X. If no k-point is known, one may ask whether the Brauer-Manin condition (existence of an adelic point M_{v} ∈ ∏_{v} X(k_{v}) such that ∑_{v} inv_{v}(A(M_{v})) = 0 for each element A of the Brauer group of X) ensures the existence of (many) rational points.

In [SwD1] Swinnerton-Dyer initiated a new technique for studying these problems. The technique was systematized in the joint work [CSS3]. Swinnerton-Dyer has since then written a number of papers on the topic: [SwD2], [SwD3], [BSwD] (joint with Bender), [SwD4]. Here are some common features of these papers.

(i) They postulate the finiteness of Tate-Shafarevich groups of elliptic curves.

(ii) With the notable exception of the very recent paper [SwD4], they build upon Schinzel’s Hypothesis (a very likely, but very unreachable conjecture).

(iii) Some assumption is made on the torsion subgroup of the jacobian J/K of the generic fibre of f (here K denotes the function field k(P^{1})). In the papers [SwD1], [CSS3], [SwD2], [SwD3], one assumes that the entire 2-torsion subgroup _{2}J of J is rational, i.e. isomorphic to the constant K-group scheme (Z/2)_{K}^{2}. In [BSwD], the assumption is that _{2}J contains a copy of the constant group (Z/2)_{K}. (In the paper [SwD4] the 3-torsion satisfies a similar property.)

The paper [BSwD] is written in the style of [SwD2]. The contribution of the present paper is merely to rewrite most of Sections 1 to 5 of [BSwD] in the style of [CSS3], i.e. in terms of the Brauer-Manin obstruction (I have not however produced the analogue of Section 4 of [CSS3]).

Apart from gaining more practice in the new technique, my immediate motivation for writing the present paper was the hope to produce a more satisfactory statement than Theorem 3 in Section 6 of [BSwD]. Namely, under the Schinzel hypothesis and the finiteness of Tate-Shafarevich group, one would hope to prove that the Brauer-Manin obstruction to the Hasse principle is the only obstruction for smooth complete intersections of two quadrics in P_{k}^{4}. It has been known for some time (Sansuc and the author, 1986; Harari [Ha1], Prop. 5.2.3) that this last statement implies the Hasse principle for smooth complete intersections of two quadrics in P_{k}^{n} for n ≥ 5 (for n ≥ 8, the Hasse principle for such intersections of two quadrics is proved in [CTSaSwD]). Section 6 of [BSwD] contains an interesting geometric construction which one could have tried to combine with the techniques in Harari’s papers [Ha1] [Ha2]. But for the time being I still feel that serious difficulties are in the way.

Theorem A below is a reformulation of Theorem 2 of [BSwD]. Theorem B below is a reformulation of Theorem 1 of [BSwD]. Conditions (3.a) and (3.b) in the statements of Theorems A and B are purely algebraic renditions of Conditions 7 and 8 of [BSwD]. The slightly technical Conditions 5 and 6 of [BSwD] (which involved the quadratic reciprocity law, hence had to do with the Brauer-Manin obstruction, but in a rather involved manner) are replaced by the sheer hypothesis that there is no vertical Brauer-Manin obstruction. Out of habit, I have used the one variable, inhomogeneous version of Schinzel’s hypothesis. In his papers, Swinnerton-Dyer uses the two variables, homogeneous version of that hypothesis. The latter hypothesis has now been proved in nontrivial cases (Heath-Brown for x^{3} + 2y^{3} over the integers, Heath-Brown and Moroz for any binary cubic form ax^{3} + by^{3} over the integers), hence the homogeneous version should be preferred in the long run. As Swinerton-Dyer points out, the homogeneous version also has the advantage that the point at infinity on the projective line does not play a special rôle.

The reader of the present paper is supposed to be acquainted with Sections 1 and 2 of the paper [CSS3].

Plan of the paper 0. Statement of the Theorems

1. Selmer groups associated to a degree 2 isogeny 1.1. A statement from class field theory, and some linear algebra 1.2. Isogenies of degree 2 over an arbitrary field 1.3. Local computations 1.4. The Selmer groups as kernels of pairings 1.5. Small Selmer groups

2. Proof of theorem A 2.1. Fibres with points everywhere locally 2.2. The groups 𝔖′, 𝔖′′ and the maps δ′ and δ′′ 2.3. Independence of the choice of λ: the spaces 2.4. Independence of the choice of λ: the pairing 2.5. Completion of the proof of Theorem A

3. Proof of Theorem B

0. Statement of the Theorems

Let k be a field, char(k)=0. Let c(t), d(t), m(t) ∈ k[t] be polynomials. We assume: (0.1) The polynomials c(t), d(t), m(t) are all of even degree, deg(d) = 2deg(c) > 0, c^{2} − d has even positive degree and d(c^{2} − d) has no square factor.

Let E be the smooth projective surface over k which is the projective regular minimal model over P_{k}^{1} of the surface given in affine 3-space by the affine equation

y^{2} = (x − c(t))(x^{2} − d(t)),

the fibration over the affine line A_{k}^{1} being given by (x, y, t) ↦ t.

Given m as above, let π : X(m) → P_{k}^{1} denote the projective regular minimal model over P_{k}^{1} of the surface given in affine 4-space by the system of equations

w_{1}^{2} = m(t)(x − c(t)), w_{2}^{2} = m(t)(x^{2} − d(t)),

the fibration over the affine line A_{k}^{1} being given by (x, w_{1}, w_{2}, t) ↦ t. The restrictions on the degrees of d, c and m made above ensure that the fibres at t = ∞ of X(m)/P_{k}^{1} and of E/P_{k}^{1} have a geometrically integral component of multiplicity one; if moreover the degree of c^{2} − d is equal to the degree of d, then these fibres are smooth.

Let U ⊂ A_{k}^{1} be the open set defined by d(c^{2} − d) ≠ 0. Let U′ ⊂ A_{k}^{1} be the open set defined by d ≠ 0. Let 𝔖′ = k[U′]^{∗}/k[U′]^{∗2}. Let 𝔖′_{0} ⊂ 𝔖′

be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials m′(t) ∈ k[t] which are of even degree and divide d. Let M′ be the set of closed points M of A_{k}^{1} such that d(M) = 0.

Let U′′ ⊂ A_{k}^{1} be the open set defined by c^{2} − d ≠ 0. Let 𝔖′′ = k[U′′]^{∗}/k[U′′]^{∗2}. Let 𝔖′′_{0} ⊂ 𝔖′′ be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials m′′(t) ∈ k[t] which are of even degree and divide c^{2} − d.

Let M′′ be the set of closed points M of A_{k}^{1} such that (c^{2} − d)(M) = 0.

For each M ∈ M = M′ ∪ M′′, let r_{M}(t) be the monic irreducible polynomial defining M. Let

δ′ : 𝔖′ → ⊕_{M∈M′′} k_{M}^{∗}/k_{M}^{∗2}

and

δ′′ : 𝔖′′ → ⊕_{M∈M′} k_{M}^{∗}/k_{M}^{∗2}

be the map given by evaluation at the relevant points M. Since d(c^{2} − d) is squarefree, d and c^{2} − d are coprime hence these evaluation maps are well-defined.

These maps induce maps

δ′_{0} : 𝔖′_{0} → ⊕_{M∈M′′} k_{M}^{∗}/k_{M}^{∗2}

and

δ′′_{0} : 𝔖′′_{0} → ⊕_{M∈M′} k_{M}^{∗}/k_{M}^{∗2},

which will sometimes be denoted simply δ′ and δ′′. Clearly d is in the kernel of the first map, and c^{2} − d is in the kernel of the second map.

We refer the reader to [CTSwD], Section 3, and [CT] for a detailed discussion of the Brauer-Manin obstruction. Given a morphism π : X → P_{k}^{1} with X/k smooth and projective, the vertical subgroup Br_{vert}(X) of the Brauer group of X with respect to π is the subgroup of Br(X) consisting of classes whose restriction to the generic fibre of π come from Br(k(P^{1})). When m ∈ 𝔖′′_{0}, each geometric fibre fibre of π : X(m) → P^{1} contains a component of multiplicity one. This implies that the quotient Br_{vert}(X(m))/Br(k) is finite (the proof of this fact given in [Sk], Cor. 4.5 holds under the mere assumption that the Galois module denoted Ĝ is torsion free; such is the case when the above multiplicity one condition is fulfilled, as proved in [Sk] 3.2.4).

Let Ω be the set of places of the number field 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 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 group Br_{vert}(X)/Br(k) is finite, hence X(𝔸_{k})^{Br_{vert}(X)} is open in X(𝔸_{k}).

For any commutative ring A, let ℌ(A) = A^{∗}/A^{∗2}.

Theorem A Let k be a number field. Let c, d, m ∈ k[t] satisfy (0.1) and let X = X(m).

(1) Assume that the polynomial m divides c^{2} − d. The class of m ∈ ℌ(k(t)) = k(t)^{∗}/k(t)^{∗2} then defines a class in 𝔖′′_{0}.

(2) Assume that the class of m in 𝔖′′_{0} ⊂ ℌ(k(t)) differs from that of 1 and of c^{2} − d.

(3.a) Assume that the kernel of δ′_{0} consists of the classes 1, d.

(3.b) Assume that the kernel of the composite map

𝔖′′_{0} \underrightarrow{δ′′_{0}} ⊕_{M∈M′} k_{M}^{∗}/k_{M}^{∗2} → ⊕_{M∈M′} k_{M}^{∗}/(k_{M}^{∗2}, δ′′_{M}(m)),

is spanned by c^{2} − d and m.

(4) Assume that the Tate-Shafarevich group of each elliptic curve occurring as a fibre of E/P_{k}^{1} is finite.

(5) Assume Schinzel’s hypothesis.

Let R ⊂ P^{1}(k) be the set of points λ ∈ P^{1}(k) such that the fibre X_{λ} = π^{−1}(λ) is smooth and has infinitely many k-points. Let R_{E} ⊂ P^{1}(k) be the set of points λ ∈ P^{1}(k) such that the fibre E_{λ} of E/P_{k}^{1} is an elliptic curve over k of rank (= dim_{Q}(E_{λ}(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 non-empty 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.

Hypothesis (1) ensures in particular that all geometric fibres of π : X → P_{k}^{1} contain a component of multiplicity one. Some condition of that kind is required, if we hope to conclude that infinitely many k-fibres of π have rational points (see Cor. 2.2, Cor. 2.4, Prop. 4.1, Prop. 4.2 of [CSS1]).

If Hypothesis (2) is not satisfied, then the generic fibre of X/P_{k}^{1} has a k(P^{1})-rational point; it can be regarded as an elliptic curve isogenous to the generic fibre of E/P_{k}^{1}. In particular X has a k-rational point and the question of (Zariski) density of k-rational points on the surface X is equivalent to that question for the surface E.

By analogy with the work done in Section 4 of [CSS3], Hypotheses (3.a) and (3.b) should be slightly stronger than the hypothesis that the 2-primary subgroup of the Brauer group of X is contained in the vertical part of the Brauer group. But we have not done the corresponding (algebraic) work here. As mentioned in [CSS3], this would fit in well with the obvious remark that the necessary condition X(𝔸_{k})^{Br_{vert}(X)} ≠ ∅ for the existence of a rational point (assumption made in (b) and (c)) is a priori weaker than the equally necessary condition X(𝔸_{k})^{Br(X)} ≠ ∅ (a condition which does not appear in the Theorem).

Let us now prepare for the statement of Theorem B.

Let k be a field, char(k)=0. Let α_{i}, β_{i}, i = 0, ⋯, 4 be elements of k[t]. Consider the variety given in A_{k}^{1} ×_{k} P_{k}^{3} by the system of equations

(0.2)  \begin{matrix}α_{0}U_{0}^{2} + α_{1}U_{1}^{2} + α_{2}U_{2}^{2} + α_{3}U_{3}^{2} + 2α_{4}U_{2}U_{3} = 0, \\ β_{0}U_{0}^{2} + β_{1}U_{1}^{2} + β_{2}U_{2}^{2} + β_{3}U_{3}^{2} + 2β_{4}U_{2}U_{3} = 0. \end{matrix}

This defines a one-parameter family of intersections of two quadrics in P_{k}^{3}.

Let d_{ij} = α_{i}β_{j} − α_{j}β_{i}. These d_{ij} satisfy a number of useful identities. For any subset {i, j, k, l} of {0, 1, 2, 3, 4}, we have the identity

(0.3)  d_{ij}d_{kℓ} + d_{ik}d_{ℓj} + d_{iℓ}d_{jk} = 0.

We have the identities ([BSwD], (23))

(0.4)  \begin{matrix}4d_{24}^{2}(d_{04}^{2} − d_{02}d_{03}) = (2d_{04}d_{24} − d_{02}d_{03})^{2} − d_{02}^{2}(d_{23}^{2} + 4d_{24}d_{34}) \\ 4d_{24}^{2}(d_{14}^{2} − d_{12}d_{13}) = (2d_{14}d_{24} − d_{12}d_{23})^{2} − d_{12}^{2}(d_{23}^{2} + 4d_{24}d_{34}). \end{matrix}

Let

Δ = d_{01}(d_{23}^{2} + 4d_{24}d_{34})(d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13}).

At any point t ∈ A^{1} where Δ does not vanish, the fibre is a smooth, pure intersection of two quadrics in P_{k}^{3}. In particular, if Δ does not identically vanish, the generic fibre is a smooth, pure intersection of two quadrics in P_{k}^{3}.

We assume: (0.5) All the α_{i}(t) are of the same degree r, all the β_{i}(t) are of the same degree s, and Δ is of degree r + s.

This assumption, which one could slightly relax (see [BSwD]), ensures that the surface given by the system (*) admits a regular proper model over P_{k}^{1} whose fibre over ∞ ∈ P_{k}^{1} is smooth.

We also assume: (0.6) Δ has no square factor. (0.7) d_{04}^{2} − d_{02}d_{03} and d_{02} are coprime; d_{14}^{2} − d_{12}d_{13} and d_{12} are coprime.

Let U ⊂ A_{k}^{1} be the open set defined by Δ ≠ 0. Let U′ ⊂ A_{k}^{1} be the open set defined by Δ′ = d_{01}(d_{23}^{2} + 4d_{24}d_{34}) ≠ 0. Let 𝔖′ = k[U′]^{∗}/k[U′]^{∗2}. Let 𝔖′_{0} ⊂ 𝔖′ be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials r(t) ∈ k[t] which are of even degree and divide Δ′. Let M′ be the set of closed points M of A_{k}^{1} such that Δ′(M) = 0. This set breaks up into the set M′_{1} consisting of closed points M such that d_{01}(M) = 0 and the set M′_{2} consisting of closed points M such that (d_{23}^{2} + 4d_{24}d_{34})(M) = 0.

Let U′′ ⊂ A_{k}^{1} be the open set defined by

Δ′′ = (d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13}) ≠ 0.

Let 𝔖′′ = k[U′′]^{∗}/k[U′′]^{∗2}. Let 𝔖′′_{0} ⊂ 𝔖′′ be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials r(t) ∈ k[t] which are of even degree and divide Δ′′.

Let M′′ be the set of closed points M of A_{k}^{1} such that Δ′′(M) = 0.

For each M ∈ M = M′ ∪ M′′, let r_{M}(t) be the monic irreducible polynomial defining M. Let

δ′ : 𝔖′ → ⊕_{M∈M′′} k_{M}^{∗}/k_{M}^{∗2}

and

δ′′ : 𝔖′′ → ⊕_{M∈M′} k_{M}^{∗}/k_{M}^{∗2}

be the map given by evaluation at the relevant points M. Since Δ = Δ′.Δ′′ is squarefree, Δ′ and Δ′′ are coprime hence these evaluation maps are well-defined.

These maps induce maps

δ′_{0} : 𝔖′_{0} → ⊕_{M∈M′′} k_{M}^{∗}/k_{M}^{∗2}

and

δ′′_{0} : 𝔖′′_{0} → ⊕_{M∈M′} k_{M}^{∗}/k_{M}^{∗2}

Using the formulas (0.4) and the various coprimality assumptions above, one checks that (the class of) d_{23}^{2} + 4d_{24}d_{34} ∈ 𝔖′_{0} is in the kernel of δ′_{0} and that (the classes of) d_{04}^{2} − d_{02}d_{03} and d_{14}^{2} − d_{12}d_{13} in 𝔖′′_{0} are in the kernel of the maps δ′′_{0,M} relative to the points M ∈ M′_{2}.

Let

c = 4d_{04}d_{14} − 2d_{02}d_{13} − 2d_{03}d_{12}.

Let d = 4d_{01}^{2}(d_{23}^{2} + 4d_{24}d_{34}). We have

(0.8)  c^{2} − d = 16(d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13}).

In particular, at any point M ∈ M′_{1}, we have δ′′_{M}(d_{04}^{2} − d_{02}d_{03}) = δ′′_{M}(d_{14}^{2} − d_{12}d_{13}).

Theorem B Let k be a number field. Let α_{i}, β_{i} be as above, and let X be a smooth projective model of the surface defined by the system (0.2), which we may choose equipped with a k-morphism X → P_{k}^{1} extending the map to A_{k}^{1} given by the t-coordinate. Let E/P_{k}^{1} be the associated jacobian fibration.

(1) Assume that all the α_{i}(t) are of the same degree r, that all the β_{i}(t) are of the same degree s, and that Δ is of degree r + s, and assume that the product

Δ = d_{01}(d_{23}^{2} + 4d_{24}d_{34})(d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13})

has no square factor.

(2) Assume that d_{04}^{2}−d_{02}d_{03} and d_{02} are coprime, and that d_{14}^{2}−d_{12}d_{13} and d_{12} are coprime. (3.a) Assume that the kernel of the composite map

𝔖′_{0} \underrightarrow{δ′_{0}} ⊕_{M∈M′′} k_{M}^{∗}/k_{M}^{∗2} → ⊕_{M∈M′′} k_{M}^{∗}/(k_{M}^{∗2}, ε_{M}),

where ε_{M} is the class of (−d_{01}d_{02})(M) if (d_{04}^{2} −d_{02}d_{03})(M) = 0 and is the class of (−d_{01}d_{21})(M) if (d_{14}^{2} − d_{12}d_{13})(M) = 0, consists of the classes 1, d_{23}^{2} + 4d_{24}d_{34}.

(3.b) Let c = 4d_{04}d_{14} − 2d_{02}d_{13} − 2d_{03}d_{12}. Assume that the kernel of the (composite) map

𝔖′′_{0} \underrightarrow{δ′′_{0}} (⊕_{M∈M′_{2}}k_{M}^{∗}/k_{M}^{∗2}) ⊕ (⊕_{M∈M′_{1}} k_{M}^{∗}/(k_{M}^{∗2}, δ′′_{M}(d_{14}^{2} − d_{12}d_{13}), δ′′_{M}(−c)))

is spanned by the classes (d_{04}^{2} − d_{02}d_{03}) and (d_{14}^{2} − d_{12}d_{13}).

(4) Assume that the Tate-Shafarevich group of each elliptic curve occurring as a fibre of E′/P_{k}^{1} is finite.

(5) Assume Schinzel’s hypothesis.

Then we have the same conclusions as in Theorem A, namely:

Let R ⊂ P^{1}(k) be the set of points λ ∈ P^{1}(k) such that the fibre X_{λ} = π^{−1}(λ) is smooth and has infinitely many k-points. Let R_{E} ⊂ P^{1}(k) be the set of points λ ∈ P^{1}(k) such that the fibre E_{λ} of E/P_{k}^{1} is an elliptic curve over k of rank (= dim_{Q}(E_{λ}(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 non-empty 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})).

Comment on (3.a). In their (more arithmetic) version (Condition 7 of [BSwD], as applied to their Theorem 1), Bender and Swinnerton-Dyer only have ⊕_{M∈M′′} k_{M}^{∗}/k_{M}^{∗2} as the target group (i.e. no ε_{M}), but I think this is an oversight on their part.

Comment on (3.b). To simplify matters, in their (more arithmetic) version (Condition 8 of [BSwD]), Bender and Swinnerton-Dyer replace this by the stronger condition where one only keeps the sum over points in M′_{2} in the target group. But they write

⊕_{M∈M′_{2}}k_{M}^{∗}/(k_{M}^{∗2}, δ′′_{M}(d_{14}^{2} − d_{12}d_{13}))

whereas the formulas (0.4) above show that for such an M the class δ′′_{M}(d_{14}^{2} − d_{12}d_{13}) is trivial.

1. Selmer groups associated to a degree 2 isogeny

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

Let S be a finite subset of the set Ω of places of the number field k. Consider the vector space ⊕_{v∈S}k_{v}^{∗}/k_{v}^{∗2} over the finite field F_{2}. We equip this space with the bilinear form ( , )_{S} = ⊕_{v∈S}( , )_{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 hand 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 ( , )_{S}, is nondegenerate. This pairing is skewsymmetric: for any a, b ∈ k_{v}^{∗}, we have (a, b)_{v} + (b, a)_{v} = 0. One also has (a, −a)_{v} = 0 for any a ∈ k_{v}^{∗}, but in general one need not have (a, a)_{v} = 0, i.e. the pairing need not be alternating.

Let 𝔬_{S} be the ring of S-integers of k.

Proposition 1.1.1 Let S 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_{S}^{∗}/𝔬_{S}^{∗2} is equal to half the dimension of the F_{2}-vector space ⊕_{v∈S}k_{v}^{∗}/k_{v}^{∗2}. If moreover S contains a set of generators of the ideal class group of k, then the natural map 𝔬_{S}^{∗}/𝔬_{S}^{∗2} → ⊕_{v∈S}k_{v}^{∗}/k_{v}^{∗2} is injective and its image is a maximal isotropic subspace of ⊕_{v∈S}k_{v}^{∗}/k_{v}^{∗2}.

This is [CSS3], Prop. 1.1.1. Let S_{0} denote a fixed family of places of k satisfying all the conditions of the proposition. Clearly, for any finite set S of places 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) = ℌ(k_{v}). We equip this F_{2}-vector space with the nondegenerate skewsymmetric bilinear form (., .)_{v}.

For any finite set S of places containing S_{0}, and any family of objects {∗_{v}}, v ∈ S, we shall consistently write ∗_{S} = ⊕_{v∈S} ∗_{v}.

The space V_{S} comes equipped with the pairing e_{S} : V_{S} × V_{S} → Z/2 sum of the pairings (., .)_{v}. Clearly e_{S} is skewsymmetric and nondegenerate (but unless −1 ∈ k^{∗2}, it need not be alternating).

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

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

may be read

𝔬_{S}^{∗}/𝔬_{S}^{∗2} → ⨁_{v∈S} k_{v}^{∗}/k_{v}^{∗2}

and it is injective by the previous proposition. By the global reciprocity law, the image I^{S} of this map is isotropic with respect to the symplectic pairing e_{S} defined above. By dimension count (using Proposition 1.1.1), we then see that it is maximal isotropic.

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

We shall actually consider two copies of V_{S}, call them V′_{S} and V′′_{S}, and consider the bilinear pairing

(., .)_{S} : V′_{S} × V′′_{S} → Z/2.

We have I^{′S} ⊂ V′_{S} and I^{′′S} ⊂ V′′_{S}. Note that I^{′S} is the exact orthogonal of I^{′′S} under the pairing V′_{S} × V′′_{S} → Z/2. The following Proposition is proved in [BSwD] (Lemma 8 and Corollary). It is linear algebra, but not entirely trivial. In the present context of degree 2 isogenies, it is the appropriate substitute for Prop. 1.1.2 of [CSS3].

Proposition 1.1.2 Let S_{0} be as above, and let S be a finite set of places containing S_{0}. There exist subspaces K′_{v} ⊂ V′_{v}, v ∈ S and K′′_{v} ⊂ V′′_{v}, v ∈ S such that:

(a) for each place v ∈ S, K′_{v} is the exact orthogonal of K′′_{v} under the pairing V′_{v}×V′′_{v} → Z/2;

(b) V′_{S} = I^{′S} ⊕ K′_{S}, where K′_{S} = ⊕_{v∈S}K′_{v}; and similarly V′′_{S} = I^{′′S} ⊕ K′′_{S}, where K′′_{S} = ⊕_{v∈S}K′′_{v};

(c) For v ∉ S_{0}, one may choose K′_{v} = ℌ(𝔬_{v}) and similarly K′′_{v} = ℌ(𝔬_{v}).

In the sequel, given S_{0}, we shall take the K′_{v} and K′′_{v} for v ∈ S_{0} as given by the Proposition, and we shall take K′_{v} and K′′_{v} to be equal to ℌ(𝔬_{v}) for v ∉ S_{0}.

1.2. Isogenies of degree 2

The object of this section, copied directly from Section 3 of [BSwD], is to recall the standard formulae for an elliptic curve with a rational 2-division point, for the associated isogeny, and for multiplication by 2 on such a curve. The notation introduced in this paragraph will be standard throughout the paper. Let k be an arbitrary field of characteristic not 2. An elliptic curve E′ defined over k and having a primitive 2-division point P′ defined over k can be written in the form

(1.2.1)  E′ : y^{2} = (x − c)(x^{2} − d)

where c, d are in k and d, (c^{2} −d) non-zero and where P′ is (c, 0). If O′ is the point at infinity on E′, which is the identity element under the group law, then there is an isogeny ϕ′ : E′ → E′′ = E′/{O′, P′} where E′′, whose point at infinity will be denoted by O′′, is given by the equation

(1.2.2)  E′′ : y_{1}^{2} = (x_{1} + 2c)(x_{1}^{2} + 4(d − c^{2})).

Explicitly, ϕ′ is given by

x_{1} = ((d − x^{2})/(c − x)) − 2c, y_{1} = y(x^{2} − 2cx + d)/(x − c)^{2}.

There is a dual isogeny ϕ′′ : E′′ → E′, and ϕ′′ ∘ ϕ′ and ϕ′ ∘ ϕ′′ are the doubling maps on E′ and E′′ respectively.

The elements of H^{1}(k, Z/2.P′) ≃ k^{∗}/k^{∗2} classify the ϕ′-coverings of E′′; the covering corresponding to the class of m′ is

(1.2.3)  v_{1}^{2} = m′(x_{1} + 2c), v_{2}^{2} = m′(x_{1}^{2} + 4d − 4c^{2})

with the obvious two-to-one map to E′′. The ϕ′-covering corresponding to P′′ is given by m′ = d. Similarly the ϕ′′-coverings of E′ are classified by the elements of H^{1}(k, Z/2.P′′) ≃ k^{∗}/k^{∗2}, the covering corresponding to the class of m′′ being

(1.2.4)  w_{1}^{2} = m′′(x − c), w_{2}^{2} = m′′(x^{2} − d).

The ϕ′′-covering corresponding to P′ is given by m′′ = c^{2} − d.

The composite map ϕ′′ ∘ ϕ′ : E′ → E′′ → E′ is multiplication by 2. There is an obvious commutative diagram

0 → _{2}E′ → E′ ⟶^{2} E′ → 0
↓  ↓ ϕ′  ↓ =
0 → Z/2 → E′′ ⟶^{ϕ′′} E′ → 0.

This diagram induces a map H^{1}(k,_{2}E′) → H^{1}(k, Z/2) which may be interpreted as sending 2-coverings of E′ to ϕ′′-coverings of E′.

Let us now discuss multiplication by 2 on E′. Let K = k[t]/(t^{2} − d). The kernel _{2}E′ of multiplication by 2 on E′ may be identified with the group R_{K/k}μ_{2}; the point P′ defines a natural map Z/2 → R_{K/k}μ_{2}. We have H^{1}(k,_{2}E′) = H^{1}(k, R_{K/k}μ_{2}) = K^{∗}/K^{∗2}.

From [BSwD], we also extract the following pieces of information.

Let a + b\sqrt{d} ∈ K^{∗}. The 2-covering of E′ associated to the class of a + b\sqrt{d} ∈ K^{∗}/K^{∗2} is given in homogeneous form by the intersection of two quadrics

(1.2.5)  \begin{matrix}Z_{2}^{2} + dZ_{3}^{2} = (a/(a^{2} − db^{2}))Z_{1}^{2} + (ac + bd)Z_{0}^{2}, \\ 2Z_{2}Z_{3} = (b/(a^{2} − db^{2}))Z_{1}^{2} + (a + bc)Z_{0}^{2}. \end{matrix}

in P_{k}^{3}. Let us denote this curve by Γ′. The covering map Γ′ → E′ can be factorized as Γ′ → C′′ → E′; here C′′ is the ϕ′′-covering of E′ given by (1.2.4), with m′′ = a^{2} − db^{2}, and the map Γ′ → C′′ is

W_{1} = Z_{1}/Z_{0}, W_{2} = (Z_{2}^{2} − dZ_{3}^{2})/Z_{0}^{2}.

From (1.2.5) it follows that if we are given an elliptic curve which has one nontrivial rational 2-division point then any 2-covering of it can be written as a smooth, complete intersection of two quadrics in P_{k}^{3} given by a system of homogeneous equations

(1.2.6)  \begin{matrix}α_{0}U_{0}^{2} + α_{1}U_{1}^{2} + α_{2}U_{2}^{2} + α_{3}U_{3}^{2} + 2α_{4}U_{2}U_{3} = 0, \\ β_{0}U_{0}^{2} + β_{1}U_{1}^{2} + β_{2}U_{2}^{2} + β_{3}U_{3}^{2} + 2β_{4}U_{2}U_{3} = 0. \end{matrix}

We refer to [BSwD] for the (simple) proof that conversely any curve of genus 1 defined over k by a system (1.2.6) is a 2-cover of an elliptic curve which has one nontrivial rational 2-division point.

As mentioned in the introduction, letting d_{ij} = α_{i}β_{j} − α_{j}β_{i}, for any subset {i, j, k, l} of {0, 1, 2, 3, 4} we have the identity

(1.2.7)  d_{ij}d_{kℓ} + d_{ik}d_{ℓj} + d_{iℓ}d_{jk} = 0.

The system (1.2.6) implies

(1.2.8)  \begin{matrix}d_{10}U_{0}^{2} + d_{12}U_{2}^{2} + 2d_{14}U_{2}U_{3} + d_{13}U_{3}^{2} = 0, \\ d_{01}U_{1}^{2} + d_{02}U_{2}^{2} + 2d_{04}U_{2}U_{3} + d_{03}U_{3}^{2} = 0. \end{matrix}

and is in fact equivalent to it, since the assumption that (1.2.6) defines a smooth curve of genus one implies d_{01} ≠ 0 ∈ k.

From [BSwD] we reproduce the formulas, some of which have already been mentioned:

a = −2(2d_{14}d_{34} + d_{13}d_{23})(d_{14}^{2} − d_{12}d_{13}),
b = −d_{01}^{−1}d_{13}(d_{14}^{2} − d_{12}d_{13}),
c = 4d_{04}d_{14} − 2d_{02}d_{13} − 2d_{03}d_{12},
d = 4d_{01}^{2}(d_{23}^{2} + 4d_{24}d_{34}),
c^{2} − d = 16(d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13}),
m′′ = a^{2} − db^{2} = 16d_{34}^{2}(d_{14}^{2} − d_{12}d_{13})^{3}.

From these formulas we read the equation y^{2} = (x − c)(x^{2} − d) of the curve E′, we read K = k[t]/(t^{2} − d), we read the class a + b\sqrt{d} ∈ K^{∗}/K^{∗2} of the 2-covering Γ′ of E′ which is isomorphic to the curve (1.2.5), and we also read the curve C′′ given by (1.2.4) through which the map Γ′ → E′ factorizes.

1.3. Local computations

Let E′, E′′, ϕ′, ϕ′′ be as above. We let V′ = H^{1}(k, {O′, P′}) = H^{1}(k, Z/2) = k^{∗}/k^{∗2} and V′′ = H^{1}(k, {O′′, P′′}) = H^{1}(k, Z/2) = k^{∗}/k^{∗2}. The isogeny ϕ′ gives rise to a natural embedding E′′(k)/ϕ′(E′(k)) ↪ V′ and the isogeny ϕ′′ gives rise to a natural embedding E′(k)/ϕ′′(E′′(k)) ↪ V′′. We let W′ ⊂ V′ denote the image of the first embedding and W′′ ⊂ V′′ be the image of the second. As mentioned above, the group V′ classifies the set of ϕ′-coverings of E′′ up to isomorphism; the subgroup W′ classifies the set of isomorphism classes of such coverings which have a k-point.

In this section k is a local field of characteristic different from 2. If k is nonarchimedean, we denote by A its ring of integers.

As with any pair of dual isogenies, the (nondegenerate) Weil pairing

ϕ′E′ ×_{ϕ′′}E′′ → μ_{2}

(for a general pair of isogenies, μ_{2} should be replaced by a big enough group of roots of unity) induces a pairing

V′ × V′′ → H^{2}(k, μ_{2}) ⊂ Br(k).

Since k is a local field, this pairing is nondegenerate.

Lemma 1.3.1 Assume that k is a local field. Under the nondegenerate pairing

V′ × V′′ → H^{2}(k, μ_{2}) ≃ Z/2

the subgroups W′ ⊂ V′ and W′′ ⊂ V′′ are each other’s orthogonal.

A proof of this well-known lemma is provided in [BSwD] (Lemma 3). Note that once the diagrams written down there have been shown to commute, the result follows from Tate’s duality theorem over a nonarchimedean local field, and from Witt’s result over the reals. For the commutativity, one may refer to [CTSa], Lemme 5.2 et Remarque 5.3.

We shall need an explicit description of the groups W′ ⊂ V′ and W′′ ⊂ V′′. When k is a nonarchimedean nondyadic local field, and E′, hence also E′′, has good reduction over the ring A of integers of k, the description is simple: W′ ⊂ V′ = k^{∗}/k^{∗2} coincides with the subgroup A^{∗}/A^{∗2} ⊂ k^{∗}/k^{∗2} and the same result holds for W′′ ⊂ V′′.

The following general lemma will enable us to compute W′ and W′′ in the nondyadic case.

Lemma 1.3.2 Let A be a discrete valuation ring, K its fraction field, v its valuation, π a uniformizing parameter of A and κ the residue field. Assume v(2) = 0. Let c, d, m ∈ A with d(c^{2} − d) ≠ 0.

Let X/Spec(A) be a regular proper integral scheme over Spec(A) whose generic fibre over Spec(K) is K-isomorphic to the smooth projective K-curve C given in P_{K}^{3} by the projective system of equations

w_{1}^{2} = m(xt − ct^{2}), w_{2}^{2} = m(x^{2} − dt^{2}).

Assume that either v(d) = 0 or v(c^{2} − d) = 0. Assume v(m) = 0 or v(m) = 1 (one may always reduce to this hypothesis). Then the special fibre of X/Spec(A), which is a κ-curve, has a geometrically integral component of multiplicity one if and only if c, d, m satisfy one of the conditions:

(i) v(d) = 0, v(c^{2} − d) = 0 and v(m) = 0.

(ii) v(m) = 0, v(d) > 0 is odd and m is a square modulo π.

(iii) v(m) = 0, v(d) > 0 is even and either m is a square modulo π or −mc is a square modulo π.

(iv) v(c^{2} − d) > 0 is odd.

(v) v(c^{2} − d) > 0 is even, v(m) = 0.

(vi) v(c^{2} − d) > 0 is even, v(m) = 1 and 2c is a square modulo π.

Proof Let us recall the following facts. The property that the special fibre of X/A has a multiplicity one component which is geometrically integral over κ does not depend on the regular proper model X/A of C/K. The genus one curve C actually admits a unique regular proper minimal model over A. If Y/A is some integral (not necessarily regular or proper) model of X/K such that Y → Spec(A) is surjective and the special fibre over κ has a geometrically integal component of multiplicity one, then a regular proper model X/A also has this property.

In case (i), the curve X ⊂ P_{A}^{3} given by the equations in the Lemma is smooth over A, its special fibre is a geometrically integral smooth curve of genus one over κ.

Suppose v(d) ≥ 0, v(c^{2}−d) = 0 and v(m) = 1. We claim that in that case any component of the special fibre of X/A has even multiplicity. Let V be a discrete valuation of rank one on the function field L of C, and assume e = V (π) > 0. From the affine equations w_{1}^{2} = m(x − c), w_{2}^{2} = m(x^{2} − d) viewed as identities in L we deduce that both e + V (x − c) and e + V (x^{2} − d) are even. Assume e odd. This now implies V (x − c) odd and V (x^{2} − d) odd. From V (x^{2} − d) odd, and V (d) = ev(d) ≥ 0 we deduce V (x) ≥ 0. We now have V (x − c) ≥ 1, V (x^{2} − d) ≥ 1 hence V (c^{2} − d) > 0, which implies v(c^{2} − d) > 0, contradicting our hypothesis.

Assume v(d) > 0, v(c^{2} − d) = 0 and v(m) = 0.

If m is a square modulo π, then the special fibre of the model Y/A given by the system of affine equations w_{1}^{2} − m(x − c) = 0, w_{2}^{2} − m(x^{2} − d) = 0 breaks up as the union of two smooth irreducible conics defined over the residue field κ.

Assume d = uπ^{2n+1} with u a unit. Suppose a regular proper model X/A of the K-curve given by the affine system of equations w_{1}^{2} = m(x − c), w_{2}^{2} = m(x^{2} − d) has a multiplicity one component which is geometrically integral over κ. On the function field of the curve, let V denote the valuation associated to the codimension one point of the model defined by that component. The multiplicity one assumption ensures V (π) = 1. We have V (x^{2} − d) = inf(2V (x), 2n + 1). From the second equation, we obtain 2V (w_{2}) = V (m(x^{2} − d)) = inf(2V (x), 2n + 1), hence 2V (w_{2}) = 2V (x) < 2n + 1 = V (d). The same equation now implies that m is a square modulo π.

For later use, let us consider the special case where v(d) = 1, v(c^{2} − d) = 0 and v(m) = 0. One easily checks that the subscheme of P_{A}^{3} defined by the homogeneous equations

w_{1}^{2} = m(xt − ct^{2}), w_{2}^{2} = m(x^{2} − dt^{2})

is the regular proper minimal model X/A of C/K. The special fibre is the union of two smooth conics meeting transversely in two points. Each conic is defined over κ[t]/(t^{2} − m). If this extension is a field, they are conjugate. The two points are defined over κ[t]/(t^{2} + mc).

Assume now that d = uπ^{2n} with u a unit, and that m is not a square modulo π.

Suppose a regular minimal model (over A) of the K-curve given by the affine system of equations w_{1}^{2} = m(x−c), w_{2}^{2} = m(x^{2}−d) has a multiplicity one component which is geometrically integral over κ. On the function field of the curve, let V denote the valuation associated to the codimension one point of the model defined by that component. The multiplicity one assumption ensures V (π) = 1. If V (x) ≤ 0, then from the second equation we deduce V (w_{2}^{2}) = v(x^{2}) ≤ 0 and m is a square modulo π, which we have excluded. Assume V (x) > 0. Then from the first equation we deduce that −mc is a square modulo π.

Conversely, assume that −mc is a square modulo π. Make the change of variables x = π^{n}X. Then affine equations for C/K read w_{1}^{2} = m(π^{n}X −c), w_{2}^{2} = m.π^{2n}.(X^{2} −u). Another change of variables tranforms this system into w_{1}^{2} = m(π^{n}X − c), w_{2}^{2} = m(X^{2} − u). Now these equations over A reduce to w_{1}^{2} = −\overline{m}\overline{c}, w_{2}^{2} = \overline{m}(X^{2} − \overline{u}) over κ, and provided −\overline{m}\overline{c} is a square, this decomposes into two smooth geometrically integral conics.

Assume v(d) = 0, v(c^{2} − d) > 0 and v(m) = 0. In this case the special fibre of the model Y/A given by w_{1}^{2} = m(x − c), w_{2}^{2} = m(x^{2} − d) is clearly irreducible and of multiplicity one.

Assume v(d) = 0, v(c^{2} − d) > 0 and v(m) = 1. Assume v(c^{2} − d) > 0 is odd. We claim that here also the special fibre of a suitable model contains a geometrically integral fibre of multiplicity one. Let E be the elliptic curve over K given by the affine equation y^{2} = (x − c)(x^{2} − d). The curve C is K-birational to the unramified double cover of E given by w^{2} = m(x − c). Let σ denote translation by the 2-torsion point P given by x = c, y = 0 on E. The divisor of x − c is a double, and the evaluation of the unramified double cover of E given by x − c = z^{2} at the point P is the class (c^{2} − d) ∈ K^{∗}/K^{∗2}. This is well-known to imply that in the function field of E there exists a formula (x − c) ∘ σ = (c^{2} − d).(x − c).g^{2} with g a suitable rational function. Thus C viewed as an unramified double cover of E after pulling-back by σ is birationally given by the set of equations (x − c) = uz^{2}, y^{2} = (x − c)(x^{2} − d), where u = m/(c^{2} − d) ∈ K^{∗} has even valuation. Multiplying z by a suitable power of the uniformizing parameter reduces to the case where u is a unit, i.e. to the situation v(m) = 0.

Assume v(d) = 0, v(c^{2} − d) > 0 even, and v(m) = 1. Note that this implies v(c) = 0. We claim that the special fibre has a geometrically integral component of multiplicity one if and only if 2c is a square modulo π.

Suppose that there exists such a component, let V be the discrete valuation associated to it. By hypothesis, π is a uniformizing parameter. In the function field of the K-curve, we have w_{1}^{2} = m(x − c) and w_{2}^{2} = m(x^{2} − d). We have V (π) = 1, hence V (m) = 1. From the second equality we deduce V (x) = 0. From the first equality we conclude x = c + m^{2r+1}u, with r ≥ 0, V (u) = 0 and u is a square modulo π. Hence x^{2} = c^{2} + 2cm^{2r+1}u + m^{4r+2}u^{2}. The second equality now gives

w_{2}^{2} = m(c^{2} − d + 2cm^{2r+1}u + m^{4r+2}u^{2}).

Since V (m) = 1 and V (c^{2} − d) is even, this equality implies V (2cu.m^{2r+2}) < V (c^{2} − d), hence 2cu is a square modulo π, hence finally 2c is a square modulo π.

Conversely, suppose that 2c is a square modulo π. By K-birational transformations we transform the system of equations defining C into

mw_{1}^{2} = x − c, mw_{2}^{2} = x^{2} − d

which is K-isomorphic to

mw_{1}^{2} = x − c, w_{2}^{2} = 2cw_{1}^{2} + (c^{2} − d)/m + mw_{1}^{4}.

By assumption, v(m) = 1 and v((c^{2} − d)/m) > 0 (since it is odd). The reduction modulo π of the above system reads 0 = x − \overline{c}, w_{2}^{2} = 2\overline{c}w_{1}^{2}. Since 2\overline{c} is a square, this has a geometrically integral component of multiplicity one.

Remark 1.3.3 Lemma 4 in [BSwD] is an arithmetic consequence of the above geometric lemma. Suppose A is a complete (or henselian) discrete valuation ring with finite residue field of characteristic not 2. If X/A has a special fibre which contains a multiplicity one geometrically integral component, then one may find on it a zero-cycle of degree one whose support lies in the smooth locus of the special fibre. By Hensel’s lemma, one may lift this zero-cycle to a zero-cycle of degree one on C/K. Since C/K is a curve of genus one, this curve admits a K-rational point. Conversely, if C/K admits a rational point, this defines an A-point of the regular proper A-scheme X/A, hence the special fibre of the latter contains a geometrically integral component of multiplicity one with a smooth κ-point (which as a matter of fact does not belong to any other component).

Let k be a p-adic local field, p ≠ 2, let c, d be in the ring A of integers. Let v denote the valuation of k. The above computation enables us to determine W′′ ⊂ k^{∗}/k^{∗2}. By duality, this also determines W′.

(i) If v(d) = 0 and v(c^{2} − d) = 0, then W′ = A^{∗}/A^{∗2} and W′′ = A^{∗}/A^{∗2}. (ii) If v(d) = 1 and v(c^{2} − d) = 0, then W′ = k^{∗}/k^{∗2} and W′′ = 1 ⊂ k^{∗}/k^{∗2}. (iii) If v(d) = 0 and v(c^{2} − d) = 1, then W′ = 1 ⊂ k^{∗}/k^{∗2} and W′′ = k^{∗}/k^{∗2}. This will be enough to prove Theorem A. But when proving theorem B, we shall also need: (iv) If v(d) > 0 is even and v(c^{2} − d) = 0 (hence v(c) = 0), then W′′ = 1 or W′′ = A^{∗}/A^{∗2}

depending on whether −c is or is not a square in A^{∗}. In the first case, W′ = k^{∗}/k^{∗2}, in the second case, W′ = A^{∗}/A^{∗2}.

We shall now study, though in less details, the analogue of Lemma 1.3.2 for a curve (1.2.6).

Lemma 1.3.4 Let A be a discrete valuation ring, K its fraction field, v its valuation, π be a uniformizing parameter of A and κ the residue field. Assume v(2) = 0. Let α_{i}, β_{i}, i = 0, ⋯, 3 be elements of A, and consider the closed subscheme of P_{A}^{3} defined by the system of homogeneous equations

(1.2.6)  \begin{matrix}α_{0}U_{0}^{2} + α_{1}U_{1}^{2} + α_{2}U_{2}^{2} + α_{3}U_{3}^{2} + 2α_{4}U_{2}U_{3} = 0, \\ β_{0}U_{0}^{2} + β_{1}U_{1}^{2} + β_{2}U_{2}^{2} + β_{3}U_{3}^{2} + 2β_{4}U_{2}U_{3} = 0. \end{matrix}

Assume that the generic fibre of this A-scheme is a smooth intersection of two quadrics in P_{K}^{3}. Noting as above d_{ij} = α_{i}β_{j} − α_{j}β_{i} ∈ A, this amounts to the assumption that the product

d_{01}(d_{23}^{2} + 4d_{24}d_{34})(d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13}) ≠ 0 ∈ A.

If the valuation of this product is zero, the above A-scheme is smooth over A, and the special fibre is a smooth intersection of two quadrics in P_{κ}^{3}.

Assume that the valuation of this product is equal to one (we shall only be interested in this case). Assume moreover that the valuation of (d_{04}^{2} − d_{02}d_{03})d_{02} is at most one, and that the valuation of (d_{14}^{2} − d_{12}d_{13})d_{12} is at most one.

Let X/A be a regular proper integral scheme over Spec(A) whose generic fibre over Spec(K) is K-isomorphic to the curve defined by (1.2.6) over K.

Then the special fibre of X/Spec(A), which is a κ-curve, has a geometrically integral component of multiplicity one if and only if one of the following conditions is satisfied:

(i) v(d_{01}) = 1 and either d_{04}^{2} − d_{02}d_{03} or d_{14}^{2} − d_{12}d_{13} is a square modulo π (in which case both are squares);

(ii) v(d_{04}^{2} − d_{02}d_{03}) = 1 and −d_{01}d_{02} is a square modulo π; (iii) v(d_{14}^{2} − d_{12}d_{13}) = 1 and d_{12}d_{01} is a square modulo π; (iv) v(d_{23}^{2} + 4d_{24}d_{34}) = 1.

Proof By assumption, we have

v(d_{01}(d_{23}^{2} + 4d_{24}d_{34})(d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13})) = 1.

Thus exactly one of v(d_{01}), v(d_{04}^{2} − d_{02}d_{03}), v(d_{14}^{2} − d_{12}d_{13}), v(d_{23}^{2} + 4d_{24}d_{34}) is equal to 1, the other ones are zero.

(i) Assume v(d_{01}) = v(α_{0}β_{1} − α_{1}β_{0}) = 1. Then each of v(d_{23}^{2} + 4d_{24}d_{34}), v(d_{04}^{2} − d_{02}d_{03}), v(d_{14}^{2} − d_{12}d_{13}) equals zero.

One of α_{0}, β_{0}, α_{1}, β_{1} has valuation zero. By the symmetry of the equation, we may assume v(α_{1}) = 0. If we had v(α_{0}) > 0, then we would have v(β_{0}) > 0. But v(α_{0}) > 0 and v(β_{0}) > 0 imply v(d_{04}^{2} − d_{02}d_{03}) > 0, which is excluded. Thus we have v(α_{1}) = 0 and v(α_{0}) = 0.

Another (possibly singular) model for (1.2.6) is given by the system

(1.3.1)  \begin{matrix}α_{0}U_{0}^{2} + α_{1}U_{1}^{2} + α_{2}U_{2}^{2} + α_{3}U_{3}^{2} + 2α_{4}U_{2}U_{3} = 0, \\ d_{10}U_{0}^{2} + d_{12}U_{2}^{2} + d_{13}U_{3}^{2} + 2d_{14}U_{2}U_{3} = 0. \end{matrix}

Suppose there exists a multiplicity one, geometrically integral component γ of the special fibre of X/A. Let V be the associated valuation on the function field of our curve. By hypothesis, V (π) = 1. Suppose d_{14}^{2} − d_{12}d_{13} is not a square mod π. Then it is not a square in the residue field of V either, because the κ-curve γ is geometrically integral. Since d_{14}^{2} − d_{12}d_{13} is not a square mod π, certainly each of d_{12}, d_{13} is a unit with respect to v, hence also with respect to V. In the function field L of X, we have the equality

V (−d_{10}(U_{0}/U_{2})^{2}) = V (d_{12} + d_{13}(U_{3}/U_{2})^{2} + 2d_{14}(U_{3}/U_{2})).

The left hand side has odd valuation. The right hand side has even valuation, because it is equal to the valuation of a norm of an element in a nontrivial quadratic extension, which is unramified and nonsplit at V (indeed, d_{14}^{2} − d_{12}d_{13} is a unit, and not a square in the residue field).

We conclude: if the special fibre of X/A contains a multiplicity one, geometrically integral component, then d_{14}^{2} − d_{12}d_{13} is a square mod π.

Conversely, suppose that d_{14}^{2} − d_{12}d_{13} is a square mod π. Equations for the special fibre of (1.3.1) are

(1.3.2)  \begin{matrix}α̃_{0}U_{0}^{2} + α̃_{1}U_{1}^{2} + α̃_{2}U_{2}^{2} + α̃_{3}U_{3}^{2} + 2α̃_{4}U_{2}U_{3} = 0, \\ d̃_{12}U_{2}^{2} + d̃_{13}U_{3}^{2} + 2d̃_{14}U_{2}U_{3} = 0. \end{matrix}

The discriminant of this last equation is a nonzero square. Hence it breaks up as the product of two nonproportional linear forms, defining two distinct planes.

The 2 by 2 determinants associated to the matrix

α̃_{2} α̃_{3} ˜2α_{4}
d̃_{12} d̃_{13} 2d̃_{14}

are α̃_{1}d_{23}, 2α̃_{1}d_{24}, 2α̃_{1}d_{34}. One of them is nonzero (recall v(d_{23}^{2} + 4d_{24}d_{34}) = 0). Thus the form α̃_{2}U_{2}^{2} + α̃_{3}U_{3}^{2} + 2α̃_{4}U_{2}U_{3} does not vanish identically on at least one of the two planes. From v(α_{1}) = 0 and v(α_{0}) = 0 we conclude that the trace of

α̃_{0}U_{0}^{2} + α̃_{1}U_{1}^{2} + α̃_{2}U_{2}^{2} + α̃_{3}U_{3}^{2} + 2α̃_{4}U_{2}U_{3} = 0

on that plane is a smooth conic: this produces a geometrically integral component of multiplicity one.

(iii) (the argument for (ii) is entirely similar). Assume v(d_{14}^{2} − d_{12}d_{13}) = 1. By assumption this implies v(d_{01}) = 0 and v(d_{12}) = 0. Suppose there exists a multiplicity one, geometrically integral component γ of the special fibre of X/A. Let V be the associated valuation on the function field L of our curve. By hypothesis, V (π) = 1. Let u_{i} = U_{i}/U_{3} ∈ L. From the second equation in (1.3.1), we deduce the following equation in the function field of X:

u_{0}^{2} − (d_{12}/d_{01})(u_{2} + d_{14}/d_{12})^{2} = (d_{14}^{2} − d_{12}d_{13})/d_{10}d_{12}.

If d_{12}d_{01} were not a square in the residue field of v, hence of V, then the V -valuation of the left hand side would be even. But the valuation of the right hand side is odd.

Conversely, assume that d_{12}d_{01} is a square in the residue field of v. As a possibly singular A-model for the K-curve defined by (1.2.6) we may take Y ⊂ P_{A}^{3} defined by the system of homogeneous equations

d_{10}U_{0}^{2} + d_{12}U_{2}^{2} + 2d_{14}U_{2}u_{3} + d_{13}U_{3}^{2} = 0,
d_{01}U_{1}^{2} + d_{02}U_{2}^{2} + 2d_{04}U_{2}u_{3} + d_{03}U_{3}^{2} = 0.

The fibre of this A-scheme over the residue field κ of A is the intersection of two quadrics in P_{κ}^{3}. The first of these quadrics decomposes as the product of two planes over κ, each of them being given by an equation U_{0} = ±l(U_{2}, U_{3}) where l is a linear form. The second one is the cone over a nonsingular conic in the variables U_{1}, U_{2}, U_{3} : indeed, v(d_{01}) = 0 and v(d_{04}^{2} −d_{02}d_{03}) = 0. Thus the fibre Y_{κ} is the union of two distinct smooth conics.

(iv) One considers the possibly singular model Y ⊂ P_{A}^{3} defined by the system of homogeneous equations

d_{10}U_{0}^{2} + d_{12}U_{2}^{2} + 2d_{14}U_{2}U_{3} + d_{13}U_{3}^{2} = 0,
d_{01}U_{1}^{2} + d_{02}U_{2}^{2} + 2d_{04}U_{2}U_{3} + d_{03}U_{3}^{2} = 0.

Under the assumption v(d_{23}^{2} + 4d_{24}d_{34}) = 1, we have v(d_{01}) = 0, v(d_{04}^{2} − d_{02}d_{03}) = 0 and v(d_{14}^{2} − d_{12}d_{13}) = 0. By (1.2.7), we have d_{12}d_{04} − d_{02}d_{14} = d_{01}d_{42}, d_{12}d_{03} − d_{02}d_{13} = d_{01}d_{32}, and d_{14}d_{03} − d_{04}d_{13} = d_{01}d_{34}. Since the valuation of d_{23}^{2} + 4d_{24}d_{34} is one, one at least of d_{42}, d_{34}, d_{32} is a unit in A, and we know that d_{01} is a unit. This implies that the reduction over κ of the two quadratic forms d_{12}U_{2}^{2} + 2d_{14}U_{2}u_{3} + d_{13}U_{3}^{2} and d_{01}U_{0}^{1} + d_{02}U_{2}^{2} + 2d_{04}U_{2}u_{3} + d_{03}U_{3}^{2} are nonproportional. As the reader will check, this implies that the variety in P_{κ}^{3} obtained by the reduction of the above system of equations is geometrically irreducible (with possibly one singular point).

1.4. The Selmer groups as kernels of pairings

In this section, k is a number field. We follow the model of [CSS3], Section 1.2. We let E′, E′′, ϕ′, ϕ′′ be as in Section 1.2. Define S_{ϕ′}, the ϕ′-Selmer group of E′′, to be the set of isomorphism classes of ϕ′-coverings of E′′ which are everywhere locally soluble; and similarly for S_{ϕ′′}, the ϕ′′-Selmer group of E′.

Let S_{0} be as in Section 1.1 and let S be any finite set of places containing S_{0}.

From Section 1.1 we have the nondegenerate bilinear pairing (., .)_{S} : V′_{S} × V′′_{S} → Z/2. The group I^{S} is the image of 𝔬_{S}^{∗}/𝔬_{S}^{∗2} in V_{S}, and I^{′S}, I^{′′S} are its images in V′_{S}, V′′_{S} respectively.

By Proposition 1.1.1, for any S, I^{′S} and I^{′′S} are each other’s orthogonal under the pairing V′_{S} × V′′_{S} → Z/2.

Let K′_{v}, K′′_{v}, K′_{S}, K′′_{S} be as in Proposition 1.1.2. Recall that K′_{S} ⊂ V′_{S} and K′′_{S} ⊂ V′′_{S} are each other’s othogonal under the pairing V′_{S} ×V′′_{S} → Z/2. For v a place of k, let W′_{v} ⊂ V′_{v} be the image of E′′(k_{v})/ϕ′(E′(k_{v})) ↪ V′_{v} and similarly let W′′_{v} ⊂ V′′_{v} be the image of E′(k_{v})/ϕ′′(E′′(k_{v})) ↪ V′′_{v}.

Let W′_{S} = ⊕_{v∈S}W′_{v} ⊂ V′_{S} and let W′′_{S} = ⊕_{v∈S}W′′_{v} ⊂ V′′_{S}. By Lemma 1.3.1, W′_{S} and W′′_{S} are each other’s orthogonal under the pairing V′_{S} × V′′_{S} → Z/2.

The following Lemma is the analogue of Prop. 1.2.1 of [CSS3].

Lemma 1.4.1 Let S_{0} be as above and suppose that S contains S_{0} and all the primes of bad reduction for E′, i.e. c, d and c^{2} − d are units at any v ∉ S. Then S_{ϕ′} is isomorphic to each of the following groups:

(i) the intersection I^{′S} ∩ W′_{S};

(ii) the left kernel of the map I^{′S} × W′′_{S} → Z/2 induced by (. , .)_{S};
(iii) the left kernel of the map W′_{S} × I^{′′S} → Z/2 induced by (. , .)_{S}.

A similar result holds for S_{ϕ′′}.

Proof If v is not in S then the ϕ′-covering (1.2.3) is soluble in k_{v} if and only if m′ is in ℌ(𝔬_{v}); hence S_{ϕ′} can be identified with the subgroup of I^{′S} for which (1.2.3) is soluble at every place of S. This proves (i). Since W′_{S} and W′′_{S} are orthogonal complements with respect to (. , .)_{S}, we deduce (ii). As for (iii), it follows from the corresponding fact for I^{′S} and I^{′′S}.

For any finite set S of places containing S_{0}, but not necessarily the primes of bad reduction for E′ and E′′, write

I^{′S} = I^{′S} ∩ (W′_{S} + K′_{S}), W′_{S} = W′_{S}/(W′_{S} ∩ K′_{S}) = ⊕_{v∈S}W′_{v}/(W′_{v} ∩ K′_{v})

and similarly for I^{′′S} and W′′_{S}. Define t′_{S} : V′_{S} → I^{′S} to be the projection along K′_{S} in V′_{S}, and similarly for t′′_{S}.

The image of t′_{S} lies in I^{′S}, and the image of 1 − t′_{S} lies in K′_{S}, and similarly for t′′_{S}. For x ∈ V′_{S} and y ∈ V′′_{S}, we have

(t′_{S}(x), t′′_{S}(y))_{S} = 0 = (x − t′_{S}(x), y − t′′_{S}(y))_{S},

hence

(1.4.1)  (x, y)_{S} = (t′_{S}(x), y)_{S} + (x, t′′_{S}(y))_{S}

The kernel of t′_{S} is K′_{S}, so t′_{S} induces a map W′_{S} → I^{′S} whose kernel is trivial and whose image is easily seen to be I^{′S}; in other words it induces an isomorphism

τ′_{S} : W′_{S} → I^{′S}.

There is an analogous isomorphism τ′′_{S} : W′′_{S} → I^{′′S}. We shall denote the inverse isomorphisms by σ′_{S}, σ′′_{S} respectively.

Proposition 1.4.2 The pairing (. , .)_{S} induces pairings

I^{′S} × W′′_{S} → Z/2, W′_{S} × I^{′′S} → Z/2.

The action of τ′_{S} × σ′′_{S} takes the first pairing into the second.

Proof To prove the existence of the first pairing it is enough to show that I^{′S} is orthogonal to W′′_{S} ∩ K′′_{S} with respect to (. , .)_{S}. But I^{′S} ⊂ W′_{S} + K′_{S} which is the orthogonal complement of W′′_{S} ∩ K′′_{S}. The argument for the second pairing is similar. To prove the last statement, let α′ + β′ be any element of I^{′S}, where α′ is in W′_{S} and β′ in K′_{S}, and let α′′ be any element of W′′_{S}; by abuse of language we also use α′′ to denote the corresponding element of W′′_{S}. Then σ′_{S}(α′ + β′) is the class of α′, and τ′′_{S}α′′ has the form α′′ + β′′ for some β′′ in K′′_{S}. Since we are in characteristic 2, what we need to prove is

(α′ + β′, α′′)_{S} + (α′, α′′ + β′′)_{S} = 0.

But t′_{S}(α′) = α′ + β′ because β′ is in K′_{S} and α′ + β′ is in I^{′S}, and similarly t′′_{S}(α′′) = α′′ + β′′; the left hand side of the displayed equation is equal to (α′ + β′, α′′ + β′′)_{S} by (1.4.1) and this vanishes because I^{′S} is orthogonal to I^{′′S}.

We shall denote the pairings in Prop. 1.4.2 by (. , .)′_{S} and (. , .)′′_{S} respectively.

Proposition 1.4.3 Suppose that S also contains all the primes of bad reduction for E′. Then the left kernel of either of the pairings in Proposition 1.4.2 is isomorphic to S_{ϕ′} and the right kernel to S_{ϕ′′}.

Proof By Lemma 1.4.1, S_{ϕ′} can be identified with I^{′S} ∩ W′_{S} and is therefore contained in I^{′S}; also it is orthogonal to W′′_{S} and therefore to W′′_{S}. Conversely, anything in I^{′S} which is orthogonal to W′′_{S} must be orthogonal to W′′_{S} and therefore lies in W′_{S}; since I^{′S} lies in I^{′S}, such an element lies in I^{′S} ∩ W′_{S} = S_{ϕ′}.

The left kernel of the second pairing in Prop. 1.4.2 is isomorphic through σ′_{S} to the left kernel of the first pairing. The proof for the right kernels starts from the second pairing in Prop. 1.4.2 and is similar.

What we shall actually use is the pairing W′_{S} × W′′_{S} → Z/2 given by

(x′, x′′) ↦ (τ′_{S}(x′), x′′)′_{S} = (x′, τ′′_{S}(x′′))′′_{S}.

The equality here follows from the last sentence of Prop. 1.4.2, and Prop. 1.4.3 asserts that the left kernel of this map is isomorphic to S_{ϕ′} and the right kernel to S_{ϕ′′}.

1.5. Small Selmer groups

In [CSS3], one considers multiplication by 2 on an elliptic curve E/k whose 2-torsion is rational over k and whose 2-Selmer group S_{2}(E) is of order at most 8. On the Tate-Shafarevich group of E we have the alternating Cassels-Tate pairing. If the Tate-Shararevich group is finite, then the order of the 2-torsion subgroup of �(E) must be a square. Since it is of order at most 2, it has to be trivial and E(k)/2 ≃ S_{2}(E) which forces any 2-covering of E with points everywhere locally to have a k-rational point. Here we must give an argument taking into account the two isogenies E′ → E′′ and E′′ → E′. This is Lemma 11 of [BSwD], for which I give a slightly revised proof.

Let k be a number field. Let ϕ′ : E′ → E′′ be a degree 2 isogeny with kernel Z/2.P′. Let ϕ′′ : E′′ → E′ be the dual isogeny, the kernel of which is Z/2.P′′. The composite map ϕ′ ∘ ϕ′′ is multiplication by 2 on E′′. Similarly, the composite map ϕ′′ ∘ ϕ′ is multiplication by 2 on E′.

From the isogenies ϕ′ et ϕ′′ one obtains exact sequences

0 → E′′(k)/ϕ′(E′(k)) → S_{ϕ′} → _{ϕ′}�(k, E′) → 0

and

0 → E′(k)/ϕ′′(E′′(k)) → S_{ϕ′′} → _{ϕ′′}�(k, E′′) → 0,

where S_{ϕ′} is the Selmer group classifying ϕ′-coverings of E′′ which have rational points everywhere locally, S_{ϕ′′} is the Selmer group classifying ϕ′′-coverings of E′ which have rational points everywhere locally, and _{ϕ′}�(k, E′) is the kernel of the map �(k, E′) → �(k, E′′) induced by ϕ′, and similarly for _{ϕ′′}�(k, E′′).

Lemma 1.5.1 If P′ is the unique 2-torsion point of E′(k), then P′′ ∉ ϕ′(E′(k)). If P′′ is the unique 2-torsion point of E′′(k), then P′ ∉ ϕ′′(E′′(k)).

Proof Assume there exists M ∈ E′(k) such that ϕ′(M) = P′′. Then 2M = ϕ′′ ∘ ϕ′(M) = ϕ′′(P′′) = 0, hence M is a 2-torsion point on E′. Then M = 0 or M = P′, hence P′′ = ϕ′(M) = 0, which contradicts our hypothesis.

Proposition 1.5.2 Assume that P′ is the only 2-torsion point of E′(k) and that P′′ is the only 2-torsion point of E′′(k). Assume that S_{ϕ′} has order 2 and that S_{ϕ′′} has order at most 4. Then _{ϕ′}�(E′) = 0 and S_{2}(E′) has order at most 4. If we moreover assume that the groups �(E′) and �(E′′) are finite, then the groups _{ϕ′′}�(E′′), _{2}�(E′) and _{2}�(E′′) are all zero. In particular the curves corresponding to elements in the Selmer groups S_{ϕ′}, S_{ϕ′′}, S_{2}(E′), S_{2}(E′′) all have rational points. The finitely generated abelian groups E′(k) and E′′(k) have (Mordell-Weil) rank r, where 0 ≤ r ≤ 1 and 2^{r+1} is the order of the group S_{ϕ′′}.

Proof From Lemma 1.5.1 we have an injection Z/2.P′′ ↪ E′′(k)/ϕ′(E′(k)). The hypothesis on the order of S_{ϕ′} thus implies Z/2.P′′ = E′′(k)/ϕ′(E′(k)) and _{ϕ′}�(E′) = 0. From the first equality we deduce ϕ′′(E′′(k)) = ϕ′′ ∘ ϕ′(E′(k)) = 2E′(k) ⊂ E′(k).

From the commutative diagram

0 → _{2}E′ → E′ ⟶^{2} E′ → 0
↓  ↓ ϕ′  ↓ =
0 → _{ϕ′′}E′′ → E′′ ⟶^{ϕ′′} E′ → 0

we obtain the commutative diagram of exact sequences

0 → E′(k)/2E′(k) → S_{2}(E′) → _{2}�(E′)  → 0
↓  ↓  ↓
0 → E′(k)/ϕ′′(E′′(k)) → S_{ϕ′′} → _{ϕ′′}�(E′′) → 0,

where the middle and right vertical maps are induced by ϕ′ and the left vertical map is the natural projection map, which in the case in point is an isomorphism, as noticed earlier on.

The kernel of the map _{2}�(E′) → _{ϕ′′}�(E′′) is the group _{ϕ′}�(E′), and we have shown that this group vanishes. Thus the left and right vertical maps are one-to-one, hence so is the map S_{2}(E′) → S_{ϕ′′}, and the order of S_{2}(E′) is at most 4.

According to the lemma, the order of E′(k)/ϕ′′(E′′(k)) is at least 2. The group S_{ϕ′′} has order at most 4. Thus _{ϕ′′}�(E′′) has order at most 2. Since _{2}�(E′) → _{ϕ′′}�(E′′) is one-to-one, we conclude that _{2}�(E′) has order at most 2. If the group �(E′) is finite, then the (alternating) Cassels-Tate pairing on that group is nondegenerate. This implies that the group �(E′) is a direct sum of groups (Z/n)^{2} for various n’s, in particular the order of _{2}�(E′) is a square. Thus _{2}�(E′) = 0 and E′(k)/2E′(k) ≃ S_{2}(E′), this last group being of order 2 or 4.

The isogeny ϕ′′ induces a homomorphism _{2}�(E′′) → _{ϕ′}�(E′) whose kernel is _{ϕ′′}�(E′′), a group of order at most 2. From _{ϕ′}�(E′) = 0 we now deduce that _{2}�(E′′) is of order at most 2. The hypothesis that �(E′′) is finite implies that the order of _{2}�(E′′) is a square. Hence 2�(E′′) = 0, and _{ϕ′′}�(E′′) = 0. From the commutative diagram above we then conclude that the groups E′(k)/2E′(k), E′(k)/ϕ′′(E′′(k)) and S_{ϕ′′} are isomorphic. The last one has order 2^{r+1}, with 0 ≤ r ≤ 1. Thus E′(k), whose 2-torsion subgroup is of order 2, has Mordell-Weil rank r. Hence also E′′(k).

2. Proof of Theorem A

2.1. Fibres with points everywhere locally

Let k be a number field and let c(t), d(t), m(t) ∈ k[t] satisfy the hypothesis of the theorem. Thus all have even degree, deg(d) = 2deg(c) > 0, d(c^{2} − d) has no square factor and m divides c^{2} − d, hence is coprime with d. We let r(t) = d(t)(c^{2}(t) − d(t)) and we let r_{M}(t) be the monic irreducible factors of r(t), where M denotes the associated closed point of A_{k}^{1}. The polynomial m(t) is the product of a constant ρ_{m} ∈ k^{∗} by a product of some of the r_{M}(t).

We refer the reader to the introduction for the notation. The surface X = X(m) is equipped with the fibration π : X → P_{k}^{1}.

Recall that M′ is the set of closed points M such that d(M) = 0 and that M′′ is the set of closed points where c^{2} − d vanishes. We let r′(t) be the product of the r_{M}(t) for M ∈ M′ and similarly for r′′(t). Note that r′(t) up to a scalar coincides with d(t) and that r′′(t) up to a scalar coincides with c^{2}(t) − d(t).

The assumptions imply (Lemma 1.3.2 for M ∈ A_{k}^{1}, a direct computation for M = ∞ ∈ P_{k}^{1}) that for M ∈ P_{k}^{1}, M ∉ M′, the fibre X_{M} contains a component of multiplicity one which is geometrically integral over the residue field k_{M} (the fibres need not be smooth, in contrast with what happened in [CSS3]; however, as is by now well known ([CSS2], Lemma 1.2), the above property is just as good for the method to be applied – see Lemma 2.1.1 hereafter). At a point M ∈ M′, the fibre over k_{M} when viewed over an algebraic closure of k_{M} is the union of two smooth conics meeting transversally in two points; each conic is defined over the extension K_{M} = k_{M}[u]/(u^{2} − m(M)).

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 ([Sk], 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 [CT]).

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 Theorem A, we will find a point λ ∈ P^{1}(k), lying in 𝔘_{v} for v ∈ S_{1}, such that the fibre X_{λ} = π^{−1}(λ) is smooth and has infinitely many rational points. This will prove statement (a) in the Theorem. Statements (b) and (c) then follow easily.

A standard procedure, which uses the finiteness of Br_{vert}(X)/Br(k), and is described in full detail in [CSS2] (proof of Theorem 1.1) allows us to assume also 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 at any point of 𝔘_{v} the fibre of π is a smooth curve of genus one ; 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 point of A^{1}(k), with smooth fibre, to the point at infinity. One checks that the procedure does not affect the hypotheses. This is clear for any transformation t → t − α, where α does not belong to the zeroes of d(c^{2} − d). Assume, as we may, that t = 0 is not a zero of d(c^{2}−d). Then the statement is also clear when one sets T = 1/t, and one replaces m(t), c(t), d(t) by M(T) = T^{μ}m(1/T), D(T) = T^{δ}d(1/T), C(T) = T^{γ}c(1/T), where μ, δ, γ are the respective degrees of m, d, c, all even. Note that we then have deg(C^{2} − D) = deg(D) = 2deg(C) > 0.

From now on, we assume that conditions (a),(b),(c) above are fulfilled, as well as the improved condition (0.1), where the degree of c^{2} − d is equal to the degree of d.

Given (the new) m and its associated regular minimal proper 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. For each v ∉ S_{0}, we want all polynomials c(t), d(t), m(t) to be 𝔬_{v}-integral, we want ρ_{m} to be an 𝔬_{v}-unit, 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. This implies in particular that the leading coefficients of d(t) and of (c^{2} − d)(t) are units away from S_{0}. Given any monic irreducible factor r_{M}(t) of r(t), the closed subscheme M̃ = Spec(𝔬_{S_{0}}[t]/r_{M}(t)) of A_{𝔬_{S_{0}}}^{1}, is finite and étale over Spec(𝔬_{S_{0}}). For each M ∈ M′, 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 any finite place of k whose residue characteristic is less than or equal to the product of the degree [k : Q] and the degree of the polynomial r(t) (this will be used when applying Schinzel’s hypothesis). 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∈M′} M̃ is the set of points of P_{𝔬_{S_{0}}}^{1} whose fibre does not contain a geometrically integral component of multiplicity one, and 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 S of primes containing S_{0}, we shall be interested in the affine line A_{𝔬_{S}}^{1} over 𝔬_{S} and in the open set U_{𝔬_{S}} = Spec(𝔬_{S}[t][1/r]), and the analogous U′_{𝔬_{S}} and U′′_{𝔬_{S}}.

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 r_{M} for M ∈ M, 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}). The same results hold for 𝔬_{S}[t][1/r′]) and 𝔬_{S}[t][1/r′′]).

A point λ ∈ P^{1}(k) = P^{1}(𝔬_{S_{0}}) (resp. λ_{v} ∈ P^{1}(k_{v}) = P^{1}(𝔬_{v}) for v ∉ S_{0}) defines a section λ̃ : Spec(𝔬_{S_{0}}) → P_{𝔬_{S_{0}}}^{1} (resp. a local section λ̃_{v} : Spec(𝔬_{v}) → P_{𝔬_{v}}^{1}). To simplify the notation, we shall sometimes write λ where it would be more correct to write λ_{v}. We shall call such a point transversal if the corresponding section λ̃ (resp. local section λ̃_{v}) 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 ∈ M) and of 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 ∈ M, the element r_{M}(λ) ∈ k^{∗} (resp. r_{M}(λ_{v}) ∈ k_{v}^{∗}) is either a unit or a uniformizing parameter at each v ∉ S_{0} (resp. at v), and that if v(λ) < 0, then v(λ) = −1.

Given any point λ ∈ A^{1}(k) such that d(c^{2} − d) does not vanish at λ, the fibre X_{λ} is a smooth curve of genus one over k.

The analogue of Lemma 2.1.1 of [CSS3] (see [BSwD] Lemma 4) is here:

Lemma 2.1.1 Let S_{0} be as above. Let v ∉ S_{0} and let λ ∈ A^{1}(k_{v}) be a transversal point. Let λ̃ ⊂ P^{1}(𝔬_{v}) be its closure.

(a) If λ̃ does not intersect any of the M̃ for M ∈ M′, then X_{λ}(k_{v}) ≠ ∅.

(b) If λ̃ intersects M̃ for M ∈ M′ at some place w of k_{M} (w unramified over v, of degree one over v), then X_{λ}(k_{v}) ≠ ∅ if and only if w splits in K_{M}.

Proof Let X_{λ}/𝔬_{v} be the 𝔬_{v}-scheme which is the inverse image of X /P_{𝔬_{S_{0}}}^{1} under λ̃.

(a) By the definition of S_{0}, the reduction mod v of the fibre X_{λ}/𝔬_{v} contains a geometrically integral component of multiplicity one. As noted in Remark 1.3.3, this implies X_{λ}(k_{v}) ≠ ∅.

(b) Since X_{λ}/𝔬_{v} is proper we have X_{λ}(k_{v}) = X_{λ}(𝔬_{v}). Thus a point of X_{λ}(k_{v}) gives a local section over Spec(𝔬_{v}) which intersects just one component of the closed fibre: that component must have multiplicity one and be geometrically integral. 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_{λ}/𝔬_{v} at v are individually defined over F_{w} = F_{v}. We may then apply Remark 1.3.3. We may also argue directly: Any conic contains at least 3 points, thus we can always find a smooth rational point in such a fibre. By Hensel’s lemma this point can be lifted to a point over k_{v}.

Given M ∈ M, we may write r_{M}(t) = Norm_{k_{M} /k}(t − a_{M}), with a_{M} ∈ k_{M} the class of t in k[t]/r_{M}(t). For M ∈ M′, let

(K_{M}/k_{M}, t − a_{M}) ∈ Br(k_{M}(t))

be the class of the standard quaternion algebra associated to the element t − a_{M} ∈ k_{M}(t) and the quadratic extension K_{M}(t)/k_{M}(t). For such an M let

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

The following Lemma is the analogue of Lemma 2.1.2 of [CSS3]. The hypotheses (a),(b) and (c) for the points {P_{v}}_{v∈S_{1}} are in force.

Lemma 2.1.2 Let π : X → P_{k}^{1} and S_{0} (with S_{1} ⊂ S_{0}) be as above, let X_{U} = π^{−1}(U), 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 S of places 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}(𝔞_{M}(π(Q_{v}))) = 0

for all M ∈ M′.

Proof This is just a special case of Harari’s ‘formal lemma’ ([Ha1], Cor. 2.6.1; [CTSwD], Theorem 3.2.1).

We now have a certain finite set S of places containing S_{0}.

The analogue of Prop. 2.1.3 of [CSS3] is the following proposition (the freedom allowed at the archimedean places in (6) below should also have been allowed in [CSS3].)

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

(1) For each M ∈ M′ and each place v ∈ T_{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{m(M)})/k_{M} (i.e. m(M) is a square in (k_{M})_{w});

(2) each place v ∈ T_{∞} splits in each quadratic extension k(\sqrt{Norm_{k_{M} /k}(m(M))})/k for each M ∈ M′ (i.e. for each such M, Norm_{k_{M} /k}(m(M)) is a square in k_{v} for each v ∈ T_{∞});

(3) λ̃ is transversal over 𝔬_{S};

(4) for any M ∈ M, the (transversal) intersection of λ̃ and M̃, when viewed on M̃, consists of the places w ∈ T_{M} and just one place w_{M} of k_{M}, of degree one over a place v_{M} of k; moreover, all v_{M}’s are distinct from one another and none of them belongs to ∪_{M∈M}T_{M};

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

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

Let T′ = S ∪_{M∈M′} T_{M} and T′′ = S ∪_{M∈M′′} T_{M}, let T′(λ) = T′ ∪ (⋃_{M∈M′}{v_{M}}) and T′′(λ) = T′′ ∪ (⋃_{M∈M′′}{v_{M}}). Let T = T′ ∪ T′′ and T (λ) = T′(λ) ∪ T′′(λ).

Then one has:

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

(ii.a) the evaluation map ev_{λ} : ℌ(𝔬_{T′∪T_{∞}}[U′]) → ℌ(𝔬_{T′(λ)∪T_{∞}}) is an isomorphism, where U′ = U_{𝔬_{T′∪T∞}};

(ii.b) the evaluation map ev_{λ} : ℌ(𝔬_{T′′∪T_{∞}}[U′′]) → ℌ(𝔬_{T′′(λ)∪T_{∞}}) is an isomorphism, where U′′ = U_{𝔬_{T′′∪T∞}};

(iii.a) the image of the point P′(λ) of order 2 on E′_{λ} under the Kummer map E′_{λ}(k) → k^{∗}/k^{∗2} and the class of m(λ) in k^{∗}/k^{∗2} lie in the subgroup I^{′′T′′(λ)} = ℌ(𝔬_{T′′(λ)}) and are independent;

(iii.b) the image of the point P′′(λ) of order 2 on E′′_{λ} under the Kummer map E′′_{λ}(k) → k^{∗}/k^{∗2} lies in the subgroup I^{′T′(λ)} = ℌ(𝔬_{T′′(λ)}) and is not trivial.

Proof For (i), one uses Lemma 1.3.2 (or 2.1.1) together with Lemma 2.1.2, i.e. the hypothesis that there is no Brauer-Manin obstruction (this is what enables us to get the existence of a local solution at the ‘Schinzel primes’ v_{M}). For details, see [CSS3], p. 599/600.

To prove (ii), one uses two analogues of diagram (2.1.1) of [CSS3], namely one over U′ and one over U′′. For the first one, the right hand side group is ⊕_{M∈M′}Z/2, for the second one, it is ⊕_{M∈M′′}Z/2. These are commutative diagrams of exact sequences.

0 → ℌ(𝔬_{T′∪T_{∞}}) → ℌ(𝔬_{T′∪T_{∞}}[U′]) → ⊕_{M∈M′} Z/2 → 0
↓ = ↓ ev_{λ} ↓ =
0 → ℌ(𝔬_{T′∪T_{∞}}) → ℌ(𝔬_{T′(λ)∪T_{∞}}) → ⊕_{M∈M′} Z/2 → 0

and

0 → ℌ(𝔬_{T′′∪T_{∞}}) → ℌ(𝔬_{T′′∪T_{∞}}[U′′]) → ⊕_{M∈M′′} Z/2 → 0
↓ = ↓ ev_{λ} ↓ =
0 → ℌ(𝔬_{T′′∪T_{∞}}) → ℌ(𝔬_{T′′(λ)∪T_{∞}}) → ⊕_{M∈M′′} Z/2 → 0

In these two diagrams, the bottom right hand side arrows are given by valuation at the places v_{M}.

To prove the independence statement in (iii.a), one then uses Assumption (2) in Theorem A, which is that m and c^{2} − d are independent (over Z/2) in 𝔖′′, that is to say that the class of m is not in the image of Z/2.P′ under the Kummer map E′(k(t)) → k(t)^{∗}/k(t)^{∗2} coming form the isogeny E′′ → E′. (Recall that E′ is given by y^{2} = (x − c(t))(x^{2} − d(t)).)

Similarly in (iii.b) one uses the fact that the class of d is not trivial in k(t)^{∗}/k(t)^{∗2}. That is automatic, since d is assumed nonconstant and squarefree. That the images lie in ℌ(𝔬_{T′′(λ)}), resp. in ℌ(𝔬_{T′(λ)}), i.e. that they do not involve T_{∞}, is a consequence of the fact that c^{2} − d and m, resp. d, are of even degree, and that at any finite place v ∉ S_{0} their coefficients are integral and their leading coefficient a unit; this implies that their evaluation at any point λ with v(λ) < 0 has even v-adic valuation.

2.2. The groups 𝔖′, 𝔖′′ and the maps δ′ and δ′′

Lemma 2.2.1 The maps

δ′ : 𝔖′ → ⊕_{M∈M′′} k_{M}^{∗}/k_{M}^{∗2}

and

δ′′ : 𝔖′′ → ⊕_{M∈M′} k_{M}^{∗}/k_{M}^{∗2}

given by evaluation at the relevant points M are homomorphisms. The kernels of these maps are finite.

Proof The group 𝔖′ = k[U′]^{∗}/k[U′]^{∗2} is generated by the classes of the r_{M} for M ∈ M′ and by k^{∗}/k^{∗2}. Since c^{2} − d is not constant, M′′ is not empty. Now for any closed point M, the map k^{∗}/k^{∗2} → k_{M}^{∗}/k_{M}^{∗2} has finite kernel. Hence δ′ has a finite kernel. The argument for the kernel of δ′′ is the same (here we use the fact that d is not constant).

2.3. Independence of the choice of λ: the spaces

Given λ ∈ k as in Proposition 2.1.3, let us consider the curves E′_{λ} and E′′_{λ}. They have good reduction at all places outside T (λ) (they indeed have good reduction at places in T_{∞}). In Subsection 1.1 we defined I^{′T (λ)} = ℌ(𝔬_{T (λ)}) and in Subsection 1.4 we considered the subspace I^{′T (λ)} ⊂ I^{′T (λ)}, and we defined similarly I^{′′T (λ)} ⊂ I^{′′T (λ)} (since λ determines the curves E′_{λ} and E′′_{λ}, I only use the subscript λ).

By Lemma 1.4.1, the Selmer group S′_{λ} = (S_{ϕ′})_{λ}, resp. S′′_{λ} = (S_{ϕ′′})_{λ} (we here simplify the notation of Section 1.5), can be computed as a subgroup of I^{′T (λ)}, resp. I^{′′T (λ)}.

The group I^{′′T (λ)} is defined by local conditions, namely, it consists of elements in ℌ(𝔬_{T (λ)}) whose image under the map ℌ(𝔬_{T (λ)}) → ℌ(k_{v}) belongs to W′′_{v}(λ) + K_{v}, this for all v ∈ T (λ). Similarly for the group I^{′T (λ)}.

The spaces K_{v} do not depend on λ. We shall fix the approximation condition (6) in Proposition 2.1.3 so that for v ∈ S, with the notation of Section 1.3, the subspaces W′′_{v}(λ) and W′′_{v}(λ_{v}) of V′′_{v} are identical. For a real place v, the independence of these subspaces follows from the fact that c(c^{2} − d) does not vanish on the interval 𝔘_{v}. Hence the conditions defining I^{′′T (λ)} at the places of S do not depend on λ.

For such a λ, from Remark 1.3.3, for v ∈ T′(λ) \ S, one has W′_{v}(λ)/(W′_{v}(λ) ∩ K′_{v}) = Z/2, W′′_{v}(λ)/(W′′_{v}(λ) ∩ K′′_{v}) = 0, and W′′_{v}(λ) + K′′_{v} = K′′_{v} = ℌ(𝔬_{v}); for v ∈ T′′(λ) \ S, one has W′′_{v}(λ)/(W′′_{v}(λ) ∩ K′′_{v}) = Z/2, W′_{v}(λ)/(W′_{v}(λ) ∩ K′_{v}) = 0 and W′_{v}(λ) + K′_{v} = K′_{v} = ℌ(𝔬_{v}) ; and for v ∉ T (λ) = T′(λ)∪T′′(λ), one has W′_{v}(λ)/(W′_{v}(λ)∩K′_{v}) = 0 and W′_{v}(λ)+K′_{v} = K′_{v} = ℌ(𝔬_{v}), W′′_{v}(λ)/(W′′_{v}(λ) ∩ K′′_{v}) = 0 and W′′_{v}(λ) + K′′_{v} = K′′_{v} = ℌ(𝔬_{v}).

For such a λ, the natural embeddings I^{′T′(λ)} ⊂ I^{′T (λ)} and I^{′′T′(λ)} ⊂ I^{′′T (λ)} are isomorphisms, and the natural projection maps W′_{T (λ)} → W′_{T′(λ)} and W′′_{T (λ)} → W′′_{T′′(λ)} are isomorphisms.

In particular we have inclusions S′_{λ} ⊂ I^{′T′(λ)} ⊂ ℌ(𝔬_{T′(λ)}) and S′′_{λ} ⊂ I^{′′T′′(λ)} ⊂ ℌ(𝔬_{T′′(λ)}). From Section 1.4 recall the isomorphisms τ′_{λ} : I^{′T′(λ)} ≃ W_{T′(λ)} and τ′′_{λ} : I^{′′T′′(λ)} ≃ W_{T′′(λ)}. From the above description of the sets W′_{v}(λ), W′′_{v}(λ), K′_{v}, K′′_{v}, one deduces the following

lemma.

Lemma 2.3.1 Let T′, T′′, T and λ be as in Proposition 2.1.3.

(a) There are natural inclusions ℌ(𝔬_{S}) ⊂ ℌ(𝔬_{T′}) ⊂ ℌ(𝔬_{T′(λ)}) ⊂ ℌ(𝔬_{T (λ)}).

The intersections of the subgroup I^{′T (λ)} ⊂ ℌ(𝔬_{T (λ)}) with these various subgroups are precisely I^{′S} ⊂ I^{′T′} ⊂ I^{′T′(λ)} ⊂ I^{′T (λ)}.

The group N′_{0} := I^{′S} depends neither on λ nor on the primes in T (λ). The group I^{′T′} only depends on T′ = S ∪_{M∈M′} T_{M}, it depends neither on λ nor on the places v_{M}.

(b) We have the analogous statements for the natural inclusions

ℌ(𝔬_{S}) ⊂ ℌ(𝔬_{T′′}) ⊂ ℌ(𝔬_{T′′(λ)}) ⊂ ℌ(𝔬_{T (λ)})

for the group N′′_{0} := I^{′′S}, and for the group I^{′′T′′}.

The proof to follow will involve the construction of ‘subdiagrams’ of the diagrams appearing in the proof of Prop. 2.1.3. Basically, we want to look at the inverse images under ev_{λ} of I^{′T′(λ)} ⊂ ℌ(𝔬_{T′(λ)∪T_{∞}}) = I^{′T′(λ)∪T_{∞}} and of I^{′′T′′(λ)} ⊂ ℌ(𝔬_{T′′(λ)∪T_{∞}}) = I^{′′T′′(λ)∪T_{∞}}.

Before doing this, we shall fix T_{∞}.

Consider the finite subgroup 𝔖′′_{S} of 𝔖′′ spanned by I^{′′S} and the elements r_{M}(t) for M running through M′′. This subgroup contains m(t). For each M ∈ M′, we have the finite subgroup δ′′_{M}(𝔖′′_{S}) ⊂ k_{M}^{∗}/k_{M}^{∗2}, which defines a multiquadratic extension F_{M}/k_{M}. Consider also the finite subgroup 𝔖′_{S} of 𝔖′ spanned by I^{′S} and the elements r_{M}(t) for M running through M′. For each M ∈ M′′, 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 all the extensions F_{M}/k. It is a consequence of Tchebotarev’s theorem that there exist infinitely many prime principal ideals (π_{v}) in the ring of integers 𝔬_{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{Norm_{k_{M} /k}(m(M)})/k. Choose v_{∞} ∉ S a finite place which satisfies these properties (required in Proposition 2.1.3), and which moreover is unramified in the extension F/k (note: v_{∞} is not an archimedean place, the index ∞ refers to the point at infinity on P^{1}). Let μ = π_{v_{∞}} ∈ 𝔬_{k} 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 T_{∞} = {v_{∞}}, and we let λ_{∞} = λ_{v_{∞}} = 1/μ ∈ k_{v_{∞}}.

Lemma 2.3.2 Let notation be as in Proposition 2.1.3, with T_{∞} = {v_{∞}} and μ as above.

(a) For each M ∈ M′, there exists a uniquely defined a′_{M} ∈ I^{′S} such that for any T′ and λ as in Proposition 2.1.3, a′_{M}.μ^{deg(r_{M})}.r_{M}(λ) ∈ I^{T′(λ)∪T_{∞}} is in the subspace of I^{′T′(λ)} ⊂ I^{T′(λ)∪T_{∞}} defined by the additional conditions that the local projection at v ∈ S belongs to K′_{v}.

(b) For each M ∈ M′′, there exists a uniquely defined a′′_{M} ∈ I^{′′S} such that for any T′′ and λ as in Proposition 2.1.3, a′′_{M}.μ^{deg(r_{M})}.r_{M}(λ) ∈ I^{T′′(λ)∪T_{∞}} is in the subspace of I^{′′T′′(λ)} ⊂ I^{T′′(λ)∪T_{∞}} defined by the additional conditions that the local projection at v ∈ S belongs to K′′_{v}.

Proof For any a′_{M} ∈ I^{′S}, the valuation of a′_{M}.μ^{deg(r_{M})}.r_{M}(λ) at a place v ∉ T′(λ) is even (for v = v_{∞} and r_{M} of odd degree, this follows from the transversality conditions (3) and (5) of Prop. 2.1.3). Thus any such a′_{M}.μ^{deg(r_{M})}.r_{M}(λ) belongs to I^{′T′(λ)}. For M_{1} ∈ M′ with M_{1} ≠ M, and v ∈ T_{M_{1}} ∪ {v_{M_{1}}}, the local condition a′_{M}.μ^{deg(r_{M})}.r_{M}(λ) ∈ W′_{v}(λ) + K′_{v} is satisfied, since at any such place v the valuation v(a′_{M}.μ^{deg(r_{M})}.r_{M}(λ)) is even and for v ∉ S, K′_{v} consists of the classes in k_{v}^{∗}/k_{v}^{∗2} with even valuation. For any v ∈ T_{M} ∪ {v_{M}}, the condition a′_{M}.μ^{deg(r_{M})}.r_{M}(λ) ∈ W′_{v}(λ) + K′_{v} is also satisfied. Indeed at such a place we have W′_{v}(λ) = k_{v}^{∗}/k_{v}^{∗2} (Remark 1.3.3). Since λ is v-adically very close to λ_{v} for v ∈ S (condition (6) in Prop. 2.1.3), the class of r_{M}(λ) ∈ k_{v}^{∗}/k_{v}^{∗2} does not depend on λ. Since μ is a square in each k_{v} for v ∈ S, the image of of μ^{deg(r_{M})}.r_{M}(λ) in V′_{S} is independent of λ, T and T_{∞}. We have I^{′S} ⊕ K′_{S} = V′_{S}. There thus exists a uniquely defined a′_{M} ∈ I^{′S} such that a′_{M}.μ^{deg(r_{M})}.r_{M}(λ) ∈ I^{′T′(λ)∪T_{∞}} is in the subspace of I^{′T′(λ)} ⊂ I^{′T′(λ)∪T_{∞}} defined by the additional conditions that the local projection at each v ∈ S belongs to K′_{v}. This completes the proof of (a). Mutatis mutandis, this also proves (b).

For each M ∈ M′, resp. M ∈ M′′, let a′_{M}, resp. a′′_{M}, be the elements of I^{S} determined in the previous lemma.

We now define:

a′_{M}(t) = (a′_{M}.μ^{deg(r_{M})}.r_{M}(t)) ∈ ℌ(𝔬_{S∪T_{∞}}[U′]) ⊂ 𝔖′

for M ∈ M′ and similarly

a′′_{M}(t) = (a′′_{M}.μ^{deg(r_{M})}.r_{M}(t)) ∈ ℌ(𝔬_{S∪T_{∞}}[U′′]) ⊂ 𝔖′′

for M ∈ M′′.

Notation 2.3.3 Let A′ ⊂ ℌ(𝔬_{S∪T_{∞}}[U′]) ⊂ ℌ(k[U′]) = 𝔖′ be the subspace spanned by the a′_{M}(t) for M ∈ M′. The subgroup A′ and I^{′T′} = I^{′T′(λ)} ∩ ℌ(𝔬_{T′}) of ℌ(𝔬_{T′∪T_{∞}}[U′]) clearly have trivial intersection. Let N′ = A′ ⊕ I^{′T′} ⊂ ℌ(𝔬_{T′∪T_{∞}}[U′]) be their direct sum. Let A′′ and N′′ be defined similarly.

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 (Lemma 2.1.2) Q_{v} ∈ X(k_{v}), v ∈ S (points which determine the spaces W_{v}(λ) = W_{v}(Q_{v}) for v ∈ S), as well as on T′ and T_{∞}. But once these choices have been made, Lemma 2.3.1 shows that N′ does not depend on the particular point λ chosen as in Proposition 2.1.3. More precisely, N′ does not depend on the v_{M}’s (M ∈ M′).

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

The same considerations apply for N′′, N′′_{0}, A′′.

As we observed in the proof of part (ii) of Proposition 2.1.3, the evaluation maps ev_{λ} : ℌ(𝔬_{T′∪T_{∞}}[U′]) → ℌ(𝔬_{T′(λ)∪T_{∞}}) and ev_{λ} : ℌ(𝔬_{T′′∪T_{∞}}[U′′]) → ℌ(𝔬_{T′′(λ)∪T_{∞}}) are isomorphisms as soon as properties (3) and (4) in that Proposition hold.

We have the exact analogue of Prop. 2.3.4 of [CSS3], to which we refer for the proof.

Proposition 2.3.4 Let T′, T′′, T_{∞} and λ be as in Proposition 2.1.3, and let N′ and N′′ be as above.

(a’) The evaluation map ev_{λ} from ℌ(𝔬_{T′∪T_{∞}}[U′]) to ℌ(𝔬_{T′(λ)∪T_{∞}}) induces an isomorphism between N′ and I^{′T′(λ)}.

(a”) The evaluation map ev_{λ} from ℌ(𝔬_{T′′∪T_{∞}}[U′′]) to ℌ(𝔬_{T′′(λ)∪T_{∞}}) induces an isomorphism between N′′ and I^{′′T′′(λ)}.

(b’) Denote by φ′_{λ} the composition of this isomorphism with the isomorphism I^{′T′(λ)} ⟶^{≃} W′_{T′(λ)} in Section 1.4. There is a direct sum decomposition:

N′ = N′_{0} ⊕ A′ ⊕ φ_{λ}^{−1}(⊕_{v∈T′\S}(W′_{v}(λ)/W′_{v}(λ) ∩ K′_{v})).

(b”) We have the analogous statement for φ′′_{λ}.

The class of d(t) ∈ ℌ(𝔬_{S}[U′]) may clearly be written as d(t) = ε. ∏_{M∈M′} a′_{M}(t) with ε ∈ I^{′S∪T_{∞}}. Since the degree of d is even, one actually has ε ∈ I^{′S}. Since d(λ) ∈ k^{∗}/k^{∗2} belongs to the image W′_{λ} of E′′_{λ}(k) → k^{∗}/k^{∗2}, ε belongs to I^{′S}. From the above proposition we conclude that the class of d belongs to N′, more precisely that d belongs to N′_{0} ⊕ A′.

The same argument shows that (c^{2} − d)(t) ∈ ℌ(𝔬_{S}[U′′]) belongs to N′′_{0} ⊕ A′ ⊂ N′′.

The same argument also applies to m(t) viewed as an element in ℌ(𝔬_{S}[U′′]). Indeed, the degree of m is even, m is the product of some of the r_{M}(t) for M ∈ M′′ by an element in 𝔬_{S}^{∗}, and the curve X_{λ} = X(m)_{λ} has points in each k_{v}, which implies that m(λ) belongs to W′′_{v}(λ) for each place v, i.e. m(λ) ∈ I^{′′T′′(λ)}. By the above proposition, the class of m(t) belongs to N′′. Now m(t) divided by the appropriate product of elements a′′_{M}(t) is an element of I^{′′S}, and it belongs to N′′, thus it lies in N′′_{0}.

2.4. Independence of the choice of λ: the pairing

At this point, for any λ as in Proposition 2.1.3, we have a pairing

e : N′ × N′′ ⟶^{(ev′_{λ},ev′′_{λ})} I^{′T′(λ)} × I^{′′T′′(λ)} ⟶^{(id,τ_{T′′(λ)}^{−1})} I^{′T′(λ)} × W′′_{T′′(λ)} ⟶^{e_{T (λ)}} Z/2

which may also be read

N′ × N′′ ⟶^{(ev′_{λ},ev′′_{λ})} I^{′T′(λ)} × I^{′′T′′(λ)} → W′_{T′(λ)} × W′′_{T′′(λ)} ⟶^{e_{T (λ)}} Z/2.

(Recall that I^{′T′(λ)} = I^{′T (λ)}, I^{′′T′′(λ)} = I^{′′T (λ)}, W′_{T′(λ)} = W′_{T (λ)} and W′′_{T′′(λ)} = W′′_{T (λ)}.)

Proposition 2.4.2 Let T′, T′′, T_{∞} = {v_{∞}} and λ be as in Proposition 2.1.3, and let N′, N′′ be as above. Then provided λ is close enough to λ_{∞} = 1/μ at v_{∞}, the restriction of the pairing e to a pairing (N′_{0} ⊕ A′) × (N′′_{0} ⊕ A′′) depends neither on T′, T′′ nor on λ.

Proof It is identical to the one given in given in [CSS3], Section 2.4. The only differences are:

(1) We here have pairings k_{v}^{∗}/k_{v}^{∗2}×k_{v}^{∗}/k_{v}^{∗2} → Z/2 rather than (k_{v}^{∗}/k_{v}^{∗2})^{2}×(k_{v}^{∗}/k_{v}^{∗2})^{2} → Z/2.

(2) We have to pair a′_{P} and a′′_{Q} for P ∈ M′ and Q ∈ M′′, hence in particular P ≠ Q : so the only computation one has to reproduce is the one given on pages 610, 611 and the first lines of page 612 of [CSS3].

2.5. Completion of the proof of Theorem A

We have reached Section 2.5 of [CSS3] (page 612); I now transcribe the end of Section 2 of [BSwD].

For each r ∈ N′, if r ≠ 1, d, there exists M ∈ M′′ such that r(M) ≠ 1 ∈ k_{M}^{∗}/k_{M}^{∗2}. If r is of even degree, this is hypothesis (3.a) of the theorem. If r is of odd degree, one remarks that r is the product of an element of I^{′S} an odd power of μ and a product of r_{M}(t)’s for M ∈ M′. For any M ∈ M′′, the class δ′_{M}(r) is then nontrivial, since μ has been so chosen that it does not belong to the image under δ′_{M} of the group spanned by I^{′S} and the r_{M}’s (M ∈ M′).

For such an M there exist infinitely many primes v^{M} of k_{M} which are of degree one over k (we denote by v the induced place on k) and which remain inert in the quadratic extension k_{M}(\sqrt{r(M)})/k_{M}. We shall apply this to each element different from 1, d in the left kernel of the restriction of e to N′_{0} ⊕ A′ × N′′_{0} ⊕ A′′. In this way, we produce distinct primes of the above type (one for each element of N′_{0} ⊕ A′ under consideration). Let v^{M} be one such prime, let v be the underlying prime of k and let λ_{v} ∈ 𝔬_{v} be transversal to the corresponding M̃ at v^{M}. To such a λ_{v} are associated curves E′_{λ_{v}}, E′′_{λ_{v}} and groups W′_{v}(λ_{v}), W′′_{v}(λ_{v}).

Assertion: there exists m′′_{v} ∈ W′′_{v} ⊂ k_{v}^{∗}/k_{v}^{∗2} which has a non-zero cup-product with r(λ_{v}) ∈ k_{v}^{∗}/k_{v}^{∗2}. This follows from the computations in Remark 1.3.3 (Lemma 4 in [BSwD]): for λ_{v} as above, one has v((c^{2} −d)(λ_{v})) = 1, v(d(λ_{v})) = 0 hence v(r(λ_{v})) = 0 (all this modulo 2), the cupproduct is simply given by (r(λ_{v}), m′′_{v}), it is non-zero if v(m′′_{v}) = 1 (possible since W′′_{v} = k_{v}^{∗}/k_{v}^{∗2}) and r(λ_{v}) ∈ 𝔬_{v}^{∗} is not a square, which it is not since v^{M} is inert in the quadratic extension k_{M}(\sqrt{r(M)})/k_{M}.

For each r ∈ N′′ ⊂ ℌ(k(t)), by hypothesis (3.b) of the theorem if the degree of r is even, or by the analogue of the above argument if the degree of r is odd, if r does not belong to the group spanned by c^{2} − d and m, there exists an M ∈ M′ such that r(M) does not belong to (1, m(M)) ⊂ k_{M}^{∗}/k_{M}^{∗2}. For this M ∈ M′ there exist infinitely many places v^{M} of k_{M} of degree one over k (we denote by v the induced place on k), which are split in the quadratic extension k_{M}(\sqrt{m(M)})/k_{M} but remain inert in the quadratic extension k_{M}(\sqrt{r(M)})/k_{M}.

(Note: at this point we do want the places v^{M} to split in k_{M}(\sqrt{m(M)})/k_{M}, see Proposition 2.1.3 (1): this is to ensure that X_{λ}(k_{v}) ≠ ∅ at such places.)

We apply this to each element of N′′_{0}⊕A′′ which does not belong to the subgroup {c^{2}−d, m} and which is in the right kernel of the restriction of e to N′_{0} ⊕ A′ × N′′_{0} ⊕ A′′. In this way, we produce distinct primes of the above type (one for each element of N′′_{0}⊕A′′ under consideration). Let v^{M} be one such prime, let v be the underlying prime of k and let λ_{v} ∈ 𝔬_{v} be transversal to the corresponding M̃ at v^{M}. To such a λ_{v} are associated curves E′_{λ_{v}}, E′′_{λ_{v}} and groups W′_{v}, W′′_{v}.

Assertion: there exists m′_{v} ∈ W′_{v} ⊂ k_{v}^{∗}/k_{v}^{∗2} which has a non-zero cup-product with r(λ_{v}) ∈ k_{v}^{∗}/k_{v}^{∗2}. This follows from Remark 1.3.3: we have here v(d(λ_{v})) = 1 hence v((c^{2} − d)(λ_{v})) = 0 and v(r(λ_{v})) = 0 (all this modulo 2), the cup-product is simply given by (m′_{v}, r(λ_{v})). This is nonzero if v(m′_{v}) = 1 (which occurs, by Remark 1.3.3) and r(λ_{v}) ∈ 𝔬_{v}^{∗} is not a square, which it is not since v^{M} remains inert in the quadratic extension k_{M}(\sqrt{r(M)})/k_{M}.

Let us now consider the restriction of the pairing e to (N′_{0} ⊕ A′) × (N′′_{0} ⊕ A′′). Recall that this restriction does not depend on the (not yet made) choice of T′, T′′, λ.

The classes {d}, resp. {c^{2} − d}, {m} lie in N′_{0} ⊕ A′, resp N′′_{0} ⊕ A′′, as was pointed out after Prop. 2.3.4.

We let N′_{0} ⊕ A′ = F′_{0} ⊕ F′_{1} ⊕ F′_{2}, where F′_{0} = Z/2{d} and F′_{0} ⊕ F′_{1} is the left kernel of the pairing (N′_{0} ⊕ A′) × (N′′_{0} ⊕ A′′) → Z/2.

We let N′′_{0} ⊕ A′′ = F′′_{0} ⊕ F′′_{1} ⊕ F′′_{2}, where F′′_{0} = Z/2{c^{2} − d} ⊕ Z/2{m} and F′′_{0} ⊕ F′′_{1} is the right kernel of the pairing (N′_{0} ⊕ A′) × (N′′_{0} ⊕ A′′) → Z/2.

Note that the pairing then induces a pairing F′_{2} × F′′_{2} which has trivial kernel on both sides. We now proceed as on pages 615-616 of [CSS3]. To each nonzero element in r ∈ F′_{1} we may by the above procedure associate an M ∈ M′′, a

place v_{M} in k_{M}, of degree one over k, a transversal λ_{v} and an element m′′_{v} such that (r(λ_{v}), m′′_{v}) = 1 ∈ Z/2. Let S_{2}^{0} be the set of places of k thus produced. The bilinear pairing F′_{1}×(⊕_{v∈S_{2}^{0}}Z/2) → Z/2 sending the pair (r, c_{v}) to the cup-product (r(λ_{v}), c_{v}) ∈ Z/2 is thus nondegenerate on the left hand side, one may therefore extract from S_{2}^{0} a subset S_{2} of order exactly the rank of the Z/2-vector space F′_{1}, such that the induced bilinear pairing F′_{1} × (⊕_{v∈S_{2}}Z/2) → Z/2 (sending the pair (r, 1_{v}) to r(λ_{v}) ∈ 𝔬_{v}^{∗}/𝔬_{v}^{∗2} = Z/2) is nondegenerate on both sides. We now fix such a set S_{2} and the corresponding sets T_{M} for M ∈ M′′. We also fix the λ_{v}’s as above.

We similarly proceed with F′′_{1}. We thus get sets T_{M}’s for suitable M ∈ M′, consisting of places v^{M} of degree one over the corresponding place of k (which we may assume distinct from the places appearing in S ∪S_{2}) and associated λ_{v}’s. This defines a set S_{3} of places of k such that the bilinear pairing (⊕_{v∈S_{3}}Z/2) × F′′_{1} → Z/2 (sending the pair (1_{v}, r) to r(λ_{v}) ∈ 𝔬_{v}^{∗}/𝔬_{v}^{∗2} = Z/2) is nondegenerate on both sides. We now fix such a set S_{3} and the corresponding sets T_{M} for M ∈ M′. We also fix the λ_{v}’s as above.

If we now choose a λ as in Schinzel’s hypothesis (in the (H_{1}) version, due to Serre, see [CTSwD], p. 71), close enough to the chosen λ_{v}’s for v finite in S ∪T_{∞} ∪S_{2} ∪S_{3}, and positive at the real places of k (at the real places, only the analogous condition should have been imposed on pages 598 and 617 of [CSS3]), then the associated pairing e_{λ}, whose kernels by Prop. 1.4.3 determine the Selmer groups S′_{λ} and S′′_{λ}, breaks up as

(F′_{0} ⊕ F′_{1} ⊕ F′_{2} ⊕ F′_{3}) × (F′′_{0} ⊕ F′′_{1} ⊕ F′′_{2} ⊕ F′′_{3}) → Z/2,

and we have the following properties.

F′_{0} is in the left kernel and F′′_{0} in the right kernel.

F′_{1} is orthogonal to F′′_{0} ⊕ F′′_{1} ⊕ F′′_{2} and F′′_{1} is orthogonal to F′_{0} ⊕ F′_{1} ⊕ F′_{2}.

The pairing F′_{2} × F′′_{2} → Z/2 is nondegenerate. The pairing F′_{1} × F′′_{3} → Z/2 is nondegenerate. The pairing F′_{3} × F′′_{1} → Z/2 is nondegenerate. This implies that the left kernel of the total pairing is F′_{0} = Z/2, spanned by the image

d(λ) ∈ k^{∗}/k^{∗2} of P′′(λ) ∈ E′′_{λ}(k) under the Kummer map, and that the right kernel of the total pairing is F′′_{0} = Z/2 ⊕ Z/2, spanned by m(λ) and the class (c^{2} − d)(λ) ∈ k^{∗}/k^{∗2} of the image of P′(λ) ∈ E′_{λ}(k) under the Kummer map.

It then only remains to apply 1.5.

Remark The whole discussion with T_{∞} and μ could be avoided if one made a stronger assumption in Theorem A, namely if one had hypothesis (3.a) for the map δ′ on the whole group 𝔖′ and similarly for the hypothesis (3.b) for the map δ′′ on the whole group 𝔖′′.

3. Proof of Theorem B

The proof is very close to the proof of Theorem A. Only the points which differ will be described.

On the family X/P_{k}^{1} of Theorem B one first realizes a reduction analogous to the one described in 2.1.

We have here several sets of closed points M ∈ A_{k}^{1}. The set M is defined by the vanishing of the separable polynomial

Δ = d_{01}(d_{23}^{2} + 4d_{24}d_{34})(d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13}).

The set M′_{1} is defined by the vanishing of d_{01}. The set M′_{2} is defined by the vanishing of (d_{23}^{2} + 4d_{24}d_{34}). Their union is the set M′. The set M′′ is defined by the vanishing of the product (d_{04}^{2} − d_{02}d_{03})(d_{14}^{2} − d_{12}d_{13}).

To any closed point M ∈ M′_{1}, with residue field k_{M}, we attach two quadratic extensions. The first one, K_{M}/k_{M} is obtained by adding the square root of the value at M of (d_{04}^{2} −d_{02}d_{03}), or of (d_{14}^{2} − d_{12}d_{13}) (at a point M ∈ M′, the product of these two values is a square in k_{M}). The second extension, L_{M}/k_{M}, is the one obtained by adding the square root of the value of −c in k_{M} (the value of c is given in the Introduction and repeated in Section 1.2).

To a closed point M ∈ M′′, we associate a quadratic extension K_{M}/k_{M}. If (d_{04}^{2} − d_{02}d_{03}) vanishes at M, then K_{M} is obtained by adding the square root of the value of −d_{01}d_{02}. If (d_{14}^{2} − d_{12}d_{13}) vanishes at M, then K_{M} is obtained by adding the square root of the value of d_{01}d_{12}.

We do not attach any quadratic extension to K_{M} if M belongs to M′_{2}.

One defines a set S_{0} as in Section 2.1 (containing any preassigned finite set of places), with an associated nice model X /P_{𝔬_{S_{0}}}^{1} over the ring 𝔬_{S_{0}} of S_{0}-integers.

Lemma 3.1.1 Let v ∉ S_{0} and let λ ∈ A^{1}(k_{v}) be a transversal point. Let λ̃ ⊂ P^{1}(𝔬_{v}) be its closure.

(a) If λ̃ does not intersect any of the M̃ for M ∈ M′_{1} ∪ M′′, then X_{λ}(k_{v}) ≠ ∅.

(b) If λ̃ intersects M̃ for M ∈ M′_{1} ∪ M′′ at some place w of k_{M} (w unramified over v, of degree one over v), then X_{λ}(k_{v}) ≠ ∅ if and only if w splits in K_{M}.

Note that the extension L_{M}/k_{M} does not play a rôle at this stage.

Given M ∈ M, we write r_{M}(t) = Norm_{k_{M} /k}(t − a_{M}), with a_{M} ∈ k_{M}. For M ∈ M′_{1} ∪ M′′, let

(K_{M}/k_{M}, t − a_{M}) ∈ Br(k_{M}(t))

be the class of the standard cyclic algebra associated to the element t − a_{M} ∈ k_{M}(t) and the cyclic extension K_{M}(t)/k_{M}(t). Let

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

For M ∈ M′_{1}, let similarly

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

Lemma 3.1.2 Let π : X → P_{k}^{1} and S_{0} be as above, let X_{U} = π^{−1}(U), 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 S of places 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}, such that

∑_{v∈S} inv_{v}(𝔞_{M}(π(Q_{v}))) = 0

for all M ∈ M′_{1} ∪ M′′ and such that

∑_{v∈S} inv_{v}(𝔟_{M}(π(Q_{v}))) = 0

for all M ∈ M′_{1}.

Proposition 3.1.3. Let π : X → P_{k}^{1}, the set S of places and the Q_{v} ∈ X(k_{v}), v ∈ S, be as in the conclusion of the previous lemma. For each point M ∈ M ∪ ∞, let T_{M} be a finite set of places of k such that T_{M} ∩ S = ∅, T_{M} ∩ T_{N} = ∅ for M ≠ N. Let λ ∈ U(k) ⊂ P^{1}(k) and λ̃ the associated point in P^{1}(𝔬_{S}). Assume:

(1) For each M ∈ M′_{1} and each place v ∈ T_{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} and in the quadratic extension L_{M}/k_{M}; for each M ∈ M′′ and each place v ∈ T_{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};

(2) Write K_{M} = k_{M}(\sqrt{α_{M}}) and L_{M} = k_{M}(\sqrt{β_{M}}); each place v ∈ T_{∞} splits in each quadratic extension k(\sqrt{Norm_{k_{M} /k}(α_{M})})/k and k(\sqrt{Norm_{k_{M} /k}(β_{M})})/k;

(3) λ̃ is transversal over 𝔬_{S};

(4) for any M ∈ M, the (transversal) intersection of λ̃ and M̃, when viewed on M̃, consists of the places w ∈ T_{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 λ̃ and the section at infinity of P_{𝔬_{S}}^{1} consists exactly of the places v ∈ T_{∞} (hence for any such v, we have v(λ) = −1);

(6) λ is very close to λ_{v} = π(Q_{v}) ∈ U(k_{v}) for nonarchimedean places v ∈ S, and lies in 𝔘_{v} for v ∈ S_{1}.

Let T′ = S ∪_{M∈M′} T_{M} and T′′ = S ∪_{M∈M′′} T_{M}, let T′(λ) = T′ ∪ (⋃_{M∈M′}{v_{M}}) and T′′(λ) = T′′ ∪ (⋃_{M∈M′′}{v_{M}}). Let T = T′ ∪ T′′ and T (λ) = T′(λ) ∪ T′′(λ).

Then one has:

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

(ii.a) the evaluation map ev_{λ} : ℌ(𝔬_{T′∪T_{∞}}[U′]) → ℌ(𝔬_{T′(λ)∪T_{∞}}) is an isomorphism, where U′ = U_{𝔬_{T′∪T∞}};

(ii.b) the evaluation map ev_{λ} : ℌ(𝔬_{T′′∪T_{∞}}[U′′]) → ℌ(𝔬_{T′′(λ)∪T_{∞}}) is an isomorphism, where U′′ = U_{𝔬_{T′′∪T∞}};

(iii.a) the classes of (d_{04}^{2} − d_{02}d_{03})(λ) and (d_{14}^{2} − d_{12}d_{13})(λ) lie in the subgroup I^{′′T′′(λ)} = ℌ(𝔬_{T′′(λ)}) ⊂ k^{∗}/k^{∗2} and are independent;

(iii.b) the class of (d_{23}^{2} + 4d_{24}d_{34})(λ) lies in the subgroup I^{′T′(λ)} = ℌ(𝔬_{T′(λ)}) ⊂ k^{∗}/k^{∗2} and is not trivial.

Just as in Section 2, one uses two analogues of diagram (2.1.1) of [CSS3], namely one over U′ and one over U′′ (same open sets as in Section 2). For the first one, the right hand side group is ⊕_{M∈M′}Z/2, for the second one, it is ⊕_{M∈M′′}Z/2. These are commutative diagrams of exact sequences.

0 → ℌ(𝔬_{T′∪T_{∞}}) → ℌ(𝔬_{T′∪T_{∞}}[U′]) → ⊕_{M∈M′} Z/2 → 0
↓ = ↓ ev_{λ} ↓ =
0 → ℌ(𝔬_{T′∪T_{∞}}) → ℌ(𝔬_{T′(λ)∪T_{∞}}) → ⊕_{M∈M′} Z/2 → 0

and

0 → ℌ(𝔬_{T′′∪T_{∞}}) → ℌ(𝔬_{T′′∪T_{∞}}[U′′]) → ⊕_{M∈M′′} Z/2 → 0
↓ = ↓ ev_{λ} ↓ =
0 → ℌ(𝔬_{T′′∪T_{∞}}) → ℌ(𝔬_{T′′(λ)∪T_{∞}}) → ⊕_{M∈M′′} Z/2 → 0

In these two diagrams, the bottom right hand side arrows are given by valuation at the places v_{M}.

Statements (iii.a) and (iii.b) can be reformulated exactly as in Proposition 2.1.3, in terms of the curves E′_{λ} and E′′_{λ}.

One now looks for a λ ∈ A^{1}(k) such that 3.1.3 holds (this will be provided by Schinzel’s hypothesis) and such that the 2-torsion subgroup of the Tate-Shafarevich group of E′_{λ} is trivial. Propositions 1.5.2 and 3.1.3 then enable one to conclude.

We are nearly in the same situation as we were in Section 2.3, except that

d = d_{01}^{2}.(d_{23}^{2} + 4d_{24}d_{34}) ∈ k[t]

contains a square factor.

One first has to check the “independence of the spaces” on the choice of λ.

A key point here is to check that for each M ∈ M and each v ∈ T (λ) \ S, the spaces W′_{v} and W′′_{v} are exactly the same as they were in Section 2.

Suppose M ∈ M′_{1}, i.e. d_{01}(M) = 0. For each v ∈ T_{M} ∪ {v_{M}}, we have v(d(M)) = 2, and the extension L_{M}/k_{M} is split at v (for v = v_{M}, the last statement follows from the reciprocity law.), i.e. the class of −c(M) is a square at the completion of k_{M} at v. It then follows from Remark 1.3.3 (iv) that W′′_{v} = 1 and W′_{v} = k_{M,v}^{∗}/k_{M,v}^{∗2}, hence, since K′_{v} = 𝔬_{v}^{∗}/𝔬_{v}^{∗2} = K′′_{v} for v ∉ S, W′′_{v}/(K′′_{v} + W′′_{v}) = 0 and W′_{v}/(W′_{v} + K′_{v}) = Z/2.

Suppose M ∈ M′_{2}, i.e. (d_{23}^{2} + 4d_{24}d_{34})(M) = 0. For each v ∈ T_{M} ∪ {v_{M}}, we have v(d(M)) = 1. Remark 1.3.3 (ii) shows that we are exactly in the same situation as above.

Suppose M ∈ M′′. For each v ∈ T_{M} ∪ {v_{M}}, we have v(d(M)) = 0 and v((c^{2} − d)(M)) = v(Δ′′(M)) = 1. By Remark 1.3.3 (iii), we have W′_{v} = 1, W′′_{v} = k_{M,v}^{∗}/k_{M,v}^{∗2}, hence finally W′_{v}/(W′_{v} + K′_{v}) = 0 and W′′_{v}/(K′′_{v} + W′′_{v}) = Z/2.

Now the situation is essentially identical to the one considered at the begining of Section 2.3, and the rest of the proof of Theorem B is just the same as that of Theorem A, hence shall not be reproduced.

Acknowledgements

This paper is a close variation on the paper [BSwD] of Bender and Swinnerton-Dyer, some parts of which have been simply reproduced.

Work for the paper was started on a hike in the Gers. Most of the writing was done in January 2000, while the author was staying at the Tata Institute of Fundamental Research (Mumbai, India), under the auspices of the Centre franco-indien pour la promotion de la recherche avancée (CEFIPRA/IFCPAR, Project 1601-2).

The author conveys his thanks to the referee for numerous and insightful remarks.

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J.-L. Colliot-Thélène C.N.R.S. UMR 8628, Mathématiques Bâtiment 425 Université de Paris-Sud F-91405 Orsay France e-mail : colliot@math.u-psud.fr