THE YOGA OF THE CASSELS-TATE PAIRING

TOM FISHER, EDWARD F. SCHAEFER, AND MICHAEL STOLL

Abstract. Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels-Tate pairing. In this article, we prove that the two pairings are the same.

1. Introduction

In [3], Cassels defined a pairing on the 2-Selmer group of an elliptic curve over a number field. It shares some properties with the extension of the Cassels-Tate pairing to the 2-Selmer group of an elliptic curve over a number field. He wrote “It seems highly probable that the two definitions are always equivalent, but the present writer is no longer an adept of the relevant yoga.” (see [3, p. 115]). In this article, we prove that the two pairings are the same.

The Cassels-Tate pairing is an alternating and bilinear pairing on the Shafarevich-Tate group of an elliptic curve over a number field. The fact that it is alternating gives information on the structure of the Shafarevich-Tate group. For n ≥ 2, its extension from the n-torsion of a Shafarevich-Tate group to an n-Selmer group can be used to determine the image of the n^{2}-Selmer group in the n-Selmer group. This sometimes enables the determination of which elements of the n-Selmer group come from elements of the Mordell-Weil group and which come from elements of the Shafarevich-Tate group. The Cassels-Tate pairing is, unfortunately, quite difficult to evaluate in practice. The pairing defined by Cassels on the 2-Selmer group of an elliptic curve, however, is quite straightforward to evaluate. So it is useful to prove that the two pairings are equal on the 2-Selmer group of an elliptic curve.

In Section 2, we give the Weil-pairing definition and a new definition of the Cassels-Tate pairing extended to the n-Selmer group of an elliptic curve, under a hypothesis that is always satisfied for n a prime. In Section 3 we present the definition of the pairing defined by Cassels on 2-Selmer groups of elliptic curves. In Section 4 we present a large diagram and prove it is commutative. We also discuss why our methods do not easily generalise to n-Selmer groups for n > 2. We use this diagram to prove our main theorem in Section 5 that the pairing defined by Cassels is the same as the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve over a number field.

Date: October 8, 2007.

2000 Mathematics Subject Classification. Primary 11G05; Secondary 11G07.

Key words and phrases. Cassels-Tate pairing, elliptic curve, 2-Selmer group.

The first and second author would like to thank the hospitality of Jacobs University Bremen. The second author was supported by a National Security Agency, Standard Grant and a Fulbright Award.

Recently, Swinnerton-Dyer [13] has generalised Cassels’ pairing on the 2-Selmer group, to a pairing between the m-Selmer group and the 2-Selmer group. In parallel with the results described above, we show that this pairing is again the Cassels-Tate pairing.

2. Two definitions of the Cassels-Tate pairing

Let E be an elliptic curve defined over K, a number field. The Cassels-Tate pairing is a pairing on � (K, E) taking values in ℚ/ℤ. We refer to [2] for the original definition. In the terminology of [6] this is the homogeneous space definition.

Let m, n ≥ 2 be integers. We are interested in the restriction of this pairing to the ntorsion � (K, E)[n], or more generally to � (K, E)[m] × � (K, E)[n]. Let S^{n}(K, E) denote the n-Selmer group of E over K. The group � (K, E)[n] is isomorphic to the quotient of S^{n}(K, E) by the image of E(K)/nE(K) under the coboundary map. We write ⟨ , ⟩_{CT} for the extension of the Cassels-Tate pairing to S^{m}(K, E)×S^{n}(K, E). By definition this pairing is trivial on the images of E(K)/mE(K) and E(K)/nE(K).

If M is a Gal(\overline{K}/K)-module, then we denote Z^{i}(Gal(\overline{K}/K), M) and H^{i}(Gal(\overline{K}/K), M) by Z^{i}(K, M) and H^{i}(K, M), respectively.

We recall an alternative definition of the Cassels-Tate pairing, called in [6] the Weil-pairing definition. For simplicity we assume that the natural map

(2.1)  H^{2}(K, E[n]) → ∏_{v} H^{2}(K_{v}, E[n]),

where v runs over all places of K, is injective. This is known for n a prime [2, Lemma 5.1]. (The injectivity does not hold for E[n] replaced by an arbitrary finite Galois module. See [10, III.4.7] for a counter-example.) From Section 3 onwards we restrict to the case n = 2, so our hypothesis will be automatically satisfied.

Let a ∈ S^{m}(K, E) and a′ ∈ S^{n}(K, E). We apply Galois cohomology over K and its completions K_{v} to

[figure: 0; E[n]; E[mn]; ·n; E[m]; 0; 0; E[n]; E; ·n; E; 0]

to obtain a commutative diagram

\xymatrix{H^{1}(K, E[mn]) \ar[r]^{·n} & H^{1}(K, E[m]) \ar[r] \ar[d] & H^{2}(K, E[n]) \ar[d] \\  & ∏_{v} H^{1}(K_{v}, E) \ar[r] & ∏_{v} H^{2}(K_{v}, E[n])}

By the above hypothesis, there exists c ∈ H^{1}(K, E[mn]) with nc = a. We represent c by a cocycle γ ∈ Z^{1}(K, E[mn]); then α = nγ ∈ Z^{1}(K, E[m]) represents a. For each place v of K, the cocycle res_{v}(α) = α_{v} in Z^{1}(K_{v}, E(\overline{K}_{v})) is a coboundary. So there exists β_{v} ∈ E(\overline{K}_{v}) such that α_{v} = dβ_{v} (recall dβ_{v} is the cocycle σ ↦^{σ}β_{v} − β_{v}). Take Q_{v} ∈ E(\overline{K}_{v}) such that nQ_{v} = β_{v}. Consider dQ_{v} − γ_{v} ∈ Z^{1}(K_{v}, E[n]), where γ_{v} is the restriction of γ. Let ∪_{e} be the cup product pairing induced by the Weil pairing from H^{1}(K_{v}, E[n]) × H^{1}(K_{v}, E[n]) to H^{2}(K_{v}, μ_{n}). For s, s′ ∈ H^{1}(K_{v}, E[n]) define ⟨s, s′⟩_{inv∘∪_{e},v} to be the composition of ∪_{e} with the invariant map. We define ⟨a, a′⟩_{1} = ∑_{v}⟨[dQ_{v} − γ_{v}], a′⟩_{inv∘∪_{e},v}.

Proposition 2.2. Let a ∈ S^{m}(K, E) and a′ ∈ S^{n}(K, E). We have ⟨a, a′⟩_{1} = ⟨a, a′⟩_{CT}.

Proof. See [2, Proof of Lemma 4.1] or [4, §2.2]. □

Remark 2.3. The general form of the Weil-pairing definition, avoiding the hypothesis that (2.1) is injective, is given in [5, p. 97]. This variant is used in [6] to generalise Proposition 2.2 to abelian varieties.

Let C and D be torsors (i.e., principal homogeneous spaces) under E. A morphism π : D → C is called an n-covering if π(P + Q) = nP + π(Q) for all P ∈ E and Q ∈ D. If C = E is the trivial torsor, this coincides with the usual notion of n-covering of E. For Q_{1}, Q_{2} ∈ D we write Q_{1} − Q_{2} for the point on E determined by the fact D is a torsor under E. Following [12, Chapter 6] we define the coboundary map δ_{π} : C(K) → H^{1}(K, E[n]) by sending P ∈ C(K) to the class of dQ = (σ ↦^{σ}Q − Q) where Q ∈ D(\overline{K}) with πQ = P.

In the case C = E, there is a standard bijection between the n-coverings of E up to K-isomorphism, and the Galois cohomology group H^{1}(K, E[n]). It is defined as follows. Let ψ : D → E be an isomorphsim of curves over \overline{K} with [n] ∘ ψ = π. Then ^{σ}ψ ∘ ψ^{−1} is translation by some ξ_{σ} ∈ E[n] and we identify the K-isomorphism class of D with the class of σ ↦ ξ_{σ} in H^{1}(K, E[n]). If Q ∈ D(\overline{K}) with π(Q) = 0 then we can take ψ : P ↦ P − Q, in which case D is represented by −dQ. Note also that if C → E is an m-covering of E and D → C is an n-covering of C, then D → E is an mn-covering of E. If D → E corresponds to b ∈ H^{1}(K, E[mn]), then C → E corresponds to nb ∈ H^{1}(K, E[m]).

We give a new definition of the Cassels-Tate pairing, again under the hypothesis that (2.1) is injective. Let C be an m-covering of E over K representing a. By the hypothesis, a is divisible by n in the Weil-Châtelet group. So there is an n-covering π : D → C defined over K. Let v be a place of K. Since a is trivial in H^{1}(K_{v}, E(\overline{K}_{v})), there is a point P_{v} ∈ C(K_{v}). We define ⟨a, a′⟩_{2} = ∑_{v}⟨δ_{π}(P_{v}), a′⟩_{inv∘∪_{e},v}.

Proposition 2.4. Let a ∈ S^{m}(K, E) and a′ ∈ S^{n}(K, E). We have ⟨a, a′⟩_{2} = ⟨a, a′⟩_{1}. In particular ⟨a, a′⟩_{2} does not depend on the choice of the P_{v}.

Proof. Let R_{C} ∈ C(\overline{K}) and R_{D} ∈ D(\overline{K}) such that R_{C} covers 0 on E and R_{D} covers R_{C}. Since n(dR_{D}) = dR_{C} represents −a, we can choose γ ∈ Z^{1}(K, E[mn]), as defined above, to be −dR_{D}. For each place v of K we are given P_{v} ∈ C(K_{v}). Let β_{v} = P_{v} − R_{C}, then dβ_{v} = −dR_{C}; this represents a ∈ H^{1}(K_{v}, E[m]). Take Q_{v} ∈ E(\overline{K}_{v}) with nQ_{v} = β_{v}. Then dQ_{v} − γ_{v} = d(Q_{v} + R_{D}) and π(Q_{v} + R_{D}) = β_{v} + R_{C} = P_{v}. Hence δ_{π}(P_{v}) is represented by the cocycle dQ_{v} − γ_{v} appearing in the definition of ⟨ , ⟩_{1}. □

3. The Cassels pairing

In [3], Cassels defined a bilinear pairing ⟨ , ⟩_{Cas} on S^{2}(K, E) taking values in μ_{2} with the following properties. The element a ∈ S^{2}(K, E) is in the image of S^{4}(K, E) precisely when ⟨a, a′⟩_{Cas} = +1 for all a′ ∈ S^{2}(K, E). For all a ∈ S^{2}(K, E) we have ⟨a, a⟩ = +1. These are properties of the Cassels-Tate pairing on a 2-Selmer group as well.

A mild generalisation of Cassels’ construction, due to Swinnerton-Dyer [13], gives a pairing S^{m}(K, E) × S^{2}(K, E) → μ_{2}. We work with this generalised form of the pairing, which we continue to denote ⟨ , ⟩_{Cas}. It reduces to Cassels’ definition in the case m = 2.

We prepare to recall the definition of the pairing. The group S^{2}(K, E) is a subgroup of H^{1}(K, E[2]). Let \overline{A} be the finite étale algebra that is the Galois module of maps from E[2] \ 0 to \overline{K}. Then μ_{2}(\overline{A}) is the Galois module of maps from E[2] \ 0 to μ_{2}. Let A denote the Gal(\overline{K}/K)-invariants of \overline{A}. Let E be given by y^{2} = F (x) where F (x) = x^{3} + a_{2}x^{2} + a_{4}x + a_{6} with a_{i} ∈ K. Then A ≅ K[T]/(F (T)). Let θ_{1}, θ_{2}, θ_{3} be the three roots of F (x) in \overline{K}. We have A ≅ ∏^{♢} K(θ_{j}) where ∏^{♢} denotes taking the product over one element from each Gal(\overline{K}/K)-orbit of the set of θ_{j}’s. Let T_{j} = (θ_{j}, 0) ∈ E[2] \ 0 and define w : E[2] → μ_{2}(\overline{A}) by w(P) = (T_{j} ↦ e_{2}(P, T_{j})). Then w induces an injective homomorphism from H^{1}(K, E[2]) to H^{1}(K, μ_{2}(\overline{A})), which we also denote w. Let r_{j} be the restriction map from H^{1}(K, μ_{2}(\overline{A})) to H^{1}(K(θ_{j}), μ_{2}). Shapiro’s Lemma shows that the map r = ∏^{♢} r_{j} is an isomorphism of H^{1}(K, μ_{2}(\overline{A})) with ∏^{♢} H^{1}(K(θ_{j}), μ_{2}), which we denote H^{1}(A, μ_{2}). For each j, we have a Kummer isomorphism from H^{1}(K(θ_{j}), μ_{2}) to K(θ_{j})^{×}/(K(θ_{j})^{×})^{2}. This induces an isomorphism, which we denote k, from H^{1}(A, μ_{2}) to A^{×}/(A^{×})^{2}. Note that the image of H^{1}(K, E[2]) in A^{×}/(A^{×})^{2}, under k ∘ r ∘ w, is equal to the kernel of the norm from A^{×}/(A^{×})^{2} to K^{×}/(K^{×})^{2}.

We recall the definition of ⟨ , ⟩_{Cas}. Let a ∈ S^{m}(K, E) and a′ ∈ S^{2}(K, E). Let M = k ∘ r ∘ w(a′) be the element of A^{×}/(A^{×})^{2} representing a′. The element a ∈ S^{m}(K, E) is represented by an m-covering C (which Cassels denotes D_{Λ}) of E. Swinnerton-Dyer [13] shows that there are rational functions f_{j} on C, defined over K(θ_{j}), with the following three properties

(i) div(f_{j}) = 2D_{j} where [D_{j}] ↦ T_{j} = (θ_{j}, 0) under the isomorphism of Pic^{0}(C) and E, (ii) each K-isomorphism of K(θ_{i}) to K(θ_{j}) sending θ_{i} to θ_{j} sends f_{i} to f_{j}, (iii) the product f_{1}f_{2}f_{3} is a square in K(C).

He then shows that a 2-covering of C may be defined by setting each f_{j} equal to the square of an indeterminate. In the case m = 2, Cassels gives an explicit construction of the f_{j} (which he denotes \frac{L_{j}}{L_{0}} ) and this makes it practical to compute the pairing. We write f for the element of A ⊗_{K} K(C) given by T_{j} ↦ f_{j}.

Let v be a prime of K. Since C represents an element in S^{m}(K, E), there is a point P_{v} ∈ C(K_{v}) (which Cassels calls ℭ_{v}). For γ_{j}, δ_{j} ∈ K_{v}(θ_{j})^{×}/(K_{v}(θ_{j})^{×})^{2} we let (γ_{j}, δ_{j})_{K_{v}(θ_{j})} denote the quadratic Hilbert norm residue symbol. Let \overline{A}_{v} = A ⊗_{K} \overline{K}_{v} and A_{v} be its Gal(\overline{K}_{v}/K_{v})-invariants. Then A_{v} ≅ ∏^{♢} K_{v}(θ_{j}), where this ∏^{♢} is taken over Gal(\overline{K}_{v}/K_{v})- orbits. Let (γ, δ)_{A_{v}} = ∏^{♢}(γ_{j}, δ_{j})_{K(θ_{j})_{v}} where γ, δ ∈ A_{v}^{×}/(A_{v}^{×})^{2} and γ_{j}, δ_{j} are their images in K_{v}(θ_{j})^{×}/(K_{v}(θ_{j})^{×})^{2}. Cassels defines ⟨a, a′⟩_{Cas} = ∏_{v}(f(P_{v}), M)_{A_{v}}.

4. The main diagram

Now let us introduce Figure 4.1 which will enable us to prove that for a ∈ S^{m}(K, E), a′ ∈ S^{2}(K, E) we have ⟨a, a′⟩_{Cas} = ⟨a, a′⟩_{2}. We can define the maps w, r and k locally, in an analogous way, and it will not change the image of M, locally. So we will not change our notation for these maps.

(4.1)  [figure: H^{1}(K_{v}, E[2]); w; ×; H^{1}(K_{v}, E[2]); w; ∪_{e}; H^{2}(K_{v}, μ_{2}); (1); H^{1}(K_{v}, μ_{2}(\overline{A}_{v})); r; ≅; ×; H^{1}(K_{v}, μ_{2}(\overline{A}_{v})); r; ≅; ∪_{t}; H^{2}(K_{v}, μ_{2}(\overline{A}_{v})); r; ≅; N_{∗}; H^{2}(K_{v}, μ_{2}); inv; H^{1}(A_{v}, μ_{2}); k; ≅; ×; H^{1}(A_{v}, μ_{2}); k; ≅; ∪; (2); H^{2}(A_{v}, μ_{2}); _{∏_{♢} inv_{j}}; (3); A_{v}^{×}/(A_{v}^{×})^{2}; ×; A_{v}^{×}/(A_{v}^{×})^{2}; ∏^{♢}( , )_{Kv(θ_{j})}; (4); ∏^{♢} μ_{2}; ν; μ_{2}]

We identify μ_{2} ⊗ μ_{2} = μ_{2} via (−1)^{p} ⊗ (−1)^{q} = (−1)^{pq}. Since μ_{2}(\overline{A}_{v}) is the Galois module of maps from E[2] \ 0 to μ_{2}, this identification induces a map t : μ_{2}(\overline{A}_{v}) ⊗ μ_{2}(\overline{A}_{v}) → μ_{2}(\overline{A}_{v}). Let ∪_{t} be the cup product map via t. Define N : μ_{2}(\overline{A}_{v}) → μ_{2} by (T ↦ γ(T)) ↦ ∏_{T} γ(T), and let N_{∗} be the map it induces on H^{2}’s. Let r_{j} be the restriction map from H^{2}(K_{v}, μ_{2}(\overline{A}_{v})) to H^{2}(K_{v}(θ_{j}), μ_{2}). In the same was as for the H^{1}’s, Shapiro’s Lemma shows that the map r = ∏^{♢} r_{j} is an isomorphism of H^{2}(K_{v}, μ_{2}(\overline{A}_{v})) with ∏^{♢} H^{2}(K_{v}(θ_{j}), μ_{2}), which we denote H^{2}(A_{v}, μ_{2}). Let ∪_{j} be the cup product map from H^{1}(K_{v}(θ_{j}), μ_{2})×H^{1}(K_{v}(θ_{j}), μ_{2}) to H^{2}(K_{v}(θ_{j}), μ_{2}) and ∪ = ∏^{♢} ∪_{j}. Let inv : H^{2}(K_{v}, μ_{2}) → μ_{2} be the composition of the invariant map with the isomorphism of \frac{1}{2}ℤ/ℤ and μ_{2}, and likewise for inv_{j} : H^{2}(K_{v}(θ_{j}), μ_{2}) → μ_{2}. Finally let ν : ∏^{♢} μ_{2} → μ_{2} be the usual product in μ_{2}.

Theorem 4.2. The diagram in Figure 4.1 is commutative.

We prove this theorem using the following lemmas.

Lemma 4.3. Identify μ_{2} ⊗ μ_{2} = μ_{2} as above. Then for all P, Q ∈ E[2] we have

e_{2}(P, Q) = ∏_{T ∈E[2]\0} e_{2}(P, T) ⊗ e_{2}(Q, T).

Proof. A trivial verification. □

Lemma 4.4. Diagram (1) in Figure 4.1 is commutative.

Proof. Let ξ, ψ ∈ H^{1}(K_{v}, E[2]) be represented by cocycles which, for ease of notation, we also write as ξ and ψ. We have ξ ∪_{e} ψ : (σ, τ) ↦ e_{2}(ξ_{σ},^{σ}ψ_{τ}) ∈ H^{2}(K_{v}, μ_{2}).

Now w(ξ) : σ ↦ (T ↦ e_{2}(ξ_{σ}, T)) for T ∈ E[2] \ 0 and similarly for w(ψ). Thus

N_{∗}(w(ξ) ∪_{t} w(ψ)) : (σ, τ) ↦ N_{∗}(t((S ↦ e_{2}(ξ_{σ}, S)) ⊗^{σ}(T ↦ e_{2}(ψ_{τ}, T))))

= N_{∗}(t((S ↦ e_{2}(ξ_{σ}, S)) ⊗ (T ↦^{σ}e_{2}(ψ_{τ},^{σ^{−1}}T))))

= N_{∗}(t((S ↦ e_{2}(ξ_{σ}, S)) ⊗ (T ↦ e_{2}(^{σ}ψ_{τ}, T))))

= N_{∗}(T ↦ e_{2}(ξ_{σ}, T) ⊗ e_{2}(^{σ}ψ_{τ}, T))

= ∏_{T ∈E[2]\0} e_{2}(ξ_{σ}, T) ⊗ e_{2}(^{σ}ψ_{τ}, T) ∈ μ_{2} ⊗ μ_{2}.

By Lemma 4.3 this is the same as ξ ∪_{e} ψ. □

Lemma 4.5. Diagram (2) in Figure 4.1 is commutative

Proof. Let ξ, ψ ∈ H^{1}(K_{v}, μ_{2}(\overline{A}_{v})). As in the proof of the previous lemma, we use the same symbols for cocycles representing these classes. Let T_{j} = (θ_{j}, 0) ∈ E[2] \ 0. We must show that r_{j}(ξ ∪_{t} ψ) and r_{j}(ξ) ∪_{j} r_{j}(ψ) are equal in H^{2}(K_{v}(θ_{j}), μ_{2} ⊗ μ_{2}). We find that they are represented by cocycles (σ, τ) ↦ ξ_{σ}(T_{j}) ⊗^{σ}(ψ_{τ})(T_{j}) and (σ, τ) ↦ ξ_{σ}(T_{j}) ⊗^{σ}(ψ_{τ}(T_{j})). Since σ(T_{j}) = T_{j} for all σ ∈ Gal(\overline{K}_{v}/K_{v}(θ_{j})), these cocycles are equal. □

Lemma 4.6. Diagram (3) in Figure 4.1 is commutative.

Proof. We have \overline{A}_{v} = ∏^{♢} \overline{K_{v}(θ_{j})} where \overline{K_{v}(θ_{j})} := K_{v}(θ_{j}) ⊗_{K_{v}} \overline{K}_{v}. Let N_{j} denote the norm induced by taking the product over each element in the Gal(\overline{K}_{v}/K_{v})-orbit of θ_{j}. Recall that ν : ∏^{♢} μ_{2} → μ_{2} is the usual product in μ_{2}, and let ν_{∗} be the map it induces on H^{2}’s. Then the map N_{∗} in Figure 4.1 factors as the composite of ∏^{♢} N_{j} and ν_{∗}.

We have the following commutative diagram

[figure: H^{2}(K_{v}, μ_{2}(\overline{A}_{v})); r; =; ∏^{♢} H^{2}(K_{v}, μ_{2}(\overline{K_{v}(θ_{j})})); ∏^{♢} r_{j}; ∏^{♢} N_{j}; ∏^{♢} H^{2}(K_{v}, μ_{2}); _{∏_{♢} inv}; ν_{∗}; H^{2}(K_{v}, μ_{2}); inv; H^{2}(A_{v}, μ_{2}); =; ∏^{♢} H^{2}(K_{v}(θ_{j}), μ_{2}); _{∏_{♢} inv_{j}}; (5); ∏^{♢} μ_{2}; (6); ν; μ_{2}.]

Diagram (5) commutes by the next lemma. That Diagram (6) commutes is obvious. This proves the commutativity of Diagram (3). □

Lemma 4.7. Let X_{j} be the Gal(\overline{K}_{v}/K_{v})-orbit of T_{j}. There is a commutative diagram

\xymatrix{H^{2}(K_{v}, Map(X_{j}, μ_{2^{∞}})) \ar[r]^{N_{j}} \ar[d]_{r_{j}}^{≅} & H^{2}(K_{v}, μ_{2^{∞}}) \ar[d]^{inv} \\ H^{2}(K_{v}(θ_{j}), μ_{2^{∞}}) \ar[r]^{_{inv_{j}}} & ℚ/ℤ.}

Proof. Let ι : H^{2}(K_{v}, μ_{2^{∞}}) → H^{2}(K_{v}, Map(X_{j}, μ_{2^{∞}})) be induced by the inclusion of the constant maps. Then r_{j} ∘ ι is the restriction map Br(K_{v})[2^{∞}] → Br(K_{v}(θ_{j}))[2^{∞}]. By [9, §1 Theorem 3] it is multiplication by d_{j} on the invariants, where d_{j} = [K_{v}(θ_{j}) : K_{v}] = #X_{j}, and therefore surjective. Since r_{j} is an isomorphism, it follows that ι is surjective. Then for η ∈ H^{2}(K_{v}, μ_{2^{∞}}) we compute

(inv ∘ N_{j})(ι(η)) = d_{j}inv(η) = (inv_{j} ∘ r_{j})(ι(η)).

(Alternatively, the definitions in [1, Chapter III,§9] show that N_{j} ∘ r_{j}^{−1} is corestriction, and the lemma then reduces to a well known property of the invariant maps.) □

Lemma 4.8. Diagram (4) in Figure 4.1 is commutative.

Proof. This is [8, XIV.2 Prop. 5] applied to each constituent field of A_{v}. □

Lemmas 4.4, 4.5, 4.6, 4.8 together prove Theorem 4.2. Composing the maps in the last row of (4.1) gives the pairing ( , )_{A_{v}} defined at the end of Section 3. Identifying \frac{1}{2}ℤ/ℤ with μ_{2} we obtain

Corollary 4.9. Let s, s′ ∈ H^{1}(K_{v}, E[2]). We have ⟨s, s′⟩_{inv∘∪_{e},v} = (k∘r∘w(s), k∘r∘w(s′))_{A_{v}}.

Remark 4.10. It would be useful to have an analogue of Corollary 4.9 for elements of H^{1}(K_{v}, E[n]) for general n (or at least for n prime). Lemma 4.4 depends on the equality in Lemma 4.3, which in turn only works for n = 2. This prevents any obvious generalisation to other values of n. Another difficulty is that we use μ_{2} ⊂ K_{v} in our proofs.

5. The main theorem

Let C be a torsor under E, and f ∈ A ⊗_{K} K(C) as described in Section 3. Let π : D → C be the 2-covering obtained by setting each f_{j} equal to the square of an indeterminate. The following lemma is a variant of [7, Theorem 2.3].

Lemma 5.1. We have (k ∘ r ∘ w)(δ_{π}(P)) = f(P) mod (A^{×})^{2} for all P ∈ C(K), away from the zeroes and poles of the f_{j}.

Proof. Let Q ∈ D(\overline{K}) with π(Q) = P. It suffices to show that f_{j}(P) = k_{j}r_{j}w(dQ) mod (K(θ_{j})^{×})^{2}.

We have r_{j}w(dQ) = (σ ↦ e_{2}(^{σ}Q − Q, T_{j})) in H^{1}(K(θ_{j}), μ_{2}). The construction of D gives that f_{j} ∘ π = t_{j}^{2} for some rational function t_{j} on D, defined over K(θ_{j}). We claim that e_{2}(S, T_{j}) = t_{j}(S + X)/t_{j}(X) for any X ∈ D(\overline{K}) for which the numerator and denominator are well-defined and non-zero. Indeed, since the Weil pairing is a geometric construction we may identify D and E over \overline{K}. This is an identification as torsors under E, so the action of E on D is identified with the group law on E. Then π is the multiplication-by-2 map on E, and our claim reduces to the definition of the Weil pairing in [11, Chapter III, §8].

Putting S =^{σ}Q − Q and X = Q gives e_{2}(^{σ}Q − Q, T_{j}) = t_{j}(^{σ}Q)/t_{j}(Q) =^{σ}(t_{j}(Q))/t_{j}(Q) for any σ ∈ Gal(K(θ_{j})/K). Then r_{j}w(dQ) = (σ ↦^{σ}(t_{j}(Q))/t_{j}(Q)) and k_{j}r_{j}w(dQ) = t_{j}^{2}(Q) = f_{j}π(Q) = f_{j}(P). □

As usual we identify \frac{1}{2}ℤ/ℤ with μ_{2}.

Theorem 5.2. Let K be a number field and E an elliptic curve over K. Let a ∈ S^{m}(K, E) and a′ ∈ S^{2}(K, E). We have ⟨a, a′⟩_{Cas} = ⟨a, a′⟩_{2} = ⟨a, a′⟩_{1} = ⟨a, a′⟩_{CT}.

Proof. The identification ⟨a, a′⟩_{Cas} = ⟨a, a′⟩_{2} is immediate from Corollary 4.9 and the local analogue of Lemma 5.1. The other identifications were established in Section 2. □

References

[1] K.S. Brown, Cohomology of groups, GTM 87, Springer-Verlag, New York, 1994.

[2] J.W.S. Cassels, Arithmetic on curves of genus 1, IV. Proof of the Hauptvermutung, J. reine angew. Math. 211 (1962), 95–112.

[3] J.W.S. Cassels, Second descents for elliptic curves, J. reine angew. Math. 494 (1998), 101–127.

[4] T.A. Fisher, The Cassels-Tate pairing and the Platonic solids, J. Number Theory 98 (2003), 105-155.

[5] J.S. Milne, Arithmetic duality theorems, Academic Press, Inc., Boston, Mass, 1986.

[6] B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. 150 (1999), 1109–1149.

[7] E.F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310 (1998), no. 3, 447–471.

[8] J-P. Serre, Local fields, Springer-Verlag, New York, 1979.

[9] J-P. Serre, Local class field theory, in J.W.S. Cassels and A. Fröhlich (eds), Algebraic Number Theory, Academic Press, London, 1967, 129–161.

[10] J-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 2002.

[11] J.H. Silverman, The arithmetic of elliptic curves, GTM 106, Springer-Verlag, New York, 1992.

[12] S. Stamminger, Explicit 8-descent on elliptic curves, PhD thesis, International University Bremen, 2005.

[13] H.P.F. Swinnerton-Dyer, 2^{n}-descent on elliptic curves for all n, draught, January 2007.

DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

  E-mail address: T.A.Fisher@dpmms.cam.ac.uk

Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053, USA

  E-mail address: eschaefer@scu.edu

School of Engineering and Science, Jacobs University Bremen, P.O.Box 750561, 28725 Bremen, Germany

  E-mail address: m.stoll@jacobs-university.de