The Cassels-Tate Pairing and the Platonic Solids

Tom Fisher Department of Pure Mathematics and Mathematical Statistics University of Cambridge Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB E-mail: T.A.Fisher@dpmms.cam.ac.uk

Version: 11th October 2001

We perform descent calculations for the families of elliptic curves whose mtorsion splits as μ_{m} × Z/mZ for m = 3, 4 or 5. These curves are parametrised by the modular curve X(m) ≃ P^{1}, whose cusps are arranged as the vertices of one of the Platonic solids. Following McCallum [McC] we write the Cassels-Tate pairing as a sum of local pairings. In the case m = 5 our results extend the work of Beaver [Be].

INTRODUCTION

Let E/Q be an elliptic curve and let m ≥ 2 be an integer. The process of m-descent bounds the group E(Q)/mE(Q) and so gives an estimate for the Mordell-Weil rank. It is both convenient and instructive to work with special cases where the m-torsion of E takes a simple form. For example, many authors introduce 2-descent by considering the elliptic curves

E : y^{2} = (x − a_{1})(x − a_{2})(x − a_{3})  (1)

with rational 2-torsion. More generally we consider elliptic curves E with E[m] ≃ μ_{m} × Z/mZ. For m ≥ 3 these curves are parametrised by the modular curve X(m), or strictly speaking by its open subset Y (m) obtained by deleting the cusps. Our interest is in the cases m = 3, 4 and 5 when X(m) ≃ P^{1}. Relabelling torsion gives an action of PSL_{2}(Z/mZ) on X(m) defined over Q(μ_{m}). The quotient map j : X(m) → P^{1} is ramified above j = 0, 1728 and ∞. Under stereographic projection these points are arranged as the faces, edges and vertices of one of the Platonic solids.

j = 0 j = 1728 j = ∞ PSL_{2}(Z/mZ)

Platonic Solid #Faces #Edges #Vertices

Symmetries

m = 3 tetrahedron  4  6  4  A_{4}
m = 4 octahedron  8  12  6  S_{4}
m = 5 icosahedron  20  30  12  A_{5}

In particular the cusps, i.e. the points above j = ∞, form a single orbit under the action of PSL_{2}(Z/mZ).

In §1 we prove

Theorem 1. Let K a number field and let m = 3, 4 or 5. Then the Tate-Shafarevich group of an elliptic curve over K may contain arbitrarily many elements of order m.

Special cases of this result are proved in [CaVI], [Bö], [Kr], [McG], [L] and [F1]. Our method is that used in [F1]. Since X(m) ≃ P^{1} there are infinitely many elliptic curves E/K with E[m] ≃ μ_{m} × Z/mZ. We write α : E → E′ and β̂ : E → E′′ for the isogenies with kernel μ_{m} and Z/mZ respectively. We may estimate the Mordell-Weil rank using either the pair of isogenies α and α̂, or the pair of isogenies β and β̂. In general these estimates do not agree, and an application of Dirichlet’s theorem on primes in arithmetic progression suffices to prove the theorem. We do not make use of the Cassels-Tate pairing. The method does not extend to m = 7, since the only Q-rational points on the Klein quartic X(7) are the cusps.

In §2 we give a survey of duality results for elliptic curves, and in particular the Cassels-Tate pairing. We then develop the methods of McCallum [McC] and Beaver [Be] for computing the pairing in the split torsion case. The action of PSL_{2}(Z/mZ) is used to make explicit the parameter λ_{v} appearing in [Be, Theorems 1.1,1.2]. As in [F1] we find that our descent calculations are governed not only by the primes of bad reduction, but by the cusps we obtain when we reduce mod 𝔭. We also explain how our results may be applied to curves without split torsion, by giving an example in this direction.

In §3 we restrict to K = Q and m = 3 or 5. We make explicit our various estimates for the Mordell-Weil rank. In particular the Cassels-Tate pairing is used to give a description of S^{(m)}(E/Q) as the kernel of a skewsymmetric matrix. The explicit nature of our results enables us to compute a large amount of numerical data, and to give some interesting examples. In particular we exhibit some elliptic curves over Q whose Tate-Shafarevich group contains an element of order m^{2}.

Preliminaries on descent calculations

Let K be a field of characteristic zero. Let ϕ : C → D be an isogeny of elliptic curves over K with m = deg ϕ. The dual isogeny ϕ̂ : D → C satisfies ϕ̂∘ϕ = [m] and ϕ∘ϕ̂ = [m]. The Weil pairing e_{ϕ} : C[ϕ]×D[ϕ̂] → μ_{m} is defined in [Si1, Exercise 3.15]. We write G_{K} = Gal(\overline{K}/K) and H^{i}(K, −) for H^{i}(G_{K}, −). Taking Galois cohomology of the exact sequence

0 ⟶ C[ϕ] ⟶ C ⟶^{ϕ} D ⟶ 0

we obtain the Kummer exact sequence

C(K) ⟶^{ϕ} D(K) ⟶^{δ_{ϕ}} H^{1}(K, C[ϕ]) ⟶^{ι_{ϕ}} H^{1}(K, C) ⟶^{ϕ} H^{1}(K, D). (2)

For K a number field the Selmer group attached to ϕ is

S^{(ϕ)}(C/K) = { x ∈ H^{1}(K, C[ϕ]) | x_{v} ∈ im δ_{ϕ,v} for all places v }.

Here our convention is that for (∗) a global object we write (∗)_{v} for the corresponding local object. The exact sequence (2) becomes

0 ⟶ D(K)/ϕC(K) ⟶^{δ_{ϕ}} S^{(ϕ)}(C/K) ⟶^{ι_{ϕ}} � (C/K)[ϕ] ⟶ 0 (3)

where � (C/K) = ker (H^{1}(K, C) → ∏_{v} H^{1}(K_{v}, C)) is the Tate-Shafarevich group. The Selmer groups attached to ϕ and ϕ̂ may then be used to give an upper bound for the Mordell-Weil rank.

Let E be an elliptic curve, and let T be a smooth curve of genus 1, both defined over K. We say that T is a torsor under E if there is a simple transitive action E × T → T defined over K. Equivalently there is an isomorphism ψ : T → E defined over \overline{K} such that the cocycle σ(ψ)ψ^{−1} takes values in the translation subgroup of Aut(E). In this way we identify H^{1}(K, E) with the set of torsors T under E, and � (E/K) with those torsors that are everywhere locally soluble. From either point of view it is easy to define a map, sum : Div^{0}(T) → E, which identifies E as the Jacobian of T.

Other notation and conventions

Let K be a number field. We refer to K as a global field, and the completion K_{v} at a place v, as a local field. We abbreviate G_{v} for G_{K_{v}}. If v = 𝔭 is a prime we write ord_{𝔭} : K_{𝔭}^{∗} → Z for the normalised valuation and O_{𝔭} for the ring of integers. An object is said to be unramified if the inertia subgroup I_{𝔭} ⊂ G_{𝔭} acts trivially.

We write ζ = ζ_{m} for a primitive mth root of unity, so that ζ^{ν} runs over μ_{m} as ν runs over Z/mZ. In the case m = 4 we usually write i instead of ζ_{4}. In the case m = 5 we write ϕ = 1 + ζ + ζ^{4} for the golden ratio, and \overline{ϕ} for its conjugate. The minimal polynomials for ζ\overline{ϕ} and ζϕ are

\begin{aligned}f(t) & = & t^{4} + 3t^{3} + 4t^{2} + 2t + 1 & = & ((t + 1)^{5} + 1)/(t + 2) \\ g(t) & = & t^{4} − 2t^{3} + 4t^{2} − 3t + 1 & = & ((t − 1)^{5} + t^{5})/(2t − 1). \end{aligned} (4)

In §2 we define a number of pairings taking values in Q/Z. We write Ind_{ζ} : μ_{m} → Q/Z for the map ζ ↦ 1/m. When making explicit computations we do not hesitate to identify Z/mZ with the subgroup \frac{1}{m}Z/Z ⊂ Q/Z.

Finally, many of our results are stated for an arbitrary point on the modular curve X_{1}(m) or X(m). Even when not explicitly stated, we assume that this point is not a cusp.

1. SOME DESCENT CALCULATIONS

Let K a number field and let m = 3, 4 or 5. We discuss descent by m-isogeny for some elliptic curves parametrised by X_{1}(m) and X(m).

1.1. Elliptic curves parametrised by X_{1}(m)

Let ϕ : C → D be an isogeny of elliptic curves over K with C[ϕ] ≃ μ_{m} and D[ϕ̂] ≃ Z/mZ as Galois modules. We identify H^{1}(K, C[ϕ]) = K^{∗}/K^{∗m} and H^{1}(K, D[ϕ̂]) = Hom(G_{K}, Z/mZ). In order to compute the Selmer groups attached to ϕ and ϕ̂ we must describe the images of the local connecting maps

δ_{v} = δ_{ϕ,v} : D(K_{v}) ⟶ K_{v}^{∗}/K_{v}^{∗m}.  (5)

We consider the elliptic curves D = D_{λ} with Weierstrass equations

\begin{aligned}m = 3 & y^{2} + xy + λy & = & x^{3} \\ m = 4 & y^{2} + xy + λy & = & x^{3} + λx^{2} \\ m = 5 & y^{2} + (1 − λ)xy − λy & = & x^{3} − λx^{2}. \end{aligned}

In each case (x, y) = (0, 0) is a rational point of order m. By [Si1, Exercise 8.13] these are the universal families of elliptic curves over \overline{K} with a specified rational point of order m. Thus our parameter λ is a co-ordinate on X_{1}(m) ≃ P^{1}. For m = 4 or 5 the universal property also holds over K. However for m = 3 we have in effect excluded the infinitely many curves

y^{2} = x^{3} + d^{2}  (6)

above λ = ∞. These curves have complex multiplication by Z[ζ_{3}]. We find the cusps on X_{1}(m) by computing the discriminant Δ(D_{λ}).

Δ(D_{λ})  cusps  η
m = 3 λ^{3}(27λ − 1)  0, 1/27  λ ↦ 1/27 − λ
m = 4 −λ^{4}(16λ − 1)  0, 1/16, ∞ λ ↦ 1/16 − λ
m = 5 λ^{5}(λ^{2} − 11λ − 1) 0, ∞, ϕ^{5}, \overline{ϕ^{5}} λ ↦ (ϕ^{5}λ + 1)/(λ − ϕ^{5})

In each case η is an involution of X_{1}(m), permuting the cusps, such that

(μ_{m} ↪ C_{λ}) ≃ (Z/mZ ↪ D_{η(λ)}) over Q(μ_{m}).  (7)

There is also an action of (Z/mZ)^{∗}/{±1} on X_{1}(m) given by relabelling torsion. For m = 3 or 4 this action is trivial. For m = 5 we obtain an involution

λ ↦ −1/λ.  (8)

Following Vélu [V] the elliptic curves C_{λ} isogenous to D_{λ} have Weierstrass equations

y^{2} + xy + λy = x^{3} − 5λx − λ(7λ + 1)
y^{2} + xy + λy = x^{3} + λx^{2} − 5λ(λ + 1)x + λ(3λ^{2} − 12λ − 1)
y^{2} + (1 − λ)xy − λy = x^{3} − λx^{2} − 5λ(λ^{2} + 2λ − 1)x
−λ(λ^{4} + 10λ^{3} − 5λ^{2} + 15λ − 1).

Lemma 1.1. Let m = 3, 4 or 5. The image of D_{λ}[ϕ̂] ≃ Z/mZ under the connecting map δ = δ_{ϕ} : D_{λ}(K) → K^{∗}/K^{∗m} is generated by λ.

Proof. The connecting map δ is given by a rational function f ∈ K(D_{λ}) with div(f) = m.(0, 0) − m.0. The formal group [Si1, Chapter IV] may be used to resolve the issue of scaling. We find

f(x, y) = \{ \begin{matrix}y & m = 3 \\ −y + x^{2} & m = 4 \\ xy + y − x^{2} & m = 5. \end{matrix}

Computing f on multiples of (0, 0) we obtain powers of λ as claimed.

We describe the image of the local connecting map (5). For v = 𝔭 a prime of bad reduction, the answer depends on which cusp we obtain as the reduction of λ mod 𝔭. We do not cover certain cases where 𝔭|m.

Proposition 1.2. Let m = 3. If ord_{𝔭}(λ) ≥ 0 then

im δ_{𝔭} = \{ \begin{matrix}K_{𝔭}^{∗}/K_{𝔭}^{∗3} & if ord_{𝔭}(λ) > 0 \\ O_{𝔭}^{∗}/O_{𝔭}^{∗3} & if λ(27λ − 1) ≢ 0 (mod 𝔭) \\ 1 & if 27λ − 1 ≡ 0 (mod 𝔭). \end{matrix}

If ord_{𝔭}(λ) < 0 and 𝔭 ∤ 3 then

im δ_{𝔭} = \{ \begin{matrix}O_{𝔭}^{∗}/O_{𝔭}^{∗3} & if 3|ord_{𝔭}(λ) \\ ⟨λ⟩ & otherwise. \end{matrix}

We see that the point λ = ∞, corresponding to the infinitely many curves (6), behaves to some extent as if it were a cusp.

Proposition 1.3. Let m = 4. If ord_{𝔭}(λ) ≥ 0 then

im δ_{𝔭} = \{ \begin{matrix}K_{𝔭}^{∗}/K_{𝔭}^{∗4} & if ord_{𝔭}(λ) > 0 \\ O_{𝔭}^{∗}/O_{𝔭}^{∗4} & if λ(16λ − 1) ≢ 0 (mod 𝔭) \\ 1 or ⟨−4⟩ & if 16λ − 1 ≡ 0 (mod 𝔭). \end{matrix}

If ord_{𝔭}(λ) < 0 and 𝔭 ∤ 2 then

im δ_{𝔭} = \{ \begin{matrix}K_{𝔭}^{∗2}/K_{𝔭}^{∗4} & if λ ∈ K_{𝔭}^{∗2} \\ ⟨λ⟩ & if ord_{𝔭}(λ) is odd \\ ⟨λ, O_{𝔭}^{∗2}⟩ & otherwise. \end{matrix}

Moreover at a real place v, im δ_{v} is trivial if and only if 16λ − 1 > 0.

If μ_{4} ⊂ K_{𝔭} then the inelegant ⟨−4⟩ appearing in Proposition 1.3 vanishes. Indeed −4 = (1 + i)^{4}, so ⟨−4⟩ ⊂ K_{𝔭}^{∗}/K_{𝔭}^{∗4} is trivial.

Proposition 1.4. Let m = 5. Then

im δ_{𝔭} = \{ \begin{matrix}K_{𝔭}^{∗}/K_{𝔭}^{∗5} & if ord_{𝔭}(λ) ≠ 0 \\ O_{𝔭}^{∗}/O_{𝔭}^{∗5} & if λ(λ^{2} − 11λ − 1) ≢ 0 (mod 𝔭) \\ 1 & if λ^{2} − 11λ − 1 ≡ 0 (mod 𝔭) and 𝔭 ∤ 5. \end{matrix}

The automorphism (8) of X_{1}(5) reduces the number of cases to consider both for the statement and for the proof of Proposition 1.4.

Remark. If μ_{m} ⊂ K_{𝔭} then we may identify

H^{1}(K_{𝔭}, C[ϕ]) = H^{1}(K_{𝔭}, D[ϕ̂]) = K_{𝔭}^{∗}/K_{𝔭}^{∗m}.

Tate local duality (see §2.1) tells us that im δ_{ϕ,𝔭} and im δ_{ϕ̂,𝔭} are exact annihilators with respect to the Hilbert norm residue symbol. It is instructive to check that replacing λ by η(λ) in Propositions 1.2–1.4 has the effect of replacing δ_{𝔭} by its exact annihilator.

1.2. Torsors with a diagonal action of μ_{m}.

In this section we prove Propositions 1.2–1.4. Following [F1] we consider certain smooth curves of genus 1 in P^{m−1}. Specifically for τ_{0}, τ_{1}, …, τ_{m−1} non-zero elements of K we define T = T [τ_{0}, τ_{1}, …, τ_{m−1}] ⊂ P^{m−1} via

\begin{matrix}m = 3 & \{ τ_{0}x_{0}^{3} + τ_{1}x_{1}^{3} + τ_{2}x_{2}^{3} − x_{0}x_{1}x_{2} = 0 \} ⊂ P^{2} \\ m = 4 & \{ \begin{matrix}τ_{0}x_{0}^{2} + x_{1}x_{3} − τ_{2}x_{2}^{2} = 0 \\ τ_{1}x_{1}^{2} + x_{0}x_{2} − τ_{3}x_{3}^{2} = 0 \end{matrix} \} ⊂ P^{3} \\ m = 5 & \{ τ_{ν}x_{ν}^{2} + x_{ν−1}x_{ν+1} − τ_{ν−2}τ_{ν+2}x_{ν−2}x_{ν+2} = 0 \} ⊂ P^{4} \end{matrix}

where ν runs over Z/5Z, so that the curve in P^{4} is defined by 5 quadrics. In each case T ⊂ P^{m−1} is a curve of degree m, invariant under the diagonal action of μ_{m} given by x_{ν} ↦ ζ^{ν}x_{ν}. The claims we make about these curves for m = 3 and 4 are easy to check directly. For m = 5 we refer to [F1].

Rescaling our co-ordinates x_{0}, x_{1}, …, x_{m−1} on P^{m−1} we see that the geometry of T [τ_{0}, …, τ_{m−1}] only depends on λ := ∏ τ_{ν}. It may be shown that the elliptic curves C_{λ} introduced in §1.1 have equations

\begin{matrix}m = 3 & T [λ, 1, 1] & 0 = (0 : 1 : −1) \\ m = 4 & T [λ, 1, 1, 1] & 0 = (0 : 1 : 1 : 1) \\ m = 5 & T [λ, 1, 1, 1, 1] & 0 = (0 : 1 : 1 : −1 : −1). \end{matrix}

Thus T [τ_{0}, …, τ_{m−1}] is a smooth curve of genus 1, provided λ = ∏ τ_{ν} is not a cusp of X_{1}(m). At the cusps we obtain collections of lines arranged in an m-gon. For example T [\frac{1}{3}, \frac{1}{3}, \frac{1}{3}] has equation

(x_{0} + x_{1} + x_{2})(x_{0} + ζx_{1} + ζ^{2}x_{2})(x_{0} + ζ^{2}x_{1} + ζx_{2}) = 0.

With reference to the exact sequence (2) we define C_{λ,θ} to be the torsor under C_{λ} described by θ ∈ K^{∗}/K^{∗m}.

Lemma 1.5. Let τ_{0}, …, τ_{m−1} be non-zero elements of K with

∏ τ_{ν} = λ and ∏ τ_{ν}^{ν} ≡ θ mod K^{∗m}.  (9)

Then T [τ_{0}, …, τ_{m−1}] ≃ C_{λ,θ} and T [τ_{0}, …, τ_{m−1}] meets the co-ordinate hyperplane \{x_{ν} = 0\} in m points, each with field of definition K( \sqrt[m]{λ^{−ν}θ}).

Proof. An isomorphism between T [τ_{0}, …, τ_{m−1}] and C_{λ} = T [λ, 1, …, 1] is given by rescaling co-ordinates over \overline{K}. Comparing this isomorphism with its Galois conjugates we obtain a cocycle taking values in μ_{m}. This cocycle corresponds^{1} to θ^{−1} under H^{1}(K, μ_{m}) ≃ K^{∗}/K^{∗m}. The final statement follows by direct calculation.

We outline the proof of Propositions 1.2–1.4. First by (2) we know that θ ∈ im δ_{v} if and only if C_{λ,θ}(K_{v}) ≠ ∅. We also know that im δ_{v} is a group and by Lemma 1.1 it contains λ. These observations reduce us to considering a handful of cases depending on λ and θ. In each case we choose τ_{0}, …, τ_{m−1} satisfying (9). Then Lemma 1.5 provides equations for C_{λ,θ} as a curve in P^{m−1}. If v = 𝔭 is a prime we reduce these equations mod 𝔭 to help us decide whether C_{λ,θ}(K_{𝔭}) ≠ ∅.

We summarise the proof of Proposition 1.2 in a table. We write k for the residue field mod 𝔭 and n for a negative integer. We abbreviate ord_{𝔭}(τ_{ν}) := (ord_{𝔭}(τ_{0}), …, ord_{𝔭}(τ_{m−1})).

Condition on λ ord_{𝔭}(θ)  ord_{𝔭}(τ_{ν})  Reduction mod 𝔭

(a) ord_{𝔭}(λ) > 0 —–

all ≥ 0

rational curves defined over k

(b)  ord_{𝔭}(λ) = 0  1  (−1, 0, 1)  —–
(c)  λ(27λ − 1) ≢ 0  0  (0, 0, 0)

smooth curve of genus 1

(d)  27λ − 1 ≡ 0  0  (0, 0, 0)

3 lines defined over k(\sqrt[3]{θ})

(e)  ord_{𝔭}(λ) = 3n  1  (n−1, n, n+1)  —–
(f)  ord_{𝔭}(λ) = 3n  0  (n, n, n)

smooth curve of genus 1

(g)

\begin{matrix}ord_{𝔭}(λ) = 3n + ε \\ ε = 1 or 2 \end{matrix}
0 (n + ε, n, n)

3 lines defined over k(\sqrt[3]{θ})

In cases (a), (c), (f) we may pick a smooth k-point on the reduction. Hensel’s lemma then shows that C_{λ,θ}(K_{𝔭}) ≠ ∅. In cases (b) and (e) we assume there is a K_{𝔭}-point (x_{0} : x_{1} : x_{2}) with min ord_{𝔭}(x_{ν}) = 0 and proceed to a contradiction by repeated use of the ultrametric law.

1Let us note that C_{λ,θ} and C_{λ,θ^{−1}} are isomorphic as curves, but not as torsors.

In cases (d) and (g) the reduction is a collection of 3 distinct lines each defined over k(\sqrt[3]{θ}). The lines are distinct since by assumption 𝔭 ∤ 3. We claim C_{λ,θ}(K_{𝔭}) ≠ ∅ if and only if θ is a cube, equivalently θ is a cube mod 𝔭. So suppose given a K_{𝔭}-point on C_{λ,θ}. If it reduces to a smooth point, then one of our lines is defined over k, and θ is a cube as required. It remains to consider the singular points of the reduction, i.e. where the three lines meet. In case (d) the lines are arranged in a triangle. If a vertex is k-rational, then so is the opposite side and we are done. In case (g) all 3 lines pass through (1 : 0 : 0). But if the K_{𝔭}-point (x_{0} : x_{1} : x_{2}) has reduction (1 : 0 : 0) then ord_{𝔭}(x_{0}) = 0, ord_{𝔭}(x_{1}) > 0 and ord_{𝔭}(x_{2}) > 0. So

ord_{𝔭}(τ_{1}x_{1}^{3}), ord_{𝔭}(τ_{2}x_{2}^{3}), ord_{𝔭}(x_{0}x_{1}x_{2}) ≥ n + 3

yet ord_{𝔭}(τ_{0}x_{0}^{3}) = n + ε, and this contradicts the ultrametric law. So there are no K_{𝔭}-points above the singular point, and our claim follows as before. This completes the proof of Proposition 1.2.

Again we draw up a table for the proof of Proposition 1.3. We take n, n_{0}, n_{1} negative integers with n = n_{0} + n_{1}.

Condition on λ ord_{𝔭}(θ)  ord_{𝔭}(τ_{ν})  Reduction mod 𝔭

(a) ord_{𝔭}(λ) > 0 —–

all ≥ 0

rational curves defined over k

(b)  ord_{𝔭}(λ) = 0  2  (−2, 0, 0, 2)  —–
(c) λ(16λ − 1) ≢ 0  0  (0, 0, 0, 0)

smooth curve of genus 1

(d)  16λ − 1 ≡ 0  0  (0, 0, 0, 0)

4 lines defined over k(\sqrt[4]{θ})

(e) ord_{𝔭}(λ) = 2n  1  (n − 1, 0, n, 1)  —–
0 (0, n, 0, n)

2 conics defined over k(\sqrt{θ})

(f) ord_{𝔭}(λ) = 2n 2n_{0} (n_{0}, n_{1}, n_{0}, n_{1})

4 lines defined over k(\sqrt{θ}, \sqrt{λθ})

2n (n, 0, n, 0)

2 conics defined over k(\sqrt{λθ})

(g) ord_{𝔭}(λ) = 2n + 1  0  (1, n, 0, n)

4 lines defined over k(\sqrt[4]{θ})

All except cases (d) and (f) go through as before. In these cases our assumptions give 𝔭 ∤ 2. In case (d) the reduction is a collection of 4 lines arranged in a quadrilateral. Each line has field of definition k(\sqrt[4]{θ}). It follows by elementary Galois theory that each vertex has field of definition k(\sqrt[4]{−4θ}). So if C_{λ,θ}(K_{𝔭}) ≠ ∅ then either θ ∈ K_{𝔭}^{∗4} or −4θ ∈ K_{𝔭}^{∗4}. Thus im δ_{𝔭} = 1 or ⟨−4⟩. In fact the theory of the Néron model provides a formula for | im δ_{𝔭}|. By [Sc1, Lemma 3.8] and Tate’s algorithm [Si2, IV.9] we deduce

im δ_{𝔭} = \{ \begin{matrix}1 & if ord_{𝔭}(16λ − 1) is odd \\ ⟨−4⟩ & if ord_{𝔭}(16λ − 1) is even. \end{matrix}

In case (f) we have ord_{𝔭}(λ) = 2n for some n < 0. We write n = n_{0} + n_{1} with n_{0}, n_{1} ≤ 0. We assume that ord_{𝔭}(θ) is even. If θ satisfies one of the following conditions, then our table shows that C_{λ,θ}(K_{𝔭}) ≠ ∅

(i) θ ∈ K_{𝔭}^{∗2} and ord_{𝔭}(θ) ≡ 0 (mod 4) (ii) θ ∈ K_{𝔭}^{∗2} and λθ ∈ K_{𝔭}^{∗2} (iii) λθ ∈ K_{𝔭}^{∗2} and ord_{𝔭}(λθ) ≡ 0 (mod 4).

Conversely, we must show that if C_{λ,θ}(K_{𝔭}) ≠ ∅ then at least one of the conditions (i), (ii) or (iii) is satisfied. We write C_{λ,θ} in the form T [τ_{0}, …, τ_{3}] with ord_{𝔭}(τ_{ν}) = (n_{0}, n_{1}, n_{0}, n_{1}). Replacing θ by λθ if necessary, we may suppose λθ ∉ K_{𝔭}^{∗2} and n_{0} < 0. We take (x_{0} : …: x_{3}) a K_{𝔭}-point with min{ord_{𝔭}(x_{ν})} = 0 and recall the equations

τ_{0}x_{0}^{2} + x_{1}x_{3} − τ_{2}x_{2}^{2} = 0  (10)
τ_{1}x_{1}^{2} + x_{0}x_{2} − τ_{3}x_{3}^{2} = 0.  (11)

Since τ_{0}/τ_{2} ∉ K_{𝔭}^{∗2} the equation (10) tells us ord_{𝔭}(x_{0}), ord_{𝔭}(x_{2}) > 0. Next by (11) we have τ_{1}/τ_{3} ∈ K_{𝔭}^{∗2} and ord_{𝔭}(x_{1}) = ord_{𝔭}(x_{3}) = 0. Again using (10) we see that n_{0} is even. So condition (i) holds and we our done.

To complete the proof of Proposition 1.3, it only remains to consider v a real place. If θ < 0 we may write C_{λ,θ} in the form

\begin{aligned}(16λ − 1)x_{0}^{2} + (x_{0} + x_{2})^{2} + (x_{1} + x_{3})^{2} & = & 0 \\ (16λ − 1)x_{0}^{2} + (x_{0} − x_{2})^{2} − (x_{1} − x_{3})^{2} & = & 0. \end{aligned}

It is readily seen that C_{λ,θ}(R) = ∅ if and only if 16λ − 1 > 0. This is then the condition for im δ_{v} to be trivial.

Finally Proposition 1.4 is a restatement of [F1, Proposition 2.15]. We therefore omit details of the proof, which is in any case similar to the above.

1.3. Elliptic curves parametrised by X(m)

We recall that the K-points of the modular curve Y (m) correspond to isomorphism classes of triples (E, P, Q) where E/K is an elliptic curve, P, Q ∈ E[m], e_{m}(P, Q) = ζ and Q ∈ E(K). For m = 3, 4 or 5 it follows by Lemma 1.1 that X(m) ≃ P^{1}. We make a choice of co-ordinate t on X(m) by writing

m = 3  E_{t} = C_{t^{3}/27}  X(3) has cusps at t = 0, 1, ζ, ζ^{2}
m = 4  E_{t} = C_{t^{4}/16}  X(4) has cusps at t = 0, ∞, ±1, ±i
m = 5  E_{t} = C_{t^{5}}  X(5) has cusps at t = 0, ∞, ζ^{ν}ϕ, ζ^{ν} \overline{ϕ}.

The configuration of these cusps was described in the introduction. The work of §1.2 allows us to identify E_{t} as the Jacobian of X_{t}, where X_{t} has equations

\begin{matrix}m = 3 & \{ t(x_{0}^{3} + x_{1}^{3} + x_{2}^{3}) − 3x_{0}x_{1}x_{2} = 0 \} ⊂ P^{2} \\ m = 4 & \{ \begin{matrix}t(x_{0}^{2} + x_{2}^{2}) + 2x_{1}x_{3} = 0 \\ t(x_{1}^{2} + x_{3}^{2}) + 2x_{0}x_{2} = 0 \end{matrix} \} ⊂ P^{3} \\ m = 5 & \{ tx_{ν}^{2} + x_{ν−1}x_{ν+1} − t^{2}x_{ν−2}x_{ν+2} = 0 \} ⊂ P^{4} \end{matrix}

These curves are taken from [H, Chapter III]. They are invariant under the action of the Heisenberg group, generated by σ_{P} : x_{ν} ↦ ζ^{ν}x_{ν} and σ_{Q} : x_{ν} ↦ x_{ν−1}. We write P_{t}, Q_{t} for the basis of E_{t}[m] determined by σ_{P}, σ_{Q}. Computing the commutator of σ_{P} and σ_{Q} shows that e_{m}(P_{t}, Q_{t}) = ζ. We may therefore take (E_{t}, P_{t}, Q_{t}) as our triple above t. In the cases m = 3 and 5, it is possible to identify E_{t} = X_{t} via 0 = (0 : 1 : −1) and 0 = (0 : t : 1 : −1 : −t) respectively.

Let S = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) and T = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) be the usual generators for SL_{2}(Z).

We use the same letters to denote their images in PSL_{2}(Z/mZ). Writing M^{∗} for (M^{T})^{−1} we have S^{∗} = S and T^{∗} = ST S.

Proposition 1.6. There is an action of PSL_{2}(Z/mZ) on X(m) via

(\begin{matrix}a & b \\ c & d \end{matrix}) : (E, P, Q) ↦ (E, aP + bQ, cP + dQ).

In terms of our co-ordinate t on X(m) it is given by

\begin{matrix}m = 3 & S : t ↦ (−t + 1)/(2t + 1) & T^{∗} : t ↦ ζt \\ m = 4 & S : t ↦ (−t + 1)/(t + 1) & T^{∗} : t ↦ ζt \\ m = 5 & S : t ↦ (ϕt + 1)/(t − ϕ) & T^{∗} : t ↦ ζt. \end{matrix}

Proof. (i) There is an isomorphism X_{t} ≃ X_{ζt} given by

\begin{matrix}(x_{0} : x_{1} : x_{2}) & ↦ & (ζ_{3}x_{0} : x_{1} : x_{2}) \\ (x_{0} : x_{1} : x_{2} : x_{3}) & ↦ & (x_{0} : ζ_{8}x_{1} : −x_{2} : ζ_{8}x_{3}) \\ (x_{0} : x_{1} : x_{2} : x_{3} : x_{4}) & ↦ & (ζ_{5}^{3}x_{0} : ζ_{5}x_{1} : x_{2} : x_{3} : ζ_{5}x_{4}). \end{matrix}

where ζ_{8}^{2} = ζ_{4}. Passing to the Jacobian we obtain an isomorphism (E_{t}, P_{t}, Q_{t}) ≃ (E_{ζt}, P_{ζt}, P_{ζt} + Q_{ζt}). It follows that T^{∗} : t ↦ ζt as claimed. (ii) For suitable t′, an isomorphism X_{t′} ≃ X_{t} is given by

(x_{0} : x_{1} : …: x_{m−1}) ↦ (∑ x_{ν} : ∑ ζ^{ν}x_{ν} : …: ∑ ζ^{(m−1)ν}x_{ν})  (12)

where each sum runs over ν ∈ Z/mZ. Passing to the Jacobian we obtain an isomorphism (E_{t′}, P_{t′}, Q_{t′}) ≃ (E_{t}, −Q_{t}, P_{t}). It follows that S : t ↦ t′.

Finally we compute t′ by substituting (12) into the equations for X_{t}. For m = 3 or 5 a number of short-cuts are available since E_{t} = X_{t}.

Let E = E_{t} for some t ∈ K. Then E[m] ≃ μ_{m} × Z/mZ as a Galois module. We write E′ (respectively E′′) for the elliptic curve isogenous to E obtained as the quotient by μ_{m} (respectively Z/mZ). The isogenies

α : E → E′ and β : E′′ → E

are both of the form ϕ : C_{λ} → D_{λ} but for different values of λ ∈ K. In order to apply Propositions 1.2–1.4 we record these values of λ. In the case m = 5 the answer involves the polynomials f(t) and g(t) defined by (4).

m = 3  m = 4  m = 5
λ 27λ − 1  λ 16λ − 1  λ  λ^{2} − 11λ − 1
α t^{3}/27 t^{3} − 1 t^{4}/16 t^{4} − 1  t^{5} (t^{2} − t − 1)f(t)g(t)
β \frac{t(t^{2}+t+1)}{3(2t+1)^{3}} (\frac{t−1}{2t+1})^{3} \frac{t(t^{2}+1)}{2(t+1)^{4}} (\frac{t−1}{t+1})^{4} tf(t)/g(t) (t^{2} − t − 1)^{5}/g(t)^{2}

The entries for the isogeny α are immediate from the definition of E_{t}. The entries for the isogeny β follow from Proposition 1.6 and the involution η on X_{1}(m) defined in §1.1. For example, taking m = 3 there is an isomorphism

(μ_{3} × Z/3Z ↪ E_{t}) ≃ (Z/3Z × μ_{3} ↪ E_{t′})

where t′ = (−t + 1)/(2t + 1). So the required value of λ is η((t′)^{3}/27) = t(t^{2} + t + 1)/(3(2t + 1)^{3}).

Proposition 1.7. Let m = 3, 4 or 5. Let 𝔭 be a prime of bad reduction for E = E_{t}. Suppose that μ_{m} ⊂ K_{𝔭} and 𝔭 ∤ m. Then t reduces to a cusp mod 𝔭 and the local connecting maps have images

m = 3  \begin{matrix}t mod 𝔭 & 0 & ζ, ζ^{2} & 1 \\ im δ_{α,𝔭} & K_{𝔭}^{∗}/K_{𝔭}^{∗3} & 1 & 1 \\ im δ_{β,𝔭} & K_{𝔭}^{∗}/K_{𝔭}^{∗3} & K_{𝔭}^{∗}/K_{𝔭}^{∗3} & 1 \end{matrix}

m = 4  \begin{matrix}t mod 𝔭 & 0 & ∞ & ±i & −1 & 1 \\ im δ_{α,𝔭} & K_{𝔭}^{∗}/K_{𝔭}^{∗4} & K_{𝔭}^{∗2}/K_{𝔭}^{∗4} & 1 & 1 & 1 \\ im δ_{β,𝔭} & K_{𝔭}^{∗}/K_{𝔭}^{∗4} & K_{𝔭}^{∗}/K_{𝔭}^{∗4} & K_{𝔭}^{∗}/K_{𝔭}^{∗4} & K_{𝔭}^{∗2}/K_{𝔭}^{∗4} & 1 \end{matrix}

m = 5  \begin{matrix}t mod 𝔭 & 0 & ζ^{ν}ϕ, ζ^{ν} \overline{ϕ} (ν ≠ 0) & ϕ, \overline{ϕ} \\ im δ_{α,𝔭} & K_{𝔭}^{∗}/K_{𝔭}^{∗5} & 1 & 1 \\ im δ_{β,𝔭} & K_{𝔭}^{∗}/K_{𝔭}^{∗5} & K_{𝔭}^{∗}/K_{𝔭}^{∗5} & 1 \end{matrix}

Proof. This follows from Propositions 1.2–1.4 and the table above. The assumption μ_{m} ⊂ K_{𝔭} is made purely to simplify the statement of the proposition in the case m = 4.

1.4. Large Tate-Shafarevich groups

Let E = E_{t} as above. There are exact sequences

0 ⟶ S^{(α)}(E/K) ⟶ S^{(m)}(E/K) ⟶^{α} S^{(α̂)}(E′/K)  (13)
0 ⟶ S^{(β̂)}(E/K) ⟶ S^{(m)}(E/K) \underrightarrow{β̂} S^{(β)}(E′′/K).  (14)

Propositions 1.2–1.4 allow us the estimate the Selmer groups attached to α, α̂, β and β̂. We show that for a careful choice of t the group S^{(α̂)}(E′/K) may be large compared to both S^{(β̂)}(E/K) and S^{(β)}(E′′/K). It follows that � (E′/K) may become arbitrarily large. More precisely we prove

Theorem 1. Let K a number field and let m = 3, 4 or 5. Then the Tate-Shafarevich group of an elliptic curve over K may contain arbitrarily many elements of order m.

We stress that our theorem applies to any number field K, for example K = Q. We give details of the proof.

Lemma 1.8. We may choose S_{1} and S_{2} finite disjoint set of primes, and X ⊂ H^{1}(K, Z/mZ) a finite subgroup such that

(i) Each prime 𝔭 ∈ S_{1} ∪ S_{2} satisfies Norm 𝔭 ≡ 1 (mod m). (ii) The map X → ∏_{𝔭∈S_{1}} H^{1}(K_{𝔭}, Z/mZ) is injective. (iii) X is unramified outside S_{2}. (iv) X contains arbitrarily many elements of order m.

Proof. By class field theory there exists L/K an abelian extension whose Galois group contains arbitrarily many elements of order m, and that is unramified outside a finite set of primes S_{2} satisfying (i). We may suppose L ∩ K(μ_{m}) = K and take X = Hom(Gal(L/K), Z/mZ). By the Tchebotarev density theorem there exists a finite set of primes S_{1} satisfying the remaining conditions.

With S_{1}, S_{2} and X as above, we impose congruence conditions on t.

m = 3

t^{2} + t + 1 ≡ 0 (mod 𝔭) for all 𝔭 ∈ S_{1}
t − 1 ≡ 0 (mod 𝔭) for all 𝔭 ∈ S_{2}

m = 4

t^{2} + 1 ≡ 0 (mod 𝔭) for all 𝔭 ∈ S_{1}
t − 1 ≡ 0 (mod 𝔭) for all 𝔭 ∈ S_{2}

m = 5

t^{4} + 3t^{3} + 4t^{2} + 2t + 1 ≡ 0 (mod 𝔭) for all 𝔭 ∈ S_{1}
t^{2} − t − 1 ≡ 0 (mod 𝔭) for all 𝔭 ∈ S_{2}

By Lemma 1.8(i) and the Chinese Remainder Theorem these conditions may be replaced by a single linear congruence. Now let t ∈ K be an algebraic integer satisfying this congruence and write S for the set of primes dividing m and t together with the infinite places. By Dirichlet’s theorem on primes in arithmetic progression (or perhaps a weaker result) we may assume that |S| is bounded by a constant only depending on K.

Propositions 1.2–1.4 and Tate local duality (see §2.1) now give

(a) If 𝔭 ∉ S then im δ_{α,𝔭} ⊂ O_{𝔭}^{∗}/O_{𝔭}^{∗m} and im δ_{α̂,𝔭} ⊃ Hom(G_{𝔭}/I_{𝔭}, Z/mZ).

(b) If 𝔭 ∈ S_{1} ∪ S_{2} then im δ_{α,𝔭} = 1 and im δ_{α̂,𝔭} = Hom(G_{𝔭}, Z/mZ).

(c) If 𝔭 ∈ S_{1} then im δ_{β,𝔭} = K_{𝔭}^{∗}/K_{𝔭}^{∗m} and im δ_{β̂,𝔭} = 0.

(d) If 𝔭 ∈ S_{2} then im δ_{β,𝔭} = 1 and im δ_{β̂,𝔭} = Hom(G_{𝔭}, Z/mZ).

We make two claims, namely that the groups X/(X ∩ S^{(α̂)}(E′/K)) and X ∩ αS^{(m)}(E/K) are bounded by constants only depending on K. By (a),

(b) and Lemma 1.8(iii) we have

X ∩ S^{(α̂)}(E′/K) = { x ∈ X | x_{v} ∈ im δ_{α̂,v} for all places v }
⊃ { x ∈ X | x_{v} = 0 for all v ∈ S }.

Our first claim follows from our assumption on |S|.

Now let Y = α^{−1}X ∩ S^{(m)}(E/K). By (a) we know that S^{(α)}(E/K) is unramified outside S. Then (13) and Lemma 1.8(iii) show that Y is unramified outside S ∪ S_{2}. Next (c) and Lemma 1.8(ii) give S^{(β̂)}(E/K) ∩ X = 0. From the exact sequence (14) we learn that Y injects into S^{(β)}(E′′/K). But (d) tells us that this Selmer group is unramified at all primes in S_{2}. We deduce that Y is unramified outside S. So Y is bounded by a constant only depending on K. Our second claim follows.

Finally the cokernel of the map S^{(m)}(E/K) → S^{(α̂)}(E′/K) injects into � (E′/K) and we have shown that this cokernel may contain arbitrarily many elements of order m. This completes the proof of the theorem.

Remark. To establish the existence of arbitrarily large Selmer groups, a lesser assumption on the torsion is required. In particular for m = 3, 5 or 7 we have X_{1}(m) ≃ P^{1}, and S^{(m)}(E/K) may become arbitrarily large. We refer to [F1], [Kl] or [Sc2] for details. Kloosterman [Kl] also treats the case m = 13 using the fact X_{0}(13) ≃ P^{1}.

2. THE CASSELS-TATE PAIRING

We give a survey of duality results for elliptic curves, and in particular the Cassels-Tate pairing. One reason for including this material is that we could not find a clear reference for our Theorem 3 in the existing literature, although a proof conditional on the finiteness of � appears in the work of Cassels [CaVIII, Corollary to Theorem 1.2]. In a number of proofs it is tacitly assumed that the divisors chosen have disjoint supports. We also leave it to the reader to check that certain a priori infinite sums are in fact finite.

2.1. Local duality theorems

Let K be a local field. Let M be a finite G_{K}-module. We define the Cartier dual M^{∨} = Hom(M, \overline{K}^{∗}). The (local) Tate pairing

H^{1}(K, M) × H^{1}(K, M^{∨}) → Q/Z  (15)

is defined as the cup product map induced by the canonical pairing M × M^{∨} → \overline{K}^{∗} followed by the invariant map Br(K) → Q/Z. As a special case we take ϕ : C → D an isogeny of elliptic curves and M = C[ϕ].

H^{1}(K, C[ϕ]) × H^{1}(K, D[ϕ̂]) → Q/Z  (16)

In terms of cocycles a = {a_{σ}}_{σ} and b = {b_{σ}}_{σ} this pairing is given by

(a, b) ↦ inv{e_{ϕ}(a_{σ}, σb_{τ})}_{σ,τ}

Now let E be an elliptic curve over K. Taking Galois cohomology of the exact sequence

0 → \overline{K}(E)^{∗}/\overline{K}^{∗} ⟶^{div} Div^{0}(E) ⟶^{sum} E → 0

we obtain

H^{1}(K, E) ⟶^{δ} H^{2}(K, \overline{K}(E)^{∗}/\overline{K}^{∗}) ⟶^{div} H^{2}(K, Div^{0}(E)).

There is another local pairing due to Tate

E(K) × H^{1}(K, E) → Q/Z ; (x, y) ↦ −invf(𝔵)  (17)

where sum(𝔵) = x and f = δ(y). Weil reciprocity [Si1, Exercise 2.11] shows that this pairing is well defined. Indeed if 𝔵 = div(g) for some g in \overline{K}(E), then

f(𝔵) = f(div g) = g(div f) = 0.

Proposition 2.1. The Tate pairings (16) and (17) are compatible with the Kummer exact sequence (2). In other words, if a ∈ H^{1}(K, C[ϕ]) and b ∈ H^{1}(K, D[ϕ̂]) with δ_{ϕ} : x ↦ a and ιϕ̂ : b ↦ y then (a, b) = (x, y).

Proof. Let b = {b_{σ}}_{σ} and let 𝔵, 𝔟_{σ} ∈ Div^{0}(D) with sum(𝔵) = x and sum(𝔟_{σ}) = b_{σ}. We choose rational functions f_{σ,τ} ∈ \overline{K}(D) and g_{σ} ∈ \overline{K}(C) with

div(f_{σ,τ}) = σ𝔟_{τ} − 𝔟_{στ} + 𝔟_{σ} div(g_{σ}) = ϕ^{∗}𝔟_{σ}

and scale such that f_{σ,τ} ∘ ϕ = (σg_{τ}) g_{στ}^{−1} g_{σ}. We also choose 𝔵_{1} ∈ Div^{0}(C) with ϕ 𝔵_{1} = 𝔵. Then a = {a_{σ}}_{σ} with a_{σ} = sum(σ𝔵_{1} − 𝔵_{1}). The product of the cocycles

e_{ϕ}(a_{σ}, σb_{τ}) = \frac{σ(g_{τ} 𝔵_{1})}{(σg_{τ}) 𝔵_{1}} and f_{σ,τ} (𝔵) = (f_{σ,τ} ∘ ϕ) 𝔵_{1} = \frac{(σg_{τ}) 𝔵_{1} g_{σ} 𝔵_{1}}{g_{στ} 𝔵_{1}}

is a coboundary. Thus (a, b) = (x, y) as required.

Proposition 2.2. The Tate pairings (15) and (17) are^{2} non-degenerate.

Proof. This is due to Tate [T1], [T2]. See [Si1, §17] for a detailed statement. In fact the non-degeneracy of (15) follows from local class field theory [Se1, XIV]. The non-degeneracy of (17) may then be deduced using Proposition 2.1 and a counting argument [Mi, I.3.2].

Lemma 2.3. Let K be a finite extension of Q_{p}. Let M be an unramified finite G_{K}-module of order prime to p. Then the unramified subgroups are exact annihilators with respect to the Tate pairing (15).

Proof. See [Se2, II.5.5].

The next result is often referred to simply as Tate local duality.

Lemma 2.4. Let K be a local field. Let ϕ : C → D be an isogeny of elliptic curves over K. Then im δ_{ϕ} and im δϕ̂ are exact annihilators with respect to the Tate pairing.

Proof. This follows easily from Propositions 2.1 and 2.2.

We end this section with one global result. For K a number field there is a well known exact sequence

0 ⟶ Br(K) ⟶ ⊕_{v} Br(K_{v}) \underrightarrow{∑ inv_{v}} Q/Z ⟶ 0.  (18)

It follows that all pairings discussed in this section satisfy a product formula. For example if a ∈ H^{1}(K, C[ϕ]) and b ∈ H^{1}(K, D[ϕ̂]) then we have

∑_{v}(a_{v}, b_{v})_{v} = 0  (19)

where (·, ·)_{v} is the Tate pairing (16) at K_{v}.

2.2. Global duality theorems

From now on let K be a number field. Let E/K be an elliptic curve. We review the pairing on � (E/K) due to Cassels [CaIV]. We do not discuss the extension to abelian varieties, due to Tate, details of which may be found in [Mi].

Theorem 2. There is an alternating bilinear pairing

� (E/K) × � (E/K) → Q/Z  (20)

whose kernel is the subgroup of infinitely divisible elements.

We outline two different definitions of the pairing. In the language of [PS] these are the homogeneous space definition and the Weil pairing definition. Both appear in Cassels’ original paper [CaIV].

2In the case K = R or C we must replace E(K) by its group of connected components.

Homogeneous space definition. Let x, y ∈ � (E/K). Then x corresponds to a torsor T under E. There is an exact sequence of G_{K}-modules

0 ⟶ \overline{K}^{∗} ⟶ \overline{K}(T)^{∗} ⟶^{div} Div^{0}(T) ⟶^{sum} E ⟶ 0.

Splitting into short exact sequences and passing to the long exact sequence of Galois cohomology we obtain a diagram, which we consider both in its own right and with K replaced by K_{v}

Br(K) = H^{2}(K, \overline{K}^{∗})
↓^{ι}
H^{2}(K, \overline{K}(T)^{∗})
↓
H^{1}(K, Div^{0}(T)) ⟶^{sum} H^{1}(K, E) ⟶^{δ} H^{2}(K, \overline{K}(T)^{∗}/\overline{K}^{∗})
↓
H^{3}(K, \overline{K}^{∗}) = 0.

We choose f ∈ H^{2}(K, \overline{K}(T)^{∗}) such that f and y have the same image in H^{2}(K, \overline{K}(T)^{∗}/\overline{K}^{∗}). Since y is locally trivial f_{v} = ι(ε_{v}) for some ε_{v} ∈ Br(K_{v}). We define

⟨x, y⟩ = ∑_{v} inv_{v}(ε_{v}).  (21)

Remarks. (i) The global element f exists since H^{3}(K, \overline{K}^{∗}) = 0. Cassels [CaIV, §3] avoids appealing to this fact by relaxing the condition that f is represented by a cocycle. (ii) The definition is independent of the choice of f by (18). (iii) Since T (K_{v}) ≠ ∅ the map ι : Br(K_{v}) → H^{2}(K_{v}, \overline{K}_{v}(T)^{∗}) has a section. It follows that our choice of element ε_{v} is forced. (iv) Linearity in the second argument is clear. For linearity in the first argument we refer to Cassels [CaIV, §3].

(v) The pairing is alternating, i.e. ⟨x, x⟩ = 0 for all x ∈ � (E/K). To see this let T as above and pick P ∈ T (\overline{K}). Then the cocycle σP − P defines an element 𝔵 of H^{1}(K, Div^{0}(T)) with sum(𝔵) = x. The exact row of our diagram shows δ(x) = 0. It follows that ⟨x, x⟩ = 0 as claimed.

The non-degeneracy statement of Theorem 2 follows from

Theorem 3. Let ϕ : C → D be an isogeny of elliptic curves over K. Then there is a map ϕ : � (C/K) → � (D/K), and x ∈ � (D/K) belongs to the image of ϕ if and only if ⟨x, y⟩ = 0 for all y ∈ � (D/K)[ϕ̂].

At this point we need an alternative definition of the pairing.

Weil pairing definition. Let x, y ∈ � (D/K). Suppose that x lifts to an element x_{1} ∈ H^{1}(K, C) with ϕx_{1} = x, and that y ∈ � (D/K)[ϕ̂]. Since x is locally trivial we may choose ξ_{v} ∈ H^{1}(K_{v}, C[ϕ]) such that −ξ_{v} and x_{1} have the same image in H^{1}(K_{v}, C). Let b ∈ S^{(ϕ̂)}(D/K) with ιϕ̂ : b ↦ y. We define

⟨x, y⟩ = ∑_{v}(ξ_{v}, b_{v})_{v}  (22)

where (·, ·)_{v} is the local Tate pairing (16).

Remarks. (i) The definition is independent of the choices of x_{1} and b by the product formulae for the pairings (16) and (17) respectively. (ii) The definition is independent of the choice of ξ_{v} by Tate local duality. (iii) Linearity in both arguments is clear. (iv) If x = ϕx_{1} for some x_{1} ∈ � (C/K) it is clear that ⟨x, y⟩ = 0.

Proposition 2.5. The global pairings (21) and (22) are compatible.

Proof. Let x, x_{1}, y, and b be as in the Weil pairing definition. Then x_{1} and x correspond to torsors T_{1} and T under C and D respectively. There is a commutative diagram

T_{1} ⟶^{ϕ} T
|≀  |≀ ↘
C ⟶^{ϕ} D \underrightarrow{ϕ̂} C

with non-vertical maps defined over K. Let b = {b_{σ}}_{σ} and let 𝔟_{σ} ∈ Div^{0}(T) with sum(𝔟_{σ}) = b_{σ}. We choose rational functions f_{σ,τ} ∈ \overline{K}(T) and g_{σ} ∈ \overline{K}(T_{1}) with

div(f_{σ,τ}) = σ𝔟_{τ} − 𝔟_{στ} + 𝔟_{σ} div(g_{σ}) = ϕ^{∗}𝔟_{σ}

and scale such that f_{σ,τ} ∘ ϕ = (σg_{τ}) g_{στ}^{−1} g_{σ}. At each place v we choose P ∈ T (K_{v}) and P_{1} ∈ T_{1}(\overline{K}_{v}) with ϕ(P_{1}) = P. Then the cocycle σP_{1}−P_{1} defines an element −ξ_{v} in H^{1}(K_{v}, C[ϕ]). Following the proof of Proposition 2.1 the cocycles

e_{ϕ}(σP_{1} − P_{1}, σb_{τ})^{−1} and f_{σ,τ} (P) = (f_{σ,τ} ∘ ϕ) P_{1}

are cobounding. Thus (ξ_{v}, b_{v})_{v} = ε_{v} for all v and the pairings (21) and (22) agree.

For the proof of Theorem 3 we need an approximation theorem in the style of [CaIV, Lemma 6.2]. We temporarily suspend our convention that (∗)_{v} is the local object corresponding to (∗).

Lemma 2.6. Let M be a finite G_{K}-module. For each place v we fix W_{v}(M) ⊂ H^{1}(K_{v}, M) and write W_{v}(M^{∨}) ⊂ H^{1}(K_{v}, M^{∨}) for its exact annihilator with respect to the Tate pairing. Suppose that W_{v}(M) and W_{v}(M^{∨}) are the unramified subgroups for all but finitely many v. Then given ξ_{v} ∈ H^{1}(K_{v}, M) there exists ξ ∈ H^{1}(K, M) with ξ ≡ ξ_{v} mod W_{v}(M) for all v, if and only if

(i) ξ_{v} is unramified for all but finitely many v, (ii) ∑_{v}(ξ_{v}, b_{v})_{v} = 0 for all b ∈ H^{1}(K, M^{∨}) with b_{v} ∈ W_{v}(M^{∨}).

Proof. Necessity of (i) and (ii) is clear, the latter from the product formula. Sufficiency is a consequence of the Cassels-Poitou-Tate exact sequence. We refer to [CS, §1] or [Mi, I.4.10, I.6.15] for details.

We now give the proof of Theorem 3 under the assumption

x ∈ � (D/K) lifts to x_{1} ∈ H^{1}(K, C) with ϕx_{1} = x.  (23)

Under this assumption, x is in the image of ϕ : � (C/K) → � (D/K) if and only if we can find ξ ∈ H^{1}(K, C[ϕ]) such that x_{1} + ι_{ϕ}(ξ) is everywhere locally trivial. We choose local elements ξ_{v} ∈ H^{1}(K_{v}, C[ϕ]) such that −ξ_{v} and x_{1} have the same image in H^{1}(K_{v}, C). Lemma 2.6 with M = C[ϕ] and W_{v}(M) = im δ_{ϕ,v} tells us that the ξ_{v} may be chosen coming from a global element ξ ∈ H^{1}(K, C[ϕ]) if and only if

∑_{v}(ξ_{v}, b_{v})_{v} = 0 for all b ∈ S^{(ϕ̂)}(D/K).  (24)

According to the Weil pairing definition, (24) is precisely the condition ⟨x, y⟩ = 0 for all y ∈ � (D/K)[ϕ̂]. This completes the proof of Theorem 3 under the assumption (23).

The existence of a global lifting (23) is assured if for M = C[ϕ] the natural map

H^{2}(K, M) ⟶ ∏_{v} H^{2}(K_{v}, M) is injective.  (25)

Indeed there is an exact sequence

H^{1}(K, C) ⟶ H^{1}(K, D) ⟶ H^{2}(K, C[ϕ])

and by assumption x is everywhere locally trivial.

Lemma 2.7. Let M = C[ϕ] with either deg ϕ = p or ϕ = [p] for p a prime. Then the Hasse Principle (25) holds.

Proof. First suppose deg ϕ = p. A field extension of degree prime to p reduces us to the case M = μ_{p} and we are done by (18). The case ϕ = [p] is due to Tate [CaIV, Lemma 5.1]. See [Mi, I.9.2] for a more general statement. A counter-example for arbitrary finite M may be found in [Se2, III.4.7].

We now have a proof of Theorem 3 in the case ϕ = [p]. As Cassels [CaIV, §4] explains, this is sufficient to prove Theorem 2. To complete the proof of Theorem 3 we argue by induction on the degree of the isogeny ϕ. We factor ϕ = ϕ_{1}ϕ_{2} with ϕ_{2} satisfying one of the conditions of Lemma 2.7. Say

C ⟶^{ϕ_{2}} E ⟶^{ϕ_{1}} D.

We are given x ∈ � (D/K) with ⟨x, y⟩ = 0 for all y ∈ � (D/K)[ϕ̂]. Since deg ϕ_{1} < deg ϕ our induction hypothesis shows that there exists x_{1} ∈ � (E/K) with ϕ_{1}x_{1} = x. Then by Lemma 2.7 there exists x_{2} ∈ H^{1}(K, C) with ϕ_{2}x_{2} = x_{1}. Thus ϕx_{2} = x. Our earlier proof of Theorem 3 under the assumption (23) now applies.

Lemma 2.8. Let ϕ : C → D be an isogeny of elliptic curves over K. Then the maps ϕ : � (C/K) → � (D/K) and ϕ̂ : � (D/K) → � (C/K) are adjoints with respect to the Cassels-Tate pairing.

Proof. See [CaVIII, §2] where this result is readily deduced from the homogeneous space definition. This lemma gives yet another proof of the easier implication of Theorem 3.

Remark. The Weil pairing definition may be used more generally by relaxing the condition that x_{1} is represented by a cocycle. See [Mi, I.6.9] or [PS, §12.2] for details. If we then check that the definition does not depend on our choice of isogeny ϕ, we may avoid the homogeneous space definition altogether.

Remark. The homogeneous space definition has been used for explicit computations by McGuinness [McG]. The Weil pairing definition has been used by McCallum [McC] and Beaver [Be] in the split torsion case. Variants of the Weil pairing definition have been used by Cassels [CaI], [Ca98] to treat diagonal cubics and 2-descents respectively.

2.3. Split torsion and local pairings

We write the Cassels-Tate pairing in a form expounded by Cassels [CaI], [Ca98], and then recall from [McC] a condition for this pairing to be written as a sum of local pairings.

Proposition 2.9. (i) Let ϕ : C → D be an isogeny of elliptic curves over K with m = deg ϕ. There is an alternating bilinear pairing

S^{(ϕ̂)}(D/K) × S^{(ϕ̂)}(D/K) → Q/Z  (26)

whose kernel is precisely the image of S^{(m)}(C/K). (ii) If the exact sequence of Galois modules 0 → C[ϕ] → C[m] → D[ϕ̂] → 0 splits then the global pairing (26) is a sum of local pairings

⟨·, ·⟩_{ϕ̂,v} : im δ_{ϕ̂,v} × im δ_{ϕ̂,v} → Q/Z.  (27)

Proof. (i) The Cassels-Tate pairing on � (D/K)[ϕ̂] lifts to a pairing on S^{(ϕ̂)}(D/K). The description of the kernel is immediate from Theorem 3. (ii) Let s be a Galois equivariant section of the map ϕ : C[m] → D[ϕ̂]. Then s may be used to make the lifting (23) automatic. Working over the local field K_{v} we drop the subscripts v from our local connecting maps. There is a commutative diagram

\begin{matrix}C(K_{v}) & ⟶^{ϕ} & D(K_{v}) & ⟶^{δ_{ϕ}} & H^{1}(K_{v}, C[ϕ]) \\ || & & ↓ϕ̂ & & ↓^{ι} \\ C(K_{v}) & ⟶^{[m]} & C(K_{v}) & ⟶^{δ_{m}} & H^{1}(K_{v}, C[m]) \\ ↓^{ϕ} & & || & & ↓^{ϕ} \\ D(K_{v}) & \underrightarrow{ϕ̂} & C(K_{v}) & ⟶^{δϕ̂} & H^{1}(K_{v}, D[ϕ̂]). \end{matrix}  (28)

Given a, b ∈ im δϕ̂ we choose P ∈ C(K_{v}) with δϕ̂(P) = a. Then there exists ξ ∈ H^{1}(K_{v}, C[ϕ]) with δ_{m}(P) = ι(ξ) + s(a). We define

⟨a, b⟩_{ϕ̂,v} = (ξ, b)_{v}  (29)

where (·, ·)_{v} is the Tate pairing. By Tate local duality this definition does not depend on the choices of P and ξ. The Weil pairing definition shows that the Cassels-Tate pairing on S^{(ϕ̂)}(D/K) is the sum of these local pairings.

Since im δ_{m} ⊂ H^{1}(K_{v}, C[m]) is an isotropic subspace with respect to the Tate pairing, we find

Lemma 2.10. The local pairings ⟨·, ·⟩_{ϕ̂,v} are skew-symmetric.

Proof. See [McC, Lemma 1.10].

In the setting of Proposition 2.9(ii) we have C[m] = C[ϕ] × sD[ϕ̂]. It is helpful to make some changes to our notation. We write α : E → E′ instead of ϕ : C → D, and β̂ : E → E′′ for the isogeny with kernel sD[ϕ̂]. Then it is natural to identify E[α] = E′′[β] = M (say), E[β̂] = E′[α̂] = M^{∨} and E[m] = M × M^{∨}.

Again we work over the local field K_{v} and drop the subscripts v from our local connecting maps. From the commutative diagram

\begin{matrix}E(K_{v}) & ⟶^{α} & E′(K_{v}) & ⟶^{δ_{α}} & H^{1}(K_{v}, M) \\ ↓β̂ & & ↓α̂ & & || \\ E′′(K_{v}) & ⟶^{β} & E(K_{v}) & ⟶^{δ_{β}} & H^{1}(K_{v}, M) \\ ↓^{δβ̂} & & ↓^{δα̂} \\ H^{1}(K_{v}, M^{∨}) & = & H^{1}(K_{v}, M^{∨}) \end{matrix}  (30)

it is clear that im δ_{α} ⊂ im δ_{β} and im δβ̂ ⊂ im δα̂. The next proposition shows that im δ_{α} = im δ_{β} if and only if im δβ̂ = im δα̂. We call v a switching place if these equalities do not hold. For all but finitely many primes 𝔭 we know that im δ_{α} = im δ_{β} is the unramified subgroup of H^{1}(K_{𝔭}, M). So the set of switching places is finite.

Lemma 2.11. (i) The subgroup im δ_{m} ⊂ H^{1}(K_{v}, E[m]) may be viewed as a relation on H^{1}(K_{v}, M) × H^{1}(K_{v}, M^{∨}). It induces an isomorphism

im δ_{β}/ im δ_{α} ≃ im δα̂/ im δβ̂.  (31)

(ii) The Tate pairing (15) induces a non-degenerate pairing

(·, ·)_{v} : im δ_{β}/ im δ_{α} × im δα̂/ im δβ̂ → Q/Z.

Combining (i) and (ii) we obtain the local pairings of Proposition 2.9. More precisely, if we write both the isomorphism (31) and its inverse as a ↦ \overline{a}, then ⟨a, b⟩_{α̂,v} = (\overline{a}, b)_{v} and ⟨a, b⟩_{β,v} = −(a, \overline{b})_{v}. In particular the switching places are precisely the places for which the local pairings are non-zero.

Proof. (i) The diagram (30) shows

im δ_{β}/ im δ_{α} ≃ \frac{E(K_{v})}{α̂E′(K_{v}) βE′′(K_{v})} ≃ im δα̂/ im δβ̂.

(ii) The non-degeneracy is a consequence of Tate local duality. The description of ⟨·, ·⟩_{α̂,v} follows from the proof of Proposition 2.9 on noting that our identifications suppress the maps ι and s. The description of ⟨·, ·⟩_{β,v} follows by the symmetry properties of the pairings. The final statement is clear.

Remark. By Lemma 2.10 we know that im δ_{β}/ im δ_{α} has order either a square or twice a square. Taking M = μ_{4} and following the proof of Proposition 1.7 (with μ_{4} ⊄ K_{v}) we may give examples of the latter. We deduce that the local pairings need not be alternating.

Following Beaver [Be] we give a formula for the local pairings in terms of the Hilbert norm residue symbol. We suppose

M ≃ M^{∨} ≃ Z/mZ as G_{v}-modules.  (32)

In particular μ_{m} ⊂ K_{v}. We fix P ∈ M and Q ∈ M^{∨} with e_{m}(P, Q) = ζ. The maps x ↦ e_{m}(P, x) and x ↦ e_{m}(Q, x) induce isomorphisms

\begin{aligned}j_{P} : H^{1}(K_{v}, M^{∨}) & ≃ & K_{v}^{∗}/K_{v}^{∗m} \\ j_{Q} : H^{1}(K_{v}, M) & ≃ & K_{v}^{∗}/K_{v}^{∗m}. \end{aligned}

Let a ∈ H^{1}(K_{v}, M) and b ∈ H^{1}(K_{v}, M^{∨}). By [McC, Lemma 2.7]

(a, b)_{v} = − Ind_{ζ}(j_{Q}a, j_{P} b)_{v}  (33)

where (·, ·)_{v} on the left is the Tate pairing and (·, ·)_{v} on the right is the Hilbert norm residue symbol. Under the assumption (32) we have H^{1}(K_{v}, E[m]) = Hom(G_{v}, ⟨P, Q⟩).

Proposition 2.12. Let v be a place of K satisfying (32) and

im δ_{m} = Hom(G_{v}, ⟨rP + sQ⟩)  (34)

for some (r : s) ∈ P^{1}(Z/mZ). Then

(i) im δα̂ = Hom(G_{v}, ⟨sQ⟩) and ⟨sa, sb⟩_{α̂,v} = rs Ind_{ζ}(j_{P} a, j_{P} b)_{v} (ii) im δ_{β} = Hom(G_{v}, ⟨rP ⟩) and ⟨ra, rb⟩_{β,v} = −rs Ind_{ζ}(j_{Q}a, j_{Q}b)_{v}.

Proof. (i) We compute

⟨sa, sb⟩_{α̂,v} = (\overline{sa}, sb)_{v}  by Lemma 2.11
= − Ind_{ζ}(j_{Q}(\overline{sa}), j_{P} (sb))_{v} by (33)
= Ind_{ζ}(j_{P} (ra), j_{P} (sb))_{v}  by (34)
= rs Ind_{ζ}(j_{P} a, j_{P} b)_{v}.

(ii) This follows on interchanging P and Q. Although both pairings are skew symmetric a minus sign is introduced since we have taken e_{m}(P, Q) = ζ.

2.4. A description of the ratio (r : s).

Let m = 3, 4 or 5. Let E/K be an elliptic curve with E[m] ≃ μ_{m} × Z/mZ. In the notation of §1.3 we have E = E_{t} for some t ∈ K. In the notation of §2.3 we have taken M = μ_{m}. We aim to compute the local pairings ⟨·, ·⟩_{α̂,v} and ⟨·, ·⟩_{β,v}.

We treat an important special case, namely when v = 𝔭 is a prime with Norm 𝔭 ≡ 1 (mod m), equivalently 𝔭 ∤ m and μ_{m} ⊂ K_{𝔭}. For these primes our hypothesis (32) is satisfied. In fact E has split multiplicative reduction at 𝔭. As Beaver [Be] explains the Tate parametrisation may then be used to establish (34). Instead we make use of the action of PSL_{2}(Z/mZ) on X(m). Our method gives a simple way of computing the ratio (r : s).

Proposition 2.13. Let (E, P, Q) = (E_{t}, P_{t}, Q_{t}) with t ∈ K. Let 𝔭 be a prime of bad reduction for E with Norm 𝔭 ≡ 1 (mod m). Then hypotheses (32) and (34) are satisfied for the following values of (r : s)

m = 3  \{ \begin{matrix}t ≡ 0 (mod 𝔭) \\ t ≡ ζ^{ν} (mod 𝔭) \end{matrix}  \begin{matrix}(1 : 0) \\ (ν : 1) \end{matrix}

m = 4  \{ \begin{matrix}t ≡ 0 (mod 𝔭) \\ t ≡ i^{ν} (mod 𝔭) \\ t ≡ ∞ (mod 𝔭) \end{matrix}  \begin{matrix}(1 : 0) \\ (ν : 1) \\ (1 : 2) \end{matrix}

m = 5  \{ \begin{matrix}t ≡ 0, ∞ (mod 𝔭) \\ t ≡ ζ^{ν}ϕ, ζ^{ν} \overline{ϕ} (mod 𝔭) \end{matrix}  \begin{matrix}(1 : 0) \\ (ν : 1). \end{matrix}

Proof. Since 𝔭 ∤ m the cusps of X(m) are distinct when we reduce mod 𝔭. We first consider the case t ≡ 0 (mod 𝔭). Proposition 1.7 and Tate local duality give im δ_{α,𝔭} = K_{𝔭}^{∗}/K_{𝔭}^{∗m} and im δ_{α̂,𝔭} = 0. The diagram (28) shows that there is an exact sequence

0 ⟶ im δ_{α,𝔭} ⟶ im δ_{m,𝔭} ⟶ im δ_{α̂,𝔭} ⟶ 0.

We deduce im δ_{m,𝔭} = Hom(G_{𝔭}, ⟨P ⟩). So the hypothesis (34) is satisfied with (r : s) = (1 : 0). We use the action of PSL_{2}(Z/mZ) on X(m) to

extend to the general case. Indeed if (\begin{matrix}a & b \\ c & d \end{matrix}) : t_{1} ↦ t_{2}, then in an obvious

notation

(\begin{matrix}d & −c \\ −b & a \end{matrix}) (\begin{matrix}r_{1} \\ s_{1} \end{matrix}) = (\begin{matrix}r_{2} \\ s_{2} \end{matrix}) .

It only remains to describe the action of PSL_{2}(Z/mZ) on the cusps, and this may be read off from Proposition 1.6.

Remark. The last two propositions give an alternative and perhaps more natural proof of Proposition 1.7.

Corollary 2.14. Let (E, P, Q) = (E_{t}, P_{t}, Q_{t}) with t ∈ K. Let 𝔭 be a prime with Norm 𝔭 ≡ 1 (mod m). Suppose t ≡ ζ^{ν} (mod 𝔭) (for m = 3, 4), respectively t ≡ ζ^{ν}ϕ, ζ^{ν} \overline{ϕ} (mod 𝔭) (for m = 5), for some ν prime to m. Then

⟨a, b⟩_{α̂,𝔭} = ν Ind_{ζ}(j_{P} a, j_{P} b)_{𝔭}  (35)
⟨a, b⟩_{β,𝔭} = −1/ν Ind_{ζ}(j_{Q}a, j_{Q}b)_{𝔭}.  (36)

Proof. This is the case (r : s) = (ν : 1) of the last two propositions.

Remark. Beaver [Be] obtained the formula (36) in the case m = 5. In her notation λ_{v} = 1/ν is computed in terms of the Tate parametrisation.

We give an alternative approach to Corollary 2.14 that avoids both the Tate parametrisation and the action of PSL_{2}(Z/mZ). Putting our equations for E = E_{t} into Weierstrass form and using Lemma 1.1 we blast out the identities

\begin{aligned}j_{P} δα̂(Q) & = & \{ \begin{matrix}ζ(t − ζ)(t − ζ^{2})^{2} & m = 3 \\ −(t − i)(t + 1)^{2}(t + i)^{3} & m = 4 \\ ∏_{ν} ((t − ζ^{ν} \overline{ϕ})/(t − ζ^{ν}ϕ))^{ν} & m = 5 \end{matrix} \\ j_{Q}δ_{β}(Q)^{−1} & = & \{ \begin{matrix}t(t^{2} + t + 1)/3 & m = 3 \\ t(t^{2} + 1)/2 & m = 4 \\ tf(t)/g(t) & m = 5 \end{matrix} \end{aligned}

where f(t) and g(t) are given by (4). For ν as in Corollary 2.14 it follows

ord_{𝔭}(j_{Q}δ_{β}(Q)) ≡ −ν ord_{𝔭}(j_{P} δα̂(Q)) (mod m).  (37)

We check (35) in the case a = δα̂(Q) and ord_{𝔭}(j_{P} b) ≡ 0. Indeed

⟨δα̂(Q), b⟩_{α̂,𝔭} = (δ_{β}(Q), b)_{𝔭}  by Lemma 2.11
= − Ind_{ζ}(j_{Q}δ_{β}(Q), j_{P} b)_{𝔭} by (33)
= ν Ind_{ζ}(j_{P} δα̂(Q), j_{P} b)_{𝔭} by (37).

Similarly we may check (36) in the case ord_{𝔭}(j_{Q}a) ≡ 0 and b = δ_{β}(Q).

⟨a, δ_{β}(Q)⟩_{β,𝔭} = −(a, δα̂(Q))_{𝔭}  by Lemma 2.11
= Ind_{ζ}(j_{Q}a, j_{P} δα̂(Q))_{𝔭}  by (33)
= −1/ν Ind_{ζ}(j_{Q}a, j_{Q}δ_{β}(Q))_{𝔭} by (37).

Under the hypotheses of Corollary 2.14 the pairings ⟨·, ·⟩_{α̂,v} and ⟨·, ·⟩_{β,v} are, at the level of abelian groups, just skew symmetric pairings

(Z/mZ)^{2} × (Z/mZ)^{2} → Q/Z.

Thus for m odd the formulae (35) and (36) may be established by checking at a single pair of non-trivial values. The above calculations do this whenever ord_{𝔭}(j_{Q}δ_{β}(Q)) ≢ 0 (mod m).

2.5. An example without split torsion

We compute the Cassels-Tate pairing for one of the curves found in [F1]. This section and §3 may be read in either order.

Let m = 5 and λ = 100/9. The curve C_{λ} defined in §1.1 is labelled 570L4 in Cremona’s tables [Cr1]. In [F1] we saw

S^{(ϕ)}(C_{λ}/Q) ≃ (Z/5Z)^{3} and S^{(ϕ̂)}(D_{λ}/Q) = 0.

Descent by 2-isogeny [Cr1], or equally an L-value computation, shows that C_{λ}(Q) has rank 0. We were able to deduce � (C_{λ}/Q)[5] ≃ (Z/5Z)^{2}. In this section we prove

Proposition 2.15. Let m = 5 and λ = 100/9. Then S^{(ϕ)}(C_{λ}/Q) ⊂ Q^{∗}/Q^{∗5} is generated by the primes 2, 3 and 5. The Cassels-Tate pairing on this Selmer group is given by

2 3 5 2 0 3 3 3 2 0 3 5 2 2 0

Not only does this give another proof that C_{λ}(Q) has rank 0, but we also learn that � (C_{λ}/Q) contains no element of order 25.

The idea is that we use our work in the split torsion case to give a formula for the Cassels-Tate pairing over a certain subfield F ⊂ Q(D_{λ}[5]). We must therefore explain the relationship between the Cassels-Tate pairing over Q and the Cassels-Tate pairing over F.

Proposition 2.16. Let L/K be a finite extension of number fields. Let E/K be an elliptic curve. Then the restriction map � (E/K) → � (E/L) and the corestriction map � (E/L) → � (E/K) are adjoints with respect to the Cassels-Tate pairing.

Proof. We recall [Se1, VII.7] that corestriction is the map on cohomology corresponding to the norm in dimension 0. Let v be a place of K. For any discrete G_{K}-module A and any q ≥ 0 there is commutative diagram

H^{q}(L, A) ⟶ ⊕_{w|v}H^{q}(L_{w}, A)
↓^{Cor} ↓^{∑ Cor}
H^{q}(K, A) ⟶  H^{q}(K_{v}, A)

Indeed for q = 0 this expresses a well-known property of the norm, and the general case follows by dimension shifting. In particular the corestriction map H^{1}(L, E) → H^{1}(K, E) preserves local triviality. We obtain a map on Tate-Shafarevich groups as claimed.

We give one further preliminary to our calculation. Let w|v be places of L and K and let n = [L_{w} : K_{v}]. There are commutative diagrams

Br(K_{v}) ⟶^{inv} Q/Z  Br(K_{v}) ⟶^{inv} Q/Z
↓^{Res} ↓^{×n} ↑^{Cor} ||
Br(L_{w}) ⟶^{inv} Q/Z  Br(L_{w}) ⟶^{inv} Q/Z.

The diagram on the left is well-known [CF, VI.1]. Since the restriction map Br(K_{v}) → Br(L_{w}) is surjective and Cor ∘ Res = n, the diagram on the right is a formal consequence.

Now let x ∈ � (E/K) and y ∈ � (E/L). Following the homogeneous space definition, x corresponds to a torsor T defined over K. Then Res x corresponds to T viewed as a curve over L. We choose f ∈ H^{2}(L, \overline{K}(T)^{∗}) such that f and y have the same image in H^{2}(L, \overline{K}(T)^{∗}/\overline{K}^{∗}). Writing f_{w} = ι(ε_{w}) we have (Cor f)_{v} = ι ∑_{w|v} Cor ε_{w}. Finally we compute

⟨x, Cor y⟩ = ∑_{v} inv(∑_{w|v} Cor ε_{w}) = ∑_{w} inv(ε_{w}) = ⟨Res x, y⟩.

We return to our numerical example. By Lemma 1.1 and (7) we have Q(D_{λ}[5]) = Q(μ_{5}, \sqrt[5]{η(λ)}). We put τ = \sqrt[5]{η(λ)} and t = (ϕτ + 1)/(τ − ϕ). Then t is a root of tf(t)−λg(t) = 0 where f(t) and g(t) are the quartics (4). We work over F = Q(t), a non-Galois degree 5 subfield of Q(D_{λ}[5]).

Lemma 2.17. Let λ = 100/9. Then

S^{(ϕ)}(C_{λ}/Q) = { θ ∈ Q^{∗}/Q^{∗5} | ord_{p}(θ) ≡ 0 (mod 5) for all p ≠ 2, 3, 5}
S^{(ϕ)}(C_{λ}/F) = { θ ∈ F^{∗}/F^{∗5} | ord_{𝔭}(θ) ≡ 0 (mod 5) for all 𝔭 ∤ 2, 3, 5}.

The Cassels-Tate pairings on these Selmer groups are related via

⟨Norm_{F/Q} x, y⟩_{Q} = ⟨x, y⟩_{F} for x ∈ S^{(ϕ)}(C_{λ}/F), y ∈ S^{(ϕ)}(C_{λ}/Q).

Proof. The description of the Selmer groups is immediate from Proposition 1.4. Let us note that λ^{2} − 11λ − 1 = 19/81 and that O_{𝔭}^{∗}/O_{𝔭}^{∗5} is trivial for 𝔭 | 19. The relationship between the Cassels-Tate pairings is a restatement of Proposition 2.16.

We are fortunate in our example to find that t, t − 1 and t − 2 belong to S^{(ϕ)}(C_{λ}/F) and that their norms generate S^{(ϕ)}(C_{λ}/Q). So it only remains to compute the Cassels-Tate pairing on S^{(ϕ)}(C_{λ}/F). The primes of Q(μ_{5}) above 2, 3 and 5 are (2), (3) and 𝔩 = (1 − ζ_{5}). In each case η(λ) = (ϕ^{5}λ + 1)/(λ−ϕ^{5}) is locally a 5th power. So these primes split in Q(D_{λ}[5])/Q(μ_{5}). We label the primes upstairs

(2) = 𝔓_{0}𝔓_{1} … 𝔓_{4} with τ ≡ ζ^{ν} \overline{ϕ} (mod 𝔓_{ν})
(3) = 𝔔_{0}𝔔_{1} … 𝔔_{4} with τ ≡ ζ^{ν}ϕ (mod 𝔔_{ν})
𝔩 = 𝔏_{0} 𝔏_{1} … 𝔏_{4} with τ ≡ ζ^{ν} \overline{ϕ} (mod 𝔏_{ν}^{2}).

Our treatment of the primes above 5 makes use of [CF, Exercise 2].

Lemma 2.18. Let (E, P, Q) = (E_{t}, P_{t}, Q_{t}) with t as above. The primes 𝔓_{ν}, 𝔔_{ν}, 𝔏_{ν} of Q(D_{λ}[5]) above 2, 3 and 5 satisfy the hypotheses (32) and (34) with (r : s) = (−1 : ν).

Proof. We closely follow the proof of Proposition 2.13. Since (t) = 𝔓_{0}^{2}𝔔_{0}^{−2}𝔏_{0}^{2} the case ν = 0 may be deduced from Proposition 1.4. We then use the action of Gal(Q(D_{λ}[5])/Q(μ_{5})) to extend to the general case. In the notation of Proposition 1.6, S : τ ↦ t. So we obtain (r : s) = (−1 : ν) rather than (ν : 1).

Lemma 2.19. The Cassels-Tate pairing on S^{(ϕ)}(C_{λ}/F) is given by

⟨a, b⟩ = Ind_{ζ}(a, b)_{𝔓_{1}}(a, b)_{𝔔_{1}}(a, b)_{𝔏_{1}}.  (38)

Proof. As in §1.3 we identify ϕ : C_{λ} → D_{λ} with β : E′′ → E. We have already seen

im δ_{β,𝔭} = \{ \begin{matrix}F_{𝔭}^{∗}/F_{𝔭}^{∗5} & if 𝔭|2, 3, 5 \\ O_{𝔭}^{∗}/O_{𝔭}^{∗5} & otherwise. \end{matrix}

The Cassels-Tate pairing is a sum of local pairings ⟨·, ·⟩_{β,𝔭}. By Lemmas 2.10 and 2.11 we need only consider the local pairings at the primes above 2, 3 and 5. We have (2) = 𝔭𝔭′ with 𝔭 = 𝔓_{0} and 𝔭′ = 𝔓_{1}𝔓_{2}𝔓_{3}𝔓_{4}. Since ord_{𝔭}(t) ≠ 0 we know that im δ_{α,𝔭} = F_{𝔭}/F_{𝔭}^{∗5} and so 𝔭 is not a switching prime. On the other hand 𝔭′ splits in Q(D_{λ}[5])/F. By Proposition 2.12(ii) we have

⟨a, b⟩_{β,𝔭′} = ⟨a, b⟩_{β,𝔓_{1}} = Ind_{ζ}(a, b)_{𝔓_{1}}

where in accordance with Lemma 2.17 we have suppressed the map j_{Q}. Repeating for the primes above 3 and 5 we arrive at the formula (38).

Finally we use the congruences

τ ≡ ζ\overline{ϕ} (mod 𝔓_{1}) ⟹ t ≡ ζ^{4}\overline{ϕ} (mod 𝔓_{1})
τ ≡ ζϕ (mod 𝔔_{1}) ⟹ t ≡ ζ^{4}ϕ (mod 𝔔_{1})
τ ≡ ζ\overline{ϕ} (mod 𝔏_{1}^{2}) ⟹ t ≡ ζ^{4}\overline{ϕ} (mod 𝔏_{1}^{6})

to reduce the proof of Proposition 2.15 to a calculation of Hilbert norm residue symbols at the primes of Q(μ_{5}) above 2, 3 and 5. We assume that the reader is familiar with the rules for manipulating these symbols, in particular the product formula, the relationship with the power residue symbol, and Euler’s criterion. A good reference is [CF, Exercises 1 and 2]. We further make the observation that if a ≡ 1 (mod 𝔩^{2}) then (a, b)_{𝔩} = 1 for all b ∈ Z_{5}^{∗}. This observation is useful since ϕ ≡ \overline{ϕ} (mod 𝔩^{2}).

(i) Norm_{F/Q}(t) = 100/9. We have

⟨100/9, b⟩ = Ind_{ζ}(ζ^{4}\overline{ϕ}, b)_{2}(ζ^{4}ϕ, b)_{3}(ζ^{4}\overline{ϕ}, b)_{𝔩}.

Since ζ^{4}\overline{ϕ} and ζ^{4}ϕ are units it follows

⟨100/9, 2⟩ = ⟨100/9, 3⟩ = ⟨100/9, 5⟩ = 0.  (39)

(ii) Norm_{F/Q}(t − 1) = 1/9. We have

⟨1/9, b⟩ = Ind_{ζ}(ζ^{4}\overline{ϕ} − 1, b)_{2}(ζ^{4}ϕ − 1, b)_{3}(ζ^{4}\overline{ϕ} − 1, b)_{𝔩}.

Now ζ^{4}\overline{ϕ} − 1 = −(2 + ζ^{3}) is a prime above 11, whereas ζ^{4}ϕ − 1 = −ζϕ is a unit. We compute

\begin{aligned}⟨1/9, 2⟩ & = & Ind_{ζ}(2/2 + ζ^{3}) & = & 1 \\ ⟨1/9, 3⟩ & = & 0 \\ ⟨1/9, 5⟩ & = & Ind_{ζ}(5/2 + ζ^{3}) & = & 4. \end{aligned}  (40)

(iii) Norm_{F/Q}(t − 2) = 2/9. We have

⟨2/9, b⟩ = Ind_{ζ}(ζ^{4}\overline{ϕ} − 2, b)_{2}(ζ^{4}ϕ − 2, b)_{3}(ζ^{4}\overline{ϕ} − 2, b)_{𝔩}.

Now ζ^{4}\overline{ϕ}−2 = −(3+ζ^{3}) is a prime above 61, whereas ζ^{4}ϕ−2 = −ζϕ(2+ζ^{3}) is a prime above 11. We compute

⟨2/9, 2⟩ = Ind_{ζ}(2/3 + ζ^{3}) = 1
⟨2/9, 3⟩ = Ind_{ζ}(3/2 + ζ^{3}) = 3
⟨2/9, 5⟩ = Ind_{ζ}(5/3 + ζ^{3}) = 2.

(41)

Proposition 2.15 now follows from (39), (40) and (41) together with linearity of ⟨·, ·⟩ in the first argument. In fact our calculations are overkill, since we know ahead of time that ⟨·, ·⟩ is skew-symmetric and that λ = 100/9 lies in the kernel. So it would have sufficed to compute a single non-zero value of the pairing.

Remark. The Hasse norm theorem guarantees that the elements of S^{(ϕ)}(C_{λ}/Q) are norms from F^{∗}/F^{∗5}. However we cannot always expect the elements upstairs to belong to S^{(ϕ)}(C_{λ}/F). So although the above method applies to several more of the examples in [F1], including some with m = 7, we do not claim that it is a general method. Imposing local conditions on the elements upstairs is also undesirable for computations. By comparison with [CaI, Appendix B] we believe that there must be a more general formula for the Cassels-Tate pairing on S^{(ϕ)}(C_{λ}/Q) that does not rely on Proposition 2.16.

3. EXAMPLES OVER Q

Let E/Q be an elliptic curve with E[m] ≃ μ_{m} × Z/mZ as a Galois module. We write

E′′ ⟶^{β} E ⟶^{α} E′

for the isogenies with E[α] = E′′[β] = μ_{m} and E′[α̂] = E[β̂] = Z/mZ. We may estimate the Mordell-Weil rank using either (i) the pair of isogenies α and α̂, or (ii) the pair of isogenies β and β̂, or (iii) the multiplication-by-m map on E. Theorem 4 below makes these estimates explicit in the cases m = 3 and 5.

The reader will notice that we have dropped from discussion the case m = 4. We believe that the analogue of Theorem 4 in that case would be considerably more complicated. Not least, a proper treatment would need to explain the relationship between the Selmer groups attached to many different isogenies. Let us also note that the elliptic curves E/Q with E[4] ≃ μ_{4} × Z/4Z are just rather special cases of (1).

3.1. Statement of the main theorem

Let m = 3 or 5. Let E/Q be an elliptic curve with E[m] ≃ μ_{m} × Z/mZ. Then E = E_{t} for some t ∈ Q. We recall from §1.3 that E_{t} has equations

\begin{matrix}m = 3 & \{ t(x_{0}^{3} + x_{1}^{3} + x_{2}^{3}) − 3x_{0}x_{1}x_{2} = 0 \} ⊂ P^{2} \\ m = 5 & \{ tx_{ν}^{2} + x_{ν−1}x_{ν+1} − t^{2}x_{ν−2}x_{ν+2} = 0 \} ⊂ P^{4} \end{matrix}  (42)

with 0 = (0 : 1 : −1) and 0 = (0 : t : 1 : −1 : −t) respectively. Alternatively, an equation for E_{t} in Weierstrass form is given by substituting λ = t^{3}/27, respectively λ = t^{5}, into the equations for C_{λ} appearing in §1.1.

In the case m = 3 the modular curve X(3) has cusps at t = 0, 1, ζ, ζ^{2}. For t ≠ 0, 1 a rational number we define finite disjoint sets of rational primes

P = { p prime | ord_{p}(t/3) > 0 }
Q = \{ p prime | \begin{matrix}t^{2} + t + 1 ≡ 0 (mod p) and p ≡ 1 (mod 3) \\ or p = 3 and ord_{3}(t + 1) ≠ 0 \end{matrix} \}

R = \{ p prime | \begin{matrix}t − 1 ≡ 0 (mod p) and p ≡ 1 (mod 3) \\ or p = 3 and t ≡ 1 or 4 (mod 9) \end{matrix} \}.

In the case m = 5 the modular curve X(5) has cusps at t = 0, ∞, ζ^{ν}ϕ, ζ^{ν} \overline{ϕ}, where ν runs over Z/5Z. For t ≠ 0 a rational number we define finite disjoint sets of rational primes

P = { p prime | ord_{p}(t) ≠ 0 }
Q = { p prime | f(t)g(t) ≡ 0 (mod p) and p ≡ 1 (mod 5) }
R = \{ p prime | \begin{matrix}t^{2} − t − 1 ≡ 0 (mod p) and p ≡ 1 (mod 5) \\ or p = 5 and t ≡ 3 (mod 5) \end{matrix} \}

where f(t) and g(t) are the polynomials (4).

Each prime p ∈ Q ∪ R satisfies either p ≡ 1 (mod m) or p = m. So we may choose non-trivial characters

χ_{p} : (Z/pZ)^{∗} → Z/mZ, respectively χ_{m} : (Z/m^{2}Z)^{∗} → Z/mZ.

We write [A, B] for the matrix with entries (χ_{q}(p))_{p∈A,q∈B} and define

\begin{matrix}Ξ_{α} = ([P, Q] [P, R]) Ξ_{β} = (\begin{matrix}[P, R] \\ [Q, R] \end{matrix}) \\ Ξ_{m} = (\begin{matrix}0 & [P, Q] & [P, R] \\ −[P, Q]^{T} & [Q, Q] − [Q, Q]^{T} & [Q, R] \\ −[P, R]^{T} & −[Q, R]^{T} & 0 \end{matrix}) . \end{matrix}

It is to be understood that Ξ_{m} is a skew-symmetric matrix, and that the diagonal entries are therefore zero.

Theorem 4. Let m = 3 or 5. Let E/Q be an elliptic curve with E[m] ≃ μ_{m} × Z/mZ. Then E ≃ E_{t} for some t ∈ Q and t determines matrices Ξ_{α}, Ξ_{β} and Ξ_{m} as above.

(i) The Selmer groups attached to α and α̂ are S^{(α)}(E/Q) ≃ ker_{L}(Ξ_{α}) and S^{(α̂)}(E′/Q) ≃ ker_{R}(Ξ_{α}). The corresponding estimate for rank E(Q) is

r_{α} = |P| + |Q| + |R| − 1 − 2 rank(Ξ_{α}).

(ii) The Selmer groups attached to β and β̂ are S^{(β)}(E′′/Q) ≃ ker_{L}(Ξ_{β}) and S^{(β̂)}(E/Q) ≃ ker_{R}(Ξ_{β}). The corresponding estimate for rank E(Q) is

r_{β} = |P| + |Q| + |R| − 1 − 2 rank(Ξ_{β}).

(iii) Suppose that the characters χ_{p} for p ∈ Q are compatible in a sense to be defined below. Then S^{(m)}(E/Q) ≃ ker(Ξ_{m}) and the corresponding estimate for rank E(Q) is

r_{m} = |P| + |Q| + |R| − 1 − rank(Ξ_{m}).

Remarks. (i) The estimates r_{α}, r_{β} and r_{m} are upper bounds which, granted the finiteness of � (E/Q), differ from rank E(Q) by an even integer. Some authors call these Selmer ranks. (ii) The choice of characters χ_{p} does not affect the rank of Ξ_{α} or Ξ_{β}, but may affect the rank of Ξ_{m}. So in computing r_{m} a further condition on the characters is essential. (iii) It is clear both from theory and our explicit recipes that r_{m} ≤ min{r_{α}, r_{β}}. In §3.4 and §3.5 we give some examples where this inequality is strict. (iv) The estimates r_{α}, r_{β} and r_{m} all have the same parity. By Tate’s algorithm [Si2, IV.9] together with [R], [Co] it may be checked that the set of places for which E has root number −1 is P ∪ Q ∪ R ∪ {∞}. The parity result so obtained is a special case of a theorem of Monsky [Mo].

(v) In the case m = 5 we have E_{t} ≃ E_{−1/t}. It is instructive to check that replacing t by −1/t does not alter our estimates r_{α}, r_{β} and r_{m}.

We specify a preferred choice of characters χ_{p} for primes p ∈ Q. By definition of Q we have either p ≡ 1 (mod m) or p = m = 3. If p ≡ 1 (mod m) we may choose^{3} ζ ∈ (Z/pZ)^{∗} an element of order m. According as m = 3 or 5 we have either t ≡ ζ^{ν} (mod p) or t ≡ ζ^{ν}ϕ, ζ^{ν} \overline{ϕ} (mod p) for some ν ∈ (Z/mZ)^{∗}. The character

χ_{p} : (Z/pZ)^{∗} → Z/mZ ; x ↦ x^{(p−1)/m} followed by ζ^{ν} ↦ 1  (43)

is independent of our choice of ζ. If p = m = 3 we define

χ_{3} : (Z/9Z)^{∗} → Z/3Z ; 2 ↦ \{ \begin{matrix}1 & if ord_{3}(t + 1) < 0 \\ 2 & if ord_{3}(t + 1) > 0. \end{matrix}  (44)

The case p = m = 5 does not occur. We say that the characters χ_{p} for p ∈ Q are compatible if they are all chosen to be the same scalar multiple of the characters (43) and (44).

Remark. It would be interesting to understand the relationship between our Theorem 4 and [CSS, Proposition 1.2.3] which express the 2-Selmer group for the curves (1) as the kernel of a (skew-)symmetric pairing.

3It is equivalent to choose a prime 𝔭 of Q(μ_{m}) above p.

3.2. Proof of the main theorem

We begin the proof of Theorem 4 by computing the images of the local connecting maps attached to α : E → E′ and β : E′′ → E. Since m is odd we need only concern ourselves with the finite places.

Proposition 3.1. Let E = E_{t} for some t ∈ Q. Then

im δ_{α,p} = \{ \begin{matrix}Q_{p}^{∗}/Q_{p}^{∗m} & if p ∈ P \\ Z_{p}^{∗}/Z_{p}^{∗m} & if p ∉ P ∪ Q ∪ R \\ 1 & if p ∈ Q ∪ R \end{matrix}

im δ_{β,p} = \{ \begin{matrix}Q_{p}^{∗}/Q_{p}^{∗m} & if p ∈ P ∪ Q \\ Z_{p}^{∗}/Z_{p}^{∗m} & if p ∉ P ∪ Q ∪ R \\ 1 & if p ∈ R. \end{matrix}

Proof. Assume first that p ≠ m. We apply Propositions 1.2 and 1.4 for the two values of λ given in the table of §1.3, noting that Z_{p}/Z_{p}^{∗m} is trivial if p ≢ 1 (mod m). The case p = m = 3 is postponed to §3.3. The remaining details in the case m = 5 are recalled from [F1].

As in §1.3 we take (E, P, Q) = (E_{t}, P_{t}, Q_{t}), so that e_{m}(P, Q) = ζ and Q ∈ E(Q). In the notation of §2.3 we have taken M = μ_{m} generated by P and M^{∨} = Z/mZ generated by Q. The maps x ↦ e_{m}(Q, x) and Q ↦ 1 allow us to identify

H^{1}(Q, M) = H^{1}(Q, μ_{m})  = Q^{∗}/Q^{∗m}
H^{1}(Q, M^{∨}) = H^{1}(Q, Z/mZ) = Hom(G_{Q}, Z/mZ).

(45)

Let A and B be finite sets of rational primes and suppose that B consists of primes p with p ≡ 1 (mod m) or p = m. We define

[A] = { θ ∈ Q^{∗}/Q^{∗m} | ord_{p}(θ) ≡ 0 (mod m) for all p ∉ A }
⟨B⟩ = { χ ∈ Hom(G_{Q}, Z/mZ) | χ_{p} is unramified at all p ∉ B }.

Then [A] has basis A. For each prime p ∈ B we choose p^{∨} ∈ Hom(G_{Q}, Z/mZ) a non-trivial element ramified only at p. By class field theory, specifically the Kronecker-Weber theorem, such an element exists and is unique up to scalars. Now ⟨B⟩ has basis {p^{∨}|p ∈ B}.

By Proposition 3.1 and Tate local duality we have

S^{(α)}(E/Q) = { x ∈ H^{1}(Q, M) | x_{p} ∈ im δ_{α,p} for all primes p }
= { x ∈ [P] | x_{p} = 0 for all p ∈ Q ∪ R }
S^{(α̂)}(E′/Q) = { x ∈ H^{1}(Q, M^{∨}) | x_{p} ∈ im δ_{α̂,p} for all primes p }
= { x ∈ ⟨Q ∪ R⟩ | x_{p} = 0 for all p ∈ P }.

As in [F1, §2.4] we identify S^{(α)}(E/Q) and S^{(α̂)}(E′/Q) as the left and right kernels of a pairing

\begin{matrix}[P] × ⟨Q ∪ R⟩ & → & Z/mZ \\ (x, y) & ↦ & ∑_{p∈Q∪R}(x, y)_{p} \end{matrix}

where (·, ·)_{p} is the Tate pairing. In terms of the bases {p} and {p^{∨}} this pairing is represented by the matrix Ξ_{α}. We note that the arbitrary choice of characters χ_{p} corresponds to the arbitrary choice of elements p^{∨}. Specifically we have taken

χ_{p} : Z_{p}^{∗} → Z/mZ ; x ↦ (x, p^{∨})_{p}.  (46)

Since E(Q)[α] = 0 and E′(Q)[α̂] = Z/mZ it follows from the exact sequence (3) with ϕ = α and ϕ = α̂ that

rank E(Q) + 1 + dim_{m} � (E/Q)[α] + dim_{m} � (E′/Q)[α̂] = |P| + |Q ∪ R| − 2 rank(Ξ_{α}).

This furnishes the estimate r_{α} of Theorem 4(i) and granted Proposition 3.1 the proof of Theorem 4(ii) is entirely analogous. Theorem 4(iii) will follow from

Proposition 3.2. Let m = 3 or 5. Let E = E_{t} for some t ∈ Q and suppose that the characters χ_{p} for p ∈ Q are compatible.

(i) The Cassels-Tate pairing on S^{(α̂)}(E′/Q) is represented by

Ξ′ = − (\begin{matrix}[Q, Q] − [Q, Q]^{T} & [Q, R] \\ −[Q, R]^{T} & 0 \end{matrix}) .

(ii) The Cassels-Tate pairing on S^{(β)}(E′′/Q) is represented by

Ξ′′ = − (\begin{matrix}0 & [P, Q] \\ −[P, Q]^{T} & [Q, Q] − [Q, Q]^{T} \end{matrix} ) .

Proof. (i) We read off from Proposition 3.1 that Q is the set of switching primes. We define some global pairings by summing the local pairings of Proposition 2.9.

\begin{matrix}⟨·, ·⟩α̂ : ⟨Q ∪ R⟩ × ⟨Q ∪ R⟩ → Z/mZ ; & (x, y) ↦ ∑_{p∈Q}⟨x_{p}, y_{p}⟩_{α̂,p} \\ ⟨·, ·⟩_{β} : [P ∪ Q] × [P ∪ Q] → Z/mZ ; & (x, y) ↦ ∑_{p∈Q}⟨x_{p}, y_{p}⟩_{β,p} \end{matrix}

The Cassels-Tate pairings on S^{(α̂)}(E′/Q) and S^{(β)}(E′′/Q) are restrictions of these pairings. For p ∈ Q we now make our choice of global element p^{∨} ∈ H^{1}(Q, M^{∨}) such that locally at p the isomorphism (31) satisfies

H^{1}(Q_{p}, M) → H^{1}(Q_{p}, M^{∨}) ; p ↦ p^{∨}  (47)

Later we will check that this choice means the characters χ_{p} defined by (46) are compatible in the sense of §3.1. Subject to this, we complete the proof of Proposition 3.2 by showing that Ξ′ and Ξ′′ represent the pairings ⟨·, ·⟩α̂ and ⟨·, ·⟩_{β} with respect to the bases {p} and {p^{∨}}.

(i) For p, q ∈ Q distinct primes we compute

⟨p^{∨}, q^{∨}⟩α̂ = ⟨p^{∨}, q^{∨}⟩_{α̂,p} − ⟨q^{∨}, p^{∨}⟩_{α̂,q} by Lemma 2.10
= (p, q^{∨})_{p} − (q, p^{∨})_{q}  by Lemma 2.11 and (47)
= −(p, q^{∨})_{q} + (q, p^{∨})_{p}  by the product formula
= −χ_{q}(p) + χ_{p}(q)  by (46).

The other cases (say with p ∈ Q and q ∈ R) are similar. (ii) For p, q ∈ Q distinct primes we compute

⟨p, q⟩_{β} = −⟨q, p⟩_{β,p} + ⟨p, q⟩_{β,q} by Lemma 2.10
= (q, p^{∨})_{p} − (p, q^{∨})_{q}  by Lemma 2.11 and (47)
= χ_{p}(q) − χ_{q}(p)  by (46).

The other cases are similar.

To explain how Theorem 4(iii) follows from Proposition 3.2 we need a lemma from linear algebra.

Lemma 3.3. Let V be a finite dimensional vector space over a field F. Let V = U ⊕W and write pr : V → W for the projection map. Let ⟨·, ·⟩ be a skew-symmetric pairing on V and suppose that U is an isotropic subspace, i.e. ⟨u, u′⟩ = 0 for all u, u′ ∈ U. Let W_{1} = W ∩ U^{⊥}. Then

pr(V^{⊥}) = W_{1} ∩ W_{1}^{⊥}.

Proof. Let v = u+w ∈ V^{⊥} with u ∈ U and w ∈ W. Then w ∈ W_{1} since U is isotropic, and w ∈ W_{1}^{⊥} since u, v ∈ W_{1}^{⊥}. Thus pr(V^{⊥}) ⊂ W_{1} ∩ W_{1}^{⊥}.

Conversely, given w ∈ W_{1} ∩ W_{1}^{⊥} we seek u ∈ U such that u + w ∈ V^{⊥}. But for any choice of u, we have u + w ∈ U^{⊥}, since U is isotropic and w ∈ W_{1}. Thus it suffices to find u ∈ U such that u + w ∈ W^{⊥}. We write ψ : V → W^{∗} for the linear map induced by ⟨·, ·⟩. We must show ψ(w) ∈ ψ(U). By counting dimensions we have ψ(U) = (W/W_{1})^{∗}. Finally the assumption w ∈ W_{1}^{⊥} implies ψ(w) is trivial on W_{1} and we are done.

We apply Lemma 3.3 with U = [P], W = ⟨Q ∪ R⟩ and ⟨·, ·⟩ the pairing represented by Ξ_{m} with respect to the bases {p} and {p^{∨}}. By Theorem 4(i) we have S^{(α)}(E/Q) = U ∩ V^{⊥} and S^{(α̂)}(E′/Q) = W_{1}. Then Propositions 2.9(i) and 3.2(i) give αS^{(m)}(E/Q) = W_{1}∩W_{1}^{⊥}. So Lemma 3.3 allows us to identify the exact sequences

0 ⟶ S^{(α)}(E/Q) ⟶ S^{(m)}(E/Q) ⟶^{α} S^{(α̂)}(E′/Q)

and

0 ⟶ U ∩ V^{⊥} ⟶ V^{⊥} ⟶^{pr} W_{1}.

In particular S^{(m)}(E/Q) = V^{⊥} = ker(Ξ_{m}). Since E(Q)[m] ≃ Z/mZ it follows from the exact sequence (3) with ϕ = [m] that

rank E(Q) + 1 + dim_{m} � (E/Q)[m] = |P ∪ Q ∪ R| − rank(Ξ_{m}).

This furnishes the estimate r_{m} for rank E(Q).

Remark. Theorem 4(iii) equally follows from Proposition 3.2(ii) and the exact sequence

0 ⟶ S^{(β̂)}(E/Q) ⟶ S^{(m)}(E/Q) \underrightarrow{β̂} S^{(β)}(E′′/Q).

In this case we apply Lemma 3.3 with U = ⟨R⟩ and W = [P ∪ Q].

It remains to examine our choice of characters χ_{p} for p ∈ Q. We recall that either p ≡ 1 (mod m) or p = m = 3. If p ≡ 1 (mod m) then μ_{m} ⊂ Q_{p} and we may apply Corollary 2.14. The identifications (45) suppress the map j_{Q}. For x ∈ Z_{p}^{∗} we compute

χ_{p}(x) = (x, p^{∨})_{p}  by (46)
= −⟨x, p⟩_{β,p}  by Lemma 2.11 and (47)
= 1/ν Ind_{ζ}(x, p)_{p} by Corollary 2.14
= Ind_{ζ^{ν}} (x/p).

But Euler’s criterion tells us (x/p) ≡ x^{(p−1)/m} (mod p). So χ_{p} is the character specified in §3.1. The case p = m = 3 is the subject of the next section.

3.3. Further calculations with p = m = 3

To complete the proof of Proposition 3.1 we must show

Proposition 3.4. Let m = 3 and E = E_{t} as in §1.3. Then

im δ_{α,3} = \{ \begin{matrix}Q_{3}^{∗}/Q_{3}^{∗3} & if t ≡ 0 (mod 9) \\ Z_{3}^{∗}/Z_{3}^{∗3} & if t ≡ 3, 6, 7 (mod 9) \\ 1 & if t ≡ 1, 4 (mod 9) or ord_{3}(t + 1) ≠ 0 \end{matrix}

im δ_{β,3} = \{ \begin{matrix}Q_{3}^{∗}/Q_{3}^{∗3} & if t ≡ 0 (mod 9) or ord_{3}(t + 1) ≠ 0 \\ Z_{3}^{∗}/Z_{3}^{∗3} & if t ≡ 3, 6, 7 (mod 9) \\ 1 & if t ≡ 1, 4 (mod 9). \end{matrix}

Proof. (i) Let λ = t^{3}/27. Then for ord_{3}(t) > 0 our description of im δ_{α,3} follows from Proposition 1.2. We must modify the proof to treat the case ord_{3}(t) ≤ 0. We find im δ_{α,3} ⊂ Z_{3}^{∗}/Z_{3}^{∗3} with equality if and only if the congruence

x_{0}^{3} + 4x_{1}^{3} + 7x_{2}^{3} − 3t^{−1}x_{0}x_{1}x_{2} ≡ 0 (mod 27)

is soluble for x_{0}, x_{1}, x_{2} ∈ Z_{3} not all divisible by 3. Writing x_{ν} = 1 + 3a_{ν} this is equivalent to t ≡ 7 (mod 9). (ii) We first make the claim that if ord_{3}(λ) = −1 then im δ_{ϕ,3} = Q_{3}^{∗}/Q_{3}^{∗3}. For any θ ∈ Q_{3}^{∗}/Q_{3}^{∗3} we may arrange C_{λ,θ} ≃ T [τ_{0}, τ_{1}, τ_{2}] with ord_{p}(τ_{ν}) = (1, −1, −1). So our assertion is that for any b, c ∈ Z_{3}^{∗} the equation

9x_{0}^{3} + bx_{1}^{3} + cx_{2}^{3} − 3x_{0}x_{1}x_{2} = 0

is soluble over Z_{3}. Taking x_{1}, x_{2} = ±1 gives a solution mod 3. We then use Hensel’s lemma to solve for x_{0} ∈ Z_{3}. This proves our claim.

Now let λ = t(t^{2}+t+1)/(3(2t+1)^{3}). If ord_{3}(t) > 0 then ord_{3}(λ) ≥ 0 and our description of im δ_{β,3} follows from Proposition 1.2. If ord_{3}(t + 1) ≠ 0, then ord_{3}(λ) = −1 and we are done by the above claim. It remains to consider the case t ≡ 1 (mod 3). We recall from §1.3 the identity

27λ − 1 = (\frac{t − 1}{2t + 1})^{3}.

If t ≡ 1 or 4 (mod 9) then ord_{3}(27λ − 1) ≠ 0. It follows that λ = τ^{3}/27 for some τ ∈ Q_{3} with either ord_{3}(τ) < 0 or τ ≡ 1 (mod 9). We are done by (i). Finally if t ≡ 7 (mod 9) then 27λ ≡ 2 (mod 3). By case (e) of the proof of Proposition 1.2 we have im δ_{β,3} ⊂ Z_{3}^{∗}/Z_{3}^{∗3}. Then Lemma 1.1 shows that we have equality.

Now suppose that 3 is a switching prime, i.e. ord_{3}(t + 1) ≠ 0. In §3.2 we defined a character

χ_{3} : (Z/9Z)^{∗} → Z/3Z ; x ↦ (x, 3^{∨})_{3}  (48)

where 3^{∨} ∈ H^{1}(Q, Z/3Z) is unramified outside 3 and satisfies (47). To complete the proof of Theorem 4 it remains to give an explicit description of this character. We make use of the explicit calculations at the end of §2.4. We recall

\begin{aligned}j_{P} δα̂(Q) & = & ζ(t − ζ)/(t − ζ^{2}) \\ j_{Q}δ_{β}(Q)^{−1} & = & t(t^{2} + t + 1)/3. \end{aligned}

The identifications (45) suppress the map j_{Q}. Since ⟨·, ·⟩_{β,3} is alternating it is clear that for x ∈ Z_{3}^{∗} we have

⟨x, 3⟩_{β,3} = ⟨x, δ_{β}(Q)⟩_{β,3}.  (49)

Let 𝔭 = (1 − ζ_{3}) be the prime of Q(μ_{3}) above 3. For x ∈ Z_{3}^{∗} we compute

χ_{3}(x) = (x, 3^{∨})_{3}  by (48)
= (x, δα̂(Q))_{3}  by Lemma 2.11 and (49)
= −(x, δα̂(Q))_{𝔭}  since [Q(μ_{3})_{𝔭} : Q_{3}] = 2
= Ind_{ζ}(x, j_{P} δα̂(Q))_{𝔭}  by (33)
= \{ \begin{matrix}Ind_{ζ}(x, ζ)_{𝔭} & if ord_{3}(t + 1) < 0 \\ Ind_{ζ}(x, ζ^{2})_{𝔭} & if ord_{3}(t + 1) > 0. \end{matrix}

But (2, ζ)_{𝔭} = (ζ/2) = ζ. So χ_{3} is the character specified in §3.1. The proof of Theorem 4 is now complete.

3.4. Examples with E[3] ≃ μ_{3} × Z/3Z

We apply Theorem 4 to all elliptic curves E/Q with E[3] ≃ μ_{3} × Z/3Z and conductor N ≤ 10^{4}. With the help of Cremona’s tables [Cr2] we know that there are 264 such curves. All but 4 satisfy one of

(i) � (E/Q)(3) = � (E′/Q)(3) = � (E′′/Q)(3) = 0 (ii) � (E/Q)(3) = � (E′/Q)(3) = 0 and � (E′′/Q)(3) ≃ (Z/3Z)^{2} (iii) � (E/Q)(3) = � (E′′/Q)(3) = 0 and � (E′/Q)(3) ≃ (Z/3Z)^{2}.

We record, against the rank, the frequency with which these possibilities occur.

(i) (ii) (iii) other
rank E(Q) = 0 84 58  5  4
rank E(Q) = 1 103 5  0  0
rank E(Q) = 2  5 0  0  0

The exceptional curves are E = E_{t} for t = 1/9, −1/31, 18/17 and 105/104. Their behaviour is illustrated in Examples 1 and 2 below. Table 1 in §3.6 gives data for the first 40 of the 264 curves.

Example 1. Let E = E_{t} and t = 1/9. Then E, E′, E′′ are labelled 4914N2,1,3 in Cremona’s tables. We have P = ∅, Q = {3, 7, 13} and R = ∅. A compatible choice of characters is

χ_{3} : (Z/9Z)^{∗} → Z/3Z ; x ↦ x^{2} followed by 4 ↦ 1
χ_{7} : (Z/7Z)^{∗} → Z/3Z ; x ↦ x^{2} followed by 4 ↦ 1
χ_{13} : (Z/13Z)^{∗} → Z/3Z ; x ↦ x^{4} followed by 3 ↦ 1.

The matrix Ξ_{3} has entries in Z/3Z

3 7 13 3 0 1 2 7 2 0 2 13 1 1 0

Theorem 4 gives r_{α} = r_{β} = 2 and r_{3} = 0. So E(Q) has rank 0 and

\begin{aligned}� (E/Q)(3) & = & 0 \\ � (E′/Q)(3) & ≃ & (Z/3Z)^{2} \\ � (E′′/Q)(3) & ≃ & (Z/3Z)^{2}. \end{aligned}

Example 2. Let E = E_{t} and t = 18/17. Then E, E′, E′′ are labelled 5514A2,1,3 in Cremona’s tables. We have P = {2, 3}, Q = {919} and R = ∅. But 2^{306} ≡ 3^{306} ≡ 1 (mod 919). So Ξ_{3} is the zero matrix and Theorem 4 gives r_{α} = r_{β} = r_{3} = 2. Cremona’s tables tell us that E(Q) has rank 0. We deduce

� (E′′/Q)[β] = � (E′′/Q)[3] ≃ (Z/3Z)^{2}.

By Proposition 3.2(ii) the Cassels-Tate pairing on this space is trivial. Theorem 3 applied to the multiplication-by-3 map on E′′ then shows that � (E′′/Q) contains an element of order 9.

We give an example beyond the range of Cremona’s tables.

Example 3. Let E = E_{t} and t = 124/167. We have P = {2, 31}, Q = {3, 7, 13, 19, 37} and R = {43}. Making a compatible choice of characters the matrix Ξ_{3} is

2 31 3 7 13 19 37 43
2 0 0 2 2 1 1 1 0
31 0 0 1 1 0 0 0 1
3 1 2 0 2 0 1 2 1
7 1 2 1 0 2 1 0 2
13 2 0 0 1 0 0 1 2
19 2 0 2 2 0 0 2 1
37 2 0 1 0 2 1 0 1
43 0 2 2 1 1 2 2 0

We find r_{α} = 3, r_{β} = 5 and r_{3} = 1. Let λ = t(t^{2} + t + 1)/(3(2t + 1)^{3}) = 2^{2}.31.3^{2}.7.13.19.37/1245^{3} and θ = 3534 = 2.31.3.19. By Lemma 1.5 the torsor C_{λ,θ} has equation

x_{0}^{3} + 3534x_{1}^{3} + 20202x_{2}^{3} − 1245x_{0}x_{1}x_{2} = 0.

A solution is (x_{0} : x_{1} : x_{2}) = (12 : 1 : −1). Thus E(Q) has rank 1 and

\begin{aligned}� (E/Q)(3) & = & 0 \\ � (E′/Q)(3) & ≃ & (Z/3Z)^{2} \\ � (E′′/Q)(3) & ≃ & (Z/3Z)^{4}. \end{aligned}

3.5. Examples with E[5] ≃ μ_{5} × Z/5Z

Cremona’s tables [Cr2] tell us that there are only 3 elliptic curves E/Q with E[5] ≃ μ_{5} × Z/5Z and conductor N ≤ 10^{4}. Instead we search over all E = E_{t} with t = a/b, a, b coprime integers and |a|, |b| ≤ 10^{3}. Table 2 gives data for the first 40 of these curves, ordered by conductor. We give further details for some of the curves on our list, beginning with the example considered by Beaver [Be]. Let us note that E = E_{−5} has non-split multiplicative reduction at p = 29.

Example 4. Let E = E_{t} and t = −5. We have P = {5}, Q = {11, 31, 991} and R = ∅. Beaver made a compatible choice of characters

χ_{11} : (Z/11Z)^{∗} → Z/5Z ; x ↦ x^{2} followed by 5 ↦ 1
χ_{31} : (Z/31Z)^{∗} → Z/5Z ; x ↦ x^{6} followed by 16 ↦ 1
χ_{991} : (Z/991Z)^{∗} → Z/5Z ; x ↦ x^{198} followed by 799 ↦ 1.

The matrix Ξ_{5} has entries in Z/5Z

5 11 31 991
5 0 2 0 2
11 3 0 0 3
31 0 0 0 0
991 3 2 0 0

Theorem 4 gives r_{α} = r_{5} = 1 and r_{β} = 3. An L-value computation, details of which appear in [Be], shows that E(Q) has rank 1. We deduce

� (E/Q)(5) = � (E′/Q)(5) = 0 and � (E′′/Q)(5) ≃ (Z/5Z)^{2}.

As Beaver conjectured the hypothesis “λ_{11} ≠ 0” of [Be, Corollary 1.3] is unnecessary.

Corollary 3.5. Let E = E_{t} with t ∈ Z, ord_{5}(t) > 0, ord_{11}(t) = 0 and t^{2} −t−1 not divisible by any prime congruent to 1 mod 5. Then � (E′′/Q) contains a subgroup isomorphic to (Z/5Z)^{2}.

Proof. Since (t^{2} − t − 1)f(t)g(t) = t^{10} − 11t^{5} − 1 it is clear^{4} that 11 always belongs to one of the sets P, Q or R. Our hypotheses now give 5 ∈ P, 11 ∈ Q and R = ∅. Then Theorem 4(ii) and Proposition 3.2(ii) tell us that the Cassels-Tate pairing on S^{(β)}(E′′/Q) is non-zero. The result follows.

In fact Beaver only considered the case ord_{5}(t) > 0. So the next two examples are new.

Example 5. Let E = E_{t} and t = 7. We have P = {7}, Q = {11, 31, 61, 331} and R = {41}. Making a compatible choice of characters the matrix Ξ_{5} is

7 11 31 61 331 41
7 0 4 3 2 1 1
11 1 0 1 2 2 2
31 2 4 0 2 1 2
61 3 3 3 0 2 1
331 4 3 4 3 0 0
41 4 3 3 4 0 0

Theorem 4 gives r_{α} = r_{β} = 3 and r_{5} = 1. Table 4 in §3.6 exhibits a rational point of infinite order. Thus E(Q) has rank 1 and

\begin{aligned}� (E/Q)(5) & = & 0 \\ � (E′/Q)(5) & ≃ & (Z/5Z)^{2} \\ � (E′′/Q)(5) & ≃ & (Z/5Z)^{2}. \end{aligned}

4Alternatively, applying Hasse’s bounds to the reduction of E at a prime of Q(μ_{5})

above 11, we see that E must have bad reduction at 11.

Example 6. Let E = E_{t} and t = 5/4. We have P = {2, 5}, Q = {521, 4621} and R = {11}. Making a compatible choice of characters the matrix Ξ_{5} is

2 5 521 4621 11
2 0 0 2  2  1
5 0 0 3  3  4
521 3 2 0  4  2
4621 3 2 1  0  0
11 4 1 3  0  0

Theorem 4 gives r_{α} = r_{β} = r_{5} = 2. According to pari L(E, 1) ≠ 0. By the work of Kolyvagin, E(Q) has rank 0. We deduce

� (E′′/Q)[β] = � (E′′/Q)[5] ≃ (Z/5Z)^{2}.

By Proposition 3.2(ii) the Cassels-Tate pairing on this space is trivial. It follows that � (E′′/Q) contains an element of order 25.

3.6. Tables

Tables 1 and 2. We apply Theorem 4 to the first 40 elliptic curves E/Q with E[m] ≃ μ_{m} × Z/mZ, ordered by conductor. In the case m = 3 our list is extracted from Cremona’s tables [Cr2] and so guaranteed to be complete. In the case m = 5 we have checked that there are no gaps within the range of Table 6. Table 1 (m = 3) lists our parameter t, the conductor N, the Cremona labels # for E, E′, E′′, the sets of primes P, Q, R, the estimates r_{α}, r_{β}, r_{3} and finally r = rank E(Q) taken directly from Cremona’s tables. Table 2 (m = 5) has the same column headings, except that we omit the Cremona labels, and the entry r = rank E(Q) remains to be justified below. It is unconditional in all cases except t = −8.

Tables 3 and 4. The curves listed in Table 2 are nearly all beyond the range of Cremona’s tables. We perform some further computations whenever r_{5} > 0. Theorem 4 suggests that C_{λ,θ}(Q) ≠ ∅ for λ = tf(t)/g(t) and certain θ ∈ Q^{∗}/Q^{∗5}. Lemma 1.5 provides equations for C_{λ,θ} in the form T [τ_{0}, …, τ_{4}], and we may arrange for the τ_{ν} to be coprime. We write τ_{ν} = a_{ν}/b_{ν} with a_{ν}, b_{ν} coprime integers and b_{ν} > 0. Putting x_{ν} = b_{ν}X_{ν} we arrive at the equations

{ a_{ν}b_{ν}X_{ν}^{2} + b_{ν−1}b_{ν+1}X_{ν−1}X_{ν+1} − a_{ν−2}a_{ν+2}X_{ν−2}X_{ν+2} = 0 } ⊂ P^{4}

where as usual ν runs over Z/5Z. Table 3 records when a search for Qpoints was successful. Next we use the degree 5 map C_{λ,θ} → D_{λ}, given in [F0, Appendix C], to determine rational points of infinite order on E. We record these solutions to (42) in Table 4, together with their canonical height. In accordance with [Cr1, §3.4] our heights are twice those computed by pari. We note that cyclically permuting the x_{ν} corresponds to addition of 5-torsion. So no essentially new solutions are obtained in this way. Similarly, reversing the order of the x_{ν} corresponds to the map [−1].

We justify the entries r = rank E(Q) in Table 2. In the cases where Table 4 fails to exhibit sufficient points of infinite order, we make use of the following implications due to Wiles, Kolyvagin, Gross and Zagier.

ord_{s=1} L(E, s) = 0 ⟹ rank E(Q) = 0
ord_{s=1} L(E, s) = 1 ⟹ rank E(Q) = 1.

(50)

We also make use of the program bg.gp written by T. Womack [W] that computes L^{(r)}(E, 1) using the Buhler-Gross algorithm.

If r_{5} = 1 then Remark (iv) of §3.1 tells us that ord_{s=1} L(E, s) is odd. Running bg.gp we find L′(E, 1) ≠ 0. By (50) we have r = 1. Our 5descent then gives � (E/Q)(5) = 0, and assuming � (E/Q) = 0, the Birch Swinnerton-Dyer conjecture predicts the minimal height of a rational point of infinite order. To the accuracy of T. Womack’s program this agrees with the heights of the points listed in Table 4.

If r_{5} = 2 then Remark (iv) of §3.1 tells us that ord_{s=1} L(E, s) is even. The case t = 5/4 was treated in Example 6 of §3.5. In the remaining cases, Table 4 lists at least one point of infinite order. By (50) we have L(E, 1) = 0. Running bg.gp we find L′′(E, 1) ≠ 0. Assuming the weak Birch Swinnerton-Dyer conjecture it follows that r = 2. Our 5-descent then gives � (E/Q)(5) = 0, and assuming � (E/Q) = 0, the Birch Swinnerton-Dyer conjecture predicts a value for the regulator

t  −6  −8  −9/2 −12 ; Regulator 12.686 196.322 41.688 76.056

In the cases t = −6, −9/2, −12, Table 4 lists two independent points of infinite order. They generate a subgroup with the predicted regulator. In the case t = −8 we have found only one of the generators. The claim r = 2 remains conditional on either the weak Birch Swinnerton-Dyer conjecture, or equally on the finiteness of � (E/Q)(5). The predicted value of the regulator suggests that we are looking for a second generator whose height is approximately 66.

Tables 5 and 6. Finally we apply Theorem 4 for all rational numbers t of the form a/b with a, b coprime integers and |a|, |b| ≤ 10^{3}. For m = 3 we obtain data for 1 216 765 ≈ 2.10^{6}(6/π^{2}) curves. For m = 5 we ignore repeats of the form t and −1/t, and so obtain data for 608 383 ≈ 10^{6}(6/π^{2}) curves. Tables 5 and 6 give a frequency count for each of the estimates r_{α}, r_{β}, min{r_{α}, r_{β}} and r_{m}. The second number in each column is a percentage. For example, in the case m = 3, we find 4 101 curves with min{r_{α}, r_{β}} = 4. These amount to 4 101/1 216 765 × 100% ≈ 0.34% of the curves considered. We see that the estimate r_{m} is often an improvement on both r_{α} and r_{β}. We also see that r_{β} is usually much larger than r_{α}. It is therefore somewhat perverse that our proof of Theorem 1 works by showing that r_{α} − r_{β} may become arbitrarily large.

Table 1 Elliptic curves E/Q with E[3] ≃ μ_{3} × Z/3Z

t  N  #  P Q R r_{α} r_{β} r_{3} r
−3 14 A1, 4, 3 ∅ {7} ∅ 0 0 0 0
3/5 14 A2, 6, 5 ∅ {7} ∅ 0 0 0 0
3/2 19 A1, 3, 2 ∅ {19} ∅ 0 0 0 0
3 26 A1, 3, 2 ∅ {13} ∅ 0 0 0 0
∞ 27 A1, 3, 2 ∅ {3} ∅ 0 0 0 0
−1/2 27 A3, 4, 1 ∅  ∅ {3} 0 0 0 0
−3/2 35 A1, 3, 2 ∅ {7} ∅ 0 0 0 0
3/4 37 B1, 3, 2 ∅ {37} ∅ 0 0 0 0
−3/5 38 A1, 3, 2 ∅ {19} ∅ 0 0 0 0
−1 54 A1, 3, 2 ∅ {3} ∅ 0 0 0 0
−2 54 B1, 3, 2 {2} ∅ ∅ 0 0 0 0
−3/8 77 B1, 3, 2 ∅ {7} ∅ 0 0 0 0
−3/4 91 B2, 1, 3 ∅ {13} {7} 1 1 1 1
−4/5 126 A3, 5, 1 {2} {7} {3} 0 0 0 0
2/11 126 A4, 6, 2 {2} {7} {3} 0 0 0 0
3/7 158 D1, 3, 2 ∅ {79} ∅ 0 0 0 0
−1/8 171 B2, 3, 1 ∅ {19} {3} 1 1 1 1
6/5 182 B2, 1, 3 {2} {7, 13} ∅ 0 2 0 0
1/2 189 B2, 1, 3 ∅ {3, 7} ∅ 1 1 1 1
1/4 189 C1, 3, 2 ∅ {7} ∅ 0 0 0 0
−2/7 234 E2, 3, 1 {2} {13} {3} 0 0 0 0
6/7 254 A2, 1, 3 {2} {127} ∅ 1 1 1 1
−3/13 278 B1, 3, 2 ∅ {139} ∅ 0 0 0 0
−5/4 315 A2, 3, 1 {5} {7} {3} 0 0 0 0
3/11 326 C1, 3, 2 ∅ {163} ∅ 0 0 0 0
1/10 333 A2, 3, 1 ∅ {37} {3} 1 1 1 1
−8 342 A2, 3, 1 {2} {19} {3} 0 0 0 0
−3/7 370 C1, 3, 2 ∅ {37} ∅ 0 0 0 0
2 378 A2, 1, 3 {2} {3, 7} ∅ 0 2 0 0
−1/5 378 B1, 3, 2 ∅ {7} ∅ 0 0 0 0
4 378 E1, 3, 2 {2} {7} {3} 0 0 0 0
−1/3 378 F2, 1, 3 ∅ {3, 7} ∅ 1 1 1 1
−9/7 402 D2, 1, 3 {3} {67} ∅ 1 1 1 1
6 430 C2, 1, 3 {2} {43} ∅ 1 1 1 1
−6 434 B2, 1, 3 {2} {31} {7} 0 0 0 0
3/8 485 A1, 3, 2 ∅ {97} ∅ 0 0 0 0
9 546 D2, 1, 3 {3} {7, 13} ∅ 0 2 0 0
9/8 651 E2, 1, 3 {3} {7, 31} ∅ 0 2 0 0
−6/5 682 A2, 1, 3 {2} {31} ∅ 1 1 1 1
11/2 693 C2, 3, 1 {11} {7} {3} 0 0 0 0

Table 2 Elliptic curves E/Q with E[5] ≃ μ_{5} × Z/5Z

t  N  P  Q  R r_{α} r_{β} r_{5} r
1  11  ∅  {11}  ∅  0 0 0 0
−2  550 {2}  {11}  {5} 0 0 0 0
2  1 342 {2}  {11, 61}  ∅  0 2 0 0
3  33 825 {3}  {11, 41}  {5} 1 1 1 1
3/2  165 066 {2, 3} {11, 41, 61}  ∅  0 4 0 0
−3  185 163 {3}  {31, 181}  {11} 1 1 1 1
−4  192 698 {2}  {11, 461}  ∅  0 2 0 0
−3/2  861 366 {2, 3}  {31, 421}  {11} 0 2 0 0
−4/3  2 032 734 {2, 3}  {11, 1621}  ∅  1 3 1 1
4  2 074 622 {2}  {181, 521}  {11} 1 1 1 1
4/3  2 097 150 {2, 3} {11, 31, 41}  {5} 1 3 1 1
5/3  20 301 765 {3, 5} {11, 41, 3001}  ∅  0 4 0 0
5  48 656 245 {5} {11, 101, 461}  ∅  1 3 1 1
−5  48 999 995 {5}  {11, 31, 991}  ∅  1 3 1 1
−8/3  68 986 434 {2, 3} {11, 101, 131}  ∅  0 4 0 0
5/2  86 646 010 {2, 5} {421, 1871} {11} 0 2 0 0
−5/2 108 646 010 {2, 5} {11, 151, 211} {31} 1 3 1 1
7/4 257 915 350 {2, 7} {11, 31, 2161} {5} 1 3 1 1
5/4 264 829 510 {2, 5} {521, 4621} {11} 2 2 2 0
−5/3 270 895 515 {3, 5} {11, 211, 251} {31} 1 3 1 1
6 362 283 834 {2, 3} {11, 191, 991} ∅ 0 4 0 0
−6 363 310 266 {2, 3} {11, 31, 61, 71} {41} 2 4 2 2
−7 395 724 175 {7}  {311, 661} {5, 11} 2 0 0 0
8 429 352 550 {2}  {661, 1181} {5, 11} 2 0 0 0
−5/4 439 170 490 {2, 5} {11, 31, 4441} ∅ 0 4 0 0
8/5 624 238 010 {2, 5} {11, 31, 61, 3001} ∅  1 5 1 1
9/4 1 538 513 394 {2, 3} {11, 181, 4441} ∅  0 4 0 0
7 1 976 032 597 {7} {11, 31, 61, 331} {41} 3 3 1 1
−8 2 148 204 542 {2} {11, 491, 2801} {71} 2 2 2 2
7/2 3 871 814 254 {2, 7} {11, 41, 131, 151} {31} 2 4 0 0
−7/2 4 037 464 046 {2, 7} {11, 431, 1031} ∅  0 4 0 0
−9/2 4 209 082 350 {2, 3} {11, 31, 61, 71} {5} 2 4 2 2
7/3 4 987 312 869 {3, 7} {11, 701, 1621} ∅  0 4 0 0
8/3 5 916 563 466 {2, 3} {11, 41, 251, 281} {31} 2 4 0 0
6/5 6 497 983 470 {2, 3, 5} {11, 31, 101, 331} ∅  0 6 0 0
−7/4 6 590 370 094 {2, 7} {11, 41, 71, 241} {61} 2 4 0 0
−12 6 754 920 150 {2, 3} {11, 41, 3221} {5, 31} 2 2 2 2
−7/3 6 874 167 531 {3, 7} {11, 71, 6871} {61} 1 3 1 1
11/3 8 333 929 857 {3, 11} {31, 101, 1021} ∅  0 4 0 0
−10 9 091 909 090 {2, 5} {11, 31, 61, 401} ∅  1 5 1 1

Table 3 Rational points on T [τ_{0}, τ_{1}, τ_{2}, τ_{3}, τ_{4}]

t  (τ_{0}, τ_{1}, τ_{2}, τ_{3}, τ_{4})  (X_{0} : X_{1} : X_{2} : X_{3} : X_{4})
3 (1, 41, 1/11, 1, 3)  (2 : 1 : 1 : 7 : 3)
−3 (3, 1/181, 1, 1, −31)  (7 : 1 : −17 : 2 : −1)
4/3 (1, 4/3, 1/41, 31, 11)  (2 : 21 : 2 : −4 : −4)
−5/2 (1/151, 1/2, 1/11, 1, −1055) (1 : 86 : −7 : 3 : −1)
7/4 (1/4, 2161, 7, 1, 1/341)  (27 : 4 : 14 : −694 : −52)
−5/3 (1, 211/11, 1, 5/251, −1/3) (5666 : 149 : −9811 : −167 : −4863)
−6 (1, 1/31, 22, 1/61, −213)  (17 : 5 : −8 : 1 : −3)
(22/61, 3, 71/31, 1, −1)  (2 : 67 : −1 : 65 : −149)
7 (1, 7, 331, 11, 1/1891)  (5 : 13 : 1 : −27 : −4)
−8 (1/491, 8, 2801, 1/11, −1)  (−4 : 41 : −2 : −12 : 1448)
−9/2 (11/31, 1, 1, 549/71, −1/2) (−9 : 207 : −177 : −1 : 168)
(1/2, 1, 9/2201, 61, −11)  (90 : 519 : −1 : −21 : −9)
−12 (1/121, 1, 9663, 1, −4/41)  (211 : 175 : −3 : −169 : −68)
(3221, 1, 3, 1/121, −4/41)  (26 : 356 : 740 : 514 : −71)

Table 4 Mordell-Weil generators for E = E_{t}

t  (x_{0} : x_{1} : x_{2} : x_{3} : x_{4})  height
3 (6 : 81 : 21 : 31 : 71)  1.134
−3 (189 : −1893 : 1037 : 482 : −2433)  2.641
−4/3  generator not known  33.633
4  generator not known  22.371
4/3 (7502 : −56208 : −46608 : 37942 : 57267)  3.595
5  generator not known  34.142
−5  generator not known  29.241
−5/2 (10505 : −48848 : 39977 : 12630 : −70250)  3.776
7/4 (4118592447 : 31088470672 : 13020915922 : −20707618498 : −27516097648) 8.637
−5/3 (11545293032268586 : −6439289981637105 : −5956542269698375 :
11787905264099415 : −4210746652191867) 14.152
−6 (51 : −891 : 786 : 64 : −2050)  2.244
(331728 : −29317538 : 7117825 : 4046253 : −35387658)  6.202
8/5 generator not known 80.610
7 (175 : −304604 : −47007 : 41867 : 316589)  4.280
−8 (352 : −23168 : 372 : −8101 : 8282)  2.992
generator not known  ≈ 66
−9/2 (275581521 : 525816846 : −1540675614 : 43145111 : −1910013504)  7.362
(1798854480 : −55092369 : 1126546802 : −637931742 : −155814921)  7.471
−12 (29830412804 : −274049792 : −7483139508 : 9909498771 : −157064775)  8.531
(25841586446 : −1537085461 : 121031257722 : −455623920 : −47238156912) 9.028
−7/3  generator not known  70.563
−10  generator not known  108.806

Table 5 Elliptic curves E/Q with E[3] ≃ μ_{3} × Z/3Z

r_{α} r_{β} min{r_{α}, r_{β}} r_{3}
0 247 594 20.35 57 032 4.69 267 633 22.00 378 527 31.11
1 512 138 42.09 213 730 17.57 536 514 44.09 582 220 47.85
2 350 525 28.81 351 275 28.87 336 657 27.67 229 402 18.85
3 95 849 7.88 333 628 27.42 71 844 5.90 26 154 2.15
4 10 266 0.84 191 164 15.71  4 101 0.34  462 0.04
5  387 0.03 60 585 4.98  16 0.00  0 0.00
6  6 0.00  8 918 0.73  0 0.00  0 0.00
7  0 0.00  431 0.04  0 0.00  0 0.00
8  0 0.00  2 0.00  0 0.00  0 0.00

Table 6 Elliptic curves E/Q with E[5] ≃ μ_{5} × Z/5Z

r_{α} r_{β} min{r_{α}, r_{β}} r_{5}
0 84 598 13.91  102 0.02 84 698 13.92 219 047 36.00
1 191 544 31.48  1 128 0.19 192 107 31.58 292 742 48.12
2 174 895 28.75  5 804 0.95 175 815 28.90 84 272 13.85
3 98 506 16.19 19 049 3.13 98 832 16.25 10 943 1.80
4 42 040 6.91 46 730 7.68 41 436 6.81  1 285 0.21
5 13 284 2.18 87 395 14.37 12 524 2.06  93 0.02
6  3 029 0.50 124 089 20.40  2 634 0.43  1 0.00
7  442 0.07 130 931 21.52  314 0.05  0 0.00
8  43 0.01 102 667 16.88  22 0.00  0 0.00
9  2 0.00 58 348 9.59  1 0.00  0 0.00
10  0 0.00 23 838 3.92  0 0.00  0 0.00
11  0 0.00  6 764 1.11  0 0.00  0 0.00
12  0 0.00  1 362 0.22  0 0.00  0 0.00
13  0 0.00  162 0.03  0 0.00  0 0.00
14  0 0.00  13 0.00  0 0.00  0 0.00
15  0 0.00  1 0.00  0 0.00  0 0.00

REFERENCES

[BBBCO] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier,

  pari/gp, a computer algebra package, http://www.parigp-home.de

[Be] C.D. Beaver, 5-torsion in the Shafarevich-Tate group of a family of elliptic curves, J. Number Theory 82 (2000), 25–46.

[Bö] R. Bölling, Die Ordnung der Schafarewitsch-Tate-Gruppe kann beliebig gross werden, Math. Nachr. 67 (1975), 157–179.

[CaI] J.W.S. Cassels, Arithmetic on curves of genus 1, I. On a conjecture of Selmer, J. Reine Angew. Math. 202 (1959), 52–99.

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

[CaVI] J.W.S. Cassels, Arithmetic on curves of genus 1, VI. The Tate-Šafarevič group can be arbitrarily large, J. Reine Angew. Math. 214/215 (1964), 65–70.

[CaVIII] J.W S. Cassels, Arithmetic on curves of genus 1, VIII. On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180–199.

[Ca91] J.W.S. Cassels, Lectures on elliptic curves, LMSST 24, Cambridge University Press, Cambridge, 1991.

[Ca98] J.W.S. Cassels, Second descents for elliptic curves, J. Reine Angew. Math. 494 (1998), 101–127.

[CF] J.W.S. Cassels and A. Fröhlich (Eds.), Algebraic number theory, Academic Press, London, 1967.

[CS] J. Coates and R. Sujatha, Galois cohomology of elliptic curves, TIFR Lect. on Math. 88, Narosa, New Delhi, 2000.

[CSS] J.-L. Colliot-Thélène, A.N. Skorobogatov, H.P.F. Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579–650.

[Co] I. Connell, Calculating root numbers of elliptic curves over Q, Manuscripta Math. 82 (1994), 93–104.

[Cr1] J.E. Cremona, Algorithms for modular elliptic curves, Second edition, Cambridge University Press, Cambridge, 1997.

[Cr2] J.E. Cremona, Modular elliptic curve data for conductors up to 10^{4}.

http://www.maths.nottingham.ac.uk/personal/jec/ftp/data

[F0] T.A. Fisher, On 5 and 7 descents for elliptic curves, PhD thesis, Cambridge University, 2000.

[F1] T.A. Fisher, Some examples of 5 and 7 descent for elliptic curves over Q, J. Eur. Math. Soc. 3 (2001), 169–201.

[H] K. Hulek, Projective geometry of elliptic curves, Astérisque 137 (1986).

[Kl] R. Kloosterman, Elliptic curves with large Selmer groups, Master’s thesis, University of Groningen, 2001.

[Kr] K. Kramer, A family of semistable elliptic curves with large Tate-Shafarevich groups, Proc. Amer. Math. Soc. 89 (1983), 379–386.

[L] F. Lemmermeyer, On Tate-Shafarevich groups of some elliptic curves, Algebraic number theory and Diophantine analysis, de Gruyter, Berlin, 2000.

[McC] W.G. McCallum, On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve, Invent. Math. 93 (1988), 637–666.

[McG] F.O. McGuinness, The Cassels pairing in a family of elliptic curves, Ph.D. Dissertation, Brown University, 1982.

[Mi] J.S. Milne, Arithmetic duality theorems, Perspectives in Mathematics 1, Academic Press, Boston, 1986.

[Mo] P. Monsky, Generalizing the Birch-Stephens theorem, I. Modular curves, Math. Z. 221 (1996), 415–420.

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

[R] D.E. Rohrlich, Variation of the root number in families of elliptic curves, Compositio Math. 87 (1993), 119–151.

[Sc1] E.F. Schaefer, Class groups and Selmer groups, J. Number Theory 56 (1996), 79–114.

[Sc2] E.F. Schaefer, Can the 5-part of the Shafarevich-Tate group of an elliptic curve get arbitrarily large?

http://math.scu.edu/∼eschaefe/nt.html

[Se1] J.-P. Serre, Local fields, GTM 67, Springer-Verlag, New York-Berlin, 1979.

[Se2] J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997.

[Si1] J.H. Silverman, The arithmetic of elliptic curves, GTM 106, Springer-Verlag, New York, 1986.

[Si2] J.H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151, Springer-Verlag, New York, 1994.

[T1] J. Tate, W C-groups over 𝔭-adic fields, Séminaire Bourbaki, Exposé 156, Paris, 1958.

[T2] J. Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians, Stockholm 1962, 288–295.

[V] J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris, Sér. A-B 273 (1971), A238–A241.

[W] T. Womack, bg.gp, An implementation of the Buhler-Gross algorithm for computing L-series derivatives. Repaired and improved by J.E.

Cremona. tom@womack.net