VISUALISING ELEMENTS OF ORDER 7 IN THE TATE-SHAFAREVICH GROUP OF AN ELLIPTIC CURVE

TOM FISHER

Abstract. We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order 7 in their Tate-Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian threefold.

1. Introduction

Let E/ℚ be an elliptic curve. Two groups of particular arithmetic interest associated to E are the Mordell-Weil group E(ℚ) and the Tate-Shafarevich group � (E/ℚ). Whilst the study of these groups is intimately related, it seems much easier to write down elements of the first group than the second. In an attempt to remedy this, Mazur [M] suggested visualizing elements of � (E/ℚ) as cosets of E inside some larger abelian variety.

We recall some basic definitions. A torsor (or principal homogeneous space) under E is a pair (C, μ) where C is a genus one curve, and μ : E × C → C is a morphism inducing a simply transitive action on \overline{ℚ}-points. We may view C as a twist of E by a cocycle taking values in E (acting on itself by translations) and so as an element of the Weil-Châtelet group H^{1}(ℚ, E). The identity in this group corresponds to the torsors with a ℚ-point, equivalently those that are isomorphic to E. The Tate-Shafarevich group � (E/ℚ) is the subgroup consisting of torsors that are everywhere locally soluble, i.e. have a ℚ_{v}-point for all places v.

Let ι : E → A be an inclusion of abelian varieties. If C is the twist of E by ξ ∈ H^{1}(ℚ, E), and V is the twist of A by ι_{∗}(ξ) ∈ H^{1}(ℚ, A), then there is a natural inclusion C → V. One might say that C is visible in V. Accordingly, the subgroup of H^{1}(ℚ, E) visible in A is

Vis_{A} H^{1}(ℚ, E) = ker (H^{1}(ℚ, E) ⟶^{ι_{∗}} H^{1}(ℚ, A)) .

The visibility dimension of ξ ∈ H^{1}(ℚ, E) is the least dimension of an abelian variety A such that ξ ∈ Vis_{A} H^{1}(ℚ, E).

To construct a suitable abelian variety A, we usually start with an abelian variety F/ℚ chosen so that E and F have a common finite Galois submodule Δ.

Date: 23rd May 2016.

We then take A = (E × F)/Δ where the quotient is by the diagonal embedding of Δ.

Cremona and Mazur [CM], [AS2] gave some examples of elliptic curves E/ℚ and elements of order n ∈ {2, 3, 4, 5} in � (E/ℚ) that are visible in an abelian surface. For this they take F to be a second elliptic curve (often of the same conductor as E) with E[n] ≅ F [n] as Galois modules. An argument using restriction of scalars (see [AS1, Proposition 2.4]) shows that if ξ ∈ � (E/ℚ) has order n then it has visibility dimension at most n. Mazur [M] showed that elements of order 3 in � (E/ℚ) are always visible in an abelian surface. In [F3] we gave some examples of elements of orders 6 and 7 that are not visible in an abelian surface.

In this paper we give examples of elements of � (E/ℚ) of order 7 that are visible in an abelian surface or abelian threefold. Continuing the work of Cremona and Mazur, our data shows that, at least for curves of small conductor, the visibility dimension is often much smaller than the bound coming from restriction of scalars.

Theorem 1.1. (i) There are 222 isogeny classes of elliptic curves E/ℚ with conductor N_{E} < 10^{5}, that do not admit a rational 7-isogeny, but are predicted by the Birch–Swinnerton-Dyer conjecture to have � (E/ℚ)[7] ≠ 0.

(ii) Of these examples, at least 79 are explained by visibility in an abelian surface, and at least a further 14 are explained by visibility in an abelian threefold.

The list in Theorem 1.1(i) is taken from Cremona’s tables [C]. In each case rank E(ℚ) = 0 and the Birch–Swinnerton-Dyer conjecture predicts that � (E/ℚ) has order (7m)^{2} for some integer m coprime to 7. We chose to investigate the following example in greater detail.

Theorem 1.2. Let E/ℚ be the elliptic curve

67080r y^{2} = x^{3} − x^{2} + 782367544x + 10114340277756.

Then E(ℚ) ≅ ℤ/2ℤ and � (E/ℚ) ≅ (ℤ/7ℤ)^{2}. Moreover

(i) Every element of � (E/ℚ) is visible in an abelian threefold isogenous to E ×Jac(C) where C/ℚ is the genus 2 curve y^{2} = x(x+4)(x^{4} +2x^{3} −x−3).

(ii) Conditional on the Generalised Riemann Hypothesis, none of the non-zero elements of � (E/ℚ) are visible in an abelian surface.

The computer calculations in support of this work were carried out using Magma [BCP], Pari/GP [P] and Sage [S+].

2. Background on visibility

In this section we review some basic facts about visibility. References for this include [CM], [AS1], [AS2].

Let E and F be abelian varieties over a number field K, with common finite Galois submodule Δ. Let A = (E × F)/Δ, where the quotient is by the diagonal embedding of Δ. Let F′ = A/E and E′ = A/F. There is then a commutative diagram

\xymatrix{ &  & 0 \ar[d] \\  &  & F \ar[d] \ar[dr]^{ψ} \\ 0 \ar[r] & E \ar[r] \ar[dr]_{ϕ} & A \ar[r] \ar[d] & F′ \ar[r] & 0 \\  &  & E′ \ar[d] \\  &  & 0}

where the row and column are exact sequences of abelian varieties, and the diagonal maps ϕ and ψ are isogenies with kernel Δ.

There is a commutative diagram with exact rows

[figure: 0; Δ; F; ψ; F′; 0; 0; E; A; F′; 0.]

Taking the long exact sequence of Galois cohomology gives

(2.1)  [figure: F (K); ψ; F′(K); H^{1}(K, Δ); E(K); A(K); F′(K); π; H^{1}(K, E); H^{1}(K, A).]

It follows by a diagram chase that there are short exact sequences

0 → \frac{A(K)}{E(K) + F (K)} → \frac{F′(K)}{ψF (K)} → Vis_{A} H^{1}(K, E) → 0

and (swapping the roles of E and F)

0 → \frac{A(K)}{E(K) + F (K)} → \frac{E′(K)}{ϕE(K)} → Vis_{A} H^{1}(K, F) → 0.

These two exact sequences immediately give the following lemma.

Lemma 2.1. If E′(K)/ϕE(K) = 0 then F′(K)/ψF (K) ≅ Vis_{A} H^{1}(K, E).

Next we give some local conditions under which the visible subgroup of H^{1}(K, E) is actually a subgroup of � (E/K). Let d = |Δ| be the degree of ϕ and ψ.

Theorem 2.2. If E′(K)/ϕE(K) = 0 and (i) all the Tamagawa numbers of E and F are coprime to d, and (ii) A has good reduction at all places v | d, and (iii) writing p for the rational prime below v, e(K_{v}/ℚ_{p}) < p − 1 for all places

v | d, then F′(K)/ψF (K) ≅ Vis_{A} � (E/K).

Proof. We write A^{0}(K_{v}) for the subgroup of A(K_{v}) consisting of points whose reduction mod v belong to the identity component of the special fibre of the Néron model. The following three facts are established in [AS1].

(1) The unramified subgroup of H^{1}(K_{v}, E) has order equal to the Tamagawa number c_{v}(E) = [E(K_{v}) : E^{0}(K_{v})].

(2) If v ∤d then ψ : F^{0}(K_{v}^{nr}) → F^{′0}(K_{v}^{nr}) is surjective.

(3) If A has good reduction at v and e(K_{v}/ℚ_{p}) < p−1 then A(K_{v}^{nr}) → F′(K_{v}^{nr}) is surjective.

We consider the diagram (2.1) with K replaced by K_{v}. To prove the theorem it suffices to show that π_{v} : F′(K_{v}) → H^{1}(K_{v}, E) is the zero map for all places v of the number field K. It is clear that the image of π_{v} is killed by multiplication by d. So by fact (1) and our hypothesis on the Tamagawa numbers of E, it suffices to show that every element in the image of π_{v} is unramified. This follows by fact (2) if v ∤d and by fact (3) if v | d.

Finally we note that by hypothesis (iii), d is odd, and so there are no local conditions to check at the infinite places. □

3. Visibility in abelian surfaces

In this section we give some examples of elements of order 7 in � (E/ℚ) that are visible in an abelian surface.

Elliptic curves E and F are said to be n-congruent if E[n] ≅ F [n] as Galois modules. We write X_{E}(n), respectively X_{E}^{−}(n), for the twist of the modular curve X(n) whose non-cuspidal points parametrise the elliptic curves n-congruent to E via an isomorphism of n-torsion subgroups that preserves, respectively reverses, the Weil pairing. In the case n = 7 these curves are twists of the Klein quartic X(7) = {x^{3}y + y^{3}z + z^{3}x = 0} ⊂ ℙ^{2}. Since each element of (ℤ/7ℤ)^{×} is plus or minus a square, each elliptic curve 7-congruent to E corresponds to a rational point on either X_{E}(7) or X_{E}^{−}(7).

Theorem 3.1. [HK, Théorème 2.1] Let E be an elliptic curve with Weierstrass equation y^{2} = x^{3} + ax + b. Then X_{E}(7) ⊂ ℙ^{2} has equation

ax^{4}+7bx^{3}z+3x^{2}y^{2}−3a^{2}x^{2}z^{2}−6bxyz^{2}−5abxz^{3}+2y^{3}z+3ay^{2}z^{2}+2a^{2}yz^{3}−4b^{2}z^{4} = 0.

In [PSS, Section 7.2] an equation for X_{E}^{−}(7) is derived from that for X_{E}(7). Formulae for the families of elliptic curves parametrised by X_{E}(7) and X_{E}^{−}(7) are given in [HK] and [F2]. Example 3.2. Let E be the elliptic curve 3364c in [C]. (By convention a Cremona label without a final number refers to the first curve in the isogeny class.) We take a = −4062871 and b = −3152083138 in Theorem 3.1. We then make the change of co-ordinates

(\begin{matrix}x \\ y \\ z \end{matrix}) ← (\begin{matrix}2320 & 0 & −3509 \\ −2716430 & 1682 & 4042687 \\ −2 & 0 & 3 \end{matrix}) (\begin{matrix}x \\ y \\ z \end{matrix})

so that X_{E}(7) ⊂ ℙ^{2} has equation

x^{4} + 2x^{3}y + x^{3}z + 3x^{2}y^{2} + 3x^{2}yz + 3x^{2}z^{2} + 2xy^{3}
− 3xy^{2}z + 6xyz^{2} + 12xz^{3} − 3y^{3}z − 3y^{2}z^{2} + 6yz^{3} − 2z^{4} = 0.

The Magma function PointSearch (with height bound 10^{6}) finds only two rational points (0 : 1 : 0) and (1 : −1 : 0) on X_{E}(7), and no rational points on X_{E}^{−}(7). These rational points correspond to the elliptic curves E = 3364c and F = 10092c. In particular this proves that E and F are 7-congruent.

We performed a similar search for rational points on X_{E}(7) and X_{E}^{−}(7) for each of the elliptic curves E/ℚ in Theorem 1.1(i). Many of the 7-congruent elliptic curves we found had rank 0 or 1, but in the 79 cases listed in Table 3.1 we found an elliptic curve F/ℚ of rank 2. In the small number of cases where we found more than one such F, we just kept the one with smallest conductor. The elliptic curves F beyond the range of Cremona’s tables were as follows. (In these cases we label the curve by its conductor followed by a ∗.)

\begin{matrix}561090∗ & y^{2} + xy = x^{3} + x^{2} + 95243x − 147561011 \\ 3140928∗ & y^{2} = x^{3} − 8580204x + 8146637424 \\ 2967232150∗ & y^{2} + xy + y = x^{3} − 3223101295946x + 3747757318724534268 \\ 6659445∗ & y^{2} + y = x^{3} − x^{2} − 67728609901x + 7329794977161867 \\ 25859475∗ & y^{2} + y = x^{3} − 31493068950x − 6675516099954594 \end{matrix}

Theorem 3.3. Let (E, F) be any one of the 79 pairs of elliptic curves listed in Table 3.1. Then F (ℚ) explains a subgroup of � (E/ℚ) isomorphic to (ℤ/7ℤ)^{2}.

Table 3.1. Pairs of 7-congruent elliptic curves

E  F  E  F  E  F  E  F
3364c 10092c 31311h 31311k 60885r 60885q 84150gk  84150gl
6552y 6552ba 31800e 31800f 61200gv 61200gu 84474w  84474u
9450p  9450t 34974h 174870bi 61950q 61950o 84672kg 84672kd
9510e 561090∗ 35682k 35682j 65088be 65088bc 84681e  84681h
10800y 10800u 36270l 36270j 68805d 68805c 84966ea 84966dx
11970o 11970s 36300br 36300by 69440z 69440ba 85050s  85050x
12927e 12927d 36414u 36414y 71148bh 71148bg 88305h  88305g
18832a 1712d 40362s 40362t 72384o 72384n 88450f 2967232150∗
19350q 19350s 41616cw 41616cx 73416g 10488b 89211c  681c
20544v 20544u 43296g 43296f 74400h 74400j 91035c  18207a
21312ce 21312cd 43350q 43350p 75075f 75075d 91800be  91800bd
21696l 21696k 45494e 45494d 75712bz 10816bf 92778r  92778t
23232dv 23232dt 45738ca 45738bz 76176cd 76176cg 92862j  13266e
26600m 26600l 46704k 6672i 76362r 76362q 92950be 92950bd
26640bu 26640bt 46800ew 46800fa 76608bl 3140928∗ 93795e 6659445∗
27930bj 27930bh 47232bu 47232bp 78210k 7110h 95040cd  95040ci
28314bn 28314bp 49938i 49938h 78400dg 235200eb 96558i  5082d
29400di 29400do 51600m 51600l 79350cy 79350cx 97470bp  97470br
30276r 30276q 54450gm 54450go 79530a 15906c 98325x 25859475∗
30996b 30996a 57150g 57150e 81600ho 81600hn

Proof. By construction these elliptic curves are 7-congruent, and do not admit any rational 7-isogenies. They also have ranks 0 and 2. So by Lemma 2.1 we have

(ℤ/7ℤ)^{2} ≅ F (ℚ)/7F (ℚ) ≅ Vis_{A} H^{1}(ℚ, E).

The Tamagawa numbers of E and F (at all bad primes) are coprime to 7. So in the 50 cases where E and F have good reduction at 7, it follows by Theorem 2.2 that Vis_{A} H^{1}(ℚ, E) is a subgroup of � (E/ℚ). In a further 4 cases (27930bj, 31311h, 71148bh, 84966ea) the elliptic curves E and F attain good reduction over K = ℚ_{7}(\sqrt[4]{7}). Since [K : ℚ_{7}] is coprime to 7 and e(K/ℚ_{7}) < 6 this again suffices to check local solubility.

The right hand square in (2.1) gives a commutative diagram

[figure: 0; E(ℚ_{7})/7E(ℚ_{7}); H^{1}(ℚ_{7}, E[7]); H^{1}(ℚ_{7}, E)[7]; 0; 0; F (ℚ_{7})/7F (ℚ_{7}); π_{7}; H^{1}(ℚ_{7}, F [7]); H^{1}(ℚ_{7}, F)[7]; 0]

where the rows are the Kummer exact sequences for E and F. To complete the proof it suffices to show that π_{7} is the zero map. This follows by Theorem 3.4(i) in 20 examples, and by Theorem 3.4(ii) in 4 examples (46704k, 73416g, 75712bz, 92862j). In the remaining example with E = 84672kg we found that E[7] ≅ F [7] has a unique cyclic Gal(\overline{ℚ}_{7}/ℚ_{7})-submodule, and were able to prove that π_{7} is the zero map by a brute force calculation using Lemma 4.1, Lemma 4.3(i) and Velu’s formulae [V]. □

Theorem 3.4. Let E and F be 7-congruent elliptic curves over ℚ_{7}. Suppose that either

(i) E and F have potentially multiplicative reduction, or (ii) E has non-split multiplicative reduction and F has good reduction.

Then E(ℚ_{7})/7E(ℚ_{7}) and F (ℚ_{7})/7F (ℚ_{7}) have the same images (via the Kummer exact sequence) in H^{1}(ℚ_{7}, E[7]) ≅ H^{1}(ℚ_{7}, F [7]).

Proof. These are special cases of Theorems 4.2 and 4.4 below, applied over either ℚ_{7} or a quadratic extension. □

4. Conditions for local solubility

In this section K will be a p-adic field, i.e. a finite extension of ℚ_{p}, with discrete valuation v_{K} (normalised to take integer values) and valuation ring O_{K}. Let ℓ be an odd prime. We write ζ_{ℓ} ∈ \overline{K} for a primitive ℓth root of unity, and j_{E} for the j-invariant of an elliptic curve E.

Lemma 4.1. Let ϕ : E → E′ be an ℓ-isogeny of elliptic curves over K, with dual isogeny ϕ̂: E′ → E. Then the following are equivalent.

(i) E′(K)/ϕE(K) ≅ H^{1}(K, E[ϕ]),

(ii) E(K)/ϕ̂E′(K) = 0,

(iii) the images of E(K)/ℓE(K) and H^{1}(K, E[ϕ]) in H^{1}(K, E[ℓ]) are the same.

Proof. There is a commutative diagram with exact rows and exact right hand column

(4.1)  [figure: E(K); ϕ; E′(K); δ_{ϕ}; ϕ̂; H^{1}(K, E[ϕ]); E(K); ϕ; ℓ; E(K); δ_{ℓ}; H^{1}(K, E[ℓ]); E′(K); ϕ̂; E(K); δϕ̂; H^{1}(K, E′[ϕ̂]).]

By Tate local duality, δ_{ϕ} is surjective if and only if δϕ̂ has trivial image. The remaining statements follow by a diagram chase. □

Theorem 4.2. Let E and F be ℓ-congruent elliptic curves over K, both with split multiplicative reduction. Suppose that either ζ_{ℓ} ∉ K or v_{K}(j_{E}) ≢ 0 (mod ℓ). Then E(K)/ℓE(K) and F (K)/ℓF (K) have the same images (via the Kummer exact sequence) in H^{1}(K, E[ℓ]) ≅ H^{1}(K, F [ℓ]).

Proof. By the Tate parametrisation there is an isomorphism of Gal(\overline{K}/K)-modules E(\overline{K}) ≅ \overline{K}^{×}/q^{ℤ} for some q ∈ K^{×} with v_{K}(q) = −v_{K}(j_{E}). We consider the ℓisogenies ϕ : E → E′ and ϕ̂: E′ → E given on K-points by

\begin{matrix}K^{×}/q^{ℤ} & → K^{×}/q^{ℓℤ} \\ x & ↦ x^{ℓ}, \end{matrix}  and  \begin{matrix}K^{×}/q^{ℓℤ} & → K^{×}/q^{ℤ} \\ x & ↦ x. \end{matrix}

Since the second of these maps is surjective, the conditions in Lemma 4.1 are satisfied. As Galois modules we have E[ϕ] ≅ μ_{ℓ} and E′[ϕ̂] ≅ ℤ/ℓℤ, and so an exact sequence

(4.2)  0⟶μ_{ℓ}⟶E[ℓ]⟶ℤ/ℓℤ⟶0.

If the isomorphism E[ℓ] ≅ F [ℓ] identifies the submodules μ_{ℓ}, then applying Lemma 4.1 to both E and F proves the theorem. If the submodules μ_{ℓ} are not identified then μ_{ℓ} ≅ ℤ/ℓℤ and (4.2) splits, in which case ζ_{ℓ} ∈ K and v_{K}(j_{E}) ≡ 0 (mod ℓ), contradicting our hypotheses. □

Another way to check the conditions in Lemma 4.1 is provided by the following lemma. We write c_{K}(E) for the Tamagawa number of E/K.

Lemma 4.3. Let ϕ : E → E′ be an ℓ-isogeny of elliptic curves over K, with dual isogeny ϕ̂: E′ → E. Let ω and ω′ be Néron differentials on E and E′.

(i) We have ϕ^{∗}ω′ = αω and ϕ̂^{∗}ω = βω′ for some α, β ∈ O_{K} satisfying αβ = ℓ, and

\frac{#(E′(K)/ϕE(K))}{#E(K)[ϕ]} = |α|_{K}^{−1} \frac{c_{K}(E′)}{c_{K}(E)}

(ii) If E and E′ have good reduction, E(K)[ϕ] ≠ 0 and e(K/ℚ_{p}) < p − 1 then α is a unit.

Proof. (i) See [Sc, Lemma 3.8]. (ii) We suppose p = ℓ as otherwise α is a unit by (i). Since e(K/ℚ_{p}) < p − 1 the theory of formal groups shows that E(K)[ϕ] ≅ ℤ/pℤ cannot belong to the kernel of reduction. Therefore, arguing exactly as in the proof of [S2, Chapter X, Theorem 4.2], the image of δ_{ϕ} in (4.1) lies in the unramified subgroup. Since K has a unique unramified extension of degree p it follows that #(E′(K)/ϕE(K)) divides p, and so α is a unit by (i). □

Theorem 4.4. Let E and F be ℓ-congruent elliptic curves over K. Suppose that E has non-split multiplicative reduction and F has good reduction. If p = ℓ then further suppose that e(K/ℚ_{p}) < p−1. Then E(K)/ℓE(K) and F (K)/ℓF (K) have the same images (via the Kummer exact sequence) in H^{1}(K, E[ℓ]) ≅ H^{1}(K, F [ℓ]).

Proof. Let L be the unramified quadratic extension of K. By the Tate parametrisation there is an exact sequence of Gal(\overline{K}/K)-modules

0 → M_{1} → E[ℓ] → M_{2} → 0

with M_{1} ≅ μ_{ℓ} and M_{2} ≅ ℤ/ℓℤ over L. Since [L : K] is coprime to ℓ, the proof of Theorem 4.2 shows that E(K)/ℓE(K) and H^{1}(K, M_{1}) have the same image in H^{1}(K, E[ℓ]). Since E[ℓ] ≅ F [ℓ], there are ℓ-isogenies ϕ : F → F′ and ϕ̂: F′ → F with kernels M_{1} and M_{2}. By our assumption that E has non-split reduction, the action of Gal(L/K) on M_{2} is non-trivial. Applying Lemma 4.3(ii) to the isogeny ϕ̂ : F′ → F over L, and then Lemma 4.3(i) to the same isogeny over K, shows that F (K)/ϕ̂F′(K) = 0. We are again done by Lemma 4.1. □

5. Visibility in abelian threefolds

In this section we give some examples of elements of order 7 in � (E/ℚ) that are visible in an abelian threefold.

Let E/ℚ be one of the elliptic curves in Theorem 1.1(i). We may use the modular symbols algorithms in Magma or Sage to look for modular forms f ∈ S_{2}(Γ_{0}(N_{E})) satisfying a 𝔭-congruence with E for some prime 𝔭|7. Table 5.1 lists the first few examples we found with ℚ(f) ≠ ℚ. Each f is a normalised Hecke eigenform with ℚ(f) = ℚ(\sqrt{2}) and satisfies a_{p}(f) ≡ a_{p}(E) mod (3 + \sqrt{2}) for all primes p with p < 1000. By work of Shimura [Sh] we may associate to each f

Table 5.1. Hecke eigenvalues a_{p}(f)

\begin{matrix}E & a_{2} & a_{3} & a_{5} & a_{7} & a_{11} & a_{13} \\ 6622b & −1 & −\sqrt{2} − 1 & \sqrt{2} − 2 & −1 & 1 & 2\sqrt{2} − 2 \\ 9510e & −1 & −1 & 1 & \sqrt{2} − 3 & −2\sqrt{2} − 2 & −5 \\ 14938n & 1 & −2 & \sqrt{2} − 2 & −1 & −1 & −2 \\ 15219c & −\sqrt{2} − 1 & 0 & −1 & 2\sqrt{2} − 2 & −5 & −2\sqrt{2} + 3 \\ 20502ba & 1 & 0 & −\sqrt{2} − 3 & −\sqrt{2} − 3 & 3\sqrt{2} − 2 & \sqrt{2} − 2 \end{matrix}

a modular abelian surface A_{f} with real multiplication by \sqrt{2}. Our data suggests that A_{f}[3 + \sqrt{2}] ≅ E[7] as Galois modules. If A_{f} is isogenous over ℚ to the Jacobian of a genus 2 curve C/ℚ then for each prime p of good reduction for C we have (see [FLS+, Section 2.1] or [MS, Lemma 3])

(5.1)

Trace(a_{p}) = p + 1 − n_{1},
Norm(a_{p}) = (n_{1}^{2} + n_{2})/2 − (p + 1)n_{1} − p,

where n_{1} = #C(𝔽_{p}) and n_{2} = #C(𝔽_{p^{2}}). We used the following result to help find the corresponding genus 2 curves. (An alternative approach is described in [GGG], [Wa].)

Theorem 5.1. [B, Theorem 4.1] Let g(x) = ∏_{i=1}^{3}(x^{2} − α_{i}x + P α_{i}^{2} + Qα_{i} + R) where α_{1}, α_{2}, α_{3} are the roots of x^{3} + Ax^{2} + Bx + C = 0 and A, B, C, P, Q, R ∈ ℚ with P ≠ 0 and

R = 4P, B = (Q(P A − Q) + 4P^{2} + 1)/P^{2}, C = 4(P A − Q)/P.

If g(x) has distinct roots in \overline{ℚ} then y^{2} = g(x) defines a smooth curve of genus 2 whose Jacobian admits real multiplication by \sqrt{2}, defined over ℚ, and fixed by the Rosati involution. Moreover every such genus 2 curve arises in this way, up to quadratic twist.

By searching over A, P, Q ∈ ℚ of small height, we found for each of the modular forms in Table 5.1, a genus 2 curve satisfying (5.1) for all good primes p < 1000. We record the genus 2 curves (and Cremona labels for E) in Table 5.2 and the values of A, P, Q in Table 5.3.

At this point we abandoned the computation of modular forms (which is increasingly time consuming as the level increases) and took the following more

Table 5.2. Genus 2 curves

\begin{matrix}6622b & y^{2} = 20x^{6} + 44x^{5} − 23x^{4} − 10x^{3} + 81x^{2} − 52x + 4 \\ 9510e & y^{2} = 9x^{6} − 12x^{5} + 2x^{4} − 84x^{3} + 437x^{2} + 528x + 144 \\ 14938n & y^{2} = 4x^{6} + 36x^{5} + 37x^{4} − 150x^{3} − 47x^{2} − 72x + 16 \\ 15219c & y^{2} = −7x^{6} + 18x^{5} − 93x^{4} + 18x^{3} + 117x^{2} − 36x + 64 \\ 20502ba & y^{2} = 36x^{6} + 108x^{5} + 381x^{4} + 474x^{3} + 849x^{2} + 504x + 144 \end{matrix}

naive approach. We used Theorem 5.1 to draw up a list of genus 2 curves by looping over A, P, Q ∈ ℚ of small height and keeping only those curves whose primes of bad reduction are smaller than some threshold. For each curve C on our list, we computed the right hand sides t_{p} and n_{p} in (5.1) for all p < 100. Then for each elliptic curve E in Theorem 1.1(i) we tested to see whether

(5.2)  a_{p}(E)^{2} ± t_{p}a_{p}(E) + n_{p} ≡ 0 (mod 7)

for some choice of sign ±. If this test is passed for many primes p then it seems likely that E and the Jacobian of C satisfy a congruence up to quadratic twist. Indeed in all such cases we were able to replace C by a quadratic twist so that (5.2) holds with a minus sign for all good primes p < 1000.

The examples we found are recorded in Table 5.3, except that we list just one genus 2 curve C for each elliptic curve E. In cases where we found more than one C, it appears that the Jacobians of these curves are isogenous. The data recorded is as follows.

The elliptic curve E/ℚ: This is specified by its reference in Cremona’s tables [C]. An entry in brackets indicates that this curve already appeared in Table 3.1.

The ratio of levels: The odd part of the conductor of F = Jac(C), computed using Liu’s program genus2reduction in Sage, is (κN_{odd})^{2} where N_{odd} is the odd part of the conductor of E, and κ is recorded in the second column of the table. In most cases κ = 1, as we would expect if E and F correspond to newforms of the same level.

The genus 2 curve C/ℚ: This is specified as the quadratic twist by δ of

the curve in Theorem 5.1 with parameters A, P, Q. The Mordell-Weil group of F = Jac(C): We used Magma to compute the order of the torsion subgroup and the rank of F (ℚ). For the latter, we used Stoll’s implementation of 2-descent to bound the rank, and then the functions Points and ReducedBasis to find sufficiently many independent

12 TOM FISHER

Table 5.3. Pairs of elliptic curves and genus 2 curves

E  κ  A  P  Q  δ #tors rank 7 | c_{p} p = 7
6622b  1  41/4 1/5  0  1  1  4 {11} m
(9510e)  1  29/5 18  53  2  1  4  g
14938n  1  9  1/4  3  1  2  4  m
15219c  1  9/8 −4/7 −6/7 1  1  4  g
20502ba  1  6  9/16 7/4  1  1  4  g
21771b  1  81/8  2  12  2  1  4  g
23025c  19 −52/3 −3/4  3  5  2  2 {19} g
23085h  1  6  3/5 22/5 1  1  4  g
23744v  1  −7  −2  9  1  1  4  m
24432j  1  −8 −3/4  7  15  1  4  g
25296a  37  9  2/5 5/2  1  1  6 {31, 37} g
26166h  1 83/12 4  95/6 1  1  4 {3} g^{(3)}
29445c  1 −14/3 −1/2 5/2  1  1  4  g
39984bf  1  3  4  11  1  1  4  g^{(3)}
40764a 1/3  8  2  7  1  1  4  g
40950u  1  9  4/5 29/5 5  4  4  m
41616l  1  23/3  9  40  1  1  4  g
45770c  1  4  1  −1  1  1  4 {5}  g
54327c  1  47/7 7/6  4  42  2  4  m
54663a  1  −6 −1/8 13/8 1  2  4  m
61320v  1  53/6  1  14/3 1  1  4  m
67032cf  1  2  3/16 5/4  7  2  4  m^{∗}
67080r 1/5  7  9  37  1  2  4  g
67179c  1  6  7/4 31/4 1  2  4  m^{∗}
70950bc  1  27/5  1 29/10 1  1  4 {2}  g
72128j  1  8  8/7 31/7 14  2  4  m^{∗}
(75712bz) 17/7  5  4  17 −26 2  2 {17}
(76176cd) 1  12 −1/4  1  1  1  4  g
(76608bl) 11  2/7 7/6 −1/3 −14 1  4  m
78400le  1  8  −4/7 −29/7 1  2  4  g^{(4)}

E  κ A  P  Q  δ  #tors rank 7 | c_{p} p = 7
(84474w) 49 17 3/4  7  7  1  4
90650i  1 −8 −1  3  −35  1  4  m^{∗}
90950i  1  7  8  27  1  1  4  g
90950m 1  1  4/5  1  5  1  4  g
96300bd 1  8 10/3 43/3  1  1  4  g

points of infinite order. Since F (ℚ) is a ℤ[\sqrt{2}]-module, the rank is necessarily even. It is striking that in nearly all cases we have rank F (ℚ) = 4.

The primes with Tamagawa number divisible by 7: The Tamagawa numbers c_{p}(E) are coprime to 7 in all cases. We list the primes p for which the Tamagawa number c_{p}(F) is divisible by 7. These entries were computed using Liu’s program genus2reduction in Sage, Donnelly’s programs RegularModel and ComponentGroup in Magma, and our own calculations for curves with multiplicative reduction, following [FLS+, Section 3.4] and [BLR, Theorem 9.6.1]. There was one case (E = 76608bl and p = 2) not covered by these methods. Liu’s program nonetheless reports that the potential stable reduction is the union of two elliptic curves. The Jacobian F therefore has potential good reduction, and so by a result of Silverman [S1] the Tamagawa number is coprime to 7.

The reduction behaviour at p = 7: We write g in cases where E and F both have good reduction, and m in cases where they both have multiplicative reduction. We write m^{∗} if the \sqrt{7}-twists of E and F both have multiplicative reduction. The label g^{(n)} indicates that E and F attain good reduction over ℚ_{7}(\sqrt[n]{7}).

Theorem 5.2. Let (E, F) be any one of the 16 pairs of elliptic curves and genus 2 Jacobians listed in Table 5.3, for which the Cremona reference for E appears in bold. Then F (ℚ) explains a subgroup of � (E/ℚ) isomorphic to (ℤ/7ℤ)^{2}.

Proof. In Theorem 6.3 we show that (making an appropriate choice of sign for \sqrt{2}) we have E[7] ≅ F [3 + \sqrt{2}] as Galois modules. The ranks of E and F are 0 and 4, and they have torsion subgroups of order coprime to 7. Since F (ℚ) is a ℤ[\sqrt{2}]-module, it follows by Lemma 2.1 that

(ℤ/7ℤ)^{2} ≅ F (ℚ)/[3 + \sqrt{2}]F (ℚ) ≅ Vis_{A} H^{1}(ℚ, E).

For the examples highlighted in bold:

(i) all Tamagawa numbers of E and F are coprime to 7, and (ii) E and F have potential good reduction at 7, and this good reduction is

achieved over an extension of ℚ_{7} of degree less than 6. It follows by Theorem 2.2 that Vis_{A} H^{1}(ℚ, E) is a subgroup of � (E/ℚ). □

Remark 5.3. We expect that Theorem 5.2 is true for all the pairs (E, F) in Table 5.3 with rank F (ℚ) ≥ 4. Moreover, in all cases except the one with E = 84474w, this would follow from an appropriate generalisation of Theorem 4.2.

Remark 5.4. We checked (using the method in [CF, p158], [St]) that the genus 2 Jacobian F is absolutely simple in all cases except the one with E = 78400le. In the exceptional case the formulae in [B, Section 5] show that F is isogenous to the restriction of scalars of the ℚ-curve

y^{2} = x (x^{2} + 5(7 + \sqrt{−7})x + 70(1 + \sqrt{−7})) .

In this case we were able to check the congruence E[7] ≅ F [3 + \sqrt{2}] using Theorem 3.1.

6. Checking the congruences

In Table 5.3 we gave a list of pairs (E, F) where E/ℚ is an elliptic curve and F is the Jacobian of a genus 2 curve C/ℚ. The genus 2 curves C were constructed using Theorem 5.1, and so their Jacobians are known to have real multiplication by \sqrt{2}. We suspect (on the basis of comparing traces of Frobenius for all good primes p < 1000) that E[7] ≅ F [3 + \sqrt{2}] as Galois modules. In this section we prove that this is indeed the case.

It would of course be interesting to prove this using modularity, or by finding a result analogous to Theorem 3.1, but we take a more direct approach, based on that used in [KO, Section 5] to exhibit a pair of 7-congruent elliptic curves.

Proposition 6.1. Let E/ℚ be an elliptic curve and F/ℚ a genus 2 Jacobian with real multiplication by \sqrt{2}, defined over ℚ, and fixed by the Rosati involution. Suppose that

(i) The Galois representation \overline{ρ}_{E,7} : Gal(\overline{ℚ}/ℚ) → GL(E[7]) is surjective.

(ii) There are non-zero torsion points P ∈ E[7] and Q ∈ F [7] whose images in E/{±1} and F/{±1} have the same field of definition. (By (i) this is a degree 24 number field.)

Then for some choice of sign ± the Gal(\overline{ℚ}/ℚ)-modules E[7] and F [3 ± \sqrt{2}] are isomorphic up to quadratic twist.

Proof. We write Q = Q_{+} + Q_{−} where Q_{±} = (−3 ± \sqrt{2})Q ∈ F [3 ± \sqrt{2}]. Let L, L_{±} be the fields of definition of the images of Q, Q_{±} in F/{±1}. It is clear that L_{+} and L_{−} are subfields of L and that [L : L_{1}L_{2}] ≤ 2. By our hypotheses on E, the subfields of L correspond to the subgroups H with \{( \begin{matrix}±1 & ∗ \\ 0 & ∗ \end{matrix})\} ⊂ H ⊂ GL_{2}(𝔽_{7}). In particular L has degree 24, and its only non-trivial subfield has degree 8. So either L = L_{+} or L = L_{−}. Suppose it is the former. Then replacing Q by Q_{+} we may assume that Q ∈ F [3 + \sqrt{2}].

Since \sqrt{2} is fixed by the Rosati involution, it is self-adjoint for the Weil pairing e_{7} : F [7]×F [7] → μ_{7}. In particular F [7] = F [3+\sqrt{2}]×F [3−\sqrt{2}] is an orthogonal decomposition with respect to e_{7}. Since the Weil pairing is alternating and nondegenerate we have

∧^{2}E[7] ≅ ∧^{2}F [3 ± \sqrt{2}] ≅ μ_{7}.

The proposition is now a special case of the following lemma. □

Lemma 6.2. Let ℓ ≥ 5 be a prime, and let M_{1}, M_{2} be Galois modules each isomorphic to (ℤ/ℓℤ)^{2} as an abelian group. Suppose that ∧^{2}M_{1} ≅ ∧^{2}M_{2} and

(i) the Galois representation Gal(\overline{ℚ}/ℚ) → GL(M_{1}) is surjective, and (ii) the Galois representation Gal(\overline{ℚ}/ℚ(M_{2})) → SL(M_{1}) is not surjective.

Then M_{1} and M_{2} are isomorphic up to quadratic twist.

Proof. This is a variant of [Se, Lemme 8], the difference being that we have removed the hypothesis that Gal(\overline{ℚ}/ℚ) → GL(M_{2}) is surjective.

Let B_{1} and B_{2} be finite groups and H a subgroup of B_{1} × B_{2} with pr_{1}(H) = B_{1} and pr_{2}(H) = B_{2}. We identify B_{1} with B_{1} × {1} in B_{1} × B_{2} and let N_{1} = B_{1} ∩ H. Likewise we put N_{2} = B_{2} ∩ H. Then N_{1} is normal in B_{1}, and N_{2} is normal in B_{2}. Moreover the image of H in B_{1}/N_{1} × B_{2}/N_{2} is the graph of an isomorphism α : B_{1}/N_{1} → B_{2}/N_{2}.

We apply the above with B_{i} the image of Gal(\overline{ℚ}/ℚ) → GL(M_{i}) for i = 1, 2, and H the image of Gal(\overline{ℚ}/ℚ) → GL(M_{1}) × GL(M_{2}). Hypothesis (ii) shows that N_{1} is a proper subgroup of SL(M_{1}). Since N_{1} is normal in B_{1} ≅ GL_{2}(𝔽_{ℓ}) it follows that N_{1} ⊂ {±1}. Hence B_{1}/N_{1} ≅ GL_{2}(𝔽_{ℓ}) or GL_{2}(𝔽_{ℓ})/{±1}. The only index two subgroup of GL_{2}(𝔽_{ℓ}) is the subgroup of elements with determinant a square. By inspection of the centres we see that this subgroup is not isomorphic to GL_{2}(𝔽_{ℓ})/{±1}. Since B_{1}/N_{1} ≅ B_{2}/N_{2} it follows that B_{2} ≅ GL_{2}(𝔽_{ℓ}). In other words, we have shown that Gal(\overline{ℚ}/ℚ) → GL(M_{2}) is surjective. The proof in [Se] (using the fact that every automorphism of PGL_{2}(𝔽_{ℓ}) is inner) now applies. □

Theorem 6.3. Let (E, F) be any one of the pairs in Table 5.3. Then for some choice of sign ± we have E[7] ≅ F [3 ± \sqrt{2}] as Galois modules.

Proof. In each case it is easy to find a good prime p for which f_{p}(X) = X^{2} − a_{p}(E)X + p is irreducible mod 7, and another good prime p ≠ 7 for which f_{p}(X) has distinct roots mod 7. It follows by [Se, Section 4] that \overline{ρ}_{E,7} : Gal(\overline{ℚ}/ℚ) → GL(E[7]) is surjective.

Let L be the number field of degree 24 defined by the x-coordinate of a 7-torsion point on E. As noted in [KO] the degree 8 subfield is K = ℚ(θ) where θ is a root of

(6.1)  (X^{2} + 5X + 1)^{3}(X^{2} + 13X + 49) − j_{E}X = 0.

We specify points on F = Jac(C) by pairs of points [P_{1}, P_{2}] on C corresponding to the divisor class [P_{1} + P_{2} − Ω], where Ω is a canonical divisor. We also write P_{i} = (x_{i}, y_{i}) for i = 1, 2. Let β_{0} be the rational function specified on p18 of [CF]. Then there is a morphism

F → ℙ^{3} ; [P_{1}, P_{2}] ↦ (1 : x_{1} + x_{2} : x_{1}x_{2} : β_{0})

whose image is the Kummer surface K ⊂ ℙ^{3}. This is a quartic surface isomorphic to F/{±1}.

We use the machinery for computing with analytic Jacobians in Magma [W] to compute (numerical approximations to) 24 points [(x_{1}, y_{1}), (x_{2}, y_{2})], representing the pairs of inverse elements in F [3 + \sqrt{2}] − \{0\}. In all cases except E = 78400le (discussed in Remark 5.4) we have End_{ℂ}(F) = ℤ[\sqrt{2}], and so the only ambiguity in choosing \sqrt{2} is up to sign. We arrange the 24 points into 8 sets of 3 corresponding to the action of (ℤ/7ℤ)^{×}/{±1}. We anticipate that each set of 3 points (or rather their images in K) will be jointly defined over K.

Let [(x_{1}, y_{1}), (x_{2}, y_{2})] be one of the points. Let g and h be the minimal polynomials of x_{1} + x_{2} and x_{1}x_{2} over K. Let c be one of the coefficients of g or h. We compute the minimal polynomial of c over ℚ by first computing the coefficients numerically and then recognising them as rational numbers. We then find a root of this polynomial in K. If we are unable to find a root then we either increase the precision, or go back and change the sign of \sqrt{2}. In this way we compute cubic polynomials g, h ∈ K[X].

We then solve for a point Q = (1 : ξ_{2} : ξ_{3} : ξ_{4}) ∈ K(L) where ξ_{2} and ξ_{3} are roots of g and h. For this we try all pairs of roots of g and h and use the equation for K to solve for ξ_{4}. Since the multiplication-by-m maps on the Kummer surface are implemented in Magma it is then easy to check (now by an exact calculation) that 7Q = (0 : 0 : 0 : 1). Moreover in each case we found that L = ℚ(ξ_{2}).

Proposition 6.1 now shows that E[7] ≅ F [3±\sqrt{2}] up to quadratic twist. But any quadratic twist would be supported on the bad primes. So our earlier comparison of traces of Frobenius for all good primes p < 1000 is more than sufficient to finish the proof. □

7. Proof of Theorem 1.2

Let E be the elliptic curve 67080r. Theorem 5.2 shows that � (E/ℚ) contains a subgroup (ℤ/7ℤ)^{2} visible in an abelian threefold. Moreover the abelian threefold is as described in Theorem 1.2, since taking (A, P, Q) = (7, 9, 37) in Theorem 5.1 we have

9(x + 1)^{6}g (\frac{−x − 2}{x + 1}) = x(x + 4)(x^{4} + 2x^{3} − x − 3).

Following the methods in [GJP+], we show that this is all of � (E/ℚ). The bad primes of E split in K = ℚ(\sqrt{D}) with D = −191. Computing L-values and using the Gross-Zagier formula shows there is a point y_{K} ∈ E(K) with canonical height ĥ_{K}(y_{K}) ≈ 217958.077. Following [GJP+, Section 2.1] we find that \overline{ρ}_{E,p} : Gal(\overline{ℚ}/ℚ) → GL(E[p]) is surjective for all odd primes p. Therefore by work of Kolyvagin

v_{p}(#� (E/ℚ)) ≤ 2 v_{p}([E(K) : ℤy_{K}]).

We found a point of infinite order P ∈ E^{D}(ℚ) using 12-descent [F1]. Since the point is rather large, we record it as follows. Let C_{4} ⊂ ℙ^{3} be the 4-covering of the elliptic curve 2-isogenous to E^{D} defined by

\begin{aligned}x_{1}^{2} + 6x_{1}x_{2} + 7x_{1}x_{3} + 23x_{1}x_{4} − 3x_{2}^{2} + 12x_{2}x_{3} + 8x_{2}x_{4} − 30x_{3}^{2} + 54x_{3}x_{4} − 60x_{4}^{2} & = 0 \\ 6x_{1}^{2} − x_{1}x_{2} + 89x_{1}x_{3} − 37x_{1}x_{4} + 37x_{2}^{2} − 86x_{2}x_{3} + 22x_{2}x_{4} − 166x_{3}^{2} + 114x_{3}x_{4} + 199x_{4}^{2} & = 0 \end{aligned}

The 4-covering map is given by formulae of classical invariant theory; see [AKM+]. Then P ∈ E^{D}(ℚ) is the image, under the 4-covering map and the 2-isogeny, of the point

(847793227 : 2227947281 : 2665508726 : 1875455642) ∈ C_{4}(ℚ).

We find ĥ_{K}(P) = 2ĥ_{ℚ}(P) ≈ 1112.031 and ĥ_{K}(y_{K})/ĥ_{K}(P) ≈ 196.000. We checked using the methods in [Si] (specifically the Magma functions SiksekBound, Saturation and Points) that P is a generator for E^{D}(ℚ) modulo torsion. It was useful to exploit here that E has a rational 2-torsion point.

Since E(K)_{tors} ≅ ℤ/2ℤ it follows that [E(K) : ℤy_{K}] = 2^{a} · 7 for some integer a. A 2-descent shows that � (E/ℚ)[2] = 0, and we have already shown that � (E/ℚ) contains a copy of (ℤ/7ℤ)^{2}. Therefore � (E/ℚ) ≅ (ℤ/7ℤ)^{2}.

To prove Theorem 1.2(ii) it suffices to show (see [F3, Theorem 3.2]) that the only elliptic curves 7-congruent to E are those isogenous to E. We take a = 1013948336592 and b = 471906827379024768 in Theorem 3.1 and then make the change of co-ordinates

(\begin{matrix}x \\ y \\ z \end{matrix}) ← (\begin{matrix}102188568 & −22341576 & −103840044 \\ 73939908228192 & 94676367452256 & 135756936508464 \\ −28 & 46 & −151 \end{matrix}) (\begin{matrix}x \\ y \\ z \end{matrix})

so that C = X_{E}(7) ⊂ ℙ^{2} is defined by

F(x, y, z) = 6x^{4} + 25x^{3}y − 39x^{3}z − 3x^{2}y^{2} − 15x^{2}yz + 27x^{2}z^{2} − 42xy^{2}z
− 141xyz^{2} − 87xz^{3} + 8y^{4} − 6y^{3}z − 21y^{2}z^{2} + 127yz^{3} − 15z^{4}.

This curve has rational points P_{1} = (4 : 9 : 2) and P_{2} = (5 : −10 : 3), corresponding to E and its 2-isogenous curve. To show that these are the only rational points on C, we closely follow the methods in [PSS]. In view of this, we give just a few of the details.

Since C ⊂ ℙ^{2} is a twist of the Klein quartic it has automorphism group G ≅ PSL_{2}(𝔽_{7}). By [AR, Theorem 24.1], the G-invariant divisor classes on C form a cyclic group, generated by Λ (say) of degree 2. Let K be the degree 8 number field, defined by (6.1), arising as the field of definition of a cyclic subgroup of E[7]. A 2-descent on J = Jac(C) proves the following.

Theorem 7.1. If the class number of K is odd then J(ℚ) ≅ ℤ^{2}, and G_{1} = [Λ − 2P_{1}] and G_{2} = [P_{1} − P_{2}] generate a subgroup of finite index.

Both Magma and Pari/GP report that, conditional on the Generalised Riemann Hypothesis, K has class number 1. We will now assume this is the case.

The reduction of C mod 7 has a unique singular point at (4 : 3 : 1). Since F(4, 3, 1) ≡ 35 (mod 49) this is a regular point. The smooth \overline{𝔽}_{7}-points are parametrised by

s ↦ (4s^{4} + 2s^{2} + 2s + 4 : 3s^{4} + 3s^{2} + 5s + 5 : s^{4} + 4s + 2).

Integrating these polynomials in s, and then summing over the points of a divisor, gives an isomorphism J̃(𝔽_{7}) ≅ (ℤ/7ℤ)^{3}, where J̃ is the special fibre of the Néron model of J/ℚ_{7}. We find that the images of G_{1} and G_{2} are linearly independent. The image of J(ℚ) ≅ ℤ^{2} in J̃(𝔽_{7}) is therefore the subgroup generated by G̃_{1} and G̃_{2}. The only smooth points P ∈ C̃(𝔽_{7}) with [P − P̃_{1}] belonging to this subgroup are P̃_{1}, P̃_{2} and (1 : −1 : 0). Moreover if P ∈ C(ℚ) is a point reducing to (1 : −1 : 0) then

(7.1)  [P − P_{1}] ∈ G_{1} + 4G_{2} + 7J(ℚ).

We used Chabauty’s method to show that P_{1} and P_{2} are the only points in C(ℚ) reducing to P̃_{1} and P̃_{2} (for this we show there is a differential on C/ℚ_{7} killing J(ℚ) whose reduction mod 7 corresponds to x+y+3z), and then the Mordell-Weil sieve with p = 41 to show there are no points P ∈ C(ℚ) satisfying (7.1). Therefore C(ℚ) = {P_{1}, P_{2}}.

Since X_{E}^{−}(7) has no ℚ_{2}-points, this completes the proof of Theorem 1.2.

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

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