Visualizing elements of order two in the Weil-Châtelet group

Tomas Antonius Klenke

October 1, 2001

Abstract

Given an elliptic curve E over a number field K and a two-torsion element σ ∈ H^{1}(G_{K}, E)[2] of its Weil-Châtelet group, we prove that σ is always visible in infinitely many abelian surfaces, in the sense put forward by Cremona and Mazur in their article [Cr-M]. Our argument is a variant of Mazur’s proof for the analogous statement about three-torsion elements of the Shafarevich-Tate group given in [M], using the universal elliptic curve with full level-two-structure and an invariant differential in place of the universal elliptic curve with full level-three-structure.

1 Introduction

In their article [Cr-M], J. Cremona and B. Mazur introduce the following

Definition 1 Let K be a number field with absolute Galois-group G_{K}, let E be an elliptic curve over K, and ι : E ↪ A an embedding over K of E into an abelian variety A. An element σ ∈ H^{1}(G_{K}, E) = W C(E/K) of the Weil-Châtelet group of E is said to be visible in A if under the map ι_{∗} : H^{1}(G_{K}, E) → H^{1}(G_{K}, A) on cohomology induced by ι, the cohomology class σ is sent to zero.

To give a geometric interpretation of visibility, note that the element σ ∈ H^{1}(G_{K}, E) corresponds to a principal homogeneous space C_{σ} of E. C_{σ} is a curve of genus one over K with Jacobian E; it follows from a simple Galois cohomology computation that if σ is visible in the abelian variety A in the above sense, then C_{σ} is contained in A as a subvariety, in fact as a translate of E. The converse also holds under some mild technical hypotheses (cf. [Cr-M]). Algebraically the cocycle σ ∈ H^{1}(G_{K}, E) corresponds to a K-rational point on the quotient abelian variety B = A/E, the inverse image of which in A is the subvariety C_{σ} from above.

A. Agashe and W. Stein provide a construction in [A-S] which shows that, given an elliptic curve E/K and a cohomology class σ ∈ H^{1}(G_{K}, E), there is always some abelian variety in which σ is visible, of dimension at most the minimal degree of a field extension L of K over which σ becomes trivial. This implies in particular that, for any n, an element σ ∈ � (E/K)[n] is visible in an abelian variety of dimension at most n, and any element σ ∈ W C(E/K)[n] is visible in an abelian variety of dimension at most n^{2}.

In his article [M], Mazur shows that any three-torsion element σ of the Shafarevich-Tate group of an elliptic curve E over a number field K is, in fact, visible in infinitely many abelian surfaces, thereby improving the result of [A-S] in two ways in this special case. The proof proceeds in two steps: First the question of visualizing σ in an abelian surface is reduced to finding an elliptic curve F which satisfies a full three-congruence with E and in which a certain cohomology class vanishes. The existence of infinitely many such curves F can then be established by showing that the K-rational points are Zariski-dense in certain twists of the (generalized) universal elliptic curve with full level-three-structure, where crucial use is made of its Hesse cubic representation as the blow-up of ℙ^{2} at nine points.

In this article we prove an analogous statement for two-torsion elements of the Weil-Châtelet group, again improving on the result in [A-S]:

Theorem 1 Given an elliptic curve E over a number field K, any element σ ∈ H^{1}(G_{K}, E)[2] is visible in infinitely many (non-isomorphic) abelian surfaces.

As the “universal elliptic curve with full level-two-structure” does not exist (since the corresponding moduli problem is not rigid), we have to find a suitable substitute if we want to make Mazur’s method of proof work for our purpose. To this end we will introduce a certain family of elliptic curves E and establish that it has the properties necessary for an analogue of the above argument: We will show that it admits a birational morphism to projective three-space under which a certain subgroup of its automorphisms descends; observing that any twist of ℙ^{3} by these induced automorphisms is trivial, we can then translate Mazur’s approach to our setting, without having to impose local conditions on σ.

Acknowledgements. This article has grown out of part of my doctoral dissertation which I wrote under the supervision of B. Mazur. The problem he then suggested to me as well as his constant guidance provided invaluable help in arriving at the result presented here. I would furthermore like to thank R. Pink, whose detailed comments and questions have led to significant clarifications and simplifications in the argument.

2 Preliminaries

2.1 The family E and some automorphisms

Let E be the elliptic curve over Y = Spec(ℚ[λ, t][ \frac{1}{tλ(λ−1)} ]) given by the affine “Legendre-type” equation

E : ty^{2} = x(x − 1)(x − λ)  (1)

with structure morphism m : E → Y. Denote by P_{1}, P_{2}, P_{3} the twotorsion sections of E/Y given by x = 0, x = 1, x = λ, respectively; we will use the notation O or P_{4} for the zero-section. We write t_{P_{i}} for translation on E by the section P_{i}, for i = 1, 2, 3. The standard calculation using the secant method shows that t_{P_{1}} is given on coordinates by

t_{P_{1}} : (λ, t; x, y) ↦ (λ, t; λ/x, −λy/x^{2}).  (2)

A straightforward computation shows that the two automorphisms s_{(12)} and s_{(23)} given by the formulae

\begin{aligned}s_{(12)} & : & (λ, t; x, y) & ↦ & (1 − λ, −t; 1 − x, −y) \\ s_{(23)} & : & (λ, t; x, y) & ↦ & (1/λ, t/λ; x/λ, y/λ) \end{aligned}

generate compatible actions of S_{3} on E and on Y, permuting the three twotorsion sections P_{i}, i = 1, 2, 3, as indicated by the subscripts. Moreover one easily verifies that these actions and the translations by the twotorsion sections together generate a ℚ-rational action s : τ ↦ s_{τ} of S_{4} on E/Y, indexed by the induced permutation of the four sections P_{1}, P_{2}, P_{3} and P_{4} = O. The translation t_{P_{1}} thus corresponds to the automorphism s_{(14)(23)}, which together with s_{(12)} and s_{(23)} generates the action.

Remark. The elliptic curve E over Y is, in fact, the characteristic zero fiber of the universal elliptic curve E(2, ω) with full level-two-structure and an invariant regular non-zero relative differential over the corresponding fine moduli space Y (2, ω), as follows from a construction along the lines of [KM, Chapter 2]. The universal differential on E is just the differential ω = dx/2y, and the S_{3}-actions on E and Y correspond to the action of SL_{2}(𝔽_{2}) ≅ S_{3} on the level-two-structure in the moduli problem, fixing ω.

As opposed to Mazur’s proof in [M], we do not have to appeal to the universal property of E in our argument, since we can ensure the existence of rational points on certain twists of E by Lemma 2 below instead of having to make use of the local-to-global principle. We mention, however, that the birational map to ℙ^{3} introduced in the next subsection is a good starting point for constructing “nice” compactifications of E(2, ω), which will be the topic of a subsequent article.

2.2 A birational map to ℙ^{3}

Define the rational map π : E −−→ ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} over ℚ by the equation

(X_{0} : X_{1} : X_{2} : X_{3}) = (ty : (x − 1)(x − λ) : x(x − λ) : x(x − 1)). (3)

π is, in fact, birational: In order to construct the inverse rational map ρ : ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} −−→ E, note that x(x − λ) − (x − 1)(x − λ) = x − λ, so that x = \frac{X_{2}}{X_{2}−X_{1}} , and y = x \frac{(x−1)(x−λ)}{ty} = x \frac{X_{1}}{X_{0}} = \frac{X_{1}X_{2}}{X_{0}(X_{2}−X_{1})} ; moreover,

t = \frac{(ty)^{2}}{x(x−1)(x−λ)} = \frac{X_{0}^{2}}{X_{3}(X_{2}−X_{1})} , and it follows that λ = \frac{X_{2}(X_{3}−X_{1})}{X_{3}(X_{2}−X_{1})} .

The choice of this particular map to ℙ^{3} is justified by the following

Lemma 1 The action s_{τ} of S_{4} on E from the previous subsection descends under π to a ℚ-rational action p_{τ} on ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3}.

Proof. We will give formulae for the automorphisms of ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} induced by the automorphisms s_{(12)}, s_{(23)} and s_{(14)(23)} = t_{P_{1}}; since they generate the S_{4}-action on E, this establishes the result.

The map π ∘ s_{(14)(23)} : E −−→ ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} can be given rationally by

(X_{0} : X_{1} : X_{2} : X_{3})
= ( \frac{−λty}{x^{2}} : ( \frac{λ}{x} − 1)( \frac{λ}{x} − λ) : \frac{λ}{x} ( \frac{λ}{x} − λ) : \frac{λ}{x} ( \frac{λ}{x} − 1))
= (ty : (x − 1)(λ − x) : λ(x − 1) : x − λ),

as follows from equations (3) and (2). A simple calculation now shows that π∘s_{(14)(23)} = p_{(14)(23)}∘π, where p_{(14)(23)} : ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} → ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} is the automorphism of projective three-space given by

p_{(14)(23)} : (X_{0} : X_{1} : X_{2} : X_{3}) ↦ (X_{0} : −X_{1} : X_{3} − X_{1} : X_{2} − X_{1}). (4)

Similarly we obtain that s_{(12)} and s_{(23)} descend to automorphisms p_{(12)} and p_{(23)} of ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} given by

p_{(12)} : (X_{0} : X_{1} : X_{2} : X_{3}) ↦ (X_{0} : X_{2} : X_{1} : X_{3}),  (5)
p_{(23)} : (X_{0} : X_{1} : X_{2} : X_{3}) ↦ (X_{0} : X_{1} : X_{3} : X_{2}).  (6)

Remark. It can be shown without difficulty that the rational map π can be extended to a morphism on all of E, which collapses the four two-torsion sections to a point each and is an isomorphism elsewhere. Moreover, π embeds the elliptic curve E_{λ_{0},t_{0}} (for specializations λ_{0} ≠ 0, 1 of λ and t_{0} ≠ 0 of t in equation (1)) into ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} as a curve of degree four, as the intersection of the two quadrics

λ_{0}X_{3}(X_{2} − X_{1}) = X_{2}(X_{3} − X_{1})
t_{0}X_{3}(X_{2} − X_{1}) = X_{0}^{2}.

In terms of linear systems, this embedding corresponds to the full set of global sections of the line bundle L(O + P_{1} + P_{2} + P_{3}) on E_{λ_{0},t_{0}}.

2.3 Twists of E

Let K be a number field with algebraic closure K̄ and absolute Galoisgroup G_{K}, and let Y_{K}, E_{K} and ℙ_{K}^{3} denote the base changes to K of the varieties Y, E and ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3}, respectively. Similarly we write m_{K} : E_{K} → Y_{K} and π_{K} : E_{K} −−→ ℙ_{K}^{3} for the base changes of the structure map m : E → Y and of the rational map π from the previous subsection.

Suppose V is a two-dimensional vector space over 𝔽_{2} with a continuous G_{K}-action, giving rise to a continuous representation

ρ_{V} : G_{K} → GL_{2}(𝔽_{2}) ≅ SL_{2}(𝔽_{2}) ≅ S_{3},

uniquely determined up to conjugacy (depending on the choice of basis of V). Viewing SL_{2}(𝔽_{2}) as a subgroup of AutK̄(Y_{K}) and of AutK̄(E_{K}), the homomorphism ρ_{V} induces two cocycles c_{Y_{K} ,V} ∈ H^{1}(G_{K}, AutK̄(Y_{K})) and c_{E_{K} ,V} ∈ H^{1}(G_{K}, AutK̄(E_{K})). Denote by Y_{K,V} and E_{K,V} the twists of Y_{K} and E_{K} by c_{Y_{K} ,V} and c_{E_{K} ,V}, respectively (for twisting quasi-projective varities cf. [Ser2, Section III.1.3.]). Since these twists are compatible with the structure map and the zero-section, E_{K,V} is an elliptic curve over Y_{K,V} with structure map m_{K,V} : E_{K,V} → Y_{K,V}, say; moreover E_{K,V} /Y_{K,V} has three non-trivial two-torsion sections defined over the Galois extension of K corresponding to the kernel of ρ_{V}, and E_{K,V} [2] ≅ V ×_{K} Y_{K,V} as Y_{K,V} -group schemes (where we view V as a group scheme over K).

2.4 Twists of ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3}

The action of S_{4} on ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} defined in Lemma 1 gives rise, by base change to a number field K, to a K-rational action of S_{4} on the projective three-space ℙ_{K}^{3}. We will need the following

Lemma 2 Any twist of ℙ_{K}^{3} by the action of S_{4} is trivial.

Proof. From the theory of twisting algebraic varieties we have a bijection between the set of twists of ℙ_{K}^{3} and the (pointed) set H^{1}(G_{K}, P GL_{4}(K̄)). We will show that the S_{4}-action on ℙ_{K}^{3} induced by the action on E can be lifted to a homomorphism h : S_{4} → GL_{4}(K); using the fact that the cohomology H^{1}(G_{K}, GL_{4}(K̄)) is trivial (see, e. g., [Ser1, Section X.1.]) we deduce that the push-forward to H^{1}(G_{K}, P GL_{4}(K̄)) of any cocycle in H^{1}(G_{K}, S_{4}) = Hom(G_{K}, S_{4}) is zero, and the result follows.

To this end we define new homogeneous coordinates (Y_{1} : Y_{2} : Y_{3} : Y_{4}) for ℙ_{(X_{0}:X_{1}:X_{2}:X_{3})}^{3} by setting

Y_{1} := \frac{1}{4} (X_{0} − 3X_{1} + X_{2} + X_{3})
Y_{2} := \frac{1}{4} (X_{0} + X_{1} − 3X_{2} + X_{3})
Y_{3} := \frac{1}{4} (X_{0} + X_{1} + X_{2} − 3X_{3})
Y_{4} := \frac{1}{4} (X_{0} + X_{1} + X_{2} + X_{3}).

Straightforward calculations using equations (4), (5) and (6) then show that in terms of the new coordinates, the automorphisms p_{τ} are given by

p_{τ} : (Y_{1} : Y_{2} : Y_{3} : Y_{4}) ↦ (Y_{τ(1)} : Y_{τ(2)} : Y_{τ(3)} : Y_{τ(4)})

for τ = (14)(23), (12) and (23), and hence for all τ ∈ S_{4}. The base change to K now produces the desired lift h : S_{4} → GL_{4}(K).

3 Proof of the theorem

We now have all the ingredients ready for the proof of Theorem 1:

The first part of Mazur’s argument for visualizing three-torsion elements of � (E/K) in abelian surfaces goes through for two-torsion elements of the Weil-Châtelet group without further changes. Given the elliptic curve E over K and a lift η ∈ H^{1}(G_{K}, E[2]) of the original element σ ∈ H^{1}(G_{K}, E)[2] under the surjection H^{1}(G_{K}, E[2]) → H^{1}(G_{K}, E)[2], suppose we have another elliptic curve F such that there is an isomorphism E[2] ≅ F [2] of G_{K}-modules and such that η is sent to zero under the map H^{1}(G_{K}, E[2]) ≅ H^{1}(G_{K}, F [2]) → H^{1}(G_{K}, F)[2]. Then the Galois cohomology computation described in [M] shows that σ is visible in the abelian surface A fitting into the exact sequence

0 → Φ → E × F → A → 0,

where Φ ≅ E[2] ≅ F [2] is embedded into E × F along the diagonal.

Noting that the elliptic curve E and the cohomology class σ only appear in these considerations trough the Galois-module E[2] and the lift η ∈ H^{1}(G_{K}, E[2]), Theorem 1 now follows directly from

Proposition 1 Let K be a number field with absolute Galois group G_{K}. Let V be a two-dimensional vector-space over 𝔽_{2} on which G_{K} acts continuously, and let η ∈ H^{1}(G_{K}, V) be any element of its first Galois cohomology. Then there are infinitely many elliptic curves F over K such that V ≅ F [2] as G_{K}-modules, and such that η is sent to zero under the map

H^{1}(G_{K}, V) → H^{1}(G_{K}, F)

on cohomology induced by the embedding V ≅ F [2] ↪ F.

Proof. Recall the elliptic curve E_{K,V} over Y_{K,V} defined in Section 2.3: Since E_{K,V} [2] ≅ V ×_{K} Y_{K,V} as Y_{K,V} -group schemes, η ∈ H^{1}(G_{K}, V) induces a cohomology class η′ ∈ H^{1}(G_{K}, AutK̄(E_{K,V})) and thus a twist { \atop E_{K,V}} of E_{K,V}, along with a morphism { \atop m_{K,V}} : { \atop E_{K,V}} → Y_{K,V} over K.

Suppose we have a K-rational point P on { \atop E_{K,V}} over the K-rational point Q = { \atop m_{K,V}} (P) on Y_{K,V}, say. The fiber of E_{K,V} over Q is an elliptic curve F over K whose two-torsion F [2] is isomorphic to V as G_{K}-modules. It follows, moreover, from the construction of { \atop E_{K,V}} that its fiber over Q is the twist of F by the image of the cocycle η under the map H^{1}(G_{K}, V) ≅ H^{1}(G_{K}, F [2]) → H^{1}(G_{K}, F)[2]. This twist, however, has the K-rational point P on it and is hence trivial – so the image of η in H^{1}(G_{K}, F) must be zero, and F has all the properties we demand! In order to ensure the existence of infinitely many such curves F it thus suffices to show that the K-rational points are Zariski-dense on { \atop E_{K,V}}.

From our construction the variety { \atop E_{K,V}} can be seen to be a twist of E_{K} by the K-rational action of S_{4} defined in Section 2.1, and we showed in Lemma 1 that this action descends under π_{K} to a K-rational S_{4}-action on ℙ_{K}^{3}. We thus obtain an induced twist ℙ̃_{K}^{3} of ℙ_{K}^{3}, together with a birational map π̃_{K} : { \atop E_{K,V}} −−→ ℙ̃_{K}^{3}. But it follows from Lemma 2 that ℙ̃_{K} is isomorphic over K to projective three-space, and the K-rational points are hence dense on it. Consequently the same holds for the birationally equivalent variety { \atop E_{K,V}}, and we deduce the result.

References

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[M] Mazur, B.: Visualizing elements of order three in the Shafarevich-Tate group, Asian J. Math. 3, No. 1 (1999), 221 – 232

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[Ser2] Serre, J.-P.: Cohomologie Galoisienne (fifth ed.), Lecture Notes in Mathematics 5, Springer-Verlag, New York, 1986