VISIBILITY OF 4-COVERS OF ELLIPTIC CURVES

NILS BRUIN AND TOM FISHER

Abstract. Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in ℙ^{3}. Let E′ be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover C′ of E′ with the property that C and C′ are represented by the same cohomology class on the 4-torsion. In fact we give equations for C′ as a curve of degree 8 in ℙ^{5}.

We also study the K3-surfaces fibred by the curves C′ as we vary E′. In particular we show how to write down models for these surfaces as complete intersections of quadrics in ℙ^{5} with exactly 16 singular points. This allows us to give examples of elliptic curves over ℚ that have elements of order 4 in their Tate-Shafarevich group that are not visible in a principally polarized abelian surface.

1. Introduction

Let E and E′ be elliptic curves over a field k that are n-congruent, meaning that there is an isomorphism of k-group schemes σ: E[n] → E′[n]. We suppose that the characteristic of k does not divide n. We may use σ to transfer certain interesting arithmetic information between E and E′. For instance let k be a number field. An n-torsion element of the Tate-Shafarevich group � (E/k) may be represented by a class ξ ∈ H^{1}(k, E[n]). Let σ_{∗} : H^{1}(k, E[n]) → H^{1}(k, E′[n]) be the isomorphism induced by σ. It might happen that while ξ maps to a nontrivial element in � (E/k)[n], represented say by a curve C, the image of σ_{∗}(ξ) in H^{1}(k, E′), represented say by a curve C′, could be trivial. This is an example of Mazur’s concept of visibility (see [9, 15]): the graph Δ ⊂ E[n] × E′[n] of the n-congruence σ provides an isogeny E × E′ → (E × E′)/Δ and a model for C arises as the fibre of (E × E′)/Δ over a point in E′(k) that bears witness to the triviality of C′.

The case n = 2 is relatively special, because quadratic twists have isomorphic 2-torsion. It is true, however, that given any ξ ∈ H^{1}(k, E[2]), one can find another elliptic curve E′ with isomorphic 2-torsion such that σ_{∗}(ξ) ∈ H^{1}(k, E′[2]) represents a trivial homogeneous space under E′; see [4,14].

Date: 23rd January 2017.

2010 Mathematics Subject Classification. 11G05, 11G35, 14H10.

Key words and phrases. elliptic curves, Tate-Shafarevich groups, Mazur visibility, descent, K3 surfaces, local-global obstructions.

In order to determine if a given class ξ ∈ H^{1}(k, E[n]) can be made visible using an n-congruence, one can proceed in three steps. We assume n > 2.

(i) One parametrizes the elliptic curves n-congruent to E. This amounts to determining an appropriate twist X_{E}(n) of the modular curve of full level n. (We make no reference to the Weil pairing in the definition of X_{E}(n), so geometrically this curve has ϕ(n) components, where ϕ is Euler’s totient function.)

(ii) If n > 2 then X_{E}(n) is a fine moduli space and there is a universal elliptic curve E_{t} over X_{E}(n). One constructs a fibred surface S_{E,ξ}(n) → X_{E}(n) whose fibres are the n-covers of E_{t} corresponding to ξ ∈ H^{1}(k, E_{t}[n]).

(iii) If one can find a rational point P on S_{E,ξ}(n), and none of the cusps of X_{E}(n) are rational points, then ξ can be made visible by taking E′ to be the elliptic curve corresponding to the moduli point on X_{E}(n) below P. On the other hand, if S_{E,ξ}(n) has no rational points then ξ cannot be made visible using an elliptic curve n-congruent to E.

One can classify the n-congruence σ by the effect it has on the Weil pairing. If n = 3, 4, it can either preserve or invert it. Correspondingly, the modular curve X_{E}(n) has two components X_{E}^{+}(n) and X_{E}^{−}(n). Note that there is a tautological point on X_{E}^{+}(n) corresponding to E itself, and the fibre of S_{E,ξ}^{+}(n) → X_{E}^{+}(n) above this point represents the image of ξ in H^{1}(k, E). If this image lies in � (E/k) then the fibre, and hence also S_{E,ξ}^{+}(n) itself, has points everywhere locally. Mazur uses this in [15] to show that any element of � (E/k) of order 3 can be made visible using an elliptic curve 3-congruent to E. Indeed he shows that S_{E,ξ}^{+}(3) is a blow-up of a twist of ℙ^{2}, and hence satisfies the local-to-global principle.

From a computational point of view, it is attractive if (E×E′)/Δ can be realized as a Jacobian, or more generally, admits a principal polarization. It naturally does so if we start with σ inverting the Weil pairing. Again taking n = 3, one can show that S_{E,ξ}^{+}(3) is birational to ℙ^{2} over k if and only if the same is true for S_{E,ξ}^{−}(3). It follows (see [5]) that any element of � (E/k) of order 3 is visible in the Jacobian of a genus 2 curve.

For larger n there are major obstacles to this kind of visibility over number fields. Once n ≥ 6 the components of X_{E}(n) have positive genus, and so rational points are rare: the set of candidate elliptic curves E′ is sparse for n = 6 and finite for n ≥ 7. See [13] for explicit examples over ℚ of non-existence of such E′ for n = 6, 7.

In this article we consider the case n = 4. This case is particularly interesting for several reasons:

(i) The curve X_{E}^{+}(4) is of genus 0, but the surface S_{E,ξ}^{+}(4) is a K3-surface. Much is conjectured, but little is known about the rational points on K3surfaces.

(ii) Given ξ ∈ H^{1}(k, E[n]), representing a genus 1 curve of degree n, there is another fibred surface T_{E,ξ}(n) → X_{E}(n) whose fibres are n-covers of E_{t}, but now sharing the same action of E[n] on a suitable linear system. For n = 3, 4, 5, explicit invariant-theoretic constructions of these surfaces are given in [11], [12]. When n is odd the surfaces S_{E,ξ}^{±}(n) and T_{E,ξ}^{±}(n) are the same, but when n is even a correction to this idea is needed.

Taking n = 4, we may identify X_{E}^{±}(4) ≅ ℙ^{1}. We start with D = {Q_{1} = Q_{2} = 0} ⊂ ℙ^{3} a quadric intersection representing ξ ∈ H^{1}(k, E[4]). The invariant theory in [11] allows us to write down T_{E,ξ}^{±}(4) → ℙ^{1} as a family of quadric intersections in ℙ^{3}. However the fibres of S_{E,ξ}^{±}(4) → ℙ^{1} cannot always be written as quadric intersections. In this article we show how to write down a singular model for S_{E,ξ}^{±}(4) as a complete intersection of quadrics in ℙ^{5}. On this model, the nonsingular fibres are genus 1 curves of degree 8.

In fact the surfaces S_{E,ξ}^{±}(4) and T_{E,ξ}^{±}(4) are twists of surfaces S(4) and T (4) that may be defined as follows. Let X(n) be the modular curve whose noncuspidal points parametrise elliptic curves E together with a symplectic isomorphism E[n] ≅ ℤ/nℤ × μ_{n}. We write

E_{t}: y^{2} = x(x − 1)(x − \frac{(1 − t^{2})^{2}}{(1 + t^{2})^{2}} )

for the universal elliptic curve over X(4) ≅ ℙ^{1}. The Shioda modular surface of level 4 is the minimal fibred surface S(4) → X(4) with generic fibre E_{t}. The theta modular surface of level 4 is the minimal fibred surface T (4) → X(4) with generic fibre

D_{t}: \{ \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}.

The relation between the fibres is that D_{t} has Jacobian E_{t}.

We refer to [1] for many interesting facts about the geometry of the surfaces S(4) and T (4). For example, they are both K3-surfaces (in fact Kummer surfaces) with Picard number 20. Working over ℂ, the surface T (4) is isomorphic to the diagonal quartic surface in ℙ^{3}. Moreover the surfaces S(4) and T (4) are related by generically 2-to-1 rational maps (in either direction) but are not birational.

1.1. Outline of the article. In Section 2 we review some of the interpretations of H^{1}(k, E[n]), most notably in terms of n-covers and theta groups. We also define the (twisted) Shioda and theta modular surfaces.

In Section 3 we look at methods for computing 4-covers of elliptic curves. A central notion is that of a second 2-cover: any 4-cover D → E factors through a 2-cover C → E. The cover D → C is referred to as a second 2-cover. In terms of Galois cohomology this corresponds to the sequence

H^{1}(k, E[2]) → H^{1}(k, E[4]) → H^{1}(k, E[2]),

where the first H^{1}(k, E[2]) classifies the cover D → C and the second H^{1}(k, E[2]) classifies C → E.

In Section 3.1 we review classical 4-descent. It has the drawback for us that it only gives those 4-covers with a degree 4 model in ℙ^{3}. In Section 3.2 we describe a variant of this method. In particular, given D → C → E where D has a degree 4 model in ℙ^{3}, we show how to twist the second 2-cover D → C by an arbitrary element of H^{1}(k, E[2]). The new 4-cover has a degree 8 model in ℙ^{7}. However, for our purposes, it is convenient to project this to a curve in ℙ^{5}, still of degree 8.

In Section 4 we quantify the difference between S_{E,ξ}^{±}(n) and T_{E,ξ}^{±}(n) for arbitrary n. Indeed the fibres differ by a cohomology class ν = ν(t), which we call the shift. It was already shown in [7, Lemma 3.11] that the shift is trivial when n is odd. We show that when n is even the shift takes values in H^{1}(k, E[2]).

Section 5 reviews the geometry of the surfaces S(4) and T (4). In Section 6 we describe how to compute the twists of these surfaces so that a prescribed 4-cover D → E appears as a fibre. We assume that D is given as a quadric intersection in ℙ^{3}. We use the invariant theory in [11] to write down the required twist of T (4). By finding an explicit formula for the shift, and then using the method in Section 3.2, we are then able to compute the required twist of S(4).

The methods in Section 6 for computing S_{E,ξ}^{+}(4) and T_{E,ξ}^{+}(4) are modified in Section 7 to compute S_{E,ξ}^{−}(4) and T_{E,ξ}^{−}(4). The arguments here are somewhat simplified by the observation that an elliptic curve and its quadratic twist by its discriminant are reverse 4-congruent.

In Section 9 we give several examples. We exhibit some elliptic curves E/ℚ such that for the elements ξ ∈ H^{1}(ℚ, E[4]) representing elements of order 4 in � (E/ℚ), the surface S_{E,ξ}^{−}(4) has no rational points. We show this by computing an explicit model of the surface and checking that the surface has no p-adic points for some prime p. If E(ℚ)/2E(ℚ) is trivial, it follows that visibility in a surface can only happen via a rational point on S_{E,ξ}^{+}(4). We prove in Proposition 8.2 that if the Galois action on E[4] is large enough, then the resulting abelian surface does not admit a principal polarization.

We note that if ξ ∈ H^{1}(k, E[4]) represents an element in � (E/k) then S_{E,ξ}^{+}(4) has points everywhere locally. Thus, any failure for S_{E,ξ}^{+}(4) to have rational points constitutes a failure of the Hasse Principle. We plan to investigate this possibility further in future work.

2. Preliminaries

2.1. Notation. For a field k, we write k^{sep} for its separable closure. We write t for the generic point on various modular curves we consider. When these curves are isomorphic to ℙ^{1}, then t will instead denote a coordinate on ℙ^{1}. The identity element on an elliptic curve E will be written as either 0 or 0_{E}.

2.2. Geometric interpretations of H^{1}(k, E[n]). Let k be a field of characteristic not dividing n and let E be an elliptic curve over k.

Definition 2.1. An n-cover of E over k is a pair (C, π), where C is a nonsingular complete irreducible curve over k and π: C → E is a morphism such that there exists an isomorphism ψ: (C ×_{k} k^{sep}) → (E ×_{k} k^{sep}) satisfying π = [n] ∘ ψ. Two n-covers (C_{1}, π_{1}) and (C_{2}, π_{2}) are isomorphic over k if there is an isomorphism α : C_{1} → C_{2} over k such that π_{1} = π_{2} ∘ α.

The isomorphism classes of n-covers are naturally parametrized by H^{1}(k, E[n]). This means that given ξ ∈ H^{1}(k, E[n]) there is an n-cover C_{E,ξ} → E, any n-cover is isomorphic to one of this form, and two n-covers are isomorphic if and only if they arise from the same class ξ. Restriction of cocycle classes corresponds to base extending the cover.

If (C, π) is an n-cover of E then C itself is a twist of E (as a curve, not as an elliptic curve). In fact C has the structure of homogeneous space under E, and so represents an element in H^{1}(k, E). The map H^{1}(k, E[n]) → H^{1}(k, E) may be interpretted as forgetting the covering map π. In particular its kernel consists of those n-covers (C, π) for which C(k) is non-empty.

Definition 2.2. A theta group for E[n] is a central extension 0 → 𝔾_{m} → Θ → E[n] → 0 of k-group schemes such that the commutator pairing on Θ agrees with the Weil-pairing on E[n]. An isomorphism of theta groups is an isomorphism of central extensions as k-group schemes.

The lift of the (translation) action of E[n] on the linear system |n.0_{E}| to the Riemann-Roch space L(n.0_{E}) gives a theta group Θ_{E}. If n ≥ 3 then L(n.0_{E}) is the space of global sections of a very ample line bundle L_{E,n}. Choosing a basis for this space provides a map E → ℙ^{n−1} that gives a model for E as an elliptic normal curve of degree n. The theta group Θ_{E} is then the full inverse image in GL_{n} of the group of projective linear transformations describing the action of E[n] on E by translation.

As observed in [7], there is an action of E[n] on Θ_{E} by conjugation, and every automorphism of Θ_{E} arises in this way. Therefore the isomorphism classes of theta groups for E[n], viewed as twists of Θ_{E}, are parametrised by H^{1}(k, E[n]).

Since an n-cover C_{E,ξ} (as a curve) is a twist of E by a cocycle taking values in E[n], we see that C_{E,ξ} comes equipped with a degree n line bundle L_{E,ξ} with a theta group Θ_{E,ξ} acting on it. It may be checked that Θ_{E,ξ} is indeed the twist of Θ_{E} by ξ (in the sense of the last paragraph). The line bundle L_{E,ξ} provides a model of C_{E,ξ}, but now only in an (n − 1)-dimensional Brauer-Severi variety, i.e. a possibly non-trivial twist of ℙ^{n−1}. The k-isomorphism class of the Brauer-Severi variety gives a class in Br(k)[n].

Definition 2.3. We write Ob_{E,n}: H^{1}(k, E[n]) → Br(k) for the map that sends ξ ∈ H^{1}(k, E[n]) to the class of the Brauer-Severi variety corresponding to the global sections of L_{E,ξ}. In particular Ob_{E,n}(ξ) is trivial if and only if C_{E,ξ} admits a degree n model in ℙ^{n−1} with L_{E,ξ} the pull back of O(1). In later sections we write Ob_{n}(C_{E,ξ}) = Ob_{E,n}(ξ).

It is shown in [7] that Ob_{E,n}(ξ) is determined by the isomorphism class of Θ_{E,ξ} (as a theta group for E[n]) without reference to E itself.

2.3. Twists of full level modular curves. An n-congruence between two elliptic curves E, E′ over k is an isomorphism of k-group schemes σ: E[n] → E′[n]. We only concern ourselves with the case that the characteristic of k does not divide n. The torsion subgroup scheme E[n] of an elliptic curve comes equipped with a Weil pairing

e_{n}: E[n] × E[n] → μ_{n}.

When classifying n-congruences one should take into account what happens to the Weil pairing. The following proposition is trivial to prove and shows that Aut(μ_{n}) ≅ (ℤ/nℤ)^{×} classifies the possible types of n-congruences.

Proposition 2.4. Let σ: E[n] → E′[n] be an n-congruence. Then there exists τ_{σ} ∈ Aut(μ_{n}) such that

e_{n} ∘ (σ × σ) = τ_{σ} ∘ e_{n}.

We say that σ is a direct n-congruence if τ_{σ}(ζ) = ζ, and that σ is a reverse n-congruence if τ_{σ}(ζ) = ζ^{−1}.

In our case, for n = 4, any n-congruence is either direct or reverse.

Fixing an elliptic curve E, we consider the moduli space Y_{E}^{+}(n)(k) of pairs (E′, σ), where σ: E[n] → E′[n] is a direct n-congruence. This moduli space is represented by a curve Y_{E}^{+}(n) over k, whose non-singular completion X_{E}^{+}(n) is a twist of the modular curve X(n) of full level n. Similarly, we write X_{E}^{−}(n) for the twist of X(n) corresponding to the moduli space of pairs (E′, σ) where σ: E[n] → E′[n] is a reverse n-congruence.

If σ is a direct or reverse n-congruence, then the automorphism τ_{σ} from Proposition 2.4 can be extended to an automorphism 𝔾_{m} → 𝔾_{m}. In this case, we see that σ provides a way of comparing theta groups.

Definition 2.5. Let σ: E[n] → E′[n] be a direct or reverse n-congruence. Given theta groups Θ, Θ′ for E[n] and E′[n], we say a morphism ψ: Θ → Θ′ (as k-group schemes) is a σ-isomorphism if it makes the diagram below commutative.

\xymatrix{0 \ar[r] & 𝔾_{m} \ar[r] \ar[d]^{τ_{σ}} & Θ \ar[d]^{ψ} \ar[r] & E[n] \ar[d]^{σ} \ar[r] & 0 \\ 0 \ar[r] & 𝔾_{m} \ar[r] & Θ′ \ar[r] & E′[n] \ar[r] & 0}

If σ is a direct n-congruence, this is the normal notion of isomorphism for theta groups upon identifying E[n] and E′[n] via σ.

2.4. Shioda modular surfaces. For n ≥ 3, the moduli space Y_{E}^{+}(n) is fine, so there is a universal elliptic curve E_{t} with a direct n-congruence σ_{t}: E[n] → E_{t}[n] over Y_{E}^{+}(n). The relevant property for us is that any direct n-congruence between E and another elliptic curve can be obtained by specializing t at the relevant moduli point on Y_{E}^{+}(n). We write S_{E}^{+}(n) → X_{E}^{+}(n) for the minimal fibred surface with generic fibre E_{t}. This is a twist of Shioda’s modular surface of full level n.

Given ξ ∈ H^{1}(k, E[n]), we can twist this surface further. Writing k_{t} for the function field of X_{E}^{+}(n), we view ξ as an element of H^{1}(k_{t}, E_{t}[n]) and take the n-cover C_{t,ξ} → E_{t} representing it. Let S_{E,ξ}^{+}(n) → X_{E}^{+}(n) be the minimal fibred surface with generic fibre C_{t,ξ}. This is again a twist of Shioda’s modular surface, and is isomorphic to S_{E}^{+}(n) over the splitting field of ξ. In [15], the surfaces S_{E}^{+}(n) and S_{E,ξ}^{+}(n) are called first and second twists.

We define S_{E,ξ}^{−}(n) similarly, by using a universal reverse n-congruence σ_{t}.

2.5. Theta modular surfaces. Alternatively, given an elliptic curve E over k and ξ ∈ H^{1}(k, E[n]), we base extend Θ_{E,ξ} to a theta group over k_{t}, the function field of X_{E}^{+}(n). Then Θ_{E,ξ} ×_{k} k_{t} = Θ_{E_{t},ξ_{t}} for some ξ_{t} ∈ H^{1}(k_{t}, E_{t}[n]). Just as in Section 2.4, we take the n-cover C_{t,ξ_{t}} → E_{t} representing ξ_{t}. Then we define the theta modular surface to be the minimal fibred surface T_{E,ξ}^{+}(n) → X_{E}^{+}(n) with generic fibre C_{t,ξ_{t}}.

By construction, the fibers of T_{E,ξ}^{+}(n) → X_{E}^{+}(n) are n-covers with a prescribed theta group. Since Ob_{E,n}(ξ) is a function of Θ_{E,ξ}, we see that Ob_{E_{t},n}(ξ_{t}) is the base change of Ob_{E,n}(ξ) to k_{t}. In particular, if Ob_{E,n}(ξ) = 0 then C_{t,ξ_{t}} admits a degree n model in ℙ^{n−1}, with a linear action of E_{t}[n]. In that case, it follows that T_{E,ξ}^{+}(n) is birational to a surface in ℙ^{n−1}, with an action of E[n] through Θ_{E,ξ}. This allows us to use invariant theory to write down models of T_{E,ξ}^{+}(n).

Interestingly enough, ν(t) = ξ_{t}−ξ is not necessarily trivial. In fact, Theorem 4.1 proves that ν(t) is 2-torsion (as is seen by applying the result to the elliptic curves E ×_{k} k_{t} and E_{t}). In particular, if n is odd then ν(t) = 0 and the surfaces S_{E,ξ}^{+}(n) and T_{E,ξ}^{+}(n) are isomorphic.

We define T_{E,ξ}^{−}(n) similarly, by reversing the order of multiplication on Θ_{E,ξ}. Since (ℤ/4ℤ)^{×} = {±1}, this is sufficient to define T_{E,ξ}(4).

3. Computing 4-covers

Let E be an elliptic curve and let ξ ∈ H^{1}(k, E[4]). In this section, we write D_{E,ξ} for the corresponding 4-cover of E. We have 2ξ ∈ H^{1}(k, E[2]) and write C_{E,2ξ} for the corresponding 2-cover of E. Note that the 4-cover D_{E,ξ} → E naturally factors as D_{E,ξ} → C_{E,2ξ} → E. It turns out to be advantageous to study 4-covers via this intermediate structure.

Definition 3.1. Let C → E be a 2-cover. A 2-cover of C is a cover D → C such that the composition of covers D → C → E is a 4-cover. If we want to emphasize that D is a 2-cover of a 2-cover, and not of an elliptic curve directly, we say that D → C is a second 2-cover.

In this section we are concerned with finding models of second 2-covers. We fix a 2-cover C → E over a field k with char k ≠ 2, 3. We assume Ob_{E,2}(C) = 0, so that C has a model

(1)  C: Y^{2} = G(X, Z),

where G is a binary quartic form, say

G(X, Z) = aX^{4} + bX^{3}Z + cX^{2}Z^{2} + dXZ^{3} + eZ^{4}.

The classical invariants of G are

(2)  \begin{aligned}I & = 12ae − 3bd + c^{2}, \\ J & = 72ace − 27ad^{2} − 27b^{2}e + 9bcd − 2c^{3}. \end{aligned}

As observed by Weil, a model for E is given by

(3)  E: y^{2} = x^{3} + Ax + B where A = −I/48 and B = −J/1728.

In the next two sections we assume for simplicity that ae ≠ 0.

3.1. Models of 2- and 4-covers with trivial obstruction. In this section we review classical 4-descent, as described in [10, 16, 19, 20] and implemented in Magma [2].

If D → C is a second 2-cover such that Ob_{4}(D) = 0 then D admits a degree 4 model in ℙ^{3}, say

D: Q_{1} = Q_{2} = 0,

where Q_{1}, Q_{2} ∈ k[x] = k[x_{1}, x_{2}, x_{3}, x_{4}] are quadratic forms. Conversely, any such smooth quadric intersection is a 4-cover of an elliptic curve.

Let A_{i} be the symmetric matrix such that Q_{i}(x) = \frac{1}{2}x^{T} A_{i}x. Then the intermediate 2-cover has a model (1) in weighted projective space, with G(X, Z) = det(XA_{1} + ZA_{2}). In particular Ob_{2}(C) = 0. Let T_{1}, T_{2} ∈ k[x] be the quadratic forms given by T_{i}(x) = \frac{1}{2}x^{T} B_{i}x where

adj((adj A_{1})X + (adj A_{2})Z) = a^{2}A_{1}X^{3} + aB_{1}X^{2}Z − eB_{2}XZ^{2} + e^{2}A_{2}Z^{3}.

Then the covering map D → C is given by (X : Z : Y) = (T_{1} : T_{2} : J) where

J = (1/4)\frac{∂(Q_{1}, Q_{2}, T_{1}, T_{2})}{∂(x_{1}, x_{2}, x_{3}, x_{4})}.

Let F = k[θ] be the étale algebra over k generated by a root θ of g(x) = G(x, 1). A generic calculation shows that the quadratic form

(4)  Ξ = θ^{−1}eQ_{1} + T_{1} − θT_{2} + θ^{2}aQ_{2}

in F [x] has rank 1 and satisfies aN_{F/k}(Ξ) = J^{2}. Specialising at any sufficiently general x ∈ k^{4} shows that there exist α ∈ F^{×} and r ∈ k^{×} with N_{F/k}(α) = ar^{2}.

Moreover the class of α in F^{×}/F^{×2}k^{×} only depends on the isomorphism class of the 2-cover D → C.

Conversely, given a 2-cover C of the form (1), we can construct its 2-covers D with Ob_{4}(D) = 0 in the following way. Let F = k[θ] as above, and suppose that α ∈ F^{×} and r ∈ k^{×} satisfy N_{F/k}(α) = ar^{2}. We consider the equation

(5)  α(X − θZ) = (x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3})^{2}.

Expanding in powers of θ gives 4 equations of the form

(linear form in X and Z) = (quadratic form in x_{1}, …, x_{4}).

Taking the norm N_{F/k} and then extracting a square root gives, upon choosing the sign of r, a further equation

rY = N_{F/k}(x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3}).

Taking linear combinations of these equations gives expressions for X, Y, Z in terms of x_{1}, …, x_{4}, and two further quadratic equations in x_{1}, …, x_{4} only. These define a 2-cover D_{α} → C. Moreover the isomorphism class of this 2-cover only depends on the class of α in F^{×}/F^{×2}k^{×}.

The two constructions just presented are inverse to one another. We thus obtain the following proposition. In stating it we use our freedom to multiply G(X, Z) by a square to reduce to the case r = 1.

Proposition 3.2. Let C → E be a 2-cover. If there is a second 2-cover D → C with Ob_{4}(D) = 0 then C has a model of the form Y^{2} = N_{F/k}(α(X−θZ)). Moreover if C takes this form then the collection of all 2-covers D of C with Ob_{4}(D) = 0 is given by

(6)  ker(N_{F/k}: F^{×}/F^{×2}k^{×} → k^{×}/k^{×2})

via the map δ ↦ D_{αδ}.

Remark 3.3. (i) Strictly speaking we should specify a choice of square root of N_{F/k}(δ), otherwise the 2-covers D_{E,ξ} and D_{E,−ξ} of C_{E,2ξ}, differing by the automorphism Y ↦ −Y of C_{E,2ξ}, cannot be distinguished. (ii) Let g′(x) be the derivative of g(x) = G(x, 1). It is sometimes convenient to write the equations for D_{α} as

(7)  tr_{F/k} (\frac{x^{2}}{αg′(θ)}) = tr_{F/k} (\frac{θx^{2}}{αg′(θ)}) = 0,

where x = x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3}. (iii) The group (6) may be identified with a certain subgroup of H^{1}(k, E[2])/⟨2ξ⟩ where 2ξ is the class of the 2-cover C → E. See [10] for further details.

3.2. Models for twists of second 2-covers. Let D → C be a second 2-cover with Ob_{4}(D) = 0. By Proposition 3.2 we may assume that C has a model

(8)  C : Y^{2} = G(X, Z) = N_{F/k}(α(X − θZ)),

and that D = D_{α}. In this section we are interested in models for any 2-cover of C; not just the ones with trivial obstruction (as a 4-cover). These covers are parametrized by H^{1}(k, E[2]).

We recall that C is a 2-cover of the elliptic curve E: y^{2} = f(x) = x^{3} + Ax + B given by (3). Let L = k[φ] be the étale algebra over k generated by a root φ of f(x). It is well known that

(9)  H^{1}(k, E[2]) ≅ ker(N_{L/k}: L^{×}/L^{×2} → k^{×}/k^{×2}).

In fact this is the special case of Remark 3.3(iii) with C = E.

The algebra L is related to F in the following way. We have that g(x) = (x − θ)h(x), for some cubic h(x) ∈ F [x]. Then F [x]/(h(x)) = L ⊗_{k} F, which we denote by LF. As an algebra over k, it is obtained by formally adjoining two roots, say θ and θ̃, of g(x). Let σ ∈ Aut_{k}(LF) be the involution that swaps θ and θ̃. We write M for the subalgebra of LF fixed by σ. This is the étale algebra of unordered pairs of roots of g(x) and it has degree [M : k] = 6. We may identify L as a subalgebra of M via

(10)  φ = −(aθθ̃ − c/3 + e/(θθ̃))/4.

We fix a basis m_{1}, …, m_{6} for M over k, and put α̃ = σ(α). Let ν ∈ L^{×} with N_{L/k}(ν) = s^{2} for some s ∈ k^{×}. We show how to construct a twist D_{ν} → C of D → C. This will turn out to be the twist by the element of H^{1}(k, E[2]) corresponding to ν under the isomorphism (9).

We consider the equation

(11) N_{LF/M}(α(X − θZ)) = αα̃(X − θZ)(X − θ̃Z) = ν(y_{1}m_{1} + … + y_{6}m_{6})^{2}.

Expanding and taking coefficients of m_{1}, …, m_{6} gives 6 equations of the form

(quadratic form in X and Z) = (quadratic form in y_{1}, …, y_{6}).

Taking the norm N_{M/L} and then extracting a square root gives

(12)  Y = ν N_{M/L}(y_{1}m_{1} + … + y_{6}m_{6})

and hence 3 equations of the form

(linear form in Y) = (quadratic form in y_{1}, …, y_{6}).

Taking linear combinations to eliminate X^{2}, XZ, Z^{2} and Y leaves 5 quadratic forms in y_{1}, …, y_{6}. These define D_{ν} ⊂ ℙ^{5}, a genus 1 curve of degree 8. In fact D_{ν} is a 2-cover of C. Equations for the covering map D_{ν} → C may be computed as follows. Again starting from (11), we take the norm N_{LF/F} and then extract a square root to give

±α(X − θZ)Y = s N_{LF/F} (y_{1}m_{1} + … + y_{6}m_{6}).

The sign choice here may be absorbed into the cubic norm. Using (12) to eliminate Y and then cancelling the common factor y_{1}m_{1} + … + y_{6}m_{6} gives 12 equations of the form

(bilinear form in X, Z and y_{1}, …, y_{6}) = (quadratic form in y_{1}, …, y_{6}).

These equations together with (12) define the covering map D_{ν} → C.

Remark 3.4. (i) If we just eliminate Y from the 6+3+12 equations listed above, then we get 20 quadrics in X, Z, y_{1}, …, y_{6}. These define a genus 1 curve embedded in ℙ^{7} via a complete linear system of degree 8. However we will see that working with D_{ν} ⊂ ℙ^{5} has some advantages. (ii) Taking ν = 1 gives a 2-cover D_{1} → E that is isomorphic to D → C. Indeed on comparing (5) and (11) we see that an isomorphism is given by

y_{1}m_{1} + ⋯ + y_{6}m_{6} = (x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3})(x_{1} + x_{2}θ̃ + x_{3}θ̃^{2} + x_{4}θ̃^{3}).

In fact the y_{i} span the same space as the 2×2 minors of the 2×4 matrix of partial derivatives of the quadratic forms defining D. (iii) A generic calculation shows that D_{ν} ⊂ ℙ^{5} has degree 8 and its homogeneous ideal is (minimally) generated by 5 quadrics and 2 cubics. However the 5 quadrics are sufficient to define the curve set-theoretically. (iv) Let z ∈ L^{×} correspond under the isomorphism (9) to the class of C → E. By [6, Equation (3.1)] we have z ∈ M^{×2}, and so z is a Kummer generator for the quadratic extension M/L. Absorbing z into the squared factor on the right of (11) we see that D_{ν} and D_{νz} are isomorphic as curves. However as 2-covers of C they differ by the automorphism Y ↦ −Y.

We prove an analogue of Proposition 3.2.

Proposition 3.5. Let C be the 2-cover (8). Then the collection of all 2-covers of C is given by ker(N_{L/k}: L^{×}/L^{×2} → k^{×}/k^{×2}) via the map ν ↦ D_{ν}.

Proof: Let η ∈ H^{1}(k, E[2]) map to the class of ν ∈ L^{×} under the isomorphism (9). To prove the proposition, we show that D_{ν} → C is the twist of D_{1} → C by η.

Let \overline{L} = L ⊗_{k} k^{sep}. Note that L is the coordinate ring of E[2] \ \{0_{E}\}, so \overline{L} = Map(E[2](k^{sep}) \ \{0_{E}\}, k^{sep}). There is a homomorphism of Galois modules

w : E[2] → μ_{2}(\overline{L})
S ↦ (T ↦ e_{2}(S, T)).

Noting that M is an L-algebra and a 6-dimensional k-vector space, we see that L^{×} acts linearly on ℙ(M) = ℙ^{5}. In particular S ∈ E[2] acts on ℙ^{5} via w(S) and this action restricts to D_{1}.

If g(x) has roots θ_{1}, …, θ_{4} then k^{sep}(D_{1}) = k^{sep}(C)(\sqrt{f_{12}}, \sqrt{f_{13}}) where f_{ij} = (X − θ_{i}Z)(X − θ_{j}Z)/Z^{2}. Since D_{1} is a homogeneous space under E, there is an action of E[2] on D_{1}. This is given by \sqrt{f_{12}} → ±\sqrt{f_{12}} and \sqrt{f_{13}} → ±\sqrt{f_{13}}. It follows from the definition of the Weil pairing that this action agrees with the one defined in the last paragraph.

Let η be represented by the cocycle σ ↦ η_{σ}. Then we have w(η_{σ}) = σ(γ)γ^{−1} and ν = γ^{2} for some γ ∈ \overline{L}. There is a commutative diagram

[figure: D_{ν}; ·γ; D_{1}; C; C]

Therefore D_{ν} → C is the twist of D_{1} → C via the cocycle σ ↦ σ(γ)γ^{−1}, and this completes the proof. □

4. Theta groups and the shift

In this section we prove the following theorem. We work over a field k of characteristic not dividing n.

Theorem 4.1. Let E, E′ be elliptic curves over k with a direct or a reverse ncongruence σ: E′[n] → E[n]. Then there exists ν ∈ H^{1}(k, E[n]), depending only on E, E′, σ, such that for any ξ ∈ H^{1}(k, E[n]) and ξ′ ∈ H^{1}(k, E′[n]) we have

(i) 2ν = 0 (in particular, if n is odd then ν = 0).

(ii) ξ = σ_{∗}(ξ′) + ν if and only if Θ_{E,ξ} and Θ_{E′,ξ′} are σ-isomorphic.

Proof: We start by comparing the trivial theta groups Θ_{E}, Θ_{E′} ⊂ GL_{n}. There is an isomorphism ψ defined over k^{sep} making the following diagram commute

(13)  \xymatrix{0 \ar[r] & 𝔾_{m} \ar[r] \ar[d]^{τ_{σ}} & Θ_{E′} \ar[r] \ar[d]^{ψ} & E′[n] \ar[r] \ar[d]^{σ} & 0 \\ 0 \ar[r] & 𝔾_{m} \ar[r] & Θ_{E} \ar[r] & E[n] \ar[r] & 0.}

Then Θ_{E′} is the twist of Θ_{E} by the cocycle ρ ↦ ρ(ψ)ψ^{−1}. This cocycle takes values in Aut(Θ_{E}) ≅ Hom(E[n], 𝔾_{m}) ≅ E[n], and so gives a class ν ∈ H^{1}(k, E[n]). If n is odd then ν = 0 by [7, Lemma 3.11]. We now adapt the argument to the case where n is even.

The automorphism [−1] of E lifts to ι ∈ GL_{n}(k). For each T ∈ E[n](k^{sep}) we pick a matrix M_{T} ∈ Θ_{E}(k^{sep}), representing translation by T, such that

(14)  ιM_{T} ι^{−1} = M_{T}^{−1}.

This condition determines the scaling of M_{T} up to a choice of sign. If M_{S+T} = λM_{S}M_{T} then conjugating by ι shows that λ^{2} = e_{n}(S, T) and so

(15)  M_{S+T} = ±e_{n}(S, T)^{1/2}M_{S}M_{T} for all S, T ∈ E[n](k^{sep}).

We claim that M_{T}^{n} = 1. Indeed for a suitable choice of co-ordinates x_{0}, …, x_{n−1}, defined over k^{sep}, we have M_{T} : x_{i} ↦ αζ^{i}x_{i} and ι : x_{i} ↦ βx_{n−i} for some α, β ∈ k^{sep} and ζ ∈ μ_{n}. By (14) we have α^{2} = 1. Since n is even it follows that M_{T}^{n} = 1. This proves the claim.

In exactly the same manner we pick M′_{T} ∈ Θ_{E′}(k^{sep}) for all T ∈ E′[n](k^{sep}). We can then choose ψ so that ψ(M′_{T}) = ±M_{σ(T)} for all T ∈ E′[n](k^{sep}). Indeed if we make this true on a basis for E′[n], then the rest follows by (15) and its analogue for E′.

Let ρ ∈ Gal(k^{sep}/k). Since ι is defined over k, it follows from (14) that

ρ(M_{T}) = ±M_{ρ(T)} for all T ∈ E[n](k^{sep}).

Likewise

ρ(M′_{T}) = ±M′_{ρ(T)} for all T ∈ E′[n](k^{sep}).

The cocycle ρ ↦ ρ(ψ)ψ^{−1} now takes values in Hom(E[n], μ_{2}) = E[2]. Therefore ν is in the image of the natural map H^{1}(k, E[2]) → H^{1}(k, E[n]), and this proves (i).

It also follows that Θ_{E′} is σ-isomorphic to Θ_{E,ν}. For the general statement, we choose a k^{sep}-isomorphism ψ′: Θ_{E′,ξ′} → Θ_{E′}. Then the cocycle ρ ↦ ρ(ψψ′)(ψψ′)^{−1} represents the class σ_{∗}(ξ′) + ν. It follows that Θ_{E′,ξ′} is σ-isomorphic to Θ_{E,σ_{∗}(ξ′)+ν}. This proves (ii). □

5. Geometry of the Shioda and theta modular surfaces

In this section we give two particular models of the Shioda and theta modular surfaces of level 4. Any other Shioda or theta modular surface of level 4 will be a twist of one of these, so these particular models are convenient for studying the geometry of these surfaces.

5.1. The universal elliptic curve of level 4. The Legendre form

y^{2} = x(x − 1)(x − λ)

provides a family of elliptic curves E_{λ} with a prescribed isomorphism (ℤ/2ℤ × μ_{2}) → E_{λ}[2]. The parameter λ gives an isomorphism Y (2) ≃ ℙ^{1}\{0, 1, ∞}. Setting λ = (1 − t^{2})^{2}/(1 + t^{2})^{2} we obtain a family of elliptic curves E_{t} with a prescribed isomorphism (ℤ/4ℤ×μ_{4}) → E_{t}[4]. The parameter t gives an isomorphism Y (4) ≃ ℙ^{1} \ {0, ∞, ±1, ±i} and the expression for λ in terms of t is an explicit realisation of the map Y (4) → Y (2).

5.2. The theta modular surface of level 4. The quadric intersection

(16)  D_{t}: \{ \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}

is a 4-cover of E_{t}. By varying t these curves give a genus 1 fibration on the surface

(17)  x_{0}x_{2}(x_{0}^{2} + x_{2}^{2}) − x_{1}x_{3}(x_{1}^{2} + x_{3}^{2}) = 0.

The action of E_{t}[4] on D_{t} is generated by the transformations x_{ν} ↦ x_{ν+1} and x_{ν} ↦ i^{ν}x_{ν}, where we read the subscripts mod 4. (One convenient way to check an automorphism of a genus 1 curve is geometrically a translation map, is to check it has no fixed points.) The quadric intersections D_{t} therefore all have the same theta group, and so (17) is a model for the theta modular surface T (4).

The surface T (4) is isomorphic to the Fermat quartic {u_{0}^{4}−u_{1}^{4}+u_{2}^{4}−u_{3}^{4} = 0} ⊂ ℙ^{3} via the change of coordinates

(u_{0} : u_{1} : u_{2} : u_{3}) = (x_{0} + x_{2} : x_{0} − x_{2} : x_{1} − x_{3} : x_{1} + x_{3}).

5.3. Shioda’s modular surface of level 4. The relative elliptic curve E_{t}/Y (4) provides us with an open part of the Shioda modular surface. In order to find a suitable completion, we construct E_{t} as a (trivial) 4-cover of itself. The intermediate 2-cover is

(18)  C_{t}: Y^{2} = XZ(X − Z)(X − λZ) with λ = \frac{(1 − t^{2})^{2}}{(1 + t^{2})^{2}} .

A small adaptation (needed since a = e = 0) of the construction in Section 3.2 leads to the second 2-cover given by

(19)  \begin{aligned}(X − Z)(X − λZ) & = y_{1}^{2} & (1 − λ)XZ & = y_{2}^{2} \\ X(X − λZ) & = y_{3}^{2} & λ(X − Z)Z & = y_{4}^{2} \\ (X − λZ)Z & = y_{5}^{2} & X(X − Z) & = y_{6}^{2} \end{aligned}

and

Y = \frac{1 + t^{2}}{2t} y_{1}y_{2} = \frac{1 + t^{2}}{1 − t^{2}} y_{3}y_{4} = y_{5}y_{6}.

Taking linear combinations to eliminate X^{2}, XZ, Z^{2} and Y gives 5 quadrics that define a degree 8 curve in ℙ^{5}:

(20)  D_{t}: \{ \begin{matrix}y_{1}^{2} + y_{4}^{2} − y_{6}^{2} & = 0 \\ y_{2}^{2} − y_{3}^{2} + y_{6}^{2} & = 0 \\ y_{2}^{2} + y_{4}^{2} − y_{5}^{2} & = 0 \\ (1 − t^{2})y_{1}y_{2} − 2ty_{3}y_{4} & = 0 \\ (1 + t^{2})y_{1}y_{2} − 2ty_{5}y_{6} & = 0 \end{matrix}\} ⊂ ℙ^{5}

The first 3 quadrics in (20) define a surface S_{0} ⊂ ℙ^{5} of degree 8 with exactly 16 ordinary double points as singularities (8 defined over ℚ and the remaining 8 over ℚ(i)). Each curve D_{t} passes through all 16 singular points. Furthermore, these are the points with Y = 0, and so make up the fibre of D_{t} → E_{t} above 0_{E_{t}}. In particular, D_{t} has a rational point over 0_{E_{t}}, so D_{t} → E_{t} is indeed the trivial 4-cover. Blowing up the singular points gives Shioda’s modular surface S(4).

The action of E_{t}[2] ≅ ℤ/2ℤ × μ_{2} on C_{t} is generated by

(X : Z : Y) ↦ (X − λZ : X − Z : (λ − 1)Y),
(X : Z : Y) ↦ (λZ : X : −λY),

and the action of E_{t}[4] ≅ ℤ/4ℤ × μ_{4} on D_{t} is generated by

(21)  \begin{matrix}(y_{1} : y_{2} : y_{3} : y_{4} : y_{5} : y_{6}) & ↦ (−y_{2} : y_{1} : −y_{3} : y_{4} : −y_{6} : y_{5}), \\ (y_{1} : y_{2} : y_{3} : y_{4} : y_{5} : y_{6}) & ↦ (iy_{1} : −iy_{2} : y_{4} : y_{3} : y_{6} : y_{5}). \end{matrix}

The subgroup of Pic S(4) invariant under the action (21) was computed in [1], and shown to be free of rank 2 generated by divisor classes I and F, where F is the class of a fibre, and 2I is linearly equivalent to the sum of the 16 sections (i.e. the blow-ups of the 16 singular points on S_{0}). Our surface S_{0} is the image of S(4) under the morphism to ℙ^{5} given by the complete linear system |I + F |.

More generally there is a natural action of G = ASL(ℤ/4ℤ × μ_{4}) on S(4). The corresponding automorphisms of S_{0} are again given by changes of co-ordinates on ℙ^{5}. The surfaces S_{E,ξ}^{±}(4) are twists of S(4) by cocycles taking values in G, and so each must admit a model in a 5-dimensional Brauer-Severi variety. Our calculations in Sections 6 and 7 show that if Ob_{E,4}(ξ) = 0 then this Brauer-Severi variety is trivial. Indeed we show how to write down a model for S_{E,ξ}^{±}(4) as a complete intersection of quadrics in ℙ^{5}.

Remark 5.1. Our calculations also give the genus 1 fibration, but in fact this may be recovered directly from the equations for the surface. Indeed, if we take a complement to the 3-dimensional space of quadrics vanishing on the surface, inside the 6-dimensional space of quadrics vanishing at the singular points, then this defines a map to ℙ^{2} with image a conic. For example, with S_{0} as above, the map is given by (X_{1} : X_{2} : X_{3}) = (y_{1}y_{2} : y_{3}y_{4} : y_{5}y_{6}) and the conic is X_{1}^{2} + X_{2}^{2} = X_{3}^{2}. Parametrising this conic gives the required map to ℙ^{1}.

6. Computing twists of S(4) and T (4) in the direct case

In this section we take ξ ∈ H^{1}(k, E[4]) with Ob_{E,4}(ξ) = 0 and compute models for S_{E,ξ}^{+}(4) and T_{E,ξ}^{+}(4).

6.1. The twisted universal elliptic curve of level 4. Let E/k be the elliptic curve y^{2} = f(x) = x^{3} + Ax + B. If P = (x_{P} , y_{P}) is a point on E then the x-coordinate of 2P is γ(x_{P}) where

γ(x) = (x^{4} − 2Ax^{2} − 8Bx + A^{2})/(4(x^{3} + Ax + B)).

The elliptic curves directly n-congruent to E are parametrised by the non-cuspidal points of X_{E}^{+}(n). This is a twist of the usual modular curve X(n).

Lemma 6.1. The elliptic curves directly 4-congruent to E are

E_{t} : y^{2} = −f(t)(x − γ(t))(x^{3} + Ax + B)

with base point (x, y) = (γ(t), 0), where t is a co-ordinate on X_{E}^{+}(4) ≅ ℙ^{1}, and the original curve E corresponds to t = ∞. Moreover the cusps of X_{E}^{+}(4) are the roots of d(t) = 0 where

d(x) = x^{6} + 5Ax^{4} + 20Bx^{3} − 5A^{2}x^{2} − 4ABx − A^{3} − 8B^{2}

is the 4-division polynomial of E (divided by 2).

Proof: Putting E_{t} in Weierstrass form, and make a change of co-ordinates on X_{E}^{+}(4), gives the family of curves computed by Silverberg. Indeed our parameter t on X_{E}^{+}(4) and the parameter t, here denoted by t_{Silverberg}, in [18, Theorem 4.1] are related by

t = \frac{−(4A^{3} + 27B^{2})}{18ABt_{Silverberg}} − \frac{3B}{2A} .

An alternative proof, also treating the cases j(E) = 0, 1728, and leading to the lemma as stated here, is given in [3, Proposition 7.2]. See also Remark 6.4(i). □

6.2. The twisted theta modular surface of level 4. Let D = {Q_{1} = Q_{2} = 0} ⊂ ℙ^{3} be a quadric intersection with Jacobian E. Then D = D_{E,ξ} for some ξ ∈ H^{1}(k, E[4]) with Ob_{E,4}(ξ) = 0. We use invariant theory to compute a model for T_{E,ξ}^{+}(4) as a quartic surface in ℙ^{3}, together with its genus 1 fibration over X_{E}^{+}(4).

We identify the quadratic forms Q_{1} and Q_{2} with 4 × 4 symmetric matrices A_{1} and A_{2} via Q_{i}(x_{1}, …, x_{4}) = \frac{1}{2}x^{T} A_{i}x. We then define G(X, Z) by

(22) G(X, Z) = det(XA_{1} + ZA_{2}) = aX^{4} + bX^{3}Z + cX^{2}Z^{2} + dXZ^{3} + eZ^{4},

so that D is a 2-cover of C : Y^{2} = G(X, Z). As in Section 3, we assume ae ≠ 0 and let F = k[θ] where θ is a root of g(x) = G(x, 1).

Let T_{1}, T_{2} ∈ k[x] be the quadratic forms defined in Section 3.1. The Hessian, as defined in [11], is an SL_{2} × SL_{4}-equivariant map from the space of quadric intersections to itself. It is given by

(Q_{1}, Q_{2}) ↦ (Q′_{1}, Q′_{2}) = (6T_{2} − cQ_{1} − 3bQ_{2}, 6T_{1} − cQ_{2} − 3dQ_{1}).

Lemma 6.2. The quadric intersection

(23)  D_{t}: {12tQ_{1} + Q′_{1} = 12tQ_{2} + Q′_{2} = 0} ⊂ ℙ^{3}

has intermediate 2-cover (after cancelling a factor 12^{4})

C_{t}: Y^{2} = G_{t}(X, Z) = aN_{F/k}((t + μ)X − (θt + λ)Z),

where

(24)  \begin{aligned}λ & = (6g′(θ) − θg′′(θ))/24 = −(cθ^{2} + 3dθ + 6e)/(12θ), \\ μ & = −g′′(θ)/24 = −(6aθ^{2} + 3bθ + c)/12. \end{aligned}

Proof: In principle this may be checked by a generic calculation. To make the calculation practical we consider the case Q_{1} = ∑_{i=1}^{4} ξ_{i}x_{i}^{2} and Q_{2} = − ∑_{i=1}^{4} ξ_{i}θ_{i}x_{i}^{2}. Then g(X) = 2^{4}(∏_{i=1}^{4} ξ_{i}) ∏_{i=1}^{4}(X − θ_{i}), and (Q_{1}, Q_{2}) has Hessian (Q′_{1}, Q′_{2}) where Q′_{1} = 12 ∑_{i=1}^{4} ξ_{i}μ_{i}x_{i}^{2} and Q′_{2} = −12 ∑_{i=1}^{4} ξ_{i}λ_{i}x_{i}^{2}. Computing the intermediate 2-cover, by the method used in (22), gives the equation for C_{t} as stated. □

Corollary 6.3. Let ξ ∈ H^{1}(k, E[4]) with D_{E,ξ} = {Q_{1} = Q_{2} = 0} ⊂ ℙ^{3}. Then the surface T_{E,ξ}^{+}(4) has a model

{Q_{1}Q′_{2} − Q_{2}Q′_{1} = 0} ⊂ ℙ^{3}

with genus 1 fibration D_{t} as given in Lemma 6.2.

Let I and J be the invariants (2) of the binary quartic G(X, Z). Then E has Weierstrass equation y^{2} = x^{3} + Ax + B where A = −I/48 and B = −J/1728. Let E_{t} be the family of elliptic curves directly 4-congruent to E, as given in Lemma 6.1.

Remark 6.4. (i) The genus 1 curves C_{t} and D_{t} have Jacobian E_{t}. As observed in [11], this gives an alternative proof of Lemma 6.1. (ii) The family of quartics G_{t}(X, Z) has constant (meaning independent of t) level 2 theta group. It should therefore be possible to write G_{t}(X, Z) as a linear combination of the binary quartic G(X, Z) and its Hessian

H(X, Z) = (8ac − 3b^{2})X^{4} + (24ad − 4bc)X^{3}Z + (48ae + 6bd − 4c^{2})X^{2}Z^{2}
+ (24be − 4cd)XZ^{3} + (8ce − 3d^{2})Z^{4}.

We find that G_{t}(X, Z) = (t^{3} + At + B)(4γ(t)G(X, Z) + \frac{1}{12}H(X, Z)).

We can also use Proposition 3.2 to describe the family of 4-covers D_{t}. Let α ∈ F^{×}/F^{×2}k^{×} such that D = D_{α}. We may compute α from D by evaluating the rank 1 quadratic form (4) at any point x ∈ k^{4} where it is non-zero (in each constituent field of F).

Theorem 6.5. The family of 2-covers D_{t} → C_{t} such that each 4-cover D_{t} → E_{t} has the same theta group as our original 4-cover D → E, is obtained by the procedure in Section 3.1, starting in place of (5) with

(25)  α((t + μ)X − (θt + λ)Z) = (x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3})^{2}.

Proof: We show that the family of curves D_{t} is identical to that considered in Lemma 6.2. In particular, the theta groups are not just constant up to isomorphism, but constant as subschemes of GL_{4}. The quartic g_{t}(x) = G_{t}(x, 1) has root θ_{t} = (θt + λ)/(t + μ). Following (7), the quadric intersection obtained from (25) is

tr_{F/k} (\frac{x^{2}}{α(t + μ)g′_{t}(θ_{t})}) = tr_{F/k} (\frac{θ_{t}x^{2}}{α(t + μ)g′_{t}(θ_{t})}) = 0,

where x = x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3}. Since

(26)  g′_{t}(θ_{t}) = g′(θ)d(t)/(t + μ)^{2}

this may be re-written as

tr_{F/k} (\frac{(t + μ)x^{2}}{αg′(θ)} ) = tr_{F/k} (\frac{(θt + λ)x^{2}}{αg′(θ)} ) = 0.

The formula for the Hessian used in the proof of Lemma 6.2 now shows that the fibre with t = ∞ has Hessian the fibre with t = 0. The proof is completed by noting that the fibre with t = ∞ is our original 4-cover D = D_{α}. □

Remark 6.6. Expanding (25) in powers of θ gives 4 equations of the form

(linear form in X, Z, tX, tZ) = (quadratic form in x_{1}, …, x_{4}).

Eliminating X, Z, t, we obtain a single quartic equation in x_{1}, …, x_{4}, describing the twist of (17) that has D_{t} as fibres. This provides an alternative way of arriving at the equation for T_{E,ξ}^{+}(4) in Corollary 6.3.

6.3. The twisted Shioda modular surface of level 4. As in Section 3.2, let L = k[φ] where φ is a root of x^{3} + Ax + B = 0, and write σ for the involution of LF swapping θ and θ̃. We write λ̃ = σ(λ) and μ̃ = σ(μ). By the construction in Section 3.2, alternative equations for the curves D_{t} in Theorem 6.5 are given by

αα̃((t + μ)X − (θt + λ)Z)((t + μ̃)X − (θ̃t + λ̃)Z) = (y_{1}m_{1} + … + y_{6}m_{6})^{2}.

We now modify this by introducing a factor ν(t) representing the shift from Theorem 4.1.

Theorem 6.7. The family of 2-covers D_{t} → C_{t} such that each 4-cover D_{t} → E_{t} has the same fibre above 0 (as an E[4]-torsor) as our original 4-cover D → E, is obtained by the procedure in Section 3.2, starting in place of (11) with

(27) αα̃((t + μ)X − (θt + λ)Z)((t + μ̃)X − (θ̃t + λ̃)Z) = ν(t)(y_{1}m_{1} + … + y_{6}m_{6})^{2}

where

(28)  ν(t) = \frac{d(t)}{t^{2} − 2tφ − 2φ^{2} − A}.

Proof: We first note that

(29)  N_{L/k}(t^{2} − 2tφ − 2φ^{2} − A) = d(t),

and so ν(t) does indeed correspond to an element of H^{1}(k, E[2]) via the isomorphism (9).

In Section 5.3 we exhibited another family of 4-covers D_{t} → E_{t}. This had constant fibre above 0, as could be checked by substituting (X : Z) = (0 : 1), (1 : 0), (1 : 1) or (λ : 1) into the equations (19), and observing that in each case the 4 solutions for (y_{1} : …: y_{6}) ∈ ℙ^{5}(k^{sep}) do not depend on λ. The argument here is similar.

A calculation using (2), (3), (10) and (24) shows that

(30)  (t + μ̃)(θt + λ) − (t + μ)(θ̃t + λ̃) = (θ − θ̃)(t^{2} − 2φt − 2φ^{2} − A).

Let ˜ θ̃ be another root of g. Substituting X = ˜ θ̃t + ˜ λ̃ and Z = t + ˜ μ̃ into the left hand side of (27), gives a quartic polynomial in t, which by (29) and (30) is a constant times ν(t), as defined in (28). Therefore the 16 points on D_{t} mapping to the points on C_{t} with Y = 0, are independent of t. This shows that the fibre above 0 is constant (with respect to t) as a k-scheme. In Section 5.3 we saw that the action of E_{t}[4] on D_{t} is given by formulae independent of t. Therefore the fibre above 0 is also constant as an E[4]-torsor.

Finally we note that taking t = ∞ gives the cover (11) with ν = 1, which by Remark 3.4(ii) is isomorphic to D. Of course, setting t = ∞ in (28) does not literally make sense. However after homogenising, and rescaling by a square, we do indeed have ν(∞) = 1. □

Corollary 6.8. Suppose that ξ ∈ H^{1}(k, E[4]) and D_{E,ξ} is given by (5). Then the surface S_{E,ξ}^{+}(4) has a singular model in ℙ^{5} defined by 3 quadrics. These quadrics are obtained from the [M : k] = 6 equations in k(t)[X, Z, y_{1}, …, y_{6}] coming from (27), by taking linear combinations to eliminate X^{2}, XZ and Z^{2}.

Proof: The key point is that the 3 quadrics are independent of t. Again the argument is best understood by comparing with the situation in Section 5.3. The same calculation as mentioned in the proof of Theorem 6.7 shows that the linear combinations of the left hand sides in (19) that vanish at (X : Z) = (0 : 1), (1 : 0), (1 : 1) and (λ : 1), and therefore vanish identically, do not depend on t. This explains why the the first 3 quadrics in (20) do not depend on t. The same idea works here. □

Remark 6.9. In Theorem 6.7 we not only made the fibre above 0 constant as a k-scheme, we made it constant as a subscheme of ℙ^{5}. For this, and the application to Corollary 6.8, we needed to know ν(t) mod L^{×2}, not just mod L(t)^{×2}.

The genus 1 fibration is given either by using (12) (which gives two further quadratic forms in y_{1}, …, y_{6}, now depending on t) or by using Remark 5.1.

7. Computing twists of S(4) and T (4) in the reverse case

In this section we take ξ ∈ H^{1}(k, E[4]) with Ob_{E,4}(ξ) = 0 and compute models for S_{E,ξ}^{−}(4) and T_{E,ξ}^{−}(4).

Let E/k be an elliptic curve y^{2} = f(x) = x^{3}+Ax+B, and Δ = −16(4A^{3}+27B^{2}) its discriminant. We write E^{Δ} for the quadratic twist of E by Δ. Likewise if C → ℙ^{1} is a double cover, then we write C^{Δ} for its quadratic twist (over ℙ^{1}) by Δ. We may summarise the results of this section by saying that everything carries over from Section 6 with the following changes.

• We replace E_{t} by E_{t}^{Δ} and C_{t} by C_{t}^{Δ}.

• The family D_{t} is computed using the contravariants instead of the covariants (these were the identity map and the Hessian).

• In Theorem 6.5 we multiply one side of the equation by g′(θ). This is an element of F whose norm is Δ (up to squares).

• In Theorem 6.7 we multiply one side of the equation by (θ − θ̃)^{2}Δ. We now go through the changes in detail.

7.1. Reverse twists of the universal elliptic curve. Let

E^{Δ} : y^{2} = Δ f(x)

be the quadratic twist of E by Δ. Then by [3, Corollary 7.4], or Remark 7.3 below, there is a reverse 4-congruence σ: E[4] → E^{Δ}[4]. We may thus identify X_{E}^{−}(4) = X_{E^{Δ}}^{+}(4). It is immediate from Lemma 6.1 that

E_{t}^{Δ} : y^{2} = −Δ f(t) (x − γ(t))(x^{3} + Ax + B),

with base point (x, y) = (γ(t), 0), is the universal elliptic curve over X_{E}^{−}(4).

7.2. Reverse twists of the theta modular surface. Let ξ ∈ H^{1}(k, E[4]) with Ob_{E,4}(ξ) = 0. Then the 4-cover D_{E,ξ} has a model as a quadric intersection D ⊂ ℙ^{3} with theta group Θ = Θ_{E,ξ} ⊂ GL_{4}. Let Θ^{∨} ⊂ GL_{4} be the subgroup of matrices inverse transpose to those in Θ. Let π : Θ^{∨} → E^{Δ}[4] be the map that makes the following diagram commute (where the superscript −T denotes inverse transpose)

\xymatrix{0 \ar[r] & 𝔾_{m} \ar[r] \ar[d]^{.^{−1}} & Θ \ar[r] \ar[d]^{.^{−T}} & E[4] \ar[r] \ar[d]^{σ} & 0 \\ 0 \ar[r] & 𝔾_{m} \ar[r] & Θ^{∨} \ar[r]^{π} & E^{Δ}[4] \ar[r] & 0}

Since σ reverses the Weil pairing, the second row gives Θ^{∨} the structure of theta group for E^{Δ}[4]. The diagram then shows that Θ and Θ^{∨} are σ-isomorphic (see Definition 2.5).

We have Θ^{∨} = Θ_{E^{Δ},ξ′} for some ξ′ ∈ H^{1}(k, E^{Δ}[4]). By [7, Theorem 5.2] there is a unique model for the 4-cover D_{E^{Δ},ξ′} as a quadric intersection D^{∨} ⊂ ℙ^{3} with theta group Θ^{∨}. The contravariants, introduced in [11], give a way of computing equations for D^{∨}. The details are as follows.

Let D = {Q_{1} = Q_{2} = 0} ⊂ ℙ^{3}, and let A_{1} and A_{2} be the corresponding symmetric matrices. Let a, b, c, d, e and I, J and A, B be as defined in Section 3. Let S_{0}, …, S_{3} be the quadratic forms corresponding to the matrices C_{0}, …, C_{3} defined by

adj(XA_{1} + ZA_{2}) = C_{0}X^{3} + C_{1}X^{2}Z + C_{2}XZ^{2} + C_{3}Z^{3}.

The contravariants are defined by

R_{1} = \frac{1}{12} (\frac{∂I}{∂a}S_{0} + \frac{∂I}{∂b} S_{1} + \frac{∂I}{∂c} S_{2} + \frac{∂I}{∂d}S_{3}) ,
R_{2} = \frac{1}{12} (\frac{∂I}{∂b} S_{0} + \frac{∂I}{∂c} S_{1} + \frac{∂I}{∂d}S_{2} + \frac{∂I}{∂e}S_{3}) ,
R′_{1} = \frac{1}{12^{2}} (\frac{∂J}{∂a} S_{0} + \frac{∂J}{∂b} S_{1} + \frac{∂J}{∂c} S_{2} + \frac{∂J}{∂d} S_{3}) ,
R′_{2} = \frac{1}{12^{2}} (\frac{∂J}{∂b} S_{0} + \frac{∂J}{∂c} S_{1} + \frac{∂J}{∂d} S_{2} + \frac{∂J}{∂e} S_{3}) .

The quadric intersection D^{∨} then has equations

(31)  D^{∨}: {−9BR_{2} + 2AR′_{2} = 9BR_{1} − 2AR′_{1} = 0} ⊂ ℙ^{3}.

Indeed the invariant theory shows that Θ^{∨} acts on D^{∨}, and that D^{∨} has Jacobian E^{Δ}. Strictly speaking, to show that D^{∨} has theta group Θ^{∨}, we need that the two actions of E^{Δ}[4] on D^{∨} (one arising from the identification Θ^{∨}/𝔾_{m} = E^{Δ}[4] and the other from the structure of D^{∨} as a homogeneous space) agree. Since they agree up to an automorphism of E^{Δ}[4] that respects the Weil pairing, the desired agreement may be achieved by adjusting our choice of σ.

Remark 7.1. It is convenient to fix our choice of σ once and for all. Following the proof of [3, Corollary 7.4] we let σ correspond to the non-trivial element in the centre of GL_{2}(ℤ/4ℤ)/{±1}. One way of seeing that this σ and the one from the previous paragraph agree (up to sign) is by observing that both are defined for the elliptic curve E: x^{3} + a_{2}x^{2} + a_{4}x + a_{6} over k(a_{2}, a_{4}, a_{6}), which is a sufficiently general elliptic curve not to admit other σ. But then any elliptic curve over k can be obtained by specializing this E, and σ specializes with it.

The following lemma follows from (31) by direct calculation.

Lemma 7.2. If D ⊂ ℙ^{3} has intermediate 2-cover C: Y^{2} = G(X, Z) then D^{∨} ⊂ ℙ^{3} has intermediate 2-cover C^{Δ} : Y^{2} = ΔG(X, Z).

Remark 7.3. Since C^{Δ} is a 2-cover of E^{Δ}, it follows that D^{∨} has Jacobian E^{Δ}. As observed in [11], this gives an alternative proof that E and E^{Δ} are reverse 4-congruent.

Let u_{1}, …, u_{4} and v_{1}, …, v_{4} be a pair of bases for F as a k-vector space, that are dual with respect to the trace form, i.e. tr_{F/k}(u_{i}v_{j}) = δ_{ij}. For example we could take u_{i} = θ^{i−1} and v_{j} = β_{j−1}/g′(θ) where

\frac{g(X)}{X − θ} = β_{3}X^{3} + β_{2}X^{2} + β_{1}X + β_{0}.

Lemma 7.4. The theta groups for the quadric intersections obtained from

(32)  α(X − θZ) = (x_{1}u_{1} + … + x_{4}u_{4})^{2}

and

(33)  \frac{1}{αg′(θ)}(X − θZ) = (x_{1}v_{1} + … + x_{4}v_{4})^{2},

are the inverse transpose of each other.

Proof: Extending our field we may assume F = k^{4}. It then suffices to prove the lemma in the case where u_{1}, …, u_{4} and v_{1}, …, v_{4} are the standard bases.

Following (7), the quadric intersections obtained from (32) and (33) are

tr_{F/k} (\frac{x^{2}}{αg′(θ)}) = tr_{F/k} (\frac{θx^{2}}{αg′(θ)}) = 0,

and

tr_{F/k} (αx^{2}) = tr_{F/k} (αθx^{2}) = 0.

The lemma reduces to showing that if D ⊂ ℙ^{3} is defined by

∑_{i=1}^{4} ξ_{i}x_{i}^{2} = ∑_{i=1}^{4} ξ_{i}θ_{i}x_{i}^{2} = 0,

then D^{∨} ⊂ ℙ^{3} is defined by

∑_{i=1}^{4} \frac{x_{i}^{2}}{ξ_{i}g′(θ_{i})} = ∑_{i=1}^{4} \frac{θ_{i}x_{i}^{2}}{ξ_{i}g′(θ_{i})} = 0.

where g(θ) = ∏_{i=1}^{4}(X − θ_{i}). This follows by direct calculation using the contravariants. □

We obtain the following analogue of Theorem 6.5. Let λ and μ be as defined in Lemma 6.2. We identify E[4] = E^{Δ}[4] via σ as specified in Remark 7.1.

Theorem 7.5. The family of 2-covers D_{t}^{∨} → C_{t}^{Δ} such that each 4-cover D_{t}^{∨} → E_{t}^{Δ} has theta group the inverse transpose of that for our original 4-cover D → E, is obtained by the procedure in Section 3.1, starting in place of (5) with

(34)  αg′(θ)((t + μ)X − (θt + λ)Z) = (x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3})^{2}.

Proof: According to Lemma 7.4 we need to add a factor g′_{t}(θ_{t}) to the formula in Theorem 6.5. However, since we only care about the image of this element in F^{×}/F^{×2}k^{×} we see by (26) that we can use g′(θ) instead. □

The analogue of Corollary 6.3 is that the surface T_{E,ξ}^{−}(4) fibered by D_{t}^{∨} is given by

{R_{1}R′_{2} − R_{2}R′_{1} = 0} ⊂ ℙ^{3}.

Alternatively, an equation for this surface may be obtained from (34), exactly as in Remark 6.6.

7.3. Reverse twists of Shioda’s modular surface. Let σ: E[4] → E^{Δ}[4] be the reverse 4-congruence specified in Remark 7.1. Given ξ ∈ H^{1}(k, E[4]) with Ob_{E,4}(ξ) = 0 we would like to write down a model for S_{E,ξ}^{−}(4) = S_{E^{Δ},σ_{∗}(ξ)}^{+}(4). If Ob_{E^{Δ},4}(σ_{∗}(ξ)) = 0, i.e. σ_{∗}(ξ) is represented by a quadric intersection (and we have these equations explicitly) then Theorem 6.7 gives equations for S_{E,ξ}^{−}(4). Unfortunately this condition is not always satisfied.

Since Ob_{E,4}(ξ) = 0 we have a model D = D_{E,ξ} ⊂ ℙ^{3}. The work in Section 7.2 gives us D^{∨} = D_{E^{Δ},ξ′} ⊂ ℙ^{3}. It remains to determine κ = σ_{∗}(ξ) − ξ′. If D is a 2-cover of C then Lemma 7.2 shows that D^{∨} is a 2-cover of C^{Δ}. Since the matrix in Remark 7.1 is congruent to the identity mod 2, we see that D_{E^{Δ},σ_{∗}(ξ)} is also a 2-cover of C^{Δ}. Therefore 2ξ′ = 2σ_{∗}(ξ), and so κ is 2-torsion.

Lemma 7.6. Suppose we have a 4-cover D = D_{E,ξ} with Ob_{E,4}(ξ) = 0 and let D^{∨} = D_{E^{Δ},ξ′}. Then σ_{∗}(ξ) = ξ′ + κ, where

κ = (3φ^{2} + A)/(4A^{3} + 27B^{2})

under the isomorphism (9).

Proof: By the same argument as in the proof of Theorem 4.1, we see that κ only depends on σ: E[4] → E^{Δ}[4], and not on ξ itself. Thus it suffices to show that the inverse transpose of Θ_{E} is the twist of Θ_{E^{Δ}} by κ.

Let E have Weierstrass equation y^{2} = x^{3} + Ax + B. We embed E → ℙ^{3} via (x_{1} : …: x_{4}) = (1 : x : y : 3x^{2} + A). The image is D ⊂ ℙ^{3} defined by

Ax_{1}^{2} − x_{1}x_{4} + 3x_{2}^{2} = 0,
3Bx_{1}^{2} + 2Ax_{1}x_{2} + x_{2}x_{4} − 3x_{3}^{2} = 0.

Then by (31), D^{∨} ⊂ ℙ^{3} has equations

(35)  \begin{aligned}3Ax_{1}^{2} − 9Bx_{1}x_{2} − A^{2}x_{2}^{2} + 6(4A^{3} + 27B^{2})x_{4}^{2} & = 0, \\ 9Bx_{1}^{2} + 4A^{2}x_{1}x_{2} − 3ABx_{2}^{2} + (4A^{3} + 27B^{2})(x_{3}^{2} + 4x_{2}x_{4}) & = 0. \end{aligned}

On the other hand the 2-cover of E^{Δ} corresponding to κ is given by

x + (4A^{3} + 27B^{2})φ = κ(u + vφ + wφ^{2})^{2}.

Expanding and taking the coefficients of φ and φ^{2} gives equations

(36)  \begin{aligned}3u^{2} − 2Av^{2} − 4Auw − 6Bvw + 2A^{2}w^{2} & = 0, \\ −4Auv − 3Bv^{2} − 6Buw + 4A^{2}vw + 5ABw^{2} & = s^{2}. \end{aligned}

The curves (35) and (36) are isomorphic via

(x_{1} : x_{2} : x_{3} : x_{4}) = (18Bv − 4A^{2}w : 12Av + 18Bw : 6s : 3u − 2Aw).  □

Using the construction in Section 3.2, alternative equations for the curves D_{t}^{∨} in Theorem 7.5 are given by

(37)  N_{LF/M}(αg′(θ)((t + μ)X − (θt + λ)Z)) = (y_{1}m_{1} + … + y_{6}m_{6})^{2}.

We now modify this by introducing a factor ν(t)κ representing the shift from Theorem 4.1. We identify E[4] = E^{Δ}[4] via σ as specified in Remark 7.1.

Theorem 7.7. The family of 2-covers D_{t}^{∨} → C_{t}^{Δ} such that each 4-cover D_{t}^{∨} → E_{t}^{Δ} has the same fibre above 0 (as an E[4]-torsor) as our original 4-cover D → E, is obtained by the procedure in Section 3.2, starting in place of (11) with

(38) N_{LF/M}(α((t + μ)X − (θt + λ)Z)) = ν(t)(θ − θ̃)^{2}Δ(y_{1}m_{1} + … + y_{6}m_{6})^{2}

where ν(t) is given by (28).

Proof: We introduce an extra factor ν(t)κ to the right hand side of (37). Since the factors g′(θ) and κ do not depend on t, the proof that we obtain a family of curves with constant fibre above 0 is exactly the same as for Theorem 6.7.

If D = D_{E,ξ} then by Lemmas 7.4 and 7.6 the fibre above t = ∞ is D_{E^{Δ},σ_{∗}(ξ)}. Finally we simplify our modified version of (37). A calculation along the same

lines as the proof of (30) shows that

g′(θ)g′(θ̃) = −16(θ − θ̃)^{2}(3φ^{2} + A).

Therefore

g′(θ)g′(θ̃)κ ≡ (θ − θ̃)^{2}Δ mod L^{×2},

and this gives the equation (38) as required. □

Exactly as in Section 6.3, we obtain the following.

Corollary 7.8. Suppose that ξ ∈ H^{1}(k, E[4]) and D_{E,ξ} is given by (5). Then the surface S_{E,ξ}^{−}(4) has a singular model in ℙ^{5} defined by 3 quadrics. These quadrics are obtained from the [M : k] = 6 equations in k(t)[X, Z, y_{1}, …, y_{6}] coming from (38), by taking linear combinations to eliminate X^{2}, XZ and Z^{2}.

For the purposes of Corollary 7.8 we may ignore the factor Δ ∈ k in (38). So compared to the direct case, we only need to multiply ν(t) by a factor (θ − θ̃)^{2}. This is a Kummer generator for the quadratic extension LF/M.

8. Polarizations

Let A be an abelian surface over a field k of characteristic 0. We write A^{∨} for the dual abelian surface. As is described in, for instance, [17, Section 13], there is an injective group homomorphism NS(A) → Hom(A, A^{∨}). A polarization is a homomorphism that lies in the image of the ample cone. These are isogenies. A principal polarization is a polarization that is an isomorphism.

If an abelian variety A has a principal polarization λ_{A}, then the map λ ↦ ψ_{λ} = λ_{A}^{−1}λ identifies the set of polarizations with a special semigroup in End(A).

Elliptic curves E have a natural principal polarization λ_{E}: E → E^{∨} and on a product of elliptic curves E × E′, the product of these gives a principal product polarization.

If E, E′ are two non-isogenous elliptic curves without complex multiplication (CM) then End(E ×E′) = End(E)×End(E′) = ℤ×ℤ. For such a surface one has NS(E×E′) ≃ ℤ×ℤ, the semigroup of ample classes is ℤ_{>0}×ℤ_{>0}, and polarizations correspond to the endomorphisms [n]_{E} × [n′]_{E′}, with n, n′ ∈ ℤ_{>0}.

An abelian surface A is called decomposable if it admits a non-constant map to an elliptic curve. In that case the Poincaré reducibility theorem [17, Proposition 12.1] gives us that there are two elliptic curves E, E′ ⊂ A, such that the natural map ϕ: E × E′ → A is an isogeny. We call such an isogeny an optimal decomposition.

In this section we are interested in determining when such a surface A may admit a principal polarization λ_{A}. If it does, we have a polarization ϕ^{∗}(λ_{A}) = ϕ^{∨}λ_{A}ϕ on E × E′ of degree deg(ϕ)^{2}.

Write e_{n} for the product of the Weil pairings on (E × E′)[n]. We paraphrase [17, Proposition 16.8].

Proposition 8.1. Let ϕ: E × E′ → A be an isogeny and Δ = ker ϕ. Let λ be a polarization on E × E′. Suppose that Δ ⊂ ker λ ⊂ (E × E′)[n]. Then λ = ϕ^{∗}(λ′) for some polarization λ′ on A if and only if the Weil pairing e_{n} restricts to the trivial pairing on Δ × ψ_{λ}(\frac{1}{n}Δ).

Proposition 8.2. Let A be a principally polarized decomposable abelian surface, with optimal decomposition ϕ: E ×E′ → A. Suppose that E, E′ are non-isogenous and have no CM. Then the kernel of ϕ is the graph of a reverse n-congruence, where deg(ϕ) = n^{2}.

Proof: Since ker ϕ intersects trivially with E × {0} and {0} × E′, we have that ker ϕ ≅ ℤ/d_{1}ℤ × ℤ/d_{2}ℤ for some positive integers d_{1} and d_{2}. The principal polarization on A pulls back to a polarization λ of degree d_{1}^{2}d_{2}^{2}. It follows that ψ_{λ} must be an endomorphism of the same degree. Therefore ψ_{λ} = [n]_{E} × [n′]_{E′} for some positive integers n and n′ with nn′ = d_{1}d_{2}. Let pr_{1} : E × E′ → E be the first projection. Since ker ϕ ⊂ ker λ we have ker ϕ ≅ pr_{1}(ker ϕ) ⊂ E[n]. Therefore d_{1} | n and d_{2} | n. The same argument shows that d_{1} | n′ and d_{2} | n′. Since nn′ = d_{1}d_{2} it follows that d_{1} = d_{2} = n = n′. Hence we have that Δ = ker ϕ is the graph of an isomorphism σ: E[n] → E′[n].

We see that Δ ⊂ ψ_{λ}(\frac{1}{n}Δ), so Proposition 8.1 implies that e_{n}(Δ, Δ) = 1. In particular, if we have points T_{1} = (t_{1}, σ(t_{1})) and T_{2} = (t_{2}, σ(t_{2})) belonging to Δ then by Proposition 2.4 we have that

1 = e_{n}(T_{1}, T_{2}) = e_{E[n]}(t_{1}, t_{2})τ_{σ}(e_{E[n]}(t_{1}, t_{2})).

Therefore τ_{σ}(ζ) = ζ^{−1}, i.e. the n-congruence σ is indeed a reverse n-congruence. □

Lemma 8.3. If σ: E[n] → E′[n] is a reverse n-congruence, then the restriction σ′: E[d] → E′[d] for any d | n is also a reverse congruence.

Proof: Let n = dm. If T_{1}, T_{2} are generators of E[n], then mT_{1}, mT_{2} are generators of E[d]. The result follows from the basic property of Weil pairings that

e_{dm}(T_{1}, T_{2})^{m} = e_{d}(mT_{1}, mT_{2}).  □

9. Examples

We first give an example showing how our methods improve on [11]. We then give an example where � (E/ℚ)[4] is made visible by a second elliptic curve E′, but our methods are needed to find E′. Finally we give some examples of 4torsion in � (E/ℚ) that cannot be made visible in a principally polarized abelian surface. We do this by exhibiting some twists of S(4) that are not everywhere locally soluble.

We refer to elliptic curves by their labels in Cremona’s tables.

Example 9.1. Let E and E′ be the elliptic curves 96266a1 and 96266b1. We have E(ℚ) = 0 and E′(ℚ) ≅ ℤ^{2}. In this case there is a direct 4-congruence, and the Mordell-Weil group of E′ explains a subgroup (ℤ/4ℤ)^{2} ⊂ � (E/ℚ). We verify this for one element of � (E/ℚ) of order 4, the other cases being similar. The element we consider is represented by C = {Q_{1} = Q_{2} = 0} ⊂ ℙ^{3} where

Q_{1} = x_{1}^{2} + 3x_{1}x_{2} + x_{1}x_{3} + x_{1}x_{4} − x_{2}^{2} + 2x_{2}x_{3} + 2x_{2}x_{4} − 2x_{3}^{2} − x_{3}x_{4} − 3x_{4}^{2},
Q_{2} = x_{1}^{2} − 3x_{1}x_{2} + 2x_{1}x_{3} − 6x_{1}x_{4} + 3x_{2}^{2} − 4x_{2}x_{3} + 2x_{2}x_{4} − 4x_{3}^{2} − 2x_{3}x_{4} − 2x_{4}^{2}.

One of the elements of E′(ℚ)/4E′(ℚ) of order 4 maps to the 4-covering C′ = {Q′_{1} = Q′_{2} = 0} ⊂ ℙ^{3} where

Q′_{1} = x_{1}x_{2} + 2x_{1}x_{3} + x_{2}x_{3} − x_{2}x_{4} + 2x_{3}x_{4} + 6x_{4}^{2},
Q′_{2} = x_{1}x_{2} + x_{1}x_{3} + 3x_{1}x_{4} + 2x_{2}^{2} − 2x_{2}x_{3} − x_{2}x_{4} + 2x_{3}^{2} − 3x_{3}x_{4} + 4x_{4}^{2}.

Starting from either C or C′, and computing a twist of Shioda’s modular surface using the method described in Corollary 6.8, we obtain

S : \{ \begin{matrix}2y_{1}y_{4} − 2y_{1}y_{5} − 2y_{2}y_{6} + y_{3}^{2} − y_{5}^{2} + y_{6}^{2} & = 0 \\ 2y_{1}^{2} − 2y_{1}y_{3} − y_{1}y_{5} + y_{1}y_{6} − 2y_{2}y_{3} \\ +2y_{2}y_{4} − y_{2}y_{5} + y_{2}y_{6} − y_{3}^{2} + y_{3}y_{5} + y_{4}y_{6} + y_{5}y_{6} − y_{6}^{2} & = 0 \\ y_{1}^{2} + 4y_{1}y_{3} − 2y_{1}y_{4} − 2y_{1}y_{5} + 2y_{1}y_{6} + 2y_{2}^{2} + 2y_{2}y_{3} + 2y_{2}y_{4} \\ −2y_{2}y_{5} + y_{3}^{2} + 2y_{3}y_{4} − 2y_{3}y_{6} + 2y_{4}y_{5} − 2y_{5}^{2} + 2y_{5}y_{6} + y_{6}^{2} & = 0 \end{matrix}\} ⊂ ℙ^{5}.

The embeddings C → S and C′ → S are given by

(\begin{matrix}y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \\ y_{5} \\ y_{6} \end{matrix}) = (\begin{matrix}30 & 16 & 12 & 22 & 10 & 8 \\ −3 & −12 & 2 & −23 & −33 & −14 \\ −7 & −4 & −6 & 37 & −13 & 6 \\ 41 & −4 & −38 & 21 & 3 & 6 \\ 22 & 40 & −36 & 30 & −14 & 4 \\ 10 & 24 & −28 & 2 & −18 & 28 \end{matrix}) (\begin{matrix}f_{12} \\ f_{13} \\ f_{14} \\ f_{23} \\ f_{24} \\ f_{34} \end{matrix})

and

(\begin{matrix}y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \\ y_{5} \\ y_{6} \end{matrix}) = (\begin{matrix}−2 & −10 & −4 & 6 & 2 & 0 \\ 10 & 3 & 3 & −2 & 0 & −5 \\ −20 & 1 & −1 & 4 & 2 & 1 \\ −16 & −11 & 7 & 8 & −6 & 1 \\ −20 & −16 & −4 & 0 & −4 & −2 \\ −8 & −16 & 4 & −2 & 2 & 0 \end{matrix}) (\begin{matrix}f′_{12} \\ f′_{13} \\ f′_{14} \\ f′_{23} \\ f′_{24} \\ f′_{34} \end{matrix})

where

f_{ij} = \frac{∂(Q_{1}, Q_{2})}{∂(x_{i}, x_{j})} and f′_{ij} = \frac{∂(Q′_{1}, Q′_{2})}{∂(x_{i}, x_{j})}.

It may be checked that these maps send the flex points on C and C′ to the singular points of S. In particular C and C′ correspond to the same element of H^{1}(ℚ, E[4]) = H^{1}(ℚ, E′[4]).

Suppose instead that we use invariant theory. Let (Q′_{1}, Q′_{2}) have Hessian (Q′′_{1}, Q′′_{2}). Then the quadric intersection {−281Q′_{1} + Q′′_{1} = −281Q′_{2} + Q′′_{2} = 0} ⊂ ℙ^{3} is a 4covering of E. However this 4-covering is not locally soluble at 2. Therefore the method in [11] for computing visible elements of � (E/ℚ) of order 4 does not apply. This is because the shift is not locally soluble at 2.

The equations for C, C′ and S in Example 9.1 were simplified by making careful choices of coordinates. This was achieved by a combination of minimisation and reduction. For quadric intersections (such as C and C′) these processes are described in [8]. We make some brief comments on how this works for S.

The minimisation step relies on defining a suitable invariant. The discriminant of a binary cubic form f(x, y) = ax^{3} + bx^{2}y + cxy^{2} + dy^{3} is

Δ(f) = −27a^{2}d^{2} + 18abcd − 4ac^{3} − 4b^{3}d + b^{2}c^{2}.

Let q_{1}, q_{2} ∈ k[z_{1}, z_{2}, z_{3}] be a pair of quadratic forms with corresponding 3 × 3 symmetric matrices B_{1}, B_{2}. Then f(x, y) = det(B_{1}x + B_{2}y) is a binary cubic form. We define Δ(q_{1}, q_{2}) = Δ(f). Now let Q_{1}, Q_{2}, Q_{3} be quadratic forms defining a twist of the surface S_{0} in Section 5.3. Writing A_{1}, A_{2}, A_{3} for the corresponding 6 × 6 symmetric matrices we find that

(39)  det(A_{1}z_{1} + A_{2}z_{2} + A_{3}z_{3}) = f(q_{1}, q_{2})

where f is a binary cubic and q_{1}, q_{2} ∈ k[z_{1}, z_{2}, z_{3}] are quadratic forms. We define Δ(Q_{1}, Q_{2}, Q_{3}) = Δ(f)Δ(q_{1}, q_{2}). This definition is independent of the choices of f, q_{1}, q_{2}, provided that they satisfy (39).

Now let S ⊂ ℙ^{5} be a twist of S_{0} defined over ℚ. Clearing denominators we may assume that S is defined by Q_{1}, Q_{2}, Q_{3} ∈ ℤ[y_{1}, …, y_{6}]. Then the discriminant Δ = Δ(Q_{1}, Q_{2}, Q_{3}) is a non-zero integer. Using the natural action of GL_{3}(ℚ) × GL_{6}(ℚ) we seek to minimise |Δ|, while preserving that the coefficients of the Q_{i} are integers. This process is carried out one prime at a time, the idea being that for each prime p dividing Δ we consider the scheme defined by the reductions of the Q_{i} mod p. We did not work out algorithms guaranteed to minimise |Δ|, but rather implemented some methods that seem to work reasonably well in practice.

The reduction step relies on defining a suitable inner product. Specifically we take the inner product (unique up to scalars) that is invariant under the action of ASL(ℤ/4ℤ × μ_{4}). For the surface S_{0} in Section 5.3 this is the standard inner product. For general S we reduce to this case by finding a change of co-ordinates over ℂ relating S and S_{0}. Performing lattice reduction on the Gram matrix of the inner product then gives a change of coordinates in GL_{6}(ℤ) that may be used to simplify our equations for S.

In preparing Example 9.1 we also had to find the change of co-ordinates relating the surfaces constructed from C and C′. However it was easy to solve for this as the unique change of co-ordinates defined over ℚ taking the singular points to the singular points.

Example 9.2. Let E be the elliptic curve 31252a1. We have E(ℚ) = 0 and � (E/ℚ)[4] ≅ (ℤ/4ℤ)^{2}. One of the elements of order 4 in � (E/ℚ) is represented by the 4-covering {Q_{1} = Q_{2} = 0} ⊂ ℙ^{3} where

Q_{1} = 2x_{1}x_{2} + 2x_{1}x_{3} + x_{3}^{2} + 2x_{4}^{2}
Q_{2} = 6x_{1}^{2} + 6x_{1}x_{2} − 14x_{1}x_{3} + 9x_{1}x_{4} + 11x_{2}^{2} + 10x_{2}x_{3} − 31x_{2}x_{4} + 3x_{3}^{2} + 22x_{3}x_{4} + 7x_{4}^{2}

Corollary 7.8 gives the following reverse twist of Shioda’s modular surface.

S : \{ \begin{matrix}2y_{1}y_{3} + 2y_{1}y_{5} + 2y_{1}y_{6} + 2y_{2}^{2} − 2y_{2}y_{4} − 2y_{3}y_{4} + y_{5}^{2} & = 0 \\ y_{1}^{2} − 2y_{1}y_{2} − y_{1}y_{4} + 2y_{1}y_{5} + y_{2}^{2} + y_{2}y_{3} \\ −y_{2}y_{4} + y_{2}y_{5} + y_{2}y_{6} − y_{3}y_{5} + y_{3}y_{6} − y_{4}^{2} − y_{4}y_{6} & = 0 \\ y_{1}^{2} − y_{1}y_{3} + 2y_{1}y_{4} − y_{1}y_{5} − y_{1}y_{6} + 2y_{2}y_{3} \\ +3y_{2}y_{4} + y_{3}^{2} + y_{3}y_{5} + y_{4}y_{5} + y_{4}y_{6} − y_{5}^{2} − y_{5}y_{6} & = 0 \end{matrix}\} ⊂ ℙ^{5}.

A useful check on our calculations is that the flex points on C and the singular points of S have the same field of definition. The genus 1 fibration on S may be computed as described in Remark 5.1. We searched for rational points on S of small height. Among the points we found were

(3 : 0 : −1 : −2 : −2 : 3), (1 : −8 : 3 : −22 : −6 : 31),
(−33 : 13 : −25 : 23 : 34 : 22), (21 : −10 : −17 : −26 : −14 : 55),

all lying on a fibre isomorphic to E′ : y^{2} = x^{3} + 10609x + 58646 with E′(ℚ) ≅ ℤ^{3}. The elliptic curves E and E′ are reverse 4-congruent. It turns out that all of � (E/ℚ)[4] is explained by E′(ℚ). The conductors of E and E′ are 31252 = 2^{2} · 13 · 601 and 2468908 = 2^{2} · 13 · 79 · 601. In particular E′ is beyond the range of any current tables of elliptic curves. (In fact E is 2-congruent to a rank 2 elliptic curve of the same conductor, but these curves are not 4-congruent.)

Finally we give some examples where our twists of S(4) are not locally soluble. As explained in the introduction, this can only happen in the reverse case. In Table 9 we list some elliptic curves E/ℚ with E(ℚ) = 0 and � (E/ℚ)[4] ≅ (ℤ/4ℤ)^{2}. In each case, for some of the elements ξ ∈ H^{1}(ℚ, E[4]), representing an element of � (E/ℚ) of order 4, the surface S_{E,ξ}^{−}(4) has no points locally at p, where p is the prime indicated.

Proposition 9.3. Each of the elliptic curves E/ℚ in Table 9 has an element of order 4 in � (E/ℚ) that cannot be made visible in a principally polarized abelian surface over ℚ.

Proof: By construction, there exists ξ ∈ H^{1}(ℚ, E[4]) such that [C_{E,ξ}] ∈ � (E/ℚ) has order 4, yet S_{E,ξ}^{−}(4)(ℚ_{p}) = ∅. Let us now assume that [C_{E,ξ}] is visible in an abelian surface A, i.e., that there is an injection E → A such that [C_{E,ξ}] lies in the kernel of the induced map on Galois cohomology H^{1}(ℚ, E) → H^{1}(ℚ, A).

As described in Section 8, there is an elliptic curve E′ ⊂ A and an optimal decomposition ϕ : E × E′ → A. In particular, the kernel of ϕ is the graph of an isomorphism between finite subgroups of E and E′.

For each of the elliptic curves on our list we have E(ℚ)/2E(ℚ) = 0, equivalently rank E(ℚ) = 0 and E[2] is irreducible. By [13, Theorem 3.1], there exists, for some l ≥ 2, a congruence σ: E[2^{l}] → E′[2^{l}] such that the graph of σ is contained in Table 1. Some elliptic curves E for which there exists ξ ∈ H^{1}(ℚ, E[4]) with [C_{E,ξ}] ∈ � (E/ℚ), yet S_{E,ξ}^{−}(4)(ℚ_{p}) = ∅. Proposition 9.3 establishes that [C_{E,ξ}] is not visibile in a principally polarized abelian surface.

p = 2 21720c1, 26712e1, 32784c1, 32816j1, 33536e1, 34560o1, 37984e1,
40328b1, 47664p1, 49176b1, 59248g1, 62328bj1, 69192f1, 69312ch1,

69312dp1, 73600bn1, 73840a1, 74368b1, 77440cl1, 77440cr1, 77600p1

p = 5 23950g1, 60725j1, 63825g1, 64975e1, 72600df1, 76175e1, 90450bs1,
105350z1, 120300n1, 121950ca1, 129850r1, 133950cy1, 137025s1,
141200bf1, 146700p1, 153425u1, 153425bd1, 154850m1, 154850m2
p = 13 56446n1, 62192t1, 70135c1, 100386g1, 104442w1, 124384g1,
132496df1, 172042o1, 200772u1, 216151f1, 226629g1, 256880dn1,
294060j1, 306735z1, 321945v1, 331240cy1, 335296dj1, 337155x1
p = 29 220342v1, 277530bc1, 277530bs1, 323785n1, 364994k1
p = 37 370999a1
p = 61 301401k1, 260470l1, 260470l2
p = 101 306030bg1, 306030bg2

ker ϕ, and [C_{E,ξ}] = π(P′) for some P′ ∈ E′(ℚ), where π is the diagonal map in the following commutative diagram

[figure: E(ℚ)/2^{l}E(ℚ); H^{1}(ℚ, E[2^{l}]); H^{1}(ℚ, E)[2^{l}]; 0; E′(ℚ)/2^{l}E′(ℚ); π; H^{1}(ℚ, E′[2^{l}]); H^{1}(ℚ, E′)[2^{l}]; 0]

Since E(ℚ)/2^{l}E(ℚ) = 0, we see that P′ ∈ E′(ℚ)/2^{l}E′(ℚ) has order 4. Since E′(ℚ)[2] = 0 it follows that rank E′(ℚ) > 0 and P′ ∈ 2^{l−2}E′(ℚ). Therefore ξ is explained (via a 4-congruence) by an element of E′(ℚ)/4E′(ℚ).

Since rank E(ℚ) = 0 and rank E′(ℚ) > 0, it is clear that E and E′ are not isogenous. Computation shows that Gal(ℚ^{sep}/ℚ) acts on E[4] via a large enough group to ensure that E has no CM. Since E and E′ are 4-congruent, this also shows that E′ has no CM.

Proposition 8.2 and Lemma 8.3 show that for A to be principally polarized, the congruence σ must be reverse.

However, as noted at the start of the proof, S_{E,ξ}^{−}(4) does not have any rational points. Therefore the 4-congruence induced by σ is not reverse, and hence neither is σ itself. This is the required contradiction. □

Remark 9.4. A curious fact about the examples in Table 9 is that the odd primes p at which we find local obstructions satisfy p ≡ 5 (mod 8). Indeed, for any one p, there are only finitely many ℚ_{p}-isomorphism classes for the surface S_{E,ξ}^{−}(4), so determining which ones have local obstructions is in principle a finite amount of work. Proposition 9.5 provides one description of an insolvability criterion, that appears to explain all the examples in Table 9 with p odd.

Proposition 9.5. Let p be a prime with p ≡ 5 (mod 8). Let E be the elliptic curve y^{2} = x^{3} + Ax for some A ∈ ℚ_{p}^{×} with v_{p}(A) odd. Let ξ be the image of P = (0, 0) under the connecting map

E(ℚ_{p})/4E(ℚ_{p})⟶H^{1}(ℚ_{p}, E[4]).

Then the surface S_{E,ξ}^{−}(4) has no ℚ_{p}-points.

Proof: We use Corollary 7.8 to show that S_{E,ξ}^{−}(4) has equations

0 = y_{2}y_{3} − Ay_{5}y_{6}
0 = y_{1}y_{4} + 2y_{2}y_{5} + y_{3}^{2} − Ay_{6}^{2}
0 = (y_{1}^{2} − 2y_{2}^{2}) + A(y_{4}^{2} + 2y_{5}^{2}) + 4Ay_{3}y_{6}

Since multiplying A by a 4th power gives the same elliptic curve we may suppose v_{p}(A) = ±1. We consider the case v_{p}(A) = 1. Suppose (y_{1} : …: y_{6}) is a ℚ_{p}-point, with y_{1}, …, y_{6} ∈ ℤ_{p}, not all in pℤ_{p}. Since (2/p) = −1 we have y_{1} ≡ y_{2} ≡ y_{3} ≡ 0 (mod p). Then since (−2/p) = −1 we have y_{4} ≡ y_{5} ≡ y_{6} ≡ 0 (mod p). This is the required contradiction. The case v_{p}(A) = −1 is similar. □

Remark 9.6. We also found 4 examples (225336k1, 271800bt1, 329536y1, 368928bj1) where for every ξ ∈ H^{1}(ℚ, E[4]), representing an element of � (E/ℚ) of order 4, the surface S_{E,ξ}^{−} has no points locally at 2, and a further 4 examples (271800bj1, 352800md1, 378400bv1, 378400by1) where each S_{E,ξ}^{−}(4) is locally insoluble either at 2 or 5.

Example 9.7. Let E/ℚ be the elliptic curve 225336k1 with Weierstrass equation

y^{2} = x^{3} − x^{2} − 453476x − 197032572.

One of the elements of � (E/ℚ) of order 4 is represented by D = D_{E,ξ} ⊂ ℙ^{3} with equations

\begin{aligned}3x_{1}x_{2} + 2x_{1}x_{3} + 4x_{1}x_{4} − 3x_{2}x_{3} − 3x_{2}x_{4} + 2x_{3}^{2} + x_{3}x_{4} + 3x_{4}^{2} & = 0 \\ 4x_{1}^{2} + 2x_{1}x_{2} + x_{1}x_{3} + x_{1}x_{4} + 4x_{2}^{2} + 2x_{2}x_{3} + 4x_{3}^{2} − 2x_{4}^{2} & = 0 \end{aligned}

In the reverse case we get a surface S = S_{E,ξ}^{−}(4) ⊂ ℙ^{5} with equations

7y_{1}y_{2} − 7y_{1}y_{3} − y_{1}y_{4} + 19y_{1}y_{5} + y_{1}y_{6} − 4y_{2}^{2} + y_{2}y_{3} + 5y_{2}y_{4} − 11y_{2}y_{5}
−9y_{2}y_{6} + 2y_{3}^{2} + 4y_{3}y_{5} + 10y_{3}y_{6} + 13y_{4}^{2} − 10y_{4}y_{5} + 2y_{4}y_{6} − 4y_{5}^{2} − 22y_{5}y_{6} + 8y_{6}^{2} = 0,

4y_{1}^{2} − 8y_{1}y_{3} − 6y_{1}y_{4} + 11y_{1}y_{5} − 18y_{1}y_{6} − y_{2}^{2} + 17y_{2}y_{3} − 8y_{2}y_{4} − 11y_{2}y_{5} − y_{2}y_{6}

−5y_{3}^{2} + 17y_{3}y_{4} + 3y_{3}y_{5} + 18y_{3}y_{6} − 15y_{4}^{2} + 4y_{4}y_{5} − 21y_{4}y_{6} − 12y_{5}^{2} − 13y_{5}y_{6} + 3y_{6}^{2} = 0,

3y_{1}^{2} + 4y_{1}y_{2} + 16y_{1}y_{3} + 4y_{1}y_{4} − 11y_{1}y_{5} + 6y_{1}y_{6} − 11y_{2}^{2} + 3y_{2}y_{3} − 4y_{2}y_{4} + 15y_{2}y_{5}

+y_{2}y_{6} + 10y_{3}^{2} + 19y_{3}y_{4} + 7y_{3}y_{5} + 28y_{3}y_{6} + 2y_{4}^{2} − 23y_{4}y_{6} − y_{5}^{2} + 31y_{5}y_{6} + 20y_{6}^{2} = 0.

As a check on our calculations we verified that the flex points on D and the singular points on S are defined over the same degree 16 number field.

We find that S(ℚ_{2}) = ∅. As indicated in Remark 9.6, exactly the same happens for the other elements of order 4 in � (E/ℚ). The argument in Proposition 9.3 now shows that none of the elements of � (E/ℚ) of order 4 are visible in a principally polarized abelian surface.

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Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

E-mail address: nbruin@sfu.ca

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

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