 Research
 Open Access
Visibility of 4covers of elliptic curves
 Nils Bruin^{1} and
 Tom Fisher^{2}Email author
 Received: 13 October 2017
 Accepted: 27 January 2018
 Published: 14 February 2018
Abstract
Let C be a 4cover of an elliptic curve E, written as a quadric intersection in \({\mathbb P}^3\). Let \(E'\) be another elliptic curve with 4torsion isomorphic to that of E. We show how to write down the 4cover \(C'\) of \(E'\) with the property that C and \(C'\) are represented by the same cohomology class on the 4torsion. In fact we give equations for \(C'\) as a curve of degree 8 in \({\mathbb P}^5\). We also study the K3surfaces 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 \({\mathbb P}^5\) with exactly 16 singular points. This allows us to give examples of elliptic curves over \({\mathbb Q}\) that have elements of order 4 in their Tate–Shafarevich group that are not visible in a principally polarized abelian surface.
Keywords
 Elliptic curves
 Tate–Shafarevich groups
 Mazur visibility
 Descent
 K3 surfaces
 Localglobal obstructions
Mathematics Subject Classification
 11G05
 11G35
 14H10
1 Introduction
Let E and \(E'\) be elliptic curves over a field k that are ncongruent, meaning that there is an isomorphism of kgroup schemes \(\sigma :E[n]\rightarrow E'[n]\). We suppose that the characteristic of k does not divide n. We may use \(\sigma \) to transfer certain interesting arithmetic information between E and \(E'\). For instance let k be a number field. An ntorsion element of the Tate–Shafarevich group \(\mathrm{III}(E/k)\) may be represented by a class \(\xi \in H^1(k,E[n])\). Let \(\sigma _* : H^1(k,E[n]) \rightarrow H^1(k,E'[n])\) be the isomorphism induced by \(\sigma \). It might happen that while \(\xi \) maps to a nontrivial element in \(\mathrm{III}(E/k)[n]\), represented say by a curve C, the image of \(\sigma _*(\xi )\) 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 [10, 16]): the graph \(\Delta \subset E[n]\times E'[n]\) of the ncongruence \(\sigma \) provides an isogeny \(E\times E'\rightarrow (E\times E')/\Delta \) and a model for C arises as the fibre of \((E\times E')/\Delta \) 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 2torsion. It is true, however, that given any \(\xi \in H^1(k,E[2])\), one can find another elliptic curve \(E'\) with isomorphic 2torsion such that \(\sigma _*(\xi ) \in H^1(k,E'[2])\) represents a trivial homogeneous space under \(E'\); see [4, 15].
 (i)
One parametrizes the elliptic curves ncongruent 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 \(\phi (n)\) components, where \(\phi \) 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 \(S_{E,\xi }(n) \rightarrow X_E(n)\) whose fibres are the ncovers of \(E_t\) corresponding to \(\xi \in H^1(k,E_t[n])\).
 (iii)
If one can find a rational point P on \(S_{E,\xi }(n)\), and none of the cusps of \(X_E(n)\) are rational points, then \(\xi \) 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 \(S_{E,\xi }(n)\) has no rational points then \(\xi \) cannot be made visible using an elliptic curve ncongruent to E.
From a computational point of view, it is attractive if \((E\times E')/\Delta \) can be realized as a Jacobian, or more generally, admits a principal polarization. It naturally does so if we start with \(\sigma \) inverting the Weil pairing. Again taking \(n=3\), one can show that \(S^+_{E,\xi }(3)\) is birational to \({\mathbb P}^2\) over k if and only if the same is true for \(S^_{E,\xi }(3)\). It follows (see [5]) that any element of \(\mathrm{III}(E/k)\) 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 \(n\ge 6\) 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 \(n\ge 7\). See [14] for explicit examples over \({\mathbb Q}\) of nonexistence of such \(E'\) for \(n=6,7\).
 (i)
The curve \(X_E^+(4)\) is of genus 0, but the surface \(S^+_{E,\xi }(4)\) is a K3surface. Much is conjectured, but little is known about the rational points on K3surfaces.
 (ii)
Given \(\xi \in H^1(k,E[n])\), representing a genus 1 curve of degree n, there is another fibred surface \(T_{E,\xi }(n) \rightarrow X_E(n)\) whose fibres are ncovers of \(E_t\), but now sharing the same action of E[n] on a suitable linear system. For \(n=3,4,5\), explicit invarianttheoretic constructions of these surfaces are given in [12, 13]. When n is odd the surfaces \(S^\pm _{E,\xi }(n)\) and \(T^\pm _{E,\xi }(n)\) are the same, but when n is even a correction to this idea is needed.
We refer to [1] for many interesting facts about the geometry of the surfaces S(4) and T(4). For example, they are both K3surfaces (in fact Kummer surfaces) with Picard number 20. Working over \({\mathbb C}\), the surface T(4) is isomorphic to the diagonal quartic surface in \({\mathbb P}^3\). Moreover the surfaces S(4) and T(4) are related by generically 2to1 rational maps (in either direction) but are not birational.
1.1 Outline of the article
In Sect. 2 we review some of the interpretations of \(H^1(k,E[n])\), most notably in terms of ncovers and theta groups. We also define the (twisted) Shioda and theta modular surfaces.
In Sect. 3.1 we review classical 4descent. It has the drawback for us that it only gives those 4covers with a degree 4 model in \({\mathbb P}^3\). In Sect. 3.2 we describe a variant of this method. In particular, given \(D\rightarrow C\rightarrow E\) where D has a degree 4 model in \({\mathbb P}^3\), we show how to twist the second 2cover \(D \rightarrow C\) by an arbitrary element of \(H^1(k,E[2])\). The new 4cover has a degree 8 model in \({\mathbb P}^7\). However, for our purposes, it is convenient to project this to a curve in \({\mathbb P}^5\), still of degree 8.
In Sect. 4 we quantify the difference between \(S^\pm _{E,\xi }(n)\) and \(T^\pm _{E,\xi }(n)\) for arbitrary n. Indeed the fibres differ by a cohomology class \(\nu = \nu (t)\), which we call the shift. It was already shown in [8] 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 Sect. 6 we describe how to compute the twists of these surfaces so that a prescribed 4cover \(D \rightarrow E\) appears as a fibre. We assume that D is given as a quadric intersection in \({\mathbb P}^3\). We use the invariant theory in [12] to write down the required twist of T(4). By finding an explicit formula for the shift, and then using the method in Sect. 3.2, we are then able to compute the required twist of S(4).
The methods in Sect. 6 for computing \(S^+_{E,\xi }(4)\) and \(T^+_{E,\xi }(4)\) are modified in Sect. 7 to compute \(S^_{E,\xi }(4)\) and \(T^_{E,\xi }(4)\). The arguments here are somewhat simplified by the observation that an elliptic curve and its quadratic twist by its discriminant are reverse 4congruent.
In Sect. 9 we give several examples. We exhibit some elliptic curves \(E/{\mathbb Q}\) such that for the elements \(\xi \in H^1({\mathbb Q},E[4])\) representing elements of order 4 in \(\mathrm{III}(E/{\mathbb Q})\), the surface \(S^_{E,\xi }(4)\) has no rational points. We show this by computing an explicit model of the surface and checking that the surface has no padic points for some prime p. If \(E({\mathbb Q})/2E({\mathbb Q})\) is trivial, it follows that visibility in a surface can only happen via a rational point on \(S^+_{E,\xi }(4)\). 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 \(\xi \in H^1(k,E[4])\) represents an element in \(\mathrm{III}(E/k)\) then \(S^+_{E,\xi }(4)\) has points everywhere locally. Thus, any failure for \(S^+_{E,\xi }(4)\) 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 \(k^\mathrm {sep}\) for its separable closure. We write t for the generic point on various modular curves we consider. When these curves are isomorphic to \({\mathbb P}^1\), then t will instead denote a coordinate on \({\mathbb P}^1\). 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 ncover of E over k is a pair \((C,\pi )\), where C is a nonsingular complete irreducible curve over k and \(\pi :C\rightarrow E\) is a morphism such that there exists an isomorphism \(\psi :(C\times _kk^\mathrm {sep})\rightarrow (E\times _kk^\mathrm {sep})\) satisfying \(\pi =[n]\circ \psi \). Two ncovers \((C_1, \pi _1)\) and \((C_2,\pi _2)\) are isomorphic over k if there is an isomorphism \(\alpha : C_1 \rightarrow C_2\) over k such that \(\pi _1 = \pi _2 \circ \alpha \).
The isomorphism classes of ncovers are naturally parametrized by \(H^1(k,E[n])\). This means that given \(\xi \in H^1(k,E[n])\) there is an ncover \(C_{E,\xi }\rightarrow E\), any ncover is isomorphic to one of this form, and two ncovers are isomorphic if and only if they arise from the same class \(\xi \). Restriction of cocycle classes corresponds to base extending the cover.
If \((C,\pi )\) is an ncover 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 \(H^1(k,E[n])\rightarrow H^1(k,E)\) may be interpreted as forgetting the covering map \(\pi \). In particular its kernel consists of those ncovers \((C,\pi )\) for which C(k) is nonempty.
Definition 2.2
A theta group for E[n] is a central extension \(0\rightarrow \mathbb {G}_{\mathrm{m}}\rightarrow \Theta \rightarrow E[n]\rightarrow 0\) of kgroup schemes such that the commutator pairing on \(\Theta \) agrees with the Weilpairing on E[n]. An isomorphism of theta groups is an isomorphism of central extensions as kgroup schemes.
The lift of the (translation) action of E[n] on the linear system \(n.0_E\) to the RiemannRoch space \(L(n.0_E)\) gives a theta group \(\Theta _E\). If \(n\ge 3\) then \(L(n.0_E)\) is the space of global sections of a very ample line bundle \({\mathcal {L}}_{E,n}\). Choosing a basis for this space provides a map \(E\rightarrow {\mathbb P}^{n1}\) that gives a model for E as an elliptic normal curve of degree n. The theta group \(\Theta _E\) is then the full inverse image in \({\text {GL}}_n\) of the group of projective linear transformations describing the action of E[n] on E by translation.
As observed in [8, Sections 1.5 and 1.6], there is an action of E[n] on \(\Theta _E\) by conjugation, and every automorphism of \(\Theta _E\) arises in this way. Therefore the isomorphism classes of theta groups for E[n], viewed as twists of \(\Theta _E\), are parametrized by \(H^1(k,E[n])\).
Since an ncover \(C_{E,\xi }\) (as a curve) is a twist of E by a cocycle taking values in E[n], we see that \(C_{E,\xi }\) comes equipped with a degree n line bundle \({\mathcal {L}}_{E,\xi }\) with a theta group \(\Theta _{E,\xi }\) acting on it. It may be checked that \(\Theta _{E,\xi }\) is indeed the twist of \(\Theta _E\) by \(\xi \) (in the sense of the last paragraph). The line bundle \({\mathcal {L}}_{E,\xi }\) provides a model of \(C_{E,\xi }\), but now only in an \((n1)\)dimensional BrauerSeveri variety, i.e. a possibly nontrivial twist of \({\mathbb P}^{n1}\). The kisomorphism class of the BrauerSeveri variety gives a class in \({{\text {Br}}}(k)[n]\).
Definition 2.3
We write \({\text {Ob}}_{E,n}:H^1(k,E[n])\rightarrow {{\text {Br}}}(k) \) for the map that sends \(\xi \in H^1(k,E[n])\) to the class of the BrauerSeveri variety corresponding to the global sections of \({\mathcal {L}}_{E,\xi }\). In particular \({\text {Ob}}_{E,n}(\xi )\) is trivial if and only if \(C_{E,\xi }\) admits a degree n model in \({\mathbb P}^{n1}\) with \({\mathcal {L}}_{E,\xi }\) the pull back of \({\mathcal O}(1)\). In later sections we write \({\text {Ob}}_n(C_{E,\xi })={\text {Ob}}_{E,n}(\xi )\).
It is shown in [8] that \({\text {Ob}}_{E,n}(\xi )\) is determined by the isomorphism class of \(\Theta _{E,\xi }\) (as a theta group for E[n]) without reference to E itself.
2.3 Twists of full level modular curves
Proposition 2.4
In our case, for \(n=4\), any ncongruence is either direct or reverse.
Fixing an elliptic curve E, we consider the moduli space \(Y_E^+(n)(k)\) of pairs \((E',\sigma )\), where \(\sigma :E[n]\rightarrow E'[n]\) is a direct ncongruence. This moduli space is represented by a curve \(Y_E^+(n)\) over k, whose nonsingular 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 \((E',\sigma )\) where \(\sigma :E[n]\rightarrow E'[n]\) is a reverse ncongruence.
If \(\sigma \) is a direct or reverse ncongruence, then the automorphism \(\tau _\sigma \) from Proposition 2.4 can be extended to an automorphism \(\mathbb {G}_{\mathrm{m}}\rightarrow \mathbb {G}_{\mathrm{m}}\). In this case, we see that \(\sigma \) provides a way of comparing theta groups.
Definition 2.5
If \(\sigma \) is a direct ncongruence, this is the normal notion of isomorphism for theta groups upon identifying E[n] and \(E'[n]\) via \(\sigma \).
2.4 Shioda modular surfaces
For \(n\ge 3\), the moduli space \(Y_E^+(n)\) is fine, so there is a universal elliptic curve \(E_t\) with a direct ncongruence \(\sigma _t:E[n]\rightarrow E_t[n]\) over \(Y_E^+(n)\). The relevant property for us is that any direct ncongruence between E and another elliptic curve can be obtained by specializing t at the relevant moduli point on \(Y_E^+(n)\). We write \(S^+_E(n) \rightarrow X_E^+(n)\) for the minimal fibred surface with generic fibre \(E_t\). This is a twist of Shioda’s modular surface of full level n.
Given \(\xi \in H^1(k,E[n])\), we can twist this surface further. Writing \(k_t\) for the function field of \(X_E^+(n)\), we view \(\xi \) as an element of \(H^1(k_t,E_t[n])\) and take the ncover \(C_{t,\xi }\rightarrow E_t\) representing it. Let \(S^+_{E,\xi }(n) \rightarrow X^+_E(n)\) be the minimal fibred surface with generic fibre \(C_{t,\xi }\). This is again a twist of Shioda’s modular surface, and is isomorphic to \(S^+_E(n)\) over the splitting field of \(\xi \). In [16], the surfaces \(S_E^+(n)\) and \(S_{E,\xi }^+(n)\) are called first and second twists.
We define \(S^_{E,\xi }(n)\) similarly, by using a universal reverse ncongruence \(\sigma _t\).
2.5 Theta modular surfaces
Alternatively, given an elliptic curve E over k and \(\xi \in H^1(k,E[n])\), we base extend \(\Theta _{E,\xi }\) to a theta group over \(k_t\), the function field of \(X^+_E(n)\). Then \(\Theta _{E,\xi }\times _k k_t=\Theta _{E_t,\xi _t}\) for some \(\xi _t\in H^1(k_t,E_t[n])\). Just as in Sect. 2.4, we take the ncover \(C_{t,\xi _t}\rightarrow E_t\) representing \(\xi _t\). Then we define the theta modular surface to be the minimal fibred surface \(T_{E,\xi }^+(n) \rightarrow X_E^+(n)\) with generic fibre \(C_{t,\xi _t}\).
By construction, the fibres of \(T_{E,\xi }^+(n) \rightarrow X_{E}^+(n)\) are ncovers with a prescribed theta group. Since \({\text {Ob}}_{E,n}(\xi )\) is a function of \(\Theta _{E,\xi }\), we see that \({\text {Ob}}_{E_t,n}(\xi _t)\) is the base change of \({\text {Ob}}_{E,n}(\xi )\) to \(k_t\). In particular, if \({\text {Ob}}_{E,n}(\xi )=0\) then \(C_{t,\xi _t}\) admits a degree n model in \({\mathbb P}^{n1}\), with a linear action of \(E_t[n]\). In that case, it follows that \(T_{E,\xi }^+(n)\) is birational to a surface in \({\mathbb P}^{n1}\), with an action of E[n] through \(\Theta _{E,\xi }\). This allows us to use invariant theory to write down models of \(T_{E,\xi }^+(n)\).
Interestingly enough, \(\nu (t)=\xi _t\xi \) is not necessarily trivial. In fact, Theorem 4.1 proves that \(\nu (t)\) is 2torsion (as is seen by applying the result to the elliptic curves \(E\times _k k_t\) and \(E_t\)). In particular, if n is odd then \(\nu (t) = 0\) and the surfaces \(S^+_{E,\xi }(n)\) and \(T^+_{E,\xi }(n)\) are isomorphic.
We define \(T^_{E,\xi }(n)\) similarly, by reversing the order of multiplication on \(\Theta _{E,\xi }\). Since \(({\mathbb Z}/4{\mathbb Z})^\times = \{\pm 1\}\), this is sufficient to define \(T_{E,\xi }(4)\).
3 Computing 4covers
Let E be an elliptic curve and let \(\xi \in H^1(k,E[4])\). In this section, we write \(D_{E,\xi }\) for the corresponding 4cover of E. We have \(2\xi \in H^1(k,E[2])\) and write \(C_{E,2\xi }\) for the corresponding 2cover of E. Note that the 4cover \(D_{E,\xi }\rightarrow E\) naturally factors as \(D_{E,\xi }\rightarrow C_{E,2\xi }\rightarrow E\). It turns out to be advantageous to study 4covers via this intermediate structure.
Definition 3.1
Let \(C\rightarrow E\) be a 2cover. A 2cover of C is a cover \(D\rightarrow C\) such that the composition of covers \(D\rightarrow C\rightarrow E\) is a 4cover. If we want to emphasize that D is a 2cover of a 2cover, and not of an elliptic curve directly, we say that \(D\rightarrow C\) is a second 2cover.
3.1 Models of 2 and 4covers with trivial obstruction
In this section we review classical 4descent, as described in [11, 17, 20, 21] and implemented in Magma [2].
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
Remark 3.3
 (i)
Strictly speaking we should specify a choice of square root of \(N_{F/k}(\delta )\), otherwise the 2covers \(D_{E,\xi }\) and \(D_{E,\xi }\) of \(C_{E,2\xi }\), differing by the automorphism \(Y \mapsto Y\) of \(C_{E,2\xi }\), 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 \(D_\alpha \) aswhere \(x=x_1+x_2\theta +x_3\theta ^2+x_4\theta ^3\).$$\begin{aligned} {\mathrm{tr}}_{F/k} \left( \frac{x^2 }{\alpha g'(\theta )} \right) = {\mathrm{tr}}_{F/k} \left( \frac{ \theta x^2}{\alpha g'(\theta )} \right) = 0, \end{aligned}$$(7)
 (iii)
The group (6) may be identified with a certain subgroup of \(H^1(k,E[2])/\langle 2\xi \rangle \) where \(2\xi \) is the class of the 2cover \(C \rightarrow E\). See [11] for further details.
3.2 Models for twists of second 2covers
Remark 3.4
 (i)
If we just eliminate Y from the \(6+3+12\) equations listed above, then we get 20 quadrics in \(X,Z,y_1, \ldots , y_6\). These define a genus 1 curve embedded in \({\mathbb P}^7\) via a complete linear system of degree 8. However we will see that working with \({\mathcal {D}}_\nu \subset {\mathbb P}^5\) has some advantages.
 (ii)Taking \(\nu =1\) gives a 2cover \({\mathcal {D}}_1 \rightarrow E\) that is isomorphic to \(D \rightarrow C\). Indeed on comparing (5) and (11) we see that an isomorphism is given byIn fact the \(y_i\) span the same space as the \(2\times 2\) minors of the \(2\times 4\) matrix of partial derivatives of the quadratic forms defining D.$$\begin{aligned} y_1m_1+\cdots +y_6m_6 =\left( x_1+x_2\theta +x_3\theta ^2+x_4\theta ^3\right) \left( x_1+x_2\tilde{\theta }+x_3\tilde{\theta }^2+x_4\tilde{\theta }^3\right) . \end{aligned}$$
 (iii)
A generic calculation shows that \({\mathcal {D}}_\nu \subset {\mathbb P}^5\) 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 settheoretically.
 (iv)
Let \(z \in L^\times \) correspond under the isomorphism (9) to the class of \(C \rightarrow E\). By [7, Equation (3.1)] we have \(z \in M^{\times 2}\), 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 \({\mathcal {D}}_\nu \) and \({\mathcal {D}}_{\nu z}\) are isomorphic as curves. However as 2covers of C they differ by the automorphism \(Y \mapsto Y\).
We prove an analogue of Proposition 3.2.
Proposition 3.5
Let C be the 2cover (8). Then the collection of all 2covers of C is given by \(\ker (N_{L/k}:L^\times /L^{\times 2}\rightarrow k^\times /k^{\times 2})\) via the map \(\nu \mapsto {\mathcal {D}}_\nu \).
Proof
Let \(\eta \in H^1(k,E[2])\) map to the class of \(\nu \in L^\times \) under the isomorphism (9). To prove the proposition, we show that \({\mathcal {D}}_\nu \rightarrow C\) is the twist of \({\mathcal {D}}_1 \rightarrow C\) by \(\eta \).
If g(x) has roots \(\theta _1, \ldots , \theta _4\) then \(k^\mathrm {sep}({\mathcal {D}}_1)=k^\mathrm {sep}(C)(\sqrt{f_{12}},\sqrt{f_{13}})\) where \(f_{ij} = (X \theta _iZ)(X\theta _jZ)/Z^2\). Since \({\mathcal {D}}_1\) is a homogeneous space under E, there is an action of E[2] on \({\mathcal {D}}_1\). This is given by \(\sqrt{f_{12}} \rightarrow \pm \, \sqrt{f_{12}}\) and \(\sqrt{f_{13}} \rightarrow \pm \, \sqrt{f_{13}}\). It follows from the definition of the Weil pairing that this action agrees with the one defined in the last paragraph.
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
 (i)
\(2\nu =0\) (in particular, if n is odd then \(\nu =0\)).
 (ii)
\(\xi =\sigma _*(\xi ')+\nu \) if and only if \(\Theta _{E,\xi }\) and \(\Theta _{E',\xi '}\) are \(\sigma \)isomorphic.
Proof
In exactly the same manner we pick \(M'_T \in \Theta _{E'}(k^\mathrm {sep})\) for all \(T \in E'[n](k^\mathrm {sep})\). We can then choose \(\psi \) so that \(\psi (M'_T) = \pm M_{\sigma (T)}\) for all \(T \in E'[n](k^\mathrm {sep})\). Indeed if we make this true on a basis for \(E'[n]\), then the rest follows by (15) and its analogue for \(E'\).
It also follows that \(\Theta _{E'}\) is \(\sigma \)isomorphic to \(\Theta _{E,\nu }\). For the general statement, we choose a \(k^\mathrm {sep}\)isomorphism \(\psi ':\Theta _{E',\xi '}\rightarrow \Theta _{E'}\). Then the cocycle \(\rho \mapsto \rho (\psi \psi ')(\psi \psi ')^{1}\) represents the class \(\sigma _*(\xi ')+\nu \). It follows that \(\Theta _{E',\xi '}\) is \(\sigma \)isomorphic to \(\Theta _{E,\sigma _*(\xi ')+\nu }\). Since, as we noted in Sect. 2.2, the twists of \(\Theta _E\) are parametrized by \(H^1(k,E[n])\), this proves (ii). \(\square \)
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. For the remainder of the paper, we revert to our assumption that \({\text {char}}\, k \not = 2,3\).
5.1 The universal elliptic curve of level 4
5.2 The theta modular surface of level 4
5.3 Shioda’s modular surface of level 4
More generally there is a natural action of the affine special linear group \({\mathcal G}= {\text {ASL}}_2({\mathbb Z}/4{\mathbb Z})\) on S(4). The corresponding automorphisms of \(S_0\) are again given by changes of coordinates on \({\mathbb P}^5\). The surfaces \(S^{\pm }_{E,\xi }(4)\) are twists of S(4) by cocycles taking values in \({\mathcal G}\), and so each must admit a model in a 5dimensional Brauer–Severi variety. Our calculations in Sects. 6 and 7 show that if \({\text {Ob}}_{E,4}(\xi ) = 0\) then this Brauer–Severi variety is trivial. Indeed we show how to write down a model for \(S^{\pm }_{E,\xi }(4)\) as a complete intersection of quadrics in \({\mathbb P}^5\).
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 3dimensional space of quadrics vanishing on the surface, inside the 6dimensional space of quadrics vanishing at the singular points, then this defines a map to \({\mathbb P}^2\) with image a conic. For example, with \(S_0\) as above, the map is given by \((X_1:X_2:X_3) = (y_1y_2:y_3y_4:y_5y_6)\) and the conic is \(X_1^2 + X_2^2 = X_3^2\). Parametrising this conic gives the required map to \({\mathbb P}^1\).
6 Computing twists of S(4) and T(4) in the direct case
In this section we take \(\xi \in H^1(k,E[4])\) with \({\text {Ob}}_{E,4}(\xi )=0\) and compute models for \(S^+_{E,\xi }(4)\) and \(T^+_{E,\xi }(4)\).
6.1 The twisted universal elliptic curve of level 4
Lemma 6.1
Proof
6.2 The twisted theta modular surface of level 4
Let \(D = \{ Q_1 = Q_2 = 0 \} \subset {\mathbb P}^3\) be a quadric intersection with Jacobian E. Then \(D=D_{E,\xi }\) for some \(\xi \in H^1(k,E[4])\) with \({\text {Ob}}_{E,4}(\xi )=0\). We use invariant theory to compute a model for \(T^+_{E,\xi }(4)\) as a quartic surface in \({\mathbb P}^3\), together with its genus 1 fibration over \(X^+_E(4)\).
Lemma 6.2
Proof
In principle this may be checked by a generic calculation. To make the calculation practical we consider the case \(Q_1 = \sum _{i=1}^4 \xi _i x_i^2\) and \(Q_2 = \sum _{i=1}^4 \xi _i \theta _i x_i^2\). Then \(g(X) = 2^4 (\prod _{i=1}^4 \xi _i) \prod _{i=1}^4(X  \theta _i)\), and \((Q_1,Q_2)\) has Hessian \((Q'_1,Q'_2)\) where \(Q'_1 = 12\sum _{i=1}^4 \xi _i \mu _i x_i^2\) and \(Q'_2 = 12\sum _{i=1}^4 \xi _i \lambda _i x_i^2\). Computing the intermediate 2cover, by the method used in (22), gives the equation for \(C_t\) as stated. \(\square \)
Corollary 6.3
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 4congruent 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 [12], 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
We can also use Proposition 3.2 to describe the family of 4covers \(D_t\). Let \(\alpha \in F^\times /F^{\times 2}k^\times \) such that \(D = D_\alpha \). We may compute \(\alpha \) from D by evaluating the rank 1 quadratic form (4) at any point \(\mathbf x \in k^4\) where it is nonzero (in each constituent field of F).
Theorem 6.5
Proof
Remark 6.6
6.3 The twisted Shioda modular surface of level 4
Theorem 6.7
Proof
In Sect. 5.3 we exhibited another family of 4covers \({\mathcal {D}}_t \rightarrow E_t\). This had constant fibre above 0, as could be checked by substituting \((X:Z) = (0:1)\), (1 : 0), (1 : 1) or \((\lambda :1)\) into the equations (19), and observing that in each case the 4 solutions for \((y_1 : \ldots : y_6) \in {\mathbb P}^5(k^\mathrm {sep})\) do not depend on \(\lambda \). The argument here is similar.
Finally we note that taking \(t = \infty \) gives the cover (11) with \(\nu = 1\), which by Remark 3.4(ii) is isomorphic to D. Of course, setting \(t = \infty \) in (28) does not literally make sense. However after homogenising, and rescaling by a square, we do indeed have \(\nu (\infty ) = 1\). \(\square \)
Corollary 6.8
Suppose that \(\xi \in H^1(k,E[4])\) and \(D_{E,\xi }\) is given by (5). Then the surface \(S^+_{E,\xi }(4)\) has a singular model in \({\mathbb P}^5\) defined by 3 quadrics. These quadrics are obtained from the \([M:k]=6\) equations in \(k(t)[X,Z,y_1, \ldots ,y_6]\) 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 Sect. 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 \((\lambda :1)\), and therefore vanish identically, do not depend on t. This explains why the first 3 quadrics in (20) do not depend on t. The same idea works here. \(\square \)
Remark 6.9
In Theorem 6.7 we not only made the fibre above 0 constant as a kscheme, we made it constant as a subscheme of \({\mathbb P}^5\). For this, and the application to Corollary 6.8, we needed to know \(\nu (t)\) mod \(L^{\times 2}\), not just mod \(L(t)^{\times 2}\).
The genus 1 fibration is given either by using (12) (which gives two further quadratic forms in \(y_1, \ldots , y_6\), 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 \(\xi \in H^1(k,E[4])\) with \({\text {Ob}}_{E,4}(\xi )=0\) and compute models for \(S^_{E,\xi }(4)\) and \(T^_{E,\xi }(4)\).

We replace \(E_t\) by \(E_t^\Delta \) and \(C_t\) by \(C_t^\Delta \).

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 \(g'(\theta )\). This is an element of F whose norm is \(\Delta \) (up to squares).

In Theorem 6.7 we multiply one side of the equation by \((\theta  \tilde{\theta })^2 \Delta \).
7.1 Reverse twists of the universal elliptic curve
7.2 Reverse twists of the theta modular surface
We have \(\Theta ^\vee = \Theta _{E^\Delta ,\xi '}\) for some \(\xi '\in H^1(k,E^\Delta [4])\). By [8, Theorem 5.2] there is a unique model for the 4cover \(D_{E^\Delta ,\xi '}\) as a quadric intersection \(D^\vee \subset {\mathbb P}^3\) with theta group \(\Theta ^\vee \). The contravariants, introduced in [12], give a way of computing equations for \(D^\vee \). The details are as follows.
Remark 7.1
The following lemma follows from (31) by direct calculation.
Lemma 7.2
If \(D \subset {\mathbb P}^3\) has intermediate 2cover \(C:Y^2 = G(X,Z)\) then \(D^\vee \subset {\mathbb P}^3\) has intermediate 2cover \(C^\Delta :Y^2 = \Delta G(X,Z)\).
Remark 7.3
Since \(C^\Delta \) is a 2cover of \(E^\Delta \), it follows that \(D^\vee \) has Jacobian \(E^\Delta \). As observed in [12], this gives an alternative proof that E and \(E^\Delta \) are reverse 4congruent.
Lemma 7.4
Proof
Extending our field we may assume \(F = k^4\). It then suffices to prove the lemma in the case where \(u_1, \ldots , u_4\) and \(v_1, \ldots , v_4\) are the standard bases.
We obtain the following analogue of Theorem 6.5. Let \(\lambda \) and \(\mu \) be as defined in Lemma 6.2. We identify \(E[4] = E^\Delta [4]\) via \(\sigma \) as specified in Remark 7.1.
Theorem 7.5
7.3 Reverse twists of Shioda’s modular surface
Let \(\sigma :E[4]\rightarrow E^\Delta [4]\) be the reverse 4congruence specified in Remark 7.1. Given \(\xi \in H^1(k,E[4])\) with \({\text {Ob}}_{E,4}(\xi )=0\) we would like to write down a model for \(S^_{E,\xi }(4)=S^+_{E^\Delta ,\sigma _*(\xi )}(4)\). If \({\text {Ob}}_{E^\Delta ,4}(\sigma _*(\xi ))=0\), i.e. \(\sigma _*(\xi )\) is represented by a quadric intersection (and we have these equations explicitly) then Theorem 6.7 gives equations for \(S^_{E,\xi }(4)\). Unfortunately this condition is not always satisfied.
Since \({\text {Ob}}_{E,4}(\xi )=0\) we have a model \(D=D_{E,\xi }\subset {\mathbb P}^3\). The work in Sect. 7.2 gives us \(D^\vee =D_{E^\Delta ,\xi '}\subset {\mathbb P}^3\). It remains to determine \(\kappa =\sigma _*(\xi )\xi '\). If D is a 2cover of C then Lemma 7.2 shows that \(D^\vee \) is a 2cover of \(C^\Delta \). Since the matrix in Remark 7.1 is congruent to the identity mod 2, we see that \(D_{E^\Delta ,\sigma _*(\xi )}\) is also a 2cover of \(C^\Delta \). Therefore \(2\xi '=2\sigma _*(\xi )\), and so \(\kappa \) is 2torsion.
Lemma 7.6
Proof
By the same argument as in the proof of Theorem 4.1, we see that \(\kappa \) only depends on \(\sigma :E[4]\rightarrow E^\Delta [4]\), and not on \(\xi \) itself. Thus it suffices to show that the inverse transpose of \(\Theta _E\) is the twist of \(\Theta _{E^\Delta }\) by \(\kappa \).
Theorem 7.7
Proof
We introduce an extra factor \(\nu (t) \kappa \) to the right hand side of (37). Since the factors \(g'(\theta )\) and \(\kappa \) 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 \(D = D_{E,\xi }\) then by Lemmas 7.4 and 7.6 the fibre above \(t = \infty \) is \(D_{E^\Delta , \sigma _*(\xi )}\).
Exactly as in Sect. 6.3, we obtain the following.
Corollary 7.8
Suppose that \(\xi \in H^1(k,E[4])\) and \(D_{E,\xi }\) is given by (5). Then the surface \(S^_{E,\xi }(4)\) has a singular model in \({\mathbb P}^5\) defined by 3 quadrics. These quadrics are obtained from the \([M:k]=6\) equations in \(k(t)[X,Z,y_1, \ldots ,y_6]\) 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 \(\Delta \in k\) in (38). So compared to the direct case, we only need to multiply \(\nu (t)\) by a factor \((\theta  \tilde{\theta })^2\). 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 \(A^\vee \) for the dual abelian surface. As is described in, for instance, [18, Section 13], there is an injective group homomorphism \({\text {NS}}(A)\rightarrow {\text {Hom}}(A,A^\vee )\). 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 \(\lambda _A\), then the map \(\lambda \mapsto \psi _\lambda =\lambda _A^{1}\lambda \) identifies the set of polarizations with a special semigroup in \({{\mathrm{End}}}(A)\).
Elliptic curves E have a natural principal polarization \(\lambda _E:E\rightarrow E^\vee \) and on a product of elliptic curves \(E\times E'\), the product of these gives a principal product polarization.
If \(E, E'\) are two nonisogenous elliptic curves without complex multiplication (CM) then \({{\mathrm{End}}}(E\times E')={{\mathrm{End}}}(E)\times {{\mathrm{End}}}(E')={\mathbb Z}\times {\mathbb Z}\). For such a surface one has \({\text {NS}}(E\times E')\simeq {\mathbb Z}\times {\mathbb Z}\), the semigroup of ample classes is \({\mathbb Z}_{>0}\times {\mathbb Z}_{>0}\), and polarizations correspond to the endomorphisms \([n]_E\times [n']_{E'}\), with \(n, n'\in {\mathbb Z}_{>0}\).
An abelian surface A is called decomposable if it admits a nonconstant map to an elliptic curve. In that case the Poincaré reducibility theorem [18, Proposition 12.1] gives us that there are two elliptic curves \(E,E'\subset A\), such that the natural map \(\phi :E\times E'\rightarrow A\) 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 \(\lambda _A\). If it does, we have a polarization \(\phi ^*(\lambda _A)=\phi ^\vee \lambda _A\phi \) on \(E\times E'\) of degree \(\deg (\phi )^2\).
On a principally polarized abelian variety A we write \(e_n\) for the Weil pairing on A[n] and \(e_{A[n]}\) if we want to emphasize the abelian variety. We paraphrase [18, Proposition 16.8].
Proposition 8.1
Let \(\phi :E\times E'\rightarrow A\) be an isogeny and \(\Delta = \ker \phi \). Let \(\lambda \) be a polarization on \(E\times E'\). Suppose that \(\Delta \subset \ker \lambda \subset (E\times E')[n]\). Then \(\lambda =\phi ^*(\lambda ')\) for some polarization \(\lambda '\) on A if and only if the Weil pairing \(e_n\) on \((E \times E')[n]\) restricts to the trivial pairing on \(\Delta \times \psi _\lambda (\frac{1}{n}\Delta )\).
Proposition 8.2
Let A be a principally polarized decomposable abelian surface, with optimal decomposition \(\phi :E\times E'\rightarrow A\). Suppose that \(E, E'\) are nonisogenous and have no CM. Then the kernel of \(\phi \) is the graph of a reverse ncongruence, where \(\deg (\phi )=n^2\).
Proof
Since \(\ker \phi \) intersects trivially with \(E\times \{0\}\) and \(\{0\}\times E'\), we have that \(\ker \phi {\, \cong \,}{\mathbb Z}/d_1{\mathbb Z}\times {\mathbb Z}/d_2{\mathbb Z}\) for some positive integers \(d_1\) and \(d_2\). The principal polarization on A pulls back to a polarization \(\lambda \) of degree \(d_1^2 d_2^2\). It follows that \(\psi _\lambda \) must be an endomorphism of the same degree. Therefore \(\psi _\lambda =[n]_E\times [n']_{E'}\) for some positive integers n and \(n'\) with \(n n' = d_1 d_2\). Let \({\text {pr}}_1: E \times E' \rightarrow E\) be the first projection. Since \(\ker \phi \subset \ker \lambda \) we have \(\ker \phi {\, \cong \,}{\text {pr}}_1(\ker \phi ) \subset E[n]\). Therefore \(d_1 \mid n\) and \(d_2 \mid n\). The same argument shows that \(d_1 \mid n'\) and \(d_2 \mid n'\). Since \(n n'= d_1 d_2\) it follows that \(d_1 = d_2 = n = n'\). Hence we have that \(\Delta = \ker \phi \) is the graph of an isomorphism \(\sigma :E[n]\rightarrow E'[n]\).
Lemma 8.3
If \(\sigma :E[n]\rightarrow E'[n]\) is a reverse ncongruence, then the restriction \(\sigma ':E[d]\rightarrow E'[d]\) for any \(d\mid n\) is also a reverse congruence.
Proof
9 Examples
We first give an example showing how our methods improve on [12]. We then give an example where \(\mathrm{III}(E/{\mathbb Q})[4]\) 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 \(\mathrm{III}(E/{\mathbb Q})\) 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 [6].
Example 9.1
Suppose instead that we use invariant theory. Let \((Q'_1,Q'_2)\) have Hessian \((Q''_1,Q''_2)\). Then the quadric intersection \(\{\,281 Q'_1 + Q''_1 = 281 Q'_2 + Q''_2 = 0\} \subset {\mathbb P}^3\) is a 4covering of E. However this 4covering is not locally soluble at 2. Therefore the method in [12] for computing visible elements of \(\mathrm{III}(E/{\mathbb Q})\) 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 [9]. We make some brief comments on how this works for S.
Now let \(S \subset {\mathbb P}^5\) be a twist of \(S_0\) defined over \({\mathbb Q}\). Clearing denominators we may assume that S is defined by \(Q_1,Q_2,Q_3 \in {\mathbb Z}[y_1, \ldots ,y_6]\). Then the discriminant \(\Delta = \Delta (Q_1,Q_2,Q_3)\) is a nonzero integer. Using the natural action of \({\text {GL}}_3({\mathbb Q}) \times {\text {GL}}_6({\mathbb Q})\) we seek to minimise \(\Delta \), 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 \(\Delta \) we consider the scheme defined by the reductions of the \(Q_i\) mod p. We did not work out algorithms guaranteed to minimise \(\Delta \), 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 \({\text {ASL}}_2({\mathbb Z}/4{\mathbb Z})\). For the surface \(S_0\) in Sect. 5.3 this is the standard inner product. For general S we reduce to this case by finding a change of coordinates over \({\mathbb C}\) relating S and \(S_0\). Performing lattice reduction on the Gram matrix of the inner product then gives a change of coordinates in \({\text {GL}}_6({\mathbb Z})\) that may be used to simplify our equations for S.
In preparing Example 9.1 we also had to find the change of coordinates relating the surfaces constructed from C and \(C'\). However it was easy to solve for this as the unique change of coordinates defined over \({\mathbb Q}\) taking the singular points to the singular points.
Example 9.2
Some elliptic curves E for which there exists \(\xi \in H^1({\mathbb Q},E[4])\) with \([C_{E,\xi }] \in \mathrm{III}(E/{\mathbb Q})\), yet \(S^_{E,\xi }(4)({\mathbb Q}_p)=\emptyset \). Proposition 9.3 establishes that \([C_{E,\xi }]\) is not visible in a principally polarized abelian surface
\(\begin{array}{ll} p = 2 &{} 21720c1,\, 26712e1, 32784c1, 32816j1, 33536e1, 34560o1, 37984e1, 40328b1, 47664p1, 49176b1,59248g1,\\ &{} 62328bj1, 69192f1, 69312ch1, 69312dp1, 73600bn1, 73840a1, 74368b1, 77440cl1, 77440cr1, 77600p1 \\ p = 5 &{} 23950g1, 60725j1, 63825g1, 64975e1, 72600df1, 76175e1, 90450bs1, 105350z1, 120300n1, 121950ca1, \\ &{}129850r1, 133950cy1,137025s1, 141200bf1, 146700p1, 153425u1, 153425bd1, 154850m1, 154850m2\\ p = 13 &{} 56446n1, 62192t1, 70135c1, 100386g1, 104442w1, 124384g1, 132496df1, 172042o1, 200772u1, 216151f1, \\ &{} 226629g1,256880dn1, 294060j1, 306735z1, 321945v1, 331240cy1, 335296dj1, 337155x1 \\ p = 29 &{} 220342v1, 277530bc1, 277530bs1, 323785n1, 364994k1 \\ p = 37 &{} 370999a1 \\ p = 61 &{} 301401k1, 260470l1, 260470l2 \\ p = 101 &{} 306030bg1, 306030bg2 \\ \end{array}\) 
Proposition 9.3
Each of the elliptic curves \(E/{\mathbb Q}\) in Table 1 has an element of order 4 in \(\mathrm{III}(E/{\mathbb Q})\) that cannot be made visible in a principally polarized abelian surface over \({\mathbb Q}\).
Proof
By construction, there exists \(\xi \in H^1({\mathbb Q},E[4])\) such that \([C_{E, \xi }] \in \mathrm{III}(E/{\mathbb Q})\) is an element of order 4, yet \(S^_{E,\xi }(4)({\mathbb Q}_p)= \emptyset \). Let us now assume that \([C_{E,\xi }]\) is visible in an abelian surface A, i.e., that there is an injection \(E\rightarrow A\) such that \([C_{E,\xi }]\) lies in the kernel of the induced map on Galois cohomology \(H^1({\mathbb Q},E)\rightarrow H^1({\mathbb Q},A)\).
As described in Sect. 8, there is an elliptic curve \(E' \subset A\) and an optimal decomposition \(\phi : E \times E' \rightarrow A\). In particular, the kernel of \(\phi \) is the graph of an isomorphism between finite subgroups of E and \(E'\).
Since \({\text {rank}}\, E({\mathbb Q})= 0\) and \({\text {rank}}\, E'({\mathbb Q}) >0\), it is clear that E and \(E'\) are not isogenous. Computation shows that the 4division polynomial of E is irreducible with Galois group of order 48. Since the largest abelian subgroup of \({\text {GL}}_2({\mathbb Z}/4{\mathbb Z})\) has order 16, it follows that \({\text {Gal}}({\mathbb Q}^\mathrm {sep}/{\mathbb Q})\) acts on E[4] via a large enough group to ensure that E has no CM. It follows that any elliptic curve 4congruent to E, in particular \(E'\), has no CM.
Proposition 8.2 and Lemma 8.3 show that for A to be principally polarized, the congruence \(\sigma \) must be reverse.
However, as noted at the start of the proof, \(S^_{E,\xi }(4)\) does not have any rational points. Therefore the 4congruence induced by \(\sigma \) is not reverse, and hence neither is \(\sigma \) itself. This is the required contradiction. \(\square \)
Remark 9.4
A curious fact about the examples in Table 1 is that the odd primes p at which we find local obstructions satisfy \(p\equiv 5\pmod {8}\). Indeed, for any one p, there are only finitely many \({\mathbb Q}_p\)isomorphism classes for the surface \(S^_{E,\xi }(4)\), 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 1 with p odd. Specifically, we have checked in each of these cases that the elliptic curve E is directly 4congruent over \({\mathbb Q}_p\) to an elliptic curve of the form considered in the proposition.
Proposition 9.5
Proof
Remark 9.6
We also found 4 examples (225336k1, 271800bt1, 329536y1, 368928bj1) where for every \(\xi \in H^1({\mathbb Q},E[4])\), representing an element of \(\mathrm{III}(E/{\mathbb Q})\) of order 4, the surface \(S_{E,\xi }^(4)\) has no points locally at 2, and a further 4 examples (271800bj1, 352800md1, 378400bv1, 378400by1) where each \(S_{E,\xi }^(4)\) is locally insoluble either at 2 or 5.
Example 9.7
We find that \(S({\mathbb Q}_2) = \emptyset \). As indicated in Remark 9.6, exactly the same happens for the other elements of order 4 in \(\mathrm{III}(E/{\mathbb Q})\). The argument in Proposition 9.3 now shows that none of the elements of \(\mathrm{III}(E/{\mathbb Q})\) of order 4 are visible in a principally polarized abelian surface.
Declarations
Authors’ Affiliations
References
 Barth, W.: Projective models of Shioda modular surfaces. Manuscripta Math. 50(1), 73–132 (1985)MathSciNetView ArticleMATHGoogle Scholar
 Bosma, W., Cannon, J., Cannon, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3), 235–265 (1997)MathSciNetView ArticleMATHGoogle Scholar
 Bruin, N., Doerksen, K.: The arithmetic of genus two curves with \((4,4)\)split Jacobians. Canad. J. Math 63(5), 992–1024 (2011)MathSciNetView ArticleMATHGoogle Scholar
 Bruin, N.: Visualising Sha[2] in abelian surfaces. Math. Comp. 73(247), 1459–1476 (2004)MathSciNetView ArticleMATHGoogle Scholar
 Bruin, N., Dahmen, S.R.: Visualizing elements of Sha[3] in genus 2 jacobians. In: Hanrot, G., Morain, F., Thomé, E. (eds.) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol. 6197. Springer, Berlin (2010). https://doi.org/10.1007/9783642145186_12
 Cremona, J. E.: Algorithms for modular elliptic curves, 2nd edn. Cambridge University Press, Cambridge. http://www.warwick.ac.uk/~masgaj/ftp/data/ (1997)
 Cremona, J.E.: Classical invariants and 2descent on elliptic curves. J. Symbolic Comput. 31(1–2), 71–87 (2001)MathSciNetView ArticleMATHGoogle Scholar
 Cremona, J.E., Fisher, T.A., O’Neil, C., Simon, D., Stoll, M.: Explicit \(n\)descent on elliptic curves. I. Algebra. J. Reine Angew. Math. 615(121–155), 0075–4102 (2008)MathSciNetMATHGoogle Scholar
 Cremona, J.E., Fisher, T.A., Stoll, M.: Minimisation and reduction of 2, 3 and 4coverings of elliptic curves. Algebra Number Theory 4(6), 763–820 (2010)MathSciNetView ArticleMATHGoogle Scholar
 Cremona, J.E., Mazur, B.: Visualizing elements in the Shafarevich–Tate group. Exp. Math 9(1), 13–28 (2000)MathSciNetView ArticleMATHGoogle Scholar
 Fisher, T.: Some improvements to 4descent on an elliptic curve. In: van der Poorten, A.J., Stein, A. (eds.) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol. 5011. Springer, Berlin (2008). https://doi.org/10.1007/9783540794561_8
 Fisher, T.: The Hessian of a genus one curve. Proc. Lond. Math. Soc 104(3), 613–648 (2012)MathSciNetView ArticleMATHGoogle Scholar
 Fisher, T.: Invariant theory for the elliptic normal quintic I. Twists of X(5). Math. Ann. 356(2), 589–616 (2013)MathSciNetView ArticleMATHGoogle Scholar
 Fisher, T.: Invisibility of Tate–Shafarevich groups in abelian surfaces. IMRN 15, 4085–4099 (2014)MathSciNetView ArticleMATHGoogle Scholar
 Klenke, T.A.: Visualizing elements of order two in the Weil–Châtelet group. J. Number Theory 110(2), 387–395 (2005)MathSciNetView ArticleMATHGoogle Scholar
 Mazur, B.: Visualizing elements of order three in the Shafarevich–Tate group. Asian J. Math. 3(1), 221–232 (1999)MathSciNetView ArticleMATHGoogle Scholar
 Merriman, J.R., Siksek, S., Smart, N.P.: Explicit \(4\)descents on an elliptic curve. Acta Arith 77(4), 385–404 (1996)MathSciNetView ArticleMATHGoogle Scholar
 Milne, J.S.: Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), pp. 103–150. Springer, New York (1986)Google Scholar
 Silverberg, A.: Explicit families of elliptic curves with prescribed mod N representations. In: Modular forms and Fermat’s last theorem (Boston, MA, 1995), pp. 447–461. Springer, New York (1997)Google Scholar
 Stamminger, S.K.M.: Explicit 8descent on elliptic curves, International University Bremen, http://nbnresolving.de/urn:nbn:de:101:1201305171186, (PhD thesis) (2005)
 Womack, T.: Explicit descent on elliptic curves, University of Nottingham, (PhD thesis) (2003)Google Scholar