The Brauer-Manin obstruction on del Pezzo surfaces of degree 2 branched along a plane section of a Kummer surface

By ADAM LOGAN

Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, CANADA N2L 3G1

                         e-mail: a5logan@math.uwaterloo.ca

Abstract This paper discusses the Brauer-Manin obstruction on double covers of the projective plane

branched along a plane section of a Kummer surface from both the practical and theoretical points of view. Theoretical highlights include the determination of a complete set of generators for the Brauer group; on the practical side, we give several surfaces with a Brauer-Manin obstruction and verify that the Brauer-Manin obstruction is the only one for a collection of several thousand surfaces of this type.

1. Introduction

One of the most important problems in arithmetic geometry is to develop techniques for deciding whether a variety over a number field has a rational point. There are several techniques that can be used in attempting to prove a negative answer. Most simply, if the variety has no points over some completion of the number field, then it cannot possibly have any rational points over the field itself. Manin pointed out that having no rational points in a completion was equivalent to having no points over the adèle ring of the number field. More generally, he observed that previous proofs that varieties with points in all completions of a number field do not have any global points could be interpreted as showing that all of the adèlic points were contained in some set that did not contain any of the diagonal adèles, and in particular that the Brauer group of the variety could be used for constructing such a set. In his honour, we say that the Brauer-Manin obstruction blocks the existence of rational points on a variety V when the set of points shown by Manin’s techniques to contain all the adèlic points does not contain any diagonal adèles. Likewise, we say that the Brauer-Manin obstruction to the Hasse principle is the only one for a class of varieties if, for every variety V in the class, one of three possibilities holds: V has a rational point, V does not have any points in some completion, or the Brauer-Manin obstruction blocks the existence of rational points on V.

Colliot-Thélène and Sansuc have conjectured that the Brauer-Manin obstruction may be the only one for the class of geometrically connected, geometrically rational varieties over a number field, and have suggested that perhaps “geometrically rational” may even be weakened to “geometrically rationally connected” [7, Introduction]. In this paper, we will study a special case of this conjecture in the context of del Pezzo surfaces of degree 2. Our main result is the following.

Theorem 1.1 Let C be the hyperelliptic curve y^{2} = (x^{2} + 1)(x^{2} + 2)(x^{2} + x + 1), let K be the Kummer surface of its Jacobian (a quartic surface in ℙ^{3}, smooth except at sixteen ordinary double points) and let P be the set of hyperplanes H in ℙ^{3} defined by a linear form with integer coefficients of absolute value at most 5 such that K ∩ H is a nonsingular curve. For each such H, let Q_{H} be the projection of K ∩ H to ℙ^{2}, let f_{H} be the defining equation of Q_{H}, normalized so that the lexicographically first coefficient is 1, and let V_{H} be the set of surfaces in the weighted projective space ℙ(2, 1, 1, 1) defined by w^{2} + f_{H}(x, y, z) = 0. Then the Brauer-Manin obstruction for the set of V_{H} is the only one.

The proof of this theorem divides into several parts. For most of the V, either a short and simple search or a proof that V is not locally solvable can be given. For a few of them, a more serious search is necessary, and finally, on three of them the Brauer-Manin obstruction is decisive. We give the details of the calculation of the obstruction, as they demonstrate the important technique of constructing elements of the Brauer group from elliptic fibrations. To the best of my knowledge, this method has not previously been used for explicit computations on rational surfaces.

The paper also discusses some general properties of curves of genus 3 which are plane sections of Kummer surfaces, and of the double covers branched at these. In particular, we show that, of the two basic Brauer classes of order 2 for del Pezzo surfaces of degree 2 discussed by Corn [5, definition 4.2.2], the restriction of the difficult one is entirely tractable. Thus we are able to show that it is always practical to compute generators for the Brauer group of a double cover of ℙ^{2} branched along a plane section of a Kummer surface.

2. Basics Let C_{2} be a curve of genus 2. The Kummer surface K of C_{2} is a quartic surface in ℙ^{3} with

sixteen nodes. By the Bertini theorem, a general plane section of K is a smooth curve of genus 3. The 28 bitangents possessed by every such curve can be divided into families of six and 10 coming from special planes called tropes that pass through six of the nodes of K, and another family of degree 12 coming from certain conics. For details see [6], [1]. Definition. A del Pezzo surface of degree 2 is a smooth surface with K^{2} = 2 on which −K is an ample divisor.

As shown in, for example, [14, page 142], such a surface is a double cover of ℙ^{2} branched along a quartic curve. Equivalently, it is a hypersurface of degree 4 in the weighted projective space ℙ(2, 1, 1, 1), embedded by the three sections of −K and a section of −2K not in the subspace spanned by products of two sections of −K. By completing the square, we reduce to the case w^{2} = f(x, y, z), where deg f = 4.

Among previous work on such surfaces, the contributions of Kresch-Tschinkel [9] and Corn [5] are particularly notable. The former describes a complete procedure for calculating the Brauer group of a diagonal del Pezzo surface of degree 2, that is, one of the form aw^{2} +bx^{4} +cy^{4} +dz^{4} = 0, and gives examples. The latter proves, among other things, that every nontrivial element of the Brauer group of a del Pezzo surface of degree 2 is a restriction of one of two special kinds of elements, one of which can be represented by an easily described Azumaya algebra and the other of which cannot be in general, as far as is known at present.

In this paper, we will study the Brauer obstruction on del Pezzo surfaces of degree 2 that are double covers of ℙ^{2} branched along smooth plane sections of Kummer surfaces. (Note that every smooth plane quartic in ℙ^{2} is isomorphic to a plane section of a Kummer surface after a suitable field extension; in contrast, the curves of the form ax^{4} + by^{4} + cz^{4} are all isomorphic over \overline{ℚ}.) We completely characterize the Brauer elements that can occur and give several examples of Brauer obstruction to the existence of rational points. We also simplify some of Corn’s methods for constructing Brauer elements.

The reason for choosing smooth plane sections of Kummer surfaces is that we can construct curves with many different Galois actions on the 2-torsion points. We get a wide variety of possible actions with relatively small Galois groups, and thus a better chance of finding interesting Brauer elements.

3. Group theory For the time being, let C be a general curve of genus 3 and H a del Pezzo surface of degree 2

that is a double cover of ℙ^{2} branched along C. As is well-known, there are 56 exceptional curves on H, which are the irreducible components of the inverse images of the 28 bitangents on C. The monodromy of the exceptional curves is given by the Weyl group W(E_{7}) of order 2903040. The intersection number of two exceptional curves mapping to the same bitangent of C is 2; all other pairs of distinct exceptional curves have intersection number 0 or 1. Therefore, the Weyl group maps to the automorphism group of the configuration of bitangents, which is Sp_{6}(𝔽_{2}).

Proposition 3.1 This map is surjective and its kernel is of order 2.

Proof. From the description above, the map is induced by the action of W(E_{7}) on the set of 28 pairs of exceptional curves on H that intersect each other with multiplicity 2. The assertions of this proposition are now easily checked by computer calculation.

Note that the kernel, being a normal subgroup of order 2, is central (in fact, this is the only nontrivial normal subgroup of W(E_{7}) whose index exceeds 2). Since the kernel has trivial intersection with the subgroup of W(E_{7}) consisting of all words of even length, which is isomorphic to Sp_{6}(𝔽_{2}), it follows that W(E_{7}) is isomorphic to a direct sum ℤ/2 ⊕ Sp_{6}(𝔽_{2}). In particular, its subgroups are easily described in terms of those of Sp_{6}(𝔽_{2}): for each subgroup G of Sp_{6}(𝔽_{2}), the subgroups of W(E_{7}) mapping to it are first the inverse image of G under the map described above, and then for every homomorphism ϕ : G → ℤ/2, the subgroup {(a, g)|ϕ(g) = a}. Further, two subgroups of W(E_{7}) are conjugate if and only if their images are conjugate, and either they are both full inverse images or the one homomorphism is taken to the other by an automorphism of G induced by conjugation by an element of Sp_{6}(𝔽_{2}).

W(E_{7}) acts not only on the lines but also on the Picard group of H, which is generated by their classes. Thus we obtain an action of the Galois group of the field of definition of the lines on the Picard group. The first cohomology group of this Galois module is isomorphic to the algebraic part of the Brauer group of the surface H (see, for example, [5, proposition 1.3.7]). As H is geometrically rational, this is in fact the whole Brauer group.

Now let us restrict our attention to surfaces branched along a plane section of the Kummer surface of the Jacobian of y^{2} = f(x).

Proposition 3.2 The bitangents of such a curve are defined over an extension of ℚ whose Galois group is a subgroup of S_{6} ⊕ ℤ/2, and in the generic case they are in orbits of sizes 6, 10, 12. In addition, there is a rational 2-torsion divisor on the Jacobian of such a curve.

Proof. From [6, section 3.7] we see that the tropes of such a Kummer surface are parametrized by the roots of f and by divisions of the roots of f into two triplets. As pointed out in [1, page 16], the intersections of the tropes with the plane of C give bitangents, and there are 6 roots and 10 partitions of the roots into two triplets. Likewise, the description of the other 12 bitangents in [1, page 16] shows that they are defined after an additional quadratic extension of ℚ.

The last statement can be seen in several different ways. For example, the orbit of size 12 is in fact a Steiner system. Alternatively, consider the subgroup of Jac C[2] spanned by differences of two bitangents, one in the orbit of order 6, the other in the orbit of order 10. It is of index 2 and its orthogonal complement is spanned by the rational 2-torsion divisor.

Actually, the Galois group of a generic plane section is the stabilizer of a 2-torsion point and a nonisotropic subgroup of order 4 containing it. This subgroup is the orthogonal complement of the subgroup generated by differences of two bitangents in the orbit of order 6. It can also be seen as the subgroup generated by the 2-torsion divisor already described and the two conjugate Steiner systems consisting of the orbit of order 6 and six of the elements of the orbit of order 12.

4. Elements of the first type

In this section we will describe an alternative approach to Corn’s “elements of the first type” [5, definition 4.2.2] in the Brauer group of a del Pezzo surface of degree 2. The starting point is the following standard geometric lemma:

Lemma 4.1 Let C be a reduced plane curve of even degree over an algebraically closed field K of characteristic not equal to 2, let H be the double cover of the plane branched along C, let π be the map from H to ℙ^{2}, and let D be a conic in the plane which is not a component of C. Then π^{−1}(D) decomposes into two components switched by the automorphism of the double cover if and only if all points of intersection of C and D have even multiplicity.

Proof. One direction is easy: if π^{−1}D = E_{1} ∪ E_{2}, then the intersections of E_{1} and of E_{2} with the branch locus have the same multiplicity, and by the push-pull formula the intersection multiplicity of C and D at a point is the sum of those of E_{1} and E_{2} with the branch locus at its inverse image.

In the other direction, let us take C and D to be defined by equations f = 0 and g = 0 respectively. The double cover of C is defined by the equation w^{2} = f in a suitable weighted projective space, and the inverse image of D by the the additional equation g = 0. Now, f cuts out a divisor on g in which all points have even multiplicity, and this is the same as the divisor cut out by the square of a polynomial (since D is a conic, all divisors of even degree are cut out by polynomials). In other words, f is a square in K[x, y, z]/(g), say of h. It follows that the scheme w^{2} = f, g = 0 decomposes into the union of the components V (w − h, g) and V (w + h, g).

Recall Corn’s definition of elements of the first type.

Definition. [5, definition 4.2.2, proposition 4.2.4] Let S be the subgroup of W(E_{7}) that stabilizes the inverse image of a Steiner system (equivalently, that stabilizes an element of order 2 in the Picard group of the branch locus) and let G be any subgroup of W(E_{7}), thought of as embedded in S. (The reason for this form of the statement is that different embeddings of G in S up to conjugacy may give different elements of the first type.) It turns out that H^{1}(S, Pic V̄) is isomorphic to ℤ/2, and that a generator is given by the cocycle f(s) = ([k]^{s} −[k])/2, where [k] is the sum of the classes of lines that map to two bitangents of the branch locus, the difference of whose loci of bitangency is the given 2-torsion point. The restriction of this cohomology class to G is an element of the first type.

Let H be a del Pezzo surface of degree 2 branched over C, and let p ∈ Jac C[2]. By the standard recipe [15, lemma 1], to construct an Azumaya algebra representing the corresponding element of the first type, it suffices to construct a quadratic extension K/F with automorphism σ and a K-rational divisor D in the class [k]. Indeed, let f be a function whose divisor is D+D^{σ} −2L, where L is the divisor of any hyperplane section of H. Then the necessary Azumaya algebra is the quaternion algebra (K/F, f).

In order to construct these elements in a simple way, we use Lemma 4.1. We know that the a component of the inverse image of the union of two lines whose difference is the given 2-torsion point constitute a divisor of the right class; however, this divisor will not be rational. Let D_{1}, …, D_{6} be the six curves of degree 2 in the plane that are the unions of opposite pairs of lines in the Steiner system (so that the set {D_{1}, …, D_{6}} is rational, even though the curves themselves will not be). Clearly, these curves do decompose into components in the double cover of ℙ^{2} branched along C.

We now introduce a fact from enumerative geometry.

Theorem 4.2 The locus of (reduced) conics four times tangent to a curve X of degree 4 in ℙ^{2} consists of the union of 63 conics in ℙ^{5}.

Proof. (I thank Jason Starr for supplying the following argument.) Let C be such a conic. Then C · X is twice an effective divisor D, and 2D is linearly equivalent to 2K, so D − K is a 2-torsion divisor. This divisor cannot be 0, for the canonical class consists of intersections of lines in ℙ^{2} with the conic, and so C would be a double line.

Let L be one of the 63 nontrivial divisor classes of order 2. Then h^{0}(K + L) = 2 by Riemann-Roch, so the family of divisors of the form 2D, where D ∈ |K + L|, is of dimension 1. There is a natural map from H^{0}(K + L) to H^{0}(2K) given by D → 2D; since each point appears in one divisor in |K + L|, each point appears twice in one divisor in the image, which therefore has degree 2.

In particular, this shows that the locus of conics whose inverse image in H decomposes into two curves, one of the class [k], is a curve of degree 2 in ℙ^{5}. We know six points on this curve, corresponding to D_{1}, …, D_{6}. Since by [8, exercise IV.3.4] all curves of degree 2 are in fact plane conics, they are determined by five points, so we can determine the locus completely by linear algebra.

Assume that H is everywhere locally solvable. Then the rational divisor class [k] must be represented by an rational divisor, whose image in ℙ^{2} must be one of the conics parametrized by the conic described above. In particular this conic has infinitely many rational points, and so we can find infinitely many conics C_{1}, C_{2}, ... that pull back to curves in the desired class.

The inverse image of each of these conics decomposes into two curves over ℚ̄, and hence over some quadratic field. On the other hand, the divisor class on H represented by one of these curves is assumed to be a quadratic extension of the base, and so any field over which the inverse image decomposes must contain this quadratic field. In particular, these two fields are the same.

5. Elliptic fibrations In this section we show how to adapt the standard construction of Brauer elements from

elliptic fibrations [2] to our situation.

Our surface H has no elliptic fibrations. To see this, let C be the class of a putative fibre, and observe that we would have C · (K + C) = 2g − 2 = 0, but on the other hand C · C would be equal to 0, so C · K = 0 as well. This is impossible, because −K is an ample divisor.

On the other hand, consider a pencil P of lines on ℙ^{2}. It determines a a pencil of effective divisors of degree 4 on C, and so when pulled back to H it determines a pencil of double covers of ℙ^{1} branched in four points: that is to say, of elliptic curves. However, they all meet in the two points lying above the base point of P. It follows that blowing up these two points on H creates an elliptic fibration, whose fibre is of class −K − E_{1} − E_{2}, where E_{1}, E_{2} are the exceptional divisors. Definition. The vertical Picard group Pic_{vert} of a fibration on H is the subgroup of the Picard group generated by the geometrically irreducible components of fibres.

By results of Bright [2] to be cited below, a fibration can be used to construct nontrivial elements of the Brauer group of H̃ only if there are geometrically reducible fibres—in other words, only if Pic_{vert} is not generated by the class of a fibre. The obvious way to arrange for this is to choose the base point of P to lie on a bitangent, or, better still, for it to be the point of intersection of two bitangents. In other words, if there are two bitangents defined and conjugate over a quadratic field, we should take the elliptic fibration of H determined by the pullbacks of lines through their point of intersection. Then there are two fibres of Kodaira type I_{2}, and so, assuming that no other fibre is reducible, the vertical Picard group of H̃ will be generated by the four components of these fibres, modulo the relation that the classes of the two fibres are equal. If other fibres are reducible, the classes constructed below will still represent elements of the Brauer group, though there may be other elements as well.

In general the field of definition of the lines of H is the compositum of the field of definition of the bitangents of C with a quadratic extension of ℚ. It follows that the Galois group of these four lines is in general (ℤ/2)^{2}, where one generator exchanges the fibres and the other interchanges the components of each fibre. In other words, letting l_{1}, l_{2}, m_{1}, m_{2} be the four lines, viewed as generators of Pic_{vert} modulo the relation l_{1} + l_{2} = m_{1} + m_{2}, one of the generators of (ℤ/2)^{2} switches l_{1} with l_{2} and m_{1} with m_{2}, while the other switches l_{1} with m_{1} and l_{2} with m_{2}. Proposition 5.1 The first cohomology group of (ℤ/2)^{2} with coefficients in Pic_{vert} is isomorphic to ℤ_{2}. Proof. Using a computer this is trivial, but it is also not difficult to give a direct proof. According to the Künneth formula, this group is isomorphic to

(H^{1}(G_{1}, Pic_{vert}) ⊗ Ĥ^{0}(G_{2}, Pic_{vert})) ⊕ ˆ(H^{0}(G_{1}, Pic_{vert}) ⊗ H^{1}(G_{2}, Pic_{vert})),

where G_{1} and G_{2} are the two factors of (ℤ/2)^{2}. Then Ĥ^{0}(G_{1}, Pic_{vert}) is trivial; indeed, this is H^{0}(G_{1}, Pic_{vert})/N_{G_{1}} Pic_{vert}, and it is immediate that both of these are generated by l_{1} +l_{2}. On the other hand, N_{G_{2}} Pic_{vert} has index 2 in its saturation (being generated by l_{1} +m_{1} and l_{1} +2l_{2} −m_{1}, and therefore containing 2l_{1} + 2l_{2} but not l_{1} + l_{2}), and so Ĥ^{0}(G_{2}, Pic_{vert}) is isomorphic to ℤ/2. Similarly, H^{1}(G_{1}, Pic_{vert}) = ker N_{G_{1}} Pic_{vert} /(s − 1) Pic_{vert}, where s is the generator of G_{1}. Again, (s−1) Pic_{vert} is generated by l_{1} −l_{2}, l_{1} +l_{2} +2m_{1}, so it has index 2 in its saturation ker N_{G_{1}} Pic_{vert}. The result follows.

Now we consider which Azumaya algebras on ℙ^{1} pull back to give Azumaya algebras on H̃. Recall Bright’s characterization: Proposition 5.2 Let A ∈ Br ℚ(t) be given in the form ∑ n_{P} cores_{ℚ(P)/ℚ}(t − t_{P} , c_{P}) + a, where ℚ(P) is the field of definition of ℚ, c_{P} is a cyclic Galois character on ℚ(P), and a ∈ Br ℚ. For each P occurring in the definition of A, let D̄ run over one ℚ̄-component of each ℚ(P)-component of the fibre at t_{P}. Then A pulls back to an Azumaya algebra on H̃ if and only if:

(i) ∑ n_{P} cores_{ℚ(P)/ℚ} c_{P} is trivial; and (ii) the restriction of c_{P} to the field of definition of D̄ is trivial.

Proof. Except for minor changes of notation, this is identical to [2, proposition 4.28].

To interpret (i) more concretely, note that if c_{P} is a quadratic character corresponding to the extension ℚ(P)(\sqrt{α}_{P})/ℚ(P), then cores c_{P} corresponds to the extension ℚ(Nα_{P})/ℚ, so that ∑ n_{P} cores_{ℚ(P)/ℚ} c_{P} is trivial if and only if ∏_{P} (Nα_{P})_{P}^{n} is a square. As for (ii), it states simply that for each P, the fields of definition of the geometric components of the fibre above P contain the field cut out by c_{P}. In particular, if the fibre above P is geometrically irreducible, then c_{P} must be trivial.

In particular, in our situation where we have only one Gal ℚ̄/ℚ-orbit of geometrically reducible fibres, each fibre having two geometric components, the only possible Azumaya algebra is cores(t− t_{P} , c_{P}), where c_{P} is the character corresponding to the field of definition of the components. These components are defined over an extension of ℚ(t_{P}) obtained by adjoining the square root of a rational number, namely the square root which generates the field of definition of the 56 lines on the surface over the field of definition of the 28 bitangents (compare Proposition 3.1), so condition (i) holds automatically.

It is now appropriate to mention the relation between the Brauer groups of H and H̃: Proposition 5.3 ([14, theorem 29.1]) The Brauer groups of H and H̃ are canonically isomorphic.

It turns out that this gives a construction of Corn’s “elements of the second kind” in the case of the del Pezzo surfaces of degree 2 we are studying. In particular: Theorem 5.4 Let H be a del Pezzo surface of degree 2 branched along a smooth plane section C of a Kummer surface. Then the Brauer group of H is generated by two types of elements: elements of the first kind, which correspond to rational 2-torsion points in the Jacobian of C; and elliptic elements as described above, which correspond to rational pairs of Weierstrass points of the hyperelliptic curve used to define the Kummer surface. In particular, elements of order greater than 2 do not occur in the Brauer group of such a del Pezzo surface. Proof. This is done by computer calculation, using code originally written by Ronald van Luijk for use in [12]. One lists the possible Galois groups, constructs the two given 2-torsion elements, and verifies that for every group its cohomology is generated by restrictions of the two elements described above. See [11] for the code and a transcript of the calculations.

6. The examples In this section, we discuss some of the basic building blocks of the calculation: some subsidiary

algorithms that must be available, as well as the varieties that are used and the reason for their choice. A number of auxiliary functions must be available: for example, we need to calculate the bitangents of plane sections of Kummer surfaces, for without this the algorithm cannot get started. In the present context, this can be read off from [6] and [1], but I instead used the general-purpose code already available from [10].

A more serious matter is to calculate the invariants. The problem lies not in evaluating the norm residue symbol in finite extensions of ℚ_{p}, which is fairly easy even when the residue characteristic is 2; rather, it lies in reducing to the calculation of finitely many such symbols. The method presented here follows [12] fairly closely, though with one additional wrinkle to deal with the elements of the second type.

Recall that our goal is to evaluate the invariants of algebras of the form cores(t−t_{P} , c_{P}), where t is the fibration from S to ℙ^{1}(ℚ_{p}). The key insight is that, rather than attempting to classify the fibres that have p-adic rational points, we should attempt to divide ℙ^{1}(ℚ_{p}) into discs on which the invariant is constant, and then determine which of these discs have rational points lying over them.

Lemma 6.1 Suppose that the minimal polynomial of t_{P} has no roots in ℚ_{p}. Then, for all n, the open set p^{−n}ℤ_{p} can be divided into finitely many discs on each of which the invariant is constant.

Proof. The point is that, if (t − t_{P})/(t′ − t_{P}) is square, then the invariants at t and t′ are equal. In particular, this will happen when t and t′ are sufficiently close, compared with the distance from t_{P} to the nearest element of ℚ_{p}; it will also happen when t_{P} is small in comparison to t and t′, and t/t′ is a square. Since p^{−n}ℤ_{p} is compact, this completes the argument.

It follows immediately that if there is a disc around infinity in which the invariant is constant, then conclusion of this lemma holds for all of ℚ_{p}. Furthermore, if the fibre at infinity has no ℚ_{p}-points, then there is a disc near infinity where that is the case, and this disc can be ignored, reducing the problem to that treated above.

If the minimal polynomial of t_{P} does factor in ℚ_{p}, there is still hope. For example, it is possible for there to be no ℚ_{p}-rational points on fibres sufficiently close to the roots, or for the local invariant to be constant near t_{P}. (This could be proved by using the other fibration that gives the same element of the Brauer group; if there is a neighbourhood of t_{P} such that all points mapped by the one fibration to that neighbourhood should happen to be mapped by the other one to a region where its invariant is constant, the same holds for the original.) It is genuinely possible, though, for the invariant not to be locally constant near a bad fibre. (This happens if the bad fibre has two rational singular points where the components intersect and these points have different invariants.) This problem did not need to be dealt with in our examples, for in all such cases a small point could be found, so there is no obstruction.

In order to use Lemma 6.1 effectively, it is necessary to be able to decide which of the discs have rational points. For greater generality, we state this for several fibrations at once.

Lemma 6.2 (cf. [12]) Let S be a smooth projective surface over ℚ_{p}, let π_{1}, …, π_{k} be fibrations, and let D_{1}, …, D_{k} be open discs in ℙ^{1}(ℤ_{p}). It can be effectively determined whether there are ℚ_{p}-points of S mapping by π_{i} to D_{i} for all i = 1, …, k.

Proof. By changing coordinates, we may assume that each D_{i} is actually ℤ_{p}. Consider the graph of all of the fibrations as a subscheme of S × ℙ^{1} × … × ℙ^{1} (one factor of ℙ^{1} for each fibration). The statement of the theorem is true if we can decide whether there is an affine patch of the form S_{i} × 𝔸^{1} × … × 𝔸^{1} that has integral points; however, by Hensel’s lemma it is routine (at least in principle) to decide whether a smooth affine variety over ℤ_{p} has integral points. In practice, this was only needed for k = 1, 2, and only for small primes, as in other cases the invariants needed could be evaluated more efficiently using results of [3].

For the purposes of this paper, we need a slightly stronger result, allowing us to evaluate the conics as well.

Lemma 6.3 In the situation of Lemma 6.2, let ℙ^{n} be the projective space containing S, let R be the set of ℚ_{p}-rational points of S mapped by the π_{i} to the D_{i}, let c be a section of O_{ℙ^{n}}(2) not vanishing on any point of R, and suppose additionally that there is a section l of O_{ℙ^{n}}(1) that do not vanish on any ℚ_{p}-rational point of S mapped by the π_{i} to the D_{i}. Then, for any class k in ℚ_{p}^{∗}/(ℚ_{p}^{∗})^{2}, it can be decided whether there is a ℚ_{p}-point of S that maps by π_{1}, …, π_{k} to D_{1}, …, D_{k} respectively and at which c evaluates to an element of ℚ_{p} in the class k modulo squares. Proof. Consider instead the graph of the π in the product S × ℙ^{1} × … × ℙ^{1} × ℙ^{2} together with the equations cx − l^{2}y and xy − kz^{2} (where x, y, z are coordinates on ℙ^{2}). Again, it suffices to decide whether this graph has an integral point in one of a given set of affine patches.

It follows immediately that we can decide whether a given set of local invariants occurs, at least when that can be decided for the elliptic fibrations. In fact, the hypothesis on c and l can be replaced with weaker hypotheses on the existence of several forms, not all vanishing simultaneously, but for the calculations in this paper it was not necessary to do so.

Finally, let us explain how to bound the set of primes that needs to be checked. Proposition 6.4 Let E be an Azumaya algebra on S arising from an elliptic fibration f. Then E is trivial at all primes at which S has good reduction and the fibre at ∞ of f is locally solvable. Similarly, if C = (f, s) is an Azumaya algebra described by a conic f with p-adically integral coefficients whose inverse image in S splits over ℚ(\sqrt{s}), then C is trivial at all primes at which S has good reduction, ℚ(\sqrt{s})/ℚ is unramified, and there is a point on S at which f ≠ 0 mod p. This last condition is automatically satisfied for good primes greater than or equal to 53. Proof. By [3, theorem 1] the local invariant of any Azumaya algebra at a prime of good reduction is constant, so it suffices to prove that the other conditions imply that there is a single p-adic point at which the invariant is trivial. In the elliptic fibration case, if the fibre at ∞ is locally solvable, then all sufficiently nearby fibres are as well, including the fibre at 1/p^{2n} for large n. As 1/p^{2n} −t_{P} is a p-adic square for large enough n, the invariant is trivial there. In the conic case, the given conditions ensure that (a, s) is trivial for ord_{p} a = 0 and that there is a point P on S defined over ℚ_{p} for which ord_{p} f(P) = 0. Since there are at most eight points at which f = 0, and since a simple calculation with the Weil bounds shows that every curve of genus 3 over F_{p} has at least 11 points for p ≥ 53, the result follows.

Let us explain how the Kummer surface of which to take plane sections was chosen. I chose somewhat arbitrarily to work with a hyperelliptic curve of the form y^{2} = f(x) = q_{1}(x)q_{2}(x)q_{3}(x), where q_{1}, q_{2}, q_{3} are quadratic polynomials whose roots generate three independent fields F_{1}, F_{2}, F_{3} whose compositum has degree 8. The 28 bitangents of the general plane section of the Kummer surface will then be acted on by Galois in the following way: six of them will be acted on in the same way as the roots of f, so there will be three pairs of bitangents defined over F_{1}, F_{2}, F_{3} respectively; ten will be acted on in the same way as partitions of the roots of f into two disjoint triples, so there will be three more pairs defined over F_{1}, F_{2}, F_{3} and one quadruple defined over the unique degree-4 subfield of F_{1}F_{2}F_{3} not containing any of the F_{i}. In addition, in general there is a quadratic field L such that the other twelve fall into three orbits, one defined over each of the fields F_{i}L: total 28. (In special situations this field L can be a subfield of the splitting field of f and the description would be different.) Definition. Let E_{1}, E_{2}, E_{3} be the Azumaya algebras coming from the elliptic fibrations corresponding to the factors q_{1}, q_{2}, q_{3} of f as in Proposition 5.2, let E′_{1}, E′_{2}, E′_{3} be the Azumaya algebras coming from the elliptic fibrations given by projection from the point of intersection of the other pairs of bitangents defined over F_{1}, F_{2}, F_{3}, and let C be the Brauer element of the first kind corresponding to the 2-torsion point given by the 12 lines defined over F_{i}L. Proposition 6.5 The Brauer group of a generic plane section of the Kummer surface of a generic hyperelliptic curve as described above is isomorphic to (ℤ/2)^{3}, and is generated by E_{1}, E_{2}, C. Up to elements of Br ℚ, we have E_{i} = E′_{i} and E_{1} + E_{2} + E_{3} = 0. Proof. In light of the description of the cohomology classes E_{i}, E′_{i}, and C, this is a routine computer calculation.

As was, I believe, first pointed out by Swinnerton-Dyer, this means that the invariants of the E_{i} are more likely to be globally constant than they might otherwise be, because their local invariants are constant on fibres of two entirely different elliptic fibrations.

For these reasons I chose q_{1} = x^{2} + x + 1, q_{2} = x^{2} + 1, q_{3} = x^{2} + 2, this being as simple an example as possible that avoids degeneracy. I chose polynomials with complex roots so as to avoid having to compute local invariants at ℝ for these algebras.

7. Results In this section we present the proof of the main theorem, repeated here for the reader’s

convenience. Theorem 7.1 Let C be the hyperelliptic curve y^{2} = (x^{2} + 1)(x^{2} + 2)(x^{2} + x + 1), let K be the Kummer surface of its Jacobian (a quartic surface in ℙ^{3}, smooth except at sixteen ordinary double points) and let P be the set of hyperplanes H in ℙ^{3} defined by a linear form with integer coefficients of absolute value at most 5 such that K ∩ H is a nonsingular curve. For each such H, let Q_{H} be the projection of K ∩H to ℙ^{2}, let f_{H} be the defining equation of Q_{H}, normalized so that the lexicographically first coefficient is 1, and let V_{H} be the set of surfaces in ℙ(2, 1, 1, 1) defined by w^{2} + f_{H}(x, y, z) = 0. Then the Brauer-Manin obstruction for the set of V_{H} is the only one. Proof. First, we may assume that the coefficient of x_{4} in the linear form defining the hyperplane is nonzero, for if it were 0 then K ∩ H would be singular at (0 : 0 : 0 : 1); and then, by scaling, we may take it to be positive. This leaves us with 11^{3} · 5 = 6655 possibilities. Of these, however, 304 do not have relatively prime coefficients, and a further 146 are singular. Therefore, the set V has 6655 − 304 − 146 = 6205 elements. It is necessary to prove, for every element S of V, one of the following statements: that S does not have points everywhere locally, that S has a rational point, or that S is prevented from having points by the Brauer-Manin obstruction.

For the first of these, if we choose a (finite) prime p, it is easy, in principle, to determine whether S has a ℚ_{p}-point. For example, considering the plane 4x_{1} + x_{3} + x_{4}, we get the surface defined by w^{2} + f(x, y, z) = 0, where

f(x, y, z) = 1/4(4x^{4} + 8x^{3}y − 12x^{3}z + 16x^{2}y^{2} − 4x^{2}yz + 23x^{2}z^{2} + 8xy^{3} − 16xy^{2}z
−2xyz^{2} − 22xz^{3} + 8y^{4} + 8y^{3}z + 19y^{2}z^{2} + 10yz^{3} + 11z^{4},

and letting x, y, z range over all possibilities in ℤ/32ℤ, we find that 4f(x, y, z) cannot be 7 mod 8, 28 mod 32, or 0 mod 32, which means that it cannot be the negative of a square in ℚ_{2}. This calculation could be time-consuming for large p, but fortunately it is never necessary.

It is also not very difficult to decide whether S has a real point. If Q_{H} is defined by f(x, y, z) = 0, we need to know whether f can be negative. This can be decided as follows: first determine whether f(x, y, 0) can be negative, which is a matter of finding the real roots of a polynomial in one variable, and if not then use standard multivariable calculus, combined with a Gröbner-basis computation, to find the local extrema of f(x, y, 1) in the affine plane. If any of these is negative, then f can be negative; otherwise, it cannot.

We now check local solvability at ∞ and 2; it turns out that 4849 elements of V are solvable at 2 but not ∞, that 46 are solvable at ∞ but not 2, and that 129 are solvable at neither place. This leaves 1181 elements still to treat.

Of these, 106 come from planes that contain small rational points on K, and it turns out they all have points of height at most 10 (here and henceforth, “height” refers to the exponential height induced by projection to ℙ^{2}). For the remainder, a simple search program finds points of height at most 100 on all but 33, and standard sieving techniques find points of height at most 500 on all but 5. One of these five, corresponding to the plane 2x_{1} − 5x_{2} − 3x_{3} + 2x_{4} = 0, has a very small region of real points, and therefore it is practical to sieve more deeply on it than on the others. In particular, we find the smallest point (35607, −229, 770, 147) by sieving successively up to 600, 700, and 800. This leaves 4 surfaces still undone. It is easily verified that these are all locally solvable everywhere. To sum up what we have found so far:

Smallest height of point  Number
(total)  6205
(not locally solvable)  5024
h ≤ 10  702
10 < h ≤ 20  182
20 < h ≤ 50  113
50 < h ≤ 100  45
100 < h ≤ 200  21
200 < h ≤ 300  4
300 < h ≤ 400  2
400 < h ≤ 500  1
500 < h ≤ 700  0
700 < h ≤ 800  1
(undecided)  4

The four surfaces S_{1}, S_{2}, S_{3}, S_{4} as yet undecided come from the hyperplanes 3x_{1}+2x_{2}+4x_{3}+x_{4}, x_{1} −4x_{2} +2x_{3} +x_{4}, 3x_{1} −3x_{2} +4x_{3} +x_{4}, −2x_{1} −4x_{2} +5x_{3} +x_{4}. They are defined by the equations w^{2} + f_{i}(x) = 0, where

f_{1} = 1/12(12x^{4} − 4x^{3}y − 44x^{3}z + 15x^{2}y^{2} − 2x^{2}yz + 43x^{2}z^{2} − 4xy^{3}
−28xy^{2}z + 14xyz^{2} − 6xz^{3} + 4y^{4} − 8y^{3}z − 4yz^{3} − z^{4}),
f_{2} = 1/28(28x^{4} + 52x^{3}y − 20x^{3}z + 47x^{2}y^{2} + 58x^{2}yz + 35x^{2}z^{2} + 16xy^{3}
+76xy^{2}z − 2xyz^{2} − 14xz^{3} − 8y^{4} + 24y^{3}z + 24y^{2}z^{2} + 24yz^{3} + 7z^{4}),
f_{3} = 1/12(12x^{4} + 36x^{3}y − 44x^{3}z + 35x^{2}y^{2} − 22x^{2}yz + 43x^{2}z^{2} + 26xy^{3}
+22xy^{2}z − 66xyz^{2} − 6xz^{3} − y^{4} + 32y^{3}z + 10y^{2}z^{2} + 16yz^{3} − z^{4}),
f_{4} = 1/52(52x^{4} + 64x^{3}y + 28x^{3}z + 44x^{2}y^{2} + 160x^{2}yz − 73x^{2}z^{2} − 8xy^{3}
+100xy^{2}z − 110xyz^{2} + 34xz^{3} − 8y^{4} + 48y^{3}z + 3y^{2}z^{2} + 18yz^{3} − 5z^{4}).

To complete the proof of the theorem, we will find a rational point on the first of these surfaces and show that the remaining surfaces have no points due to the Brauer-Manin obstruction. The following lemmas will be useful for proving that algebras have constant local invariant at certain places: Lemma 7.2 ([4, lemma 5 (i)]) The local invariant of cores_{ℚ(t_{P})/ℚ}(t − t_{P} , c_{P}) at p is the sum of those of (t − t_{P} , c_{P}) at places of ℚ(t_{P}) lying above p. Lemma 7.3 Let r, s be rational numbers, and let p be a prime such that s is a square in ℚ_{p}(\sqrt{r}) (in particular, this holds if s is a square in ℚ_{p}, if p is odd and r and s are both p-adic units and not squares mod p, or for p = 2 and r/s ≡ 1 mod 8). Then the local invariant of the algebra cores_{ℚ(\sqrt{n})/ℚ}(f, s) at p is trivial. Proof. Indeed, by Lemma 7.2, the local invariant is a sum of local invariants of algebras of the form (f, s) over fields where s is a square.

First, let us consider the simplest example, namely S_{2}. The curve f_{2} = 0 and the surface S_{2} have bad reduction at {2, 3, 7, 23, 673}. The bitangents fall into ten orbits as usual; two sets each defined over ℚ(i), ℚ(\sqrt{−2}), ℚ(\sqrt{−3}), one over ℚ(\sqrt{−2}, \sqrt{3}), two over ℚ(i, \sqrt{3}), and one over ℚ(\sqrt{2}, \sqrt{3}). Let E_{1} be the Azumaya algebra from the fibration defined by projection from [1 : −1 : 1], which has two I_{2} fibres defined over ℚ(\sqrt{−3}), let E_{2} be the Azumaya algebra coming from the elliptic fibration defined by projection from [5 : −4 : 6], which has two I_{2} fibres defined over ℚ(i), and let C be the conic class. By direct calculation, the bad fibres of E_{1} and E_{2} become defined upon adjoining \sqrt{7}, while the conics decompose over ℚ(\sqrt{21}). In particular, notice that E_{1}, E_{2}, C are all trivial at ∞ and 673.

On the other hand, for p = 2, we find that E_{2} is trivial, by Lemma 7.3, since 7 is a square in ℚ_{2}(i). Also, it turns out that C is trivial, since the value of

g(x, y, z) = 155724x^{2} + 123492xy − 138600xz − 10009y^{2} + 24420yz + 78540z^{2}

always has even valuation and is therefore a norm in the unramified extension ℚ(\sqrt{21})/ℚ. It follows that C is 2-adically trivial as well. However, E_{1} takes both values: it is nontrivial above the disc 3/2 + 2ℤ_{2} and trivial above 4ℤ_{2}. For this reason E_{1} may be ignored for the rest of the calculation.

For p = 3, the class of E_{2} is always trivial, because 7 is a square in ℚ_{3}. In fact C is always nontrivial, because g(x, y, z) is always congruent to 2 mod 3 or to 3 mod 9, and neither 2 nor 3 is a norm from ℚ_{3}(\sqrt{21}) to ℚ_{3}.

Continuing to 7, we find that S has no rational singular points defined over 𝔽_{7} (it has two conjugate singular points over 𝔽_{49}). Hence, by [3, theorem 1], it is sufficient to consider one 7-adic point above each 𝔽_{7}-point of S. (Our Azumaya algebras split only over a field ramified at 7, so we cannot conclude that they are locally constant.) The 7 in the denominator of f_{2} means that every such point has w = 0; the points correspond to 𝔽_{7}-points of the branch locus, which are [0 : 0 : 1], [3 : 2 : 1], [6 : 2 : 1], [3 : 3 : 1], [2 : 5 : 1], [1 : 0 : 0]. Evaluating the elliptic fibration at such points, we get respectively 4, 3, 5, 6, 0, ∞ mod 7, which give us the local invariants 0, 0, 0, 1/2, 1/2, 0 respectively. Evaluating the conic, we get respectively 0, 4, 4, 0, 5, 2; using the alternative conic

204x^{2} + 132xy + 311y^{2} + 720yz + 240z^{2}

gives respectively 2, 1, 1, 5, 3, 1. The point [3 : 2 : 1] gives a square for both, so the two conics give the same algebra over ℚ_{7} (rather than differing by a constant); evaluating the norm residue symbols (a, 21)_{7}, we again get 0, 0, 0, 1/2, 1/2, 0. It follows that E_{2} + C is always 0 at this place.

For p = 23, Lemma 7.3 applies to E_{2}, since −1 is not a square mod 23 and 7 is a unit. Since C splits over an unramified extension of ℚ_{23} and every point on S over 𝔽_{23} is regular (the two singular points are defined over 𝔽_{23^{2}}), by [3] it suffices to consider one point mod 23, such as [20 : 1 : 1]. Evaluating the conic g(x, y, z) there gives 10, and 10 is a norm in the unramified extension ℚ_{23}(\sqrt{21})/ℚ_{23}. It follows C is again trivial.

We need to consider the bad primes of the fibre at infinity for E_{2}, which is

y^{2} = 569/2268r^{4} − 221/378r^{3} − 155/252r^{2} + 41/21r − 1.

These are 2, 3, 7, 18587087. The first three have already been considered, and for the last there is a smooth point with r = 4, so the fibre at infinity is locally solvable. Therefore E_{2} is trivial at all primes not mentioned. Similarly, the conic

17/20x^{2} + 11/20xy + 311/240y^{2} + 3yz + z^{2}

has good reduction away from 2, 3, 5, and the conic class is trivial at 5 because 21 is a square in ℚ_{5}. It follows that C too is trivial at all primes not mentioned.

To sum up, then, C + E_{2} is equal to 1/2 at 3 and is trivial at all other places. It follows that the Brauer-Manin obstruction blocks the existence of rational points on S.

Now we consider S_{3}. The bad primes are 2, 3, 5, 7, 11, 17, 229. Let E_{1}, respectively E_{2}, be the elliptic fibrations obtained by blowing up (1 : 0 : 0), respectively (1 : 1 : 3), and projecting. The fibres of type I_{2} are defined over ℚ(\sqrt{−3}), resp. ℚ(i), and their components are defined over the field obtained by adjoining \sqrt{3}.

Our third element is defined by conics. In particular, by the technique described in the discussion following Theorem 4.2, we find that elements representing that class are parametrized by points on the conic in ℙ^{2} defined by

506532x^{2} + 561204xy − 5104xz + 153409y^{2} + 9384yz + 80784z^{2},

in such a way that the point (a : b : c) determines the conic

2248ax^{2} + (2140a − 1010b − 4488c)xy + 2248bxz+
(−5750a − 3547b + 3340c)y^{2} + (−8466a − 7549b − 9364c)yz + 2248cz^{2}.

By good fortune, we only need one conic in this class, namely that corresponding to the point (7096 : −12988 : 3247) and defined by the equation

k(x, y, z) = 7096x^{2} + 6108xy − 12988xz + 7167y^{2} + 3366yz + 3247z^{2}.

The intersection of this conic with S decomposes into two curves defined over ℚ(\sqrt{17}), so we use the Azumaya algebras given by (c/l^{2}, 17), where l is a linear form that need not be specified.

We begin to calculate the local invariants. It is certainly necessary to consider all primes of bad reduction for S, which are the same as the bad primes for C (since C already has bad reduction at 2 and 3). In addition, it may be required to deal with some other primes, for example, those at which the fibre of ϕ_{1} or ϕ_{2} at ∞ is not locally solvable. In any case, the invariants at ∞ are all trivial. The algebras E_{1}, E_{2} are corestrictions from algebras over totally imaginary fields, which are automatically trivial, and likewise C_{1} is trivial at ∞, because 17 is a square in ℝ.

Next, we consider 2. Again, C_{1} is trivial there, 17 being a square in ℚ_{2}, but E_{1} and E_{2} are not. Indeed, as discussed in Lemma 6.2, we find that all 2-adic points of S are mapped by ϕ_{1} to the discs 3 + 8ℤ_{2}, 2 + 4ℤ_{2}, and 1/2 + ℤ_{2}, on each of which the local invariant of E_{1} is checked to be 1/2, while the image of S(ℚ_{2}) under ϕ_{2} is contained in the discs 4ℤ_{2} and ord_{p} x < 0, where the local invariants are 1/2 and 0 respectively. Further, each of these discs does in fact contain the image of a ℚ_{p}-rational point. From this it follows that the sequences of local invariants (1/2, 0, 0) and (1/2, 1/2, 0) both occur. But this means that E_{2} is useless for demonstrating that the Brauer obstruction blocks the existence of rational points on S, because the local invariant of an adelic point there can be chosen without affecting the other local invariants. In what follows, E_{2} will be ignored.

We consider p = 3. All 3-adic points of S are mapped by ϕ_{1} to the discs 1/9 + 1/3ℤ_{3}, 2/9 + 1/3ℤ_{3}, and 2/3 + ℤ_{3} with local invariant 0, and ℤ_{3} with local invariant 0. Both local invariants also occur for C_{1}. However, it turns out that the local invariant of E_{1} cannot be 0 at the same time that the invariant of C_{1} is 1/2. To be more precise, in the discs 1/9 + 1/3ℤ_{3}, 2/9 + 1/3ℤ_{3}, and 2/3 + ℤ_{3}, the value of k(x, y, z) is always a 3-adic square. On the other hand, in ℤ_{3} it may be in the coset of 1, 2, or 3 of ℚ_{3}^{∗}/(ℚ_{3}^{∗})^{2}. C_{1} has local invariant 1/2 exactly when ord_{3} k(x, y, z) is odd, and it turns out that this is equivalent to ϕ_{1}(P) ∈ ℤ_{3} and ϕ_{2}(P) ∈ 1 + 9ℤ_{3}.

Note that the fact that three of the possible four pairs of local invariants occur at 3 implies that no single element of the Brauer group rules out the existence of rational points on S.

For p = 5, Lemma 7.3 allows us to conclude that the invariant of E_{1} is 0. Also, k(x, y, z) is nonzero for all points of S over 𝔽_{5}. Therefore the local invariant of C_{1} at 5 is that of (a, 17), where ord_{5}(a) = 0. This is 0.

For p = 7, all p-adic rational points of S go under ϕ to a + ℤ_{p} for some a ∈ {0, 1, 2, 3, 5, 6}. In each case the local invariant is 1/2. On the other hand, k(x, y, z) is not 0 for any point on S over 𝔽_{7}. The local invariant is 0 for the same reason as above.

For p = 11, Lemma 7.3 applies, because 3 is a square in ℚ_{11}, and so the invariant of E_{1} at 3 is 0. Again, k(x, y, z) is never 0 for any 𝔽_{11}-rational point of S, so yet again the local invariant of C_{1} at 11 is 0.

For p = 17, Lemma 7.3 again applies, because 3 is a square in ℚ_{17}(\sqrt{−3}). This time, though, the invariant of C_{1} is 1/2. Indeed, for all but two points on the reduction of S mod 17, the value of k(x, y, z) is not a square mod 17. If a is not a square mod 17, then (a, 17)_{17} = 1/2. The exceptions are (0 : 5 : 3 : 1) and (0 : 7 : 11 : 1), at both of which the value is 0. The first of these does not lift mod 17^{2}. The second does; however, k takes the value 14 · 17^{2} mod 17^{3} at all lifts mod 17^{3}, and (17, 14 · 17^{2})_{17} = 1/2, so the invariant is always 1/2.

Finally, for p = 229, the invariant of C_{1} is trivial, because 229 is a square in ℚ_{1}7. The invariant of E_{1} is also trivial, from Lemma 7.3. Adding up the invariants for all the bad primes, now, we see that (0, 0) never occurs. To conclude, we note that the only prime where the fibre at infinity of the fibration defining E_{1} has bad reduction while S_{3} has good reduction is p = 47; but the fibre there is still locally solvable. As for C, it is simple to check that for good primes less than 53, there is a point where k ≠ 0.

For S_{4}, the calculations are similar. The primes of bad reduction are 2, 3, 13, 19, 73, 157, 1033. Again letting E_{1} be the Brauer element coming from a fibration with fibres of type I_{1} defined over ℚ(\sqrt{−3}), E_{2} the element coming from a fibration with I_{1}-fibres defined over ℚ(i), and C the element given by conics, we compute the following local invariants.

Prime  Local invariants
2  (0, 0, 0), (0, 1/2, 1/2)
3  (0, 0, 1/2)
13  (1/2, 0, 0), (0, 1/2, 1/2), (1/2, 1/2, 1/2),
(0, 0, 0), (1/2, 1/2, 0), (1/2, 0, 1/2)
19  (0, 0, 0)
73  (0, 0, 0), (0, 1/2, 1/2)
157  (0, 0, 0)
1033  (0, 0, 0), (0, 1/2, 1/2)

This establishes that the set of adelic points at which the sum of local invariants of both E_{1} and E_{2} + C is both 0 is empty, and therefore proves that there are no rational points on S_{4}. The only new phenomenon to be mentioned is that at 1033, adelic points lying over the singular 𝔽_{1033}-point [271 : 795 : 1] have local invariants (0, 1/2, 1/2), while other points go to (0, 0, 0). For the second statement, it is sufficient to check one point on S over ℚ_{1033} whose reduction mod 1033 is smooth; for the first, we consider all lifts mod p^{2} and verify that the only one for which the local invariant cannot be concluded to be (0, 1/2, 1/2) does not lift mod p^{3}.

Finally, for S_{1}, we need to find a point. With E_{1}, E_{2}, C as before, we find the following local invariants:

Prime  Local invariants
2  (1/2, 0, 0), (1/2, 0, 1/2), (1/2, 1/2, 1/2)
3  (0, 0, 0), (0, 0, 1/2), (0, 1/2, 0), (0, 1/2, 1/2), (1/2, 1/2, 1/2)
5  (0, 0, 0)
7  (0, 0, 0)
281  (0, 0, 0), (0, 1/2, 1/2)

It is easy to see that (0, 1/2, 1/2) is not the sum of a 2-adic and a 3-adic local invariant, so only 281-adic points with local invariants (0, 0, 0) may come from ℚ-rational points. These are precisely the points lying above the singular 𝔽_{281}-point [66 : 73 : 1]. Hence it is sufficient to search among those points. This greatly increases the range of the sieving and allows the smallest rational point [19905 : 7468 : 14722] (with w = 101353890) to be found within a few hours.

Acknowledgments. I am grateful to Victor Flynn for encouragement and to Jason Starr for enlightening me on the subject of enumerative geometry. One of the proofs presented below is his. In addition, this project was greatly facilitated by the MAGMA computer algebra system [13].

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[12] A. Logan and R. van Luijk, Nontrivial elements of Sha explained through K3 surfaces, unpublished.

[13] MAGMA software, by John Cannon et al. Documentation available at

http://magma.maths.usyd.edu.au.

[14] Y. Manin, Cubic Forms (Elsevier 1986, translated by M. Hazewinkel).

[15] H. P. F. Swinnerton-Dyer, Brauer-Manin obstructions on some Del Pezzo surfaces, Math. Proc. Cam. Phil. Soc. 125 (2) (1999), 193–198.