Manin Obstruction Research Articles

A NOTE ON A THEOREM OF HEATH-BROWN AND SKOROBOGATOV

We generalize a result of Heath-Brown and Skorobogatov [Rational solutions of certain equations involving norms, Acta Math.189 (2002), 161–177] to show that a certain class of varieties over a number field k satisfies Weak Approximation and the Hasse Principle, provided there is no Brauer–Manin obstruction.

Integral points for groups of multiplicative type

We construct a finite subgroup of Brauer–Manin obstruction for detecting the existence of integral points on integral models of homogeneous spaces of linear algebraic groups of multiplicative type. As an application, the strong approximation theorem for linear algebraic groups of multiplicative type is established. Moreover, the sum of two integral squares over some quadratic fields is discussed.

Good reduction of the Brauer–Manin obstruction

For a smooth and projective variety over a number field with torsion-free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer–Manin set.

Descent obstructions and Brauer–Manin obstruction in positive characteristic

AbstractWe prove that the Brauer–Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic $p$. More precisely, we show that the only obstructions come from étale covers of exponent $p$ or, alternatively, from flat covers coming from torsors under connected group schemes of exponent $p$.

On the quasi-group of a cubic surface over a finite field

We study the Mordell–Weil group MW ( V ) for cubic surfaces V over finite fields that are not necessarily irreducible and smooth. We construct a surjective map from MW ( V ) to a group that can be computed explicitly. For # MW ( V ) , this yields a lower bound, which is (often but) not always trivial. To distinguish cases, we follow the classification of cubic surfaces, originally due to Schläfli and Cayley. On the other hand, we describe an algorithm that a priori gives an upper bound for MW ( V ) . We report on our experiments for “randomly” chosen surfaces of the various types, showing that in all but one case lower and upper bounds agree. Finally, we give two applications to the number field case. First, we prove that the number of generators of MW ( V ) is unbounded. A second application explains why, for many reduction types, the Brauer–Manin obstruction may not distinguish points reducing to the smooth part.

The arithmetic of certain quartic curves

We give a sufficient condition using the Brauer–Manin obstruction under which certain quartic curves have no rational points. Using this sufficient condition, we construct two families of genus one quartic curves violating the Hasse principle explained by the Brauer–Manin obstruction.

THE HASSE PRINCIPLE FOR CERTAIN HYPERELLIPTIC CURVES AND FORMS

We prove that, for each positive integer n≥2, there is an infinite arithmetic family of hyperelliptic curves of genus n violating the Hasse principle explained by the Brauer–Manin obstruction. Using these families of curves, we show that, for any positive integer k≥1, there are infinitely many algebraic and arithmetic families of forms in three variables of degree 4k+2 such that they are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

The Brauer group of Kummer surfaces and torsion of elliptic curves

In this paper we are interested in computing the Brauer group of K3 surfaces. To an element of the Brauer–Grothendieck group Br(X) of a smooth projective variety X over a number field k class field theory associates the corresponding Brauer–Manin obstruction, which is a closed condition satisfied by k-points inside the topological space of adelic points of X, see [20, Ch. 5.2]. If such a condition is non-trivial, X is a counterexample to weak approximation, and if no adelic point satisfies this condition, X is a counterexample to the Hasse principle. The computation of Br(X) is thus a first step in the computation of the Brauer–Manin obstruction on X. Let k be an arbitrary field with a separable closure k, Γ = Gal(k/k). Recall that for a variety X over k the subgroup Br0(X) ⊂ Br(X) denotes the image of Br(k) in Br(X), and Br1(X) ⊂ Br(X) denotes the kernel of the natural map Br(X) → Br(X), where X = X ×k k. In [22] we showed that if X is a K3 surface over a field k finitely generated over Q, then Br(X)/Br0(X) is finite. No general approach to the computation of Br(X)/Br0(X) seems to be known; in fact until recently there was not a single K3 surface over a number field for which Br(X)/Br0(X) was known. One of the aims of this paper is to give examples of K3 surfaces X over Q such that Br(X) = Br(Q). We study a particular kind of K3 surfaces, namely Kummer surfaces X = Kum(A) constructed from abelian surfaces A. Let Br(X)n denote the n-torsion subgroup of Br(X). Section 1 is devoted to the geometry of Kummer surfaces. We show that there is a natural isomorphism of Γ-modules Br(X)−→Br(A) (Proposition 1.3). When A is a product of two elliptic curves, the algebraic Brauer group Br1(X) often coincides with Br(k), see Proposition 1.4. Section 2 starts with a general remark on the etale cohomology of abelian varieties which may be of independent interest (Proposition 2.2). It implies that if n is an odd integer, then for any abelian variety A the group Br(A)n/Br1(A)n is canonically isomorphic to the quotient of Het(A, μn) Γ by (NS (A)/n), where NS (A) is the Neron–Severi group (Corollary 2.3). For any n ≥ 1 we prove that Br(X)n/Br1(X)n is a subgroup of Br(A)n/Br1(A)n, and this inclusion is an equality for odd n, see

Transcendental obstructions to weak approximation on general K3 surfaces

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer–Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.

Failure of the Hasse principle for Enriques surfaces

We construct an Enriques surface X over Q with empty étale-Brauer set (and hence no rational points) for which there is no algebraic Brauer–Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.

The Brauer–Manin obstruction on a general diagonal quartic surface

We prove that, on a sufficiently general diagonal quartic surface, there is a non-trivial Brauer group but no Brauer-Manin obstruction to the existence of rational points.

Cubic points on cubic curves and the Brauer–Manin obstruction on K3 surfaces

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.

Effectivity of Brauer–Manin obstructions on surfaces

We study Brauer–Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields. A technique for constructing Azumaya algebra representatives of Brauer group elements is given, and this is applied to the computation of obstructions.

The Brauer–Manin obstruction for sections of the fundamental group

We introduce the notion of a Brauer–Manin obstruction for sections of the fundamental group extension and establish Grothendieck’s section conjecture for an open subset of the Reichardt–Lind curve.

Del Pezzo surfaces of degree 4 and their relation to Kummer surfaces

In this note, which has little pretence to originality, we clarify the relation between the geometry of del Pezzo surfaces of degree 4 and their realization as the zero set of two quadratic forms in five variables. We also review the classical description of the desingularized Kummer surface K constructed from the Jacobian J of a curve C of genus 2 as the zero set of three quadratic forms in six variables (Plucker, Kummer, Klein [7], [6], see [5] or [3] for a modern treatment). If C has a rational Weierstrass point, a partial diagonalization of this system gives rise to a natural projection onto a hyperplane, defining a finite morphism π : K → X of degree 2 onto a del Pezzo surface X of degree 4 (see [9, §6]). We show that X is the blow-up of Pk in the images of the five other Weierstrass points of C under the embedding of Pk as a conic in Pk. The morphism π sends the 16 lines on K to the 16 lines on X, and is equivariant with respect to the action of the subgroup of 2-division points J [2] ⊂ J . Thus π gives rise to a morphism from the twisted Kummer surface to the twisted del Pezzo surface. In our presentation it is obvious that all del Pezzo surfaces of degree 4 can be obtained in this way, an observation made by Victor Flynn in [4]. The fact that any 2-covering of J maps to a del Pezzo surface of degree 4 was first observed in [2], and used in [2], [1] and [9] to construct and visualize elements of order 2 in the Tate–Shafarevich group of J over Q using the theory of the Brauer– Manin obstruction on del Pezzo surfaces of degree 4. It was the author’s desire to understand the geometry behind these calculations that prompted him to write this note. I would like to thank Igor Dolgachev for useful discussions.

The Brauer–Manin obstruction for integral points on curves

AbstractWe discuss the question of whether the Brauer–Manin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and we relate it to an old conjecture of Skolem. Finally, we show that for elliptic curves minus one point a strong version of the question (describing the set of integral points by local conditions) has a negative answer.

Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

AbstractAn integer may be represented by a quadratic form over each ring ofp-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

Descent obstruction is equivalent to étale Brauer–Manin obstruction

Let X be a projective variety over a number field. Completing earlier work of D. Harari, C. Demarche and M. Stoll, we prove that the obstruction to the Hasse principle and weak approximation on X given by descent on torsors under linear algebraic groups is equivalent to the Brauer–Manin obstruction applied to etale covers of X.

Weak approximation on del Pezzo surfaces of degree 1

We study del Pezzo surfaces of degree 1 of the form w 2 = z 3 + A x 6 + B y 6 in the weighted projective space P k ( 1 , 1 , 2 , 3 ) , where k is a perfect field of characteristic not 2 or 3 and A , B ∈ k ∗ . Over a number field, we exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer–Manin obstruction.

Local-global principles for 1-motives

Building upon our arithmetic duality theorems for 1-motives, we prove that the Manin obstruction related to a finite subquotient Б(X) of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the Tate-Shafarevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels-Tate-type dual exact sequence for 1-motives and give an application to weak approximation