Manin Obstruction Research Articles

The Brauer–Manin Obstruction for Zero-Cycles on K3 Surfaces

Abstract We study local–global principles for zero-cycles on $K3$ surfaces defined over number fields. We follow an idea of Liang to use the trivial fibration over the projective line.

The triviality of Brauer–Manin obstruction for subvarieties of semi-abelian varieties over function fields of characteristic zero

The triviality of Brauer–Manin obstruction for subvarieties of semi-abelian varieties over function fields of characteristic zero

Diophantine definability of nonnorms of cyclic extensions of global fields

We show that, for any square-free natural number n n and any global field K K with ( char ( K ) , n ) = 1 (\textrm {char}(K), n)=1 containing a primitive n n th root of unity, the pairs ( x , y ) ∈ K × × K × (x,y)\in K^{\times }\times K^{\times } such that x x is not a relative norm of K ( y n ) / K K(\sqrt [n]{y})/K form a diophantine set over K K . We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that, for any n ∈ N n\in {\mathbb {N}} and any global field K K with char ( K ) ≠ n \textrm {char}(K)\neq n , K × ∖ K × n K^{\times }\setminus K^{\times n} is diophantine over K K . For a number field K K , this is a result of Colliot-Thélène and Van Geel, proved using results on the Brauer–Manin obstruction. Additionally, we prove a variation of our main theorem for global fields K K without the n n th roots of unity, where we parametrize varieties arising from norm forms of cyclic extensions of K K without any rational points by a diophantine set.

Index of fibrations and Brauer classes that never obstruct the Hasse principle

Let X be a smooth projective variety over a number field with a fibration into varieties that either satisfy a certain condition. We study the classes in the Brauer group of X that never obstruct the Hasse principle for X. We prove that if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer–Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.

Brauer–Manin obstruction and families of generalised Châtelet surfaces over number fields

Over infinitely many number fields [Formula: see text] (including all finite Galois extensions [Formula: see text] of odd degree unramified at 2), we give general sufficient conditions in order for the generalised Châtelet surfaces [Formula: see text] over [Formula: see text] associated to the normic equation [Formula: see text], where [Formula: see text] has a specific form and [Formula: see text] is even and arbitrarily large, to have the property that [Formula: see text] but [Formula: see text]. We also give general sufficient conditions in order for the generalised Châtelet surfaces [Formula: see text] over [Formula: see text] of the same form as above to have the property that [Formula: see text] and [Formula: see text]. As an application, we prove that, for a certain family of generalised Châtelet surfaces over [Formula: see text], a positive proportion (but not 100[Formula: see text]) of its members exhibits a violation of the Hasse principle explained by the Brauer–Manin obstruction.

Arithmetic purity of strong approximation for homogeneous spaces

We prove that any open subset U of a semi-simple simply connected quasi-split linear algebraic group G with codim(G∖U,G)≥2 over a number field satisfies strong approximation by establishing a fibration of G over a toric variety. We also prove a similar result of strong approximation with Brauer–Manin obstruction for a partial equivariant smooth compactification of a homogeneous space where all invertible functions are constant and the semi-simple part of the linear algebraic group is quasi-split. Some semi-abelian varieties of any given dimension where the complements of a rational point do not satisfy strong approximation with Brauer–Manin obstruction are given.

Arithmetic of Rational Points and Zero-cycles on Products of Kummer Varieties and K3 Surfaces

AbstractLet $k$ be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ for Kummer varieties over $k$. For example, for any Kummer variety $X$ over $k$, we show that if the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rational points on $X$ over all finite extensions of $k$, then the ($2$-primary) Brauer–Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on $X$ over $k$. We also obtain similar results for products of Kummer varieties, K3 surfaces, and rationally connected varieties.

On the Frequency of Algebraic Brauer Classes on Certain Log K3 Surfaces

AbstractGiven systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer–Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer–Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.

Brauer–Manin obstructions on degree 2 K3 surfaces

We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over $\mathbb{Q}$ given by double covers of $\mathbb{P}^2$ ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.

Klaus Friedrich Roth, 1925-2015

Klaus Friedrich Roth, 1925-2015

The d-primary Brauer–Manin obstruction for curves

For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it is possible that there is a d-primary obstruction for infinitely many pairwise coprime d, and also an example where the odd part of the Brauer group does not obstruct although there is a Brauer–Manin obstruction. These examples demonstrate that a slightly stronger form of a conjecture of Poonen is false, despite being supported by the same heuristic.

There are No Transcendental Brauer–Manin Obstructions on Abelian Varieties

Abstract Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer–Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate–Shafarevich group. Our results are unconditional.

Strong Approximation with Brauer–Manin Obstruction for Toric Varieties

For smooth open toric varieties, we establish strong approximation off infinity with Brauer-Manin obstruction.

Embeddings of maximal tori in classical groups and explicit Brauer–Manin obstruction

We give necessary and sufficient conditions for the Hasse principle to hold for the embeddings of maximal tori into classical groups over global fields of characteristic not 2.

On integral points on degree four del Pezzo surfaces

We report on our investigations concerning algebraic and transcendental Brauer–Manin obstructions to integral points on complements of a hyperplane section in degree four del Pezzo surfaces. We discuss two concepts of an obstruction at an archimedean place. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing (0, 0) over Q and over Z p for all primes p, but not over Z. By blow-up, these yield cubic polynomials in three variables all integral solutions of which satisfy a gcd condition.

RATIONAL POINTS ON SHIMURA CURVES AND THE MANIN OBSTRUCTION

In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.

Zero-cycles of degree one on Skorobogatov's bielliptic surface

Skorobogatov constructed a bielliptic surface which is a counterexample to the Hasse principle not explained by the Brauer–Manin obstruction. We show that this surface has a 0-cycle of degree 1, as predicted by a conjecture of Colliot-Thélène.

The Brauer–Manin obstruction for certain quartic curves with negative discriminant

ABSTRACTIn this paper, we construct an infinite explicit family of genus one quartic curves with negative discriminants that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

Local–global principle for 0-cycles on fibrations over rationally connected bases

We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation for 0-cycles on algebraic varieties that possess a fibration structure. The issue is to establish the exactness of a local-to-global sequence (E) of Chow groups of 0-cycles. Recently, Harpaz and Wittenberg proved the exactness of (E) for rationally connected fibrations whose smooth fibres verify the exactness of (E) and whose base is either the projective space or a curve verifying the exactness of (E). In the present paper, we prove the exactness of (E) for fibrations whose bases are Châtelet surfaces or smooth proper models of homogeneous spaces of connected linear algebraic groups with connected stabilizers. We require that either the fibration is valuatively split in codimension 1 and most closed fibres satisfy weak approximation for 0-cycles, or the generic fibre has a 0-cycle of degree 1 and (E) is exact for most fibres.

Failures of weak approximation in families

Given a family of varieties$X\rightarrow \mathbb{P}^{n}$over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.