Brauer Manin Obstruction Research Articles

Local-global principle and integral Tate conjecture for certain varieties

We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a separably rationally connected variety defined over a global function field. We prove that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on all geometrically rational surfaces defined over a global function field, and to the Hasse principle for rational points on del Pezzo surfaces of degree four defined over a global function field of odd characteristic. Along the way, we also prove some results about the space of one-cycles on a smooth projective variety that is separably rationally connected in codimension one, which leads to the equality of the coniveau filtration and the strong coniveau filtration on degree 3 3 homology of such varieties.

Transcendental Brauer–Manin obstructions on singular K3 surfaces

Let E and E′ be elliptic curves over Q with complex multiplication by the ring of integers of an imaginary quadratic field K and let Y=Kum(E×E′) be the minimal desingularisation of the quotient of E×E′ by the action of -1. We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer–Manin obstructions to weak approximation.

Non-invariance of Weak Approximation with Brauer–Manin Obstruction

Non-invariance of Weak Approximation with Brauer–Manin Obstruction

Nonabelian descent types

We present the notion of nonabelian descent type, which classifies torsors up to twisting by a Galois cocycle. This relies on the previous construction of kernels and nonabelian Galois 2-cohomology due to Springer and Borovoi. The necessity of descent types arises in the context of the descent theory where no torsors are given a priori, for example, when we wish to study the arithmetic properties such as the Brauer–Manin obstruction to the Hasse principle on homogeneous spaces without rational points. This new definition also unifies the types by Colliot-Thélène–Sansuc, the extended types by Harari–Skorobogatov, and the finite descent types by Harpaz–Wittenberg.

Galois invariants of finite abelian descent and Brauer sets

AbstractFor a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer–Manin obstructions. Given a Galois extension of the ground field, one can consider similar sets over the extension and take Galois invariants. In this paper, we study under which circumstances the Galois invariants recover the obstruction sets over the ground field. As an application of our results, we study finite abelian descent and Brauer–Manin obstructions for isotrivial curves over function fields and extend results obtained by the first and last authors for constant curves to the isotrivial case.

Brauer–Manin obstruction for Wehler K3 surfaces of Markoff type

Following a recent work of Fuchs, Litman, Silverman, and Tran, we study the Brauer–Manin obstruction for integral points on Wehler K3 surfaces of Markoff type. In particular, we construct some families which fail the integral Hasse principle via the Brauer–Manin obstruction with some counting results of similar nature to those in recent works of Ghosh and Sarnak; Loughran and Mitankin; and Colliot-Thélène, Wei, and Xu. We also give some counterexamples to strong approximation which can be explained by the Brauer–Manin obstruction, and study a few aspects of rational points on affine surfaces.

Semi-integral Brauer–Manin obstruction and quadric orbifold pairs

Semi-integral Brauer–Manin obstruction and quadric orbifold pairs

Brauer–Manin obstructions requiring arbitrarily many Brauer classes

AbstractOn a projective variety defined over a global field, any Brauer–Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.

Weak approximation of symmetric products and norm varieties

AbstractLet be a number field. For a variety over that satisfies weak approximation with Brauer–Manin obstruction, we study the same property for smooth projective models of its symmetric products. Based on the same method, we also explore the property of weak approximation with Brauer–Manin obstruction for norm varieties.

Most odd-degree binary forms fail to primitively represent a square

Let $F$ be a separable integral binary form of odd degree $N \geq 5$ . A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree- $N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree- $N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$ , ordered by height. For every sufficiently large $N$ , we prove that among equations in the family $\mathscr {F}_N(f_0)$ , more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$ , respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points.

Brauer–Manin obstruction for integral points on Markoff-type cubic surfaces

Following [GS22], [LM20] and [CWX20], we study the Brauer–Manin obstruction for integral points on similar Markoff-type cubic surfaces. In particular, we construct a family of counterexamples to strong approximation which can be explained by the Brauer–Manin obstruction with some counting results of similar nature to those in [LM20] and [CWX20]. We also study the vanishing of the algebraic Brauer–Manin obstruction to the integral Hasse principle and give some examples of Hasse failures using the reduction theory.

Brauer-Manin obstructions on hyperelliptic curves

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does not rely on rigorous class and unit group computations of large degree number fields. We report on experiments showing it to be a very effective tool for deciding existence of rational points: Among a random samples of curves over Q of genus at least 5 we were able to decide existence of rational points for over 99% of curves. We also demonstrate its effectiveness for high genus curves, giving an example of a genus 50 hyperelliptic curve with a Brauer-Manin obstruction to the Hasse Principle. The main theoretical development allowing for this algorithm is an extension of the descent theory for abelian torsors to a framework of torsors with restricted ramification.

Noninvariance of Weak Approximation Properties Under Extension of the Ground Field

For rational points on algebraic varieties defined over a number field K, we study the behavior of the property of weak approximation with Brauer–Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension L|K such that the property holds over K (conditionally on a conjecture) whereas fails over L. The result is unconditional when K equals Q or certain quadratic number fields. We give an explicit example when K=Q.

On analogs of Cassels–Tate's exact sequence for connected reductive groups and Brauer-Manin obstruction for homogeneous spaces over global function fields

We present some new analogs of Cassels–Tate dual exact sequence for connected reductive groups over global fields. We give also some extension of some important local-global exact sequences proved by Sansuc for connected linear algebraic groups over number fields, which are analogs of Cassels–Tate dual exact sequence, to the case of connected reductive groups over global function fields. As applications, we show that the Brauer-Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global function fields with connected reductive stabilizers are the only ones, extending some of Borovoi's results over number fields in this regard.

Arithmetic of Châtelet surface bundles revisited

Abstract We study the arithmetic of algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming the finiteness of the Tate–Shafarevich group of a certain elliptic curve, we show, for Châtelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer–Manin obstruction is not invariant under an arbitrary finite extension of the ground field.

Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces

We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer–Manin obstruction to the integral Hasse principle.

Brauer–Manin obstruction for zero-cycles on certain varieties

Brauer–Manin obstruction for zero-cycles on certain varieties

Diagonal quartic surfaces with a Brauer–Manin obstruction

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

Rational points of algebraic varieties: a homotopical approach

This article, dedicated to the 100-th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.

Rational points of algebraic varieties: a homotopical approach

This article, dedicated to the 100 th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.