Manin Obstruction Research Articles

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.

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.

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.

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.

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

The Hasse principle for random Fano hypersurfaces

It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.

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.

Weak approximation for 0-cycles on a product of elliptic curves

In the 1980’s Colliot-Thélène, Sansuc, Kato and S. Saito proposed conjectures related to local-to-global principles for 0-cycles on arbitrary smooth projective varieties over a number field. We give some evidence for these conjectures for a product \(X=E_1\times E_2\) of two elliptic curves. In the special case when \(X=E\times E\) is the self-product of an elliptic curve E over \(\mathbb {Q} \) with potential complex multiplication, we show that the places of good ordinary reduction are often involved in a Brauer–Manin obstruction for 0-cycles over a finite base change. We give many examples when these 0-cycles can be lifted to global ones.

Fermat’s Last Theorem over and

AbstractWe prove Fermat’s Last Theorem over $\mathbb {Q}(\sqrt {5})$ and $\mathbb {Q}(\sqrt {17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of $\mathbb {Q}(\sqrt {5})$ is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of $\mathbb {Q}(\sqrt {17})$ , this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.

An example of a Brauer–Manin obstruction to weak approximation at a prime with good reduction

Following Bright and Newton, we construct an explicit K3 surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.

Applications of the fibration method to the Brauer–Manin obstruction to the existence of zero-cycles on certain varieties

We study the Brauer–Manin obstruction to the existence of zero-cycles of degree [Formula: see text] on certain classes of varieties over number fields. We generalize existing results in the literature and prove some results about fibrations over the projective line, where the geometric Brauer group of the generic fiber is not assumed to be finite. The idea is to assume that the Brauer–Manin obstruction to the Hasse principle is the only one for certain fibers and then deduce analogous results for zero-cycles.

Rational points on fibrations with few non-split fibres

Abstract We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is ≤ 2 {\leq 2} , as well as in various instances where it is 3. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel’s hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer–Manin obstruction in families.

Rational points on Del Pezzo surfaces of degree four

We study the distribution of the Brauer group and the frequency of the Brauer–Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals.