Related papers: Brauer-Manin obstructions on hyperelliptic curves

Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to…

Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points on X within the adelic points.…

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then…

We prove that the set of rational points on a nonisotrivial curves of genus at least 2 over a global function field is equal to the set of adelic points cut out by the Brauer-Manin obstruction.

Building upon our arithmetic duality theorems for 1-motives, we prove that the Manin obstruction related to a finite subquotient $\Be (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors…

We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves.…

A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite…

For any pencil of conics or higher-dimensional quadrics over the rationals, with all degenerate fibres defined over the rationals, we show that the Brauer-Manin obstruction controls weak approximation. The proof is based on the Hasse…

We conjecture that if C is a curve of genus >1 over a number field k such that C(k) is empty, then a method of Scharaschkin (equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that C(k) is empty. As…

Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether…

We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the…

We show that the Brauer-Manin obstruction is the only obstruction to strong approximation for all stacky curves over global fields with finite abelian fundamental groups. This includes all stacky curves of genus $g = \frac{1}{2}$, thus…

We study Brauer-Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields.

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We describe descent on families of torsors of a constant torus. A recent result of Browning and Matthiesen then implies that the Brauer--Manin obstruction controls the Hasse principle and weak approximation when the ground field is the…

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…

Given 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…

We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic…

We give large families of Shimura curves defined by congruence conditions, all of whose twists lack $p$-adic points for some $p$. For each such curve we give analytically large families of counterexamples to the Hasse principle via the…

We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the…