Related papers: Weak Approximation for $0$-cycles on a product of …

For a smooth projective variety $X$ defined over a global field $K$, one can form a notion of Weak Approximation for the Chow group of zero-cycles of $X$. There exists a Brauer-Manin obstruction to Weak Approximation here akin to that for…

Zero-cycles are conjectured to satisfy weak approximation with Brauer-Manin obstruction for proper smooth varieties defined over number fields. Roughly speaking, we prove that the conjecture is compatible for products of rationally…

We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Th\'el\`ene for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our…

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 exactness of the local-to-global sequence $(E)$ of Chow groups of 0-cycles…

We introduce new `refined' obstructions to local-global principles for 0-cycles on algebraic varieties over number fields. Assuming finiteness of relevant Tate--Shafarevich groups, we show that the Hasse principle and weak approximation for…

Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by its Brauer group. Let $X$ be a Ch\^atelet surface or a smooth compactification of a homogeneous space of…

Let $X$ be a smooth projective variety over a number field, fibered over a curve, with geometrically integral fibers. We prove that, supposing the finiteness of $\sha(Jac(C))$, if the fibers over a generalised Hilbertian subset satisfy the…

A conjecture of Colliot-Th\'{e}l\`{e}ne predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a…

Let $X$ be a rationally connected algebraic variety, defined over a number field $k$. We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following…

Let $X$ be a product of smooth projective curves over a finite unramified extension $k$ of $\mathbb{Q}_p$. Suppose that the Albanese variety of $X$ has good reduction and that $X$ has a $k$-rational point. We propose the following…

Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Th\'el\`ene, Sansuc, Kato and Saito in the 1980's. We prove that these conjectures are compatible with…

For a product $E_1\times E_2$ of two elliptic curves over a $p$-adic field with good supersingular reduction, we produce infinitely many rational equivalences in the Chow group $\mathrm{CH}_0(X)$ of zero cycles via genus 2 covers of $E_1$…

We study the local-global principle for zero-cycles of degree 1 on certain varieties fibered over the projective space. Among other applications, we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and…

We consider a product $X=E_1\times\cdots\times E_d$ of elliptic curves over a finite extension $K$ of $\mathbb{Q}_p$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has…

Recently Dasheng Wei proved that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on some fibrations over the projective line defined by bi-cyclic normic equations. In the present paper,…

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…

We study weak approximation for Ch\^{a}telet surfaces over number fields when all singular fibers are defined over rational points. We consider Ch\^{a}telet surfaces which satisfy weak approximation over every finite extension of the ground…

We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to the weak approximation for zero-cycles on certain fibrations over a smooth curve or over the projective space. The principal novelty is that…

Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Th\'el\`ene, Kato…

Classical descent theory of Colliot-Th\'el\`ene and Sansuc for rational points tells that, over a smooth variety $X$, the algebraic Brauer--Manin subset equals the descent obstruction subset defined by a universal torsor. Moreover, Harari…