相关论文: On the Hasse principle for zero-cycles on Severi-B…

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…

We investigate a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically $CH_0$-trivial. By this we mean that over any…

Let $\mathcal{X}\rightarrow C$ be a dominant morphism between smooth irreducible varieties over a finitely generated field $k$ such that the generic fiber $X$ is smooth, projective and geometrically connected. Assuming that $C$ is a curve…

We give a sufficient condition for the injectivity of the global-to-local map of the relative Chow group of zero-cycles on a quadric fibration of dimension 2 or 3 defined over a number field.

Let $k$ be a perfect field and let $p$ be a prime number different from the characteristic of $k$. Let $C$ be a smooth, projective and geometrically integral $k$-curve and let $X$ be a Severi-Brauer $C$-scheme of relative dimension $p-1$ .…

We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to…

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…

Let \(A\) be a central simple algebra over a field \(F\) with index \(n\) and let \(\mathrm{SB}_r(A)\) denote the \(r\)-th generalized Severi--Brauer variety associated with \(A\). We prove that the Chow group of zero cycles of degree zero…

Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme $G$ of finite type over the function field $K$ of $C$ we study the Tate--Shafarevich groups given by elements of $H^1(K,G)$ locally…

We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the N\'eron-Severi group and provide a proof under additional assumptions on an…

Let X be a normal connected complex algebraic variety equipped with a semisimple complex representation of its fundamental group. Then, under a maximality assumption, we prove that the covering space of X associated to the kernel of the…

Let $k$ be a number field and let $\pi \colon X \rightarrow\mathbb{P}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers and either $X(\mathbb{A}_k)\neq \emptyset$ or $X/k$ has non-trivial Brauer group,…

Let $k$ be a number field and let $T$ be a $k$-torus. Consider a fibration in torsors under $T$, i.e. a morphism $f: X \to \mathbb{P}^1_k$ from a smooth, projective $k$-variety $X$ to $\mathbb{P}^1_k$ such that the generic fibre $X_\eta \to…

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a…

Let $k$ be a number field and let $\pi \colon X \rightarrow \mathbb{P}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers with $X(\mathbb{A}_k)\neq \emptyset$ or non-trivial Brauer group, then $X$…

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…

In this article, we prove a $p$-adic analogue of the local invariant cycle theorem for $H^2$ in mixed characteristics. As a result, for a smooth projective variety $X$ over a $p$-adic local field $K$ with a proper flat regular model…

Let X_K be a proper, smooth and geometrically connected curve over a global field K. In this paper we generalize a formula of Milne relating the order of the Tate-Shafarevich group of the Jacobian of X_K to the order of the Brauer group of…

We study the Chow group of zero-cycles $\text{CH}_0(S)$ of a bielliptic surface $S=(E_1\times E_2)/G$, where $E_1, E_2$ are elliptic curves and $G$ is a finite group acting on $E_1$ by translations and on $E_2$ by automorphisms such that…

We calculate the Tate-Shafarevich group of an elliptic three-fold $f:X\rightarrow S$ when $X$ and $S$ are regular and $f$ is flat, relating it to the Brauer group of $X$ and $S$. We show that given certain hypotheses on $f$, the…