Related papers: Brauer-Manin Obstruction on Generalized Kummer Var…

For 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…

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\, Y/…

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…

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…

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from…

Let $p \geq 2$ be a prime number and let $k$ be a number field. Let $\mathcal{A}$ be an abelian variety defined over $k$. We prove that if ${\rm Gal} ( k ( {\mathcal{A}}[p] ) / k )$ contains an element $g$ of order dividing $p-1$ not fixing…

We present a method to construct explicit degenerations of higher-dimensional generalized Kummer varieties. We start with a simple degeneration $f: \mathcal Y \to C$ of abelian surfaces. Then $ \mathcal{Y} \setminus \mathcal{Y}_0$ is an…

In this paper we study principally polarized abelian varieties that admit an automorphism of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do…

For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent.…

Let $X$ be a closed subvariety of an abelian variety $A$ over a global function field $k$ such that the base change of $A$ to an algebraic closure does not have any positive dimensional isotrivial quotient. We prove that every adelic point…

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate…

We wish to give a short elementary proof of S. Saito's result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of the Jacobian…

We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.

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 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…

For a quasi-projective smooth geometrically integral variety over a number field $k$, we prove that the iterated descent obstruction is equivalent to the descent obstruction. This generalizes a result of Skorobogatov, and this answers an…

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…

Let $X$ be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\sqrt{a},\sqrt{b})/k}({x})=Q(t_{1},...,t_{m})^{2}$ over a number field $k$. We prove that : (1) the failure of the local-global principle…

In 2010, Poonen gave the first example of failure of the local-global principle that cannot be explained by Skorobogatov's \'etale Brauer-Manin obstruction. Motivated by this example, we show that the Brauer-Manin obstruction detects…

We discuss a generalization of Kummer construction which, on the base of an integral representation of a finite group and local resolution of its quotient, produces a higher dimensional variety with trivial canonical class. As an…