We prove that after inverting the Planck constant $h$, the Bezrukavnikov–Kaledin quantization $(X, {\mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {\mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant $p^\infty$-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$-divisible. We explain how the existence of these $p$-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic $p$.

We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.

We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the compactification of the stacks of hyperelliptic curves given by admissible double coverings.

Using the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves ${\mathcal {H}}_g$ over any field of characteristic $0$. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.

We discuss the kernel of the localization map from étale motivic cohomology of a variety over a number field to étale motivic cohomology of the base change to its completions. This generalizes the Hasse principle for the Brauer group, and is related to Tate–Shafarevich groups of abelian varieties.

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over $\mathbb{Q}$ whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.

In this article, 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 Néron–Severi group and provide a proof under additional assumptions on an integral model of $X$ . The proof depends on a non-degeneracy result of Brauer–Manin pairing due to Saito–Sato and on Gabber–de Jong’s comparison result of cohomological and Azumaya–Brauer groups. We will also mention the local–global problem for the Albanese cokernel; the abelian group on the ‘local side’ turns out to be a finite group.

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

A K3 category is by definition a Calabi–Yau category of dimension two. Geometrically K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.

The paper proves a result on the existence of finite flat scheme covers of Deligne–Mumford stacks. This result is used to prove that a large class of smooth Deligne–Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to a classical question on Brauer groups of schemes.Research of the first author supported in part by the Wolfgang Paul program of the Humboldt Foundation; research of the second author supported in part by the University of Bologna, funds for selected research topics.