I'll start with definitions and basic properties of  Brauer-Grothendieck groups and Brauer-Manin sets of algebraic varieties. After that I'll discuss several finiteness results for these groups with special reference to the case of abelian varieties and K3 surfaces.
This is a report on joint work with Alexei Skorobogatov.

A subset $X$ in $\{0,1\}^n$ is a called an $\epsilon$-biased set if for any nonempty subset $T\subseteq [n]$ the following condition holds. Randomly choosing an element $x\in X$, the parity of its T-coordinates sum has bias at most $\epsilon$. This concept is essentially equivalent or close to expanders, pseudorandom generators and linear codes of certain parameters. For instance, viewing $X$ as a generator matrix, an $\epsilon$-biased set is equivalent to an $[|X|, n]_2$-linear code of relative distance at least $1/2-\epsilon$. For fixed $n$ and $\epsilon$, it's a challenging problem to construct smallest $\epsilon$-biased sets. In this talk we will first introduce several known constructions of $\epsilon$-bias sets with the methods from number theory and geometrical coding theory. Then we will present a construction with a conjecture, which is closely related to the subset sum problem over prime fields.

On the occasion of the creation of a table of modular elliptic curves over Q(sqrt(5)), we review the "Remarks on isogenies" that accompanied the "Antwerp" tables (LNM 476), and outline some of the new phenomena and open questions that arise in attempting to give a similar overview of isogenies defined over Q(sqrt(5)) or other number fields. In particular, we account for some new isogeny degrees and graphs not seen over Q, and explain why the problem of proving completeness of the list over Q(sqrt(5)) is difficult but not hopeless.

What is the probability that a random abelian variety over F_q is ordinary? Using (semi)linear algebra, we will answer an analogue of this question, and explain how our method can be used to answer similar statistical questions about p-rank and a-number. The answers are perhaps surprising, and deviate from what one might expect via naive reasoning. Using these computations and numerical evidence, we formulate several ``Cohen-Lenstra" heuristics for the structure of the p-torsion on the Jacobian of a random hyperelliptic curve over F_q. These heuristics are the "l=p " analogue of Cohen-Lenstra in the function field setting. This is joint work with Jordan Ellenberg and David Zureick-Brown.