In 1971, Manin showed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the existence of points everywhere locally on X despite the lack of a global point is sometimes explained by non-trivial elements in Br(X). Since Manin's observation, the Brauer group has been the subject of a great deal of research. The 'algebraic' part of the Brauer group is the part which becomes trivial upon base change to an algebraic closure of K. It is generally easier to handle than the remaining 'transcendental' part and a substantial portion of the literature is devoted to its study. On the other hand, there are comparatively few results concerning the transcendental part of the Brauer group. The transcendental part of the Brauer group is known to have arithmetic importance – it can give non-trivial obstructions to the Hasse principle and weak approximation. I will use class field theory together with results of Ieronymou, Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the product ExE for an elliptic curve E with complex multiplication.

Inspired by the invariant of a number field given by its Dedekind zeta function we define the notion of weak arithmetic equivalence, and we show that under certain ramification hypothesis this equivalence determines the local root numbers of the number field. This is analogous to a result Rohrlich on the local root numbers of a rational elliptic curve. Additionally we prove that for tame non-totally real number fields the integral trace form is invariant under weak arithmetic equivalence.

Inspired by Scharaschkin’s approach to the Brauer–Manin obstruction for curves, Hsia, Silverman and Voloch have recently initiated the study of the “dynamical Brauer-Manin obstruction to the dynamical Hasse principle”. We shall explain what it is and make some remarks on this in the particular case of etale self-maps. This is a joint work with P. Kurlberg, K. Nguyen, A. Towsley, B. Viray and F. Voloch.

We give an account of joint work with Sara Arias-de-Reyna, Luis Dieulefait and Sug Woo Shin, in which we realise, for every even n and every d, the group PGSp_n(F_p^d) or PSp_n(F_p^d) as Galois group over the rationals, for p in a set of primes of positive density. The proof relies on compatible systems of automorphic Galois representations with special local properties.

To any variety one can associate its derived category of coherent sheaves. These categories are difficult to handle in general, but understanding their rich structure brings a big reward and their study has been subject of extensive research in recent years. Among the most interesting aspects is that one can use derived equivalences in order to translate between (commutative) algebraic geometry and (noncommutative) representation theory. This has been applied to study non-commutative resolutions and the classification of vector bundles.
An essential tool to study derived categories are so-called semi-orthogonal decompositions. By now, there are a number of results which show that such decompositions are deeply tied to the underlying geometry. Exceptional collections represent the simplest type of semi-orthogonal decompositions, where each semi-orthogonal component is generated by just one object. Though many examples are known and the existence of exceptional collections imposes strong constraints on the underlying geometry, their existence is an open problem in general. In this talk we will consider exceptional collections on smooth algebraic surfaces, and in particular on rational surfaces. We will show that here the problem of constructing exceptional collections is intrinsically related to toric geometry and singularity theory.