Brauer groups of K3 surfaces

The local-to-global principle for rational points on K3 surfaces over number fields is a difficult problem where very little progress has been achieved so far. This question involves understanding solutions of, among others, quartic equations in three variables over the rationals. Since the local-to-global behaviour of rational points on K3 surfaces is expected to be completely controlled by the Brauer-Manin obstruction, the computation of the Brauer group of K3 surfaces is an important problem too. A recent conjecture of Várilly-Alvarado predicts that only finitely many finite abelian groups can appear as transcendental Brauer groups of K3 surfaces defined over a given number field. The aim of this research is to use various recent achievements in the Tate and Mumford-Tate conjectures, André-Oort conjecture and so on, to advance in the direction of the conjecture of Várilly-Alvarado. A novel methodology involves applying isogenies of K3 surfaces to link the Brauer groups of related K3 surfaces. The similar question regarding abelian varieties is also an interesting open question. The arithmetic theory of K3 surfaces is fast becoming an active field of research due to potential impact on classical difficult questions of number theory and arithmetic geometry.