Rachel Newton: Arithmetic of rational points and zero-cycles on Kummer varieties

In 1970, Manin observed 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 lack of a K-point on X despite the existence of points over every completion of K is sometimes explained by non-trivial elements in Br(X). This so-called Brauer–Manin obstruction may not always suffice to explain the failure of the Hasse principle but it is known to be sufficient for some classes of varieties (e.g. torsors under connected algebraic groups) and conjectured to be sufficient for rationally connected varieties and K3 surfaces.

A zero-cycle on X is a formal sum of closed points of X. A rational point of X over K is a zero-cycle of degree 1. It is sometimes easier to study the zero-cycles of degree 1 on X, rather than the rational points. Yongqi Liang has shown that for rationally connected varieties, sufficiency of the Brauer–Manin obstruction to the existence of rational points over all finite extensions of K implies sufficiency of the Brauer–Manin obstruction to the existence of zero-cycles of degree 1 over K. I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties.