Arithmetic of rational points and zero-cycles on Kummer surfaces

Rachel Newton (University of Reading)

Frank Adams 1,

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 abelian varieties) and conjectured to be sufficient for rationally connected varieties and K3 surfaces.A zero-cycle on X is a formal sum of rational points over finite extensions of K. 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 X, 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 a zero-cycle of degree 1 over K. I will discuss work in progress with Francesca Balestrieri where we extend Liang's result to Kummer surfaces.

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