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Odd order transcendental obstructions to the Hasse principle on general K3 surfaces

Banff International Research Station for Mathematical Innovation and Discovery

After fixing numerical invariants such as dimension, it is natural to ask which birational classes of varieties fail the Hasse principle, and moreover whether the Brauer group (or certain distinguished subsets) explains this failure. In this talk, we will focus on K3 surfaces (e.g. a double cover of the plane branched along a smooth sextic curve), which have been a testing ground for many conjectures on rational points. In 2014, Ieronymou and Skorobogatov asked whether any odd torsion in the Brauer group of a K3 surface could obstruct the Hasse principle. We answer this question in the affirmative for transcendental classes; via a purely geometric approach, we construct a 3-torsion transcendental Brauer class on a degree 2 K3 surface over the rationals with geometric Picard rank 1 (hence with trivial algebraic Brauer group) which obstructs the Hasse principle. Moreover, we do this without needing a central simple algebra representative. This is joint work with Tony Varilly-Alvarado.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: Rice University

2019-03-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68704

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/