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

Abstract: Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures uses the Brauer group to define an obstruction set known as the Brauer-Manin set. 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 always explains this failure. In this talk, we'll focus on K3 surfaces (e.g., a double cover of the plane branched along a smooth sextic curve) which are relatively simple surfaces in terms of geometric complexity, but have rich arithmetic. Via a purely geometric approach, we construct a 3-torsion transcendental Brauer class on a degree 2 K3 surface which obstructs the Hasse principle, giving the first example of an obstruction of this type. This was joint work with Tony Varilly-Alvarado.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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