Cubic points on cubic curves and the Brauer-Manin obstruction on K3 surfaces

TL;DR

This paper demonstrates that for certain K3 surfaces, the algebraic Brauer-Manin obstruction does not fully explain the failure of the Hasse principle, using properties of specific cubic curves over number fields.

Contribution

It introduces a new counterexample showing the algebraic Brauer-Manin obstruction is not always sufficient for K3 surfaces, based on cubic curve constructions.

Findings

Existence of cubic curves with specific local and global properties

Counterexamples to the sufficiency of algebraic Brauer-Manin obstruction

Insight into the arithmetic of K3 surfaces and cubic curves

Abstract

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.