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

Topic

It is well-known that not all varieties over Q satisfy the Hasse principle. The famous Selmer curve given by 3x3 + 4y3 + 5z3 = 0 in P2, for instance, indeed has points over every completion of Q, but no points over Q itself. Though it is trivial to find points over some cubic field, it is a priori not obvious whether there are points over a cubic field that is galois. We will see that such points do exist. K3 surfaces do not satisfy the Hasse principle either, which in some cases can be explained by the so-called Brauer-Manin obstruction. It is not known whether this obstruction is the only obstruction to the existence of rational points on K3 surfaces. We relate the two problems by sketching a proof of the following fact. If there exists a smooth curve over Q given by ax3 + by3 + cz3 = 0 that is locally solvable everywhere, and that has no points over any cubic galois extension of Q, then the algebraic part of the Brauer-Manin obstruction is not the only one for K3 surfaces. No previous knowledge about Brauer-Manin obstructions or K3 surfaces will be necessary.