The Hasse norm principle for abelian extensions.

Let L/K be a normal extension of number fields. The Hasse normprinciple is a local-global principle for norms. It is satisfied if anyelement x of K is a norm from L whenever it is a norm locally at everyplace. For any fixed abelian Galois group G, we investigate the densityof G-extensions violating the Hasse norm principle, when G-extensionsare counted in order of their discriminant. This is joint work with DanLoughran and Rachel Newton.