Let \(L/K\) be an extension of number fields and let \(J_L\) and \(J_K\) be the associated groups of ideles. Using the diagonal embedding, we view \(L^*\) and \(K^*\) as subgroups of \(J_L\) and \(J_K\) respectively. The norm map \(N: J_L \to J_K\) restricts to the usual field norm \(N: L^* \to K^*\) on \(L^*\). Thus, if an element of \(K^*\) is a norm from \(L^*\), then it is a norm from \(J_L\). We say that the Hasse norm principle holds for \(L/K\) if the converse holds, i.e. if every element of \(K^*\) which is a norm from \(J_L\) is in fact a norm from \(L^*\).
The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of \(K\) fail the Hasse norm principle? More generally, for an abelian group \(G\), what proportion of extensions of K with Galois group \(G\) fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.
This is joint work with Christopher Frei and Daniel Loughran.