The proportion of failures of the Hasse norm principle (Q2810728)

Let \(K/k\) be a finite extension of number fields with associated idele groups \(J_K\) and \(J_k\), respectively. Let \(N_{K/k}\,\, : J_{K}\rightarrow J_{k}\) denote the norm map and consider the multiplicative groups \(K^{*}\) and \(k^{*}\) as subgroups of \(J_K\) and \(J_k\) respectively. The Hasse norm principle is said to hold for \(K/k\) if \(k^{*}\cap N_{K/k} J_{K} = N_{K/k} K^{*}\), viewing the \(N_{K/k} K^{*}\) of global norms as subgroup of \(k^{*}\cap N_{K/k} J_{K}\) with finite index. In the case \(k = \mathbb{Q}\), the authors calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field. In other words, the main result of this paper gives the exact proportion of element in \(\mathbb{Q^{*}}\cap N_{K/\mathbb{Q}} J_{K}\) which do not belong to \(N_{K/\mathbb{Q}} K^{*}\).