| Résume | Let $K$ be a number field and let $J_K$ be its group of ideles. If a nonzero rational number is a norm from $K^*$, then it is a norm from $J_K$. We say that the Hasse norm principle holds for $K/\mathbf Q$ if the converse holds, i.e. if every rational number which is a norm from $J_K$ is in fact a norm from $K^*$. This talk is about the proportion of rational numbers which are counterexamples to the Hasse norm principle for $K/\mathbf Q$. Using work of Odoni, we give asymptotic formulae for the counting functions for rational numbers that are norms from $J_K$ and for rational numbers that are norms from $K^*$. We calculate the proportion of rational numbers that are norms from $J_K$ which fail to be norms from $K^*$. We show that this proportion is $1-1/n$, where $n$ is the (finite) index of N($K^*$) in N($J_K$)$\cap \mathbf Q^*$.This is joint work with Tim Browning. |
