Séminaires : Séminaire Théorie des Nombres

Equipe(s) : tn,
Responsables :Ziyang Gao, Marc Hindry, Bruno Kahn, João Pedro P. dos Santos
Email des responsables : ziyang.gao@imj-prg.fr
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Orateur(s) Rachel Newton - Reading,
Titre The proportion of failures of the Hasse norm principle
Date15/12/2014
Horaire14:00 à 16:00
Diffusion
RésumeLet $K$ 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.
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