Hasse norm principle for Heisenberg extensions of degree $p^3$

Abstract

Let $k$ be a global field and $p$ be an odd prime number. We give a necessary and sufficient condition for the Hasse norm principle for separable field extensions $K/k$, i.e. the determination of the Shafarevich-Tate group $Sha(T)$ of the norm one tori $T=R^{(1)}_{K/k}(G_m)$ of $K/k$, with $[K:k]=p^3$ or $p^2$ when the Galois group of the Galois closure of $K/k$ is the Heisenberg group $E_p(p^3)\simeq (C_p)^2\rtimes C_p$ of order $p^3$, i.e. the extraspecial group of order $p^3$ with exponent $p$. As a consequence, we get the Tamagawa number $\tau(T)=p^2$, $p$ or $1$ via Ono's formula $\tau(T)=|H^1(k,\widehat{T})|/|Sha(T)|$.