Hasse norm principle for Heisenberg extensions of degree $p^3$
Akinari Hoshi, Aiichi Yamasaki
TL;DR
This work analyzes the Hasse norm principle for separable extensions $K/k$ whose Galois closure has Heisenberg type $G\cong E_p(p^3)$. By studying the norm-one torus $T=R^{(1)}_{K/k}(\,\mathbb{G}_m\,)$ and its character lattice $\widehat{T}\simeq J_{G/H}$ through the exact sequence $0\to \mathbb{Z}\to \mathbb{Z}[G/H]\to J_{G/H}\to 0$ and the connecting map $\delta: H^2(G,J_{G/H})\to H^3(G,\mathbb{Z})\simeq M(G)$, the authors compute the relevant Shafarevich-Tate groups via $\Sha^2_\omega(G,J_{G/H})$ and obtain explicit criteria for when $\Sha(T)=0$ or when $\Sha(T)\cong \mathbb{Z}/p\mathbb{Z}$ or $(\mathbb{Z}/p\mathbb{Z})^{\oplus 2}$. The main results express these obstructions in terms of ramification data, specifically the existence of ramified places with certain decomposition groups $(C_p)^2$ or containment of all ramification subgroups in $C_p$, and these lead to corresponding Tamagawa numbers $\tau(T)\in\{p^2,p,1\}$ via Ono's formula. The approach highlights how cohomological invariants of the Heisenberg group govern the Hasse norm principle in a non-abelian setting, bridging class field theory, algebraic tori, and Tamagawa-number theory.
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 $τ(T)=p^2$, $p$ or $1$ via Ono's formula $τ(T)=|H^1(k,\widehat{T})|/|Sha(T)|$.