Hasse norm principle for Heisenberg extensions of degree $p^3$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916657277960192 |
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| author | Hoshi, Akinari Yamasaki, Aiichi |
| author_facet | Hoshi, Akinari Yamasaki, Aiichi |
| contents | 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 $τ(T)=|H^1(k,\widehat{T})|/|Sha(T)|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hasse norm principle for Heisenberg extensions of degree $p^3$ Hoshi, Akinari Yamasaki, Aiichi Number Theory Algebraic Geometry 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 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 $τ(T)=|H^1(k,\widehat{T})|/|Sha(T)|$. |
| title | Hasse norm principle for Heisenberg extensions of degree $p^3$ |
| topic | Number Theory Algebraic Geometry 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 |
| url | https://arxiv.org/abs/2503.15408 |