Rationality problem for norm one tori of tensor products of étale algebras and Hasse norm principle
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| Format: | Preprint |
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2026
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| _version_ | 1866916060681207808 |
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| author | Florence, Mathieu Hoshi, Akinari Yamasaki, Aiichi |
| author_facet | Florence, Mathieu Hoshi, Akinari Yamasaki, Aiichi |
| contents | Let $k$ be a field. Let $A=\prod_{i=1}^r K_i$ and $B=\prod_{j=1}^s E_j$ be étale $k$-algebras where $K_i$ and $E_j$ are finite separable field extensions of $k$ with $[K_i:k]=m_i$ and $[E_j:k]=n_j$. Let $\mathcal{T}_A=R^{(1)}_{A/k}(\mathbb{G}_m)$ be the norm one torus of the étale $k$-algebra $A$. We prove that if $\gcd(m_i,n_j\mid 1\leq i\leq r, 1\leq j\leq s)=1$ and $\mathcal{T}_A$ and $\mathcal{T}_B$ are stably $($resp. retract$)$ $k$-rational, then the algebraic $k$-torus $\mathcal{T}_A\otimes \mathcal{T}_B$ and the norm one torus $\mathcal{T}_{A\otimes B}$ are stably $($resp. retract$)$ $k$-rational. In particular, if $k$ is a global field, then the Hasse norm principle holds for $(A\otimes B)/k$. We then give detailed applications to the case of norm one tori of field extensions. We investigate more general situations $T_1\otimes T_2$ for algebraic $k$-tori $T_1$ and $T_2$ by introducing a useful invariant of a $G$-lattice: its permutation order. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17427 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rationality problem for norm one tori of tensor products of étale algebras and Hasse norm principle Florence, Mathieu Hoshi, Akinari Yamasaki, Aiichi Algebraic Geometry Number Theory 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 Let $k$ be a field. Let $A=\prod_{i=1}^r K_i$ and $B=\prod_{j=1}^s E_j$ be étale $k$-algebras where $K_i$ and $E_j$ are finite separable field extensions of $k$ with $[K_i:k]=m_i$ and $[E_j:k]=n_j$. Let $\mathcal{T}_A=R^{(1)}_{A/k}(\mathbb{G}_m)$ be the norm one torus of the étale $k$-algebra $A$. We prove that if $\gcd(m_i,n_j\mid 1\leq i\leq r, 1\leq j\leq s)=1$ and $\mathcal{T}_A$ and $\mathcal{T}_B$ are stably $($resp. retract$)$ $k$-rational, then the algebraic $k$-torus $\mathcal{T}_A\otimes \mathcal{T}_B$ and the norm one torus $\mathcal{T}_{A\otimes B}$ are stably $($resp. retract$)$ $k$-rational. In particular, if $k$ is a global field, then the Hasse norm principle holds for $(A\otimes B)/k$. We then give detailed applications to the case of norm one tori of field extensions. We investigate more general situations $T_1\otimes T_2$ for algebraic $k$-tori $T_1$ and $T_2$ by introducing a useful invariant of a $G$-lattice: its permutation order. |
| title | Rationality problem for norm one tori of tensor products of étale algebras and Hasse norm principle |
| topic | Algebraic Geometry Number Theory 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 |
| url | https://arxiv.org/abs/2605.17427 |