Hasse norm principle for $M_{11}$ and $J_1$ extensions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910623785287680 |
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| author | Hoshi, Akinari Kanai, Kazuki Yamasaki, Aiichi |
| author_facet | Hoshi, Akinari Kanai, Kazuki Yamasaki, Aiichi |
| contents | We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K/k$ when the Galois groups ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of order $7920$ or the Janko group $J_1$ of order $175560$ by determining $H^1(k,{\rm Pic}\, \overline{X})=0$ or $\mathbb{Z}/2\mathbb{Z}$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ with a smooth $k$-compactification $X$ and $\overline{X}=X\times_k\overline{k}$. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the $26$ sporadic simple groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_09119 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hasse norm principle for $M_{11}$ and $J_1$ extensions Hoshi, Akinari Kanai, Kazuki Yamasaki, Aiichi Number Theory Algebraic Geometry 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K/k$ when the Galois groups ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of order $7920$ or the Janko group $J_1$ of order $175560$ by determining $H^1(k,{\rm Pic}\, \overline{X})=0$ or $\mathbb{Z}/2\mathbb{Z}$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ with a smooth $k$-compactification $X$ and $\overline{X}=X\times_k\overline{k}$. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the $26$ sporadic simple groups. |
| title | Hasse norm principle for $M_{11}$ and $J_1$ extensions |
| topic | Number Theory Algebraic Geometry 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 |
| url | https://arxiv.org/abs/2210.09119 |