On tamely ramified infinite Galois extensions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929207433494528 |
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| author | Hajir, Farshid Larsen, Michael Maire, Christian Ramakrishna, Ravi |
| author_facet | Hajir, Farshid Larsen, Michael Maire, Christian Ramakrishna, Ravi |
| contents | For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $μ_p \not \subset K$. Our guiding aim is to characterize the finitely generated pro-$p$ quotients of~$G^{\rm ta}$. We give a {unified point of view} by introducing the notion of {\it stably inertially generated} pro-$p$ groups~$G$, for which linear groups are archetypes. This key notion {is compatible} with local {\it tame liftings} as used in the Scholz-Reichardt Theorem. We realize every finitely generated pro-$p$ group~$G$ which is stably inertially generated as a quotient of $G^{\rm ta}$. Further examples of groups that we realize as quotients of $G^{\rm ta}$ include congruence subgroups of special linear groups over ${\mathbb Z}_p[[ T_1,\cdots, T_n ]]$. Finally, we give classes of groups which cannot be realized as quotients of $G^{\rm ta}_{\mathbb Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On tamely ramified infinite Galois extensions Hajir, Farshid Larsen, Michael Maire, Christian Ramakrishna, Ravi Number Theory 11F80, 11R37, 11R32 For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $μ_p \not \subset K$. Our guiding aim is to characterize the finitely generated pro-$p$ quotients of~$G^{\rm ta}$. We give a {unified point of view} by introducing the notion of {\it stably inertially generated} pro-$p$ groups~$G$, for which linear groups are archetypes. This key notion {is compatible} with local {\it tame liftings} as used in the Scholz-Reichardt Theorem. We realize every finitely generated pro-$p$ group~$G$ which is stably inertially generated as a quotient of $G^{\rm ta}$. Further examples of groups that we realize as quotients of $G^{\rm ta}$ include congruence subgroups of special linear groups over ${\mathbb Z}_p[[ T_1,\cdots, T_n ]]$. Finally, we give classes of groups which cannot be realized as quotients of $G^{\rm ta}_{\mathbb Q}$. |
| title | On tamely ramified infinite Galois extensions |
| topic | Number Theory 11F80, 11R37, 11R32 |
| url | https://arxiv.org/abs/2401.05927 |