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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2501.18574 |
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| _version_ | 1866912211368148992 |
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| author | Alberts, Brandon Oliver, Robert J. Lemke Wang, Jiuya Wood, Melanie Matchett |
| author_facet | Alberts, Brandon Oliver, Robert J. Lemke Wang, Jiuya Wood, Melanie Matchett |
| contents | We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group $G$. Our method relies on having asymptotic counts for $T$-extensions for some normal subgroup $T$ of $G$, uniform bounds for the number of such $T$-extensions, and possibly weak bounds on the asymptotic number of $G/T$-extensions. However, we do not require that most $T$-extensions of a $G/T$-extension are $G$-extensions. Our new results use $T$ either abelian or $S_3^m$, though our framework is general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inductive methods for counting number fields Alberts, Brandon Oliver, Robert J. Lemke Wang, Jiuya Wood, Melanie Matchett Number Theory We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group $G$. Our method relies on having asymptotic counts for $T$-extensions for some normal subgroup $T$ of $G$, uniform bounds for the number of such $T$-extensions, and possibly weak bounds on the asymptotic number of $G/T$-extensions. However, we do not require that most $T$-extensions of a $G/T$-extension are $G$-extensions. Our new results use $T$ either abelian or $S_3^m$, though our framework is general. |
| title | Inductive methods for counting number fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.18574 |