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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.08121 |
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| _version_ | 1866908873213870080 |
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| author | Widmer, Martin |
| author_facet | Widmer, Martin |
| contents | In a 2008 paper Ellenberg suggested a strategy to improve the known upper bounds for the $\ell$-torsion part of class groups of number fields of fixed degree $d$. Motivated by this he proposed a question about the number of primitive elements of small height in a number field. Here we answer Ellenberg's question. We also improve Heath-Brown's bound for the $\ell$-torsion part of class groups of purely cubic number fields, and we generalize our improvement to pure fields of arbitrary odd degree $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08121 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A short remark on the $\ell$-torsion part of class groups Widmer, Martin Number Theory Primary 11R29 Secondary 11G50 In a 2008 paper Ellenberg suggested a strategy to improve the known upper bounds for the $\ell$-torsion part of class groups of number fields of fixed degree $d$. Motivated by this he proposed a question about the number of primitive elements of small height in a number field. Here we answer Ellenberg's question. We also improve Heath-Brown's bound for the $\ell$-torsion part of class groups of purely cubic number fields, and we generalize our improvement to pure fields of arbitrary odd degree $d$. |
| title | A short remark on the $\ell$-torsion part of class groups |
| topic | Number Theory Primary 11R29 Secondary 11G50 |
| url | https://arxiv.org/abs/2603.08121 |