Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918315174133760 |
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| author | Ram, Mahesh Kumar Pandey, Prem Prakash Mahapatra, Nimish Kumar |
| author_facet | Ram, Mahesh Kumar Pandey, Prem Prakash Mahapatra, Nimish Kumar |
| contents | There are several recent works where authors have shown that number fields $K$ with `sufficiently many' units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ is absolutely abelian. In this article, we explore the latter hypothesis: how often a number field $K$ has absolutely abelian Hilbert class field? For a number field $K$ to have absolutely abelian Hilbert class field, we obtain several criteria in terms of class number of $K$, Pólya group of $K$, and genus number of $K$. We also show that for such number fields the $\ell-$torsion conjecture is true. Along with these, the article also reports some results on a theme to study class groups, developed by the authors, where primes of higher degree are used to study class groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10725 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture Ram, Mahesh Kumar Pandey, Prem Prakash Mahapatra, Nimish Kumar Number Theory 11R29, 11R37, 11R44 There are several recent works where authors have shown that number fields $K$ with `sufficiently many' units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ is absolutely abelian. In this article, we explore the latter hypothesis: how often a number field $K$ has absolutely abelian Hilbert class field? For a number field $K$ to have absolutely abelian Hilbert class field, we obtain several criteria in terms of class number of $K$, Pólya group of $K$, and genus number of $K$. We also show that for such number fields the $\ell-$torsion conjecture is true. Along with these, the article also reports some results on a theme to study class groups, developed by the authors, where primes of higher degree are used to study class groups. |
| title | Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture |
| topic | Number Theory 11R29, 11R37, 11R44 |
| url | https://arxiv.org/abs/2510.10725 |