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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2405.02740 |
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| _version_ | 1866916576452673536 |
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| author | Monnet, Sebastian |
| author_facet | Monnet, Sebastian |
| contents | Given a number field $k$, a finitely generated subgroup $\mathcal{A}\subseteq k^\times$, and an integer $n\geq 3$, we study the distribution of $S_n$-extensions of $k$ such that the elements of $\mathcal{A}$ are norms. For $n\leq 5$, and conjecturally for $n \geq 6$, we show that the density of such extensions is the product of so-called ``local masses'' at the places of $k$. When $n$ is an odd prime, we give formulas for these local masses, allowing us to express the aforementioned density as an explicit Euler product. For $n=4$, we determine almost all of these masses exactly and give an efficient algorithm for computing the rest, again yielding an explicit Euler product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02740 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $S_n$-extensions with prescribed norms Monnet, Sebastian Number Theory Given a number field $k$, a finitely generated subgroup $\mathcal{A}\subseteq k^\times$, and an integer $n\geq 3$, we study the distribution of $S_n$-extensions of $k$ such that the elements of $\mathcal{A}$ are norms. For $n\leq 5$, and conjecturally for $n \geq 6$, we show that the density of such extensions is the product of so-called ``local masses'' at the places of $k$. When $n$ is an odd prime, we give formulas for these local masses, allowing us to express the aforementioned density as an explicit Euler product. For $n=4$, we determine almost all of these masses exactly and give an efficient algorithm for computing the rest, again yielding an explicit Euler product. |
| title | $S_n$-extensions with prescribed norms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.02740 |