Orbital integrals and ideal class monoids for a Bass order
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| Format: | Preprint |
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2024
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| _version_ | 1866915520343703552 |
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| author | Cho, Sungmun Hong, Jungtaek Lee, Yuchan |
| author_facet | Cho, Sungmun Hong, Jungtaek Lee, Yuchan |
| contents | A Bass order is an order of a number field whose fractional ideals are generated by two elements. The majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is fourth-power-free in $\mathbb{Z}$, is a Bass order. In this paper, we will propose a closed formula for the number of fractional ideals of a Bass order $R$, up to its invertible ideals, using the conductor of $R$. Since $R$ is a Bass order, this is the same as the number of overorders of $R$. We will also explain the explicit enumeration of all orders containing $R$. Our method is based on the local-global argument and the exhaustion argument, using orbital integrals for $\mathfrak{gl}_n$ as a mass formula. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_16199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbital integrals and ideal class monoids for a Bass order Cho, Sungmun Hong, Jungtaek Lee, Yuchan Number Theory 11F72, 11R65 A Bass order is an order of a number field whose fractional ideals are generated by two elements. The majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is fourth-power-free in $\mathbb{Z}$, is a Bass order. In this paper, we will propose a closed formula for the number of fractional ideals of a Bass order $R$, up to its invertible ideals, using the conductor of $R$. Since $R$ is a Bass order, this is the same as the number of overorders of $R$. We will also explain the explicit enumeration of all orders containing $R$. Our method is based on the local-global argument and the exhaustion argument, using orbital integrals for $\mathfrak{gl}_n$ as a mass formula. |
| title | Orbital integrals and ideal class monoids for a Bass order |
| topic | Number Theory 11F72, 11R65 |
| url | https://arxiv.org/abs/2408.16199 |