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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2605.04792 |
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| _version_ | 1866910223244984320 |
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| author | Dixit, Anup B. Pasupulati, Sunil Kumar |
| author_facet | Dixit, Anup B. Pasupulati, Sunil Kumar |
| contents | The genus number of a number field is a fundamental invariant which measures the contribution of ramification to its ideal class group. In this paper, we establish the statistics for the genus number for $S_3\times C_q$-fields for $q\neq 3$ a prime number, $D_4$-fields and pure quartic fields. We also obtain precise results on the average and higher moments of the genus distribution within the family of $S_3\times C_q$-fields. Finally, based on heuristics, we formulate a conjecture identifying families for which one should expect the genus density to be zero, i.e., only a density zero subset of fields in the family attains any fixed genus number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04792 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Statistics of the Genus Number of $S_3 \times C_q$ and $D_4$-fields Dixit, Anup B. Pasupulati, Sunil Kumar Number Theory The genus number of a number field is a fundamental invariant which measures the contribution of ramification to its ideal class group. In this paper, we establish the statistics for the genus number for $S_3\times C_q$-fields for $q\neq 3$ a prime number, $D_4$-fields and pure quartic fields. We also obtain precise results on the average and higher moments of the genus distribution within the family of $S_3\times C_q$-fields. Finally, based on heuristics, we formulate a conjecture identifying families for which one should expect the genus density to be zero, i.e., only a density zero subset of fields in the family attains any fixed genus number. |
| title | Statistics of the Genus Number of $S_3 \times C_q$ and $D_4$-fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.04792 |