The second moment of the size of the $2$-class group of monogenized cubic fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909639657914368 |
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| author | Bhargava, Manjul Shankar, Arul Swaminathan, Ashvin |
| author_facet | Bhargava, Manjul Shankar, Arul Swaminathan, Ashvin |
| contents | We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_05539 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The second moment of the size of the $2$-class group of monogenized cubic fields Bhargava, Manjul Shankar, Arul Swaminathan, Ashvin Number Theory 11R16, 11R29, 11R45 We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates. |
| title | The second moment of the size of the $2$-class group of monogenized cubic fields |
| topic | Number Theory 11R16, 11R29, 11R45 |
| url | https://arxiv.org/abs/2506.05539 |