On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909984385662976 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | We study the distribution of $2$-torsion in class groups and narrow class groups of cubic fields and cubic orders subject to prescribed shape conditions. The \emph{shape} of a cubic order in a number field is a natural geometric invariant taking values in the modular surface $\mathbb{H}/\operatorname{GL}_2(\mathbb{Z})$. Fix a subset $W$ of the modular surface with positive hyperbolic measure and boundary of measure zero. Refining the methods of Bhargava and Varma, we prove that among cubic fields with shape in $W$, the average size of the $2$-torsion subgroup of the class group is $5/4$ for totally real fields and $3/2$ for complex fields, while the average size of the $2$-torsion subgroup of the narrow class group for totally real cubic fields is $2$. We also obtain analogous results for cubic orders satisfying prescribed local conditions at all primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04503 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape Ray, Anwesh Number Theory 11R45, 11R16 We study the distribution of $2$-torsion in class groups and narrow class groups of cubic fields and cubic orders subject to prescribed shape conditions. The \emph{shape} of a cubic order in a number field is a natural geometric invariant taking values in the modular surface $\mathbb{H}/\operatorname{GL}_2(\mathbb{Z})$. Fix a subset $W$ of the modular surface with positive hyperbolic measure and boundary of measure zero. Refining the methods of Bhargava and Varma, we prove that among cubic fields with shape in $W$, the average size of the $2$-torsion subgroup of the class group is $5/4$ for totally real fields and $3/2$ for complex fields, while the average size of the $2$-torsion subgroup of the narrow class group for totally real cubic fields is $2$. We also obtain analogous results for cubic orders satisfying prescribed local conditions at all primes. |
| title | On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape |
| topic | Number Theory 11R45, 11R16 |
| url | https://arxiv.org/abs/2601.04503 |