Secondary terms in the counting functions of quartic fields II
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909733659607040 |
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| author | Shankar, Arul Tsimerman, Jacob |
| author_facet | Shankar, Arul Tsimerman, Jacob |
| contents | We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for quartic rings (weighted by the number of cubic resolvents), deducing as a consequence that the Shintani zeta functions associated to the prehomogeneous vector space $\mathbb{C}^2\otimes\mathrm{Sym}^2(\mathbb{C}^3)$ have at most a simple pole at $s=5/6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Secondary terms in the counting functions of quartic fields II Shankar, Arul Tsimerman, Jacob Number Theory We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for quartic rings (weighted by the number of cubic resolvents), deducing as a consequence that the Shintani zeta functions associated to the prehomogeneous vector space $\mathbb{C}^2\otimes\mathrm{Sym}^2(\mathbb{C}^3)$ have at most a simple pole at $s=5/6$. |
| title | Secondary terms in the counting functions of quartic fields II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.08527 |