Counting algebraic tori over $\mathbb{Q}$ by Artin conductor
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866908623380152320 |
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| author | Lee, Jungin |
| author_facet | Lee, Jungin |
| contents | In this paper we count the number $N_n^{\text{tor}}(X)$ of $n$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor of the associated character is bounded by $X$. This can be understood as a generalization of counting number fields of given degree by discriminant. We suggest a conjecture on the asymptotics of $N_n^{\text{tor}}(X)$ and prove that this conjecture follows from Malle's conjecture for tori over $\mathbb{Q}$. We also prove that $N_2^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \varepsilon}$, and this upper bound can be improved to $N_2^{\text{tor}}(X) \ll_{\varepsilon} X (\log X)^{1 + \varepsilon}$ under the assumption of the Cohen-Lenstra heuristics for $p=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_02855 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Counting algebraic tori over $\mathbb{Q}$ by Artin conductor Lee, Jungin Number Theory In this paper we count the number $N_n^{\text{tor}}(X)$ of $n$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor of the associated character is bounded by $X$. This can be understood as a generalization of counting number fields of given degree by discriminant. We suggest a conjecture on the asymptotics of $N_n^{\text{tor}}(X)$ and prove that this conjecture follows from Malle's conjecture for tori over $\mathbb{Q}$. We also prove that $N_2^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \varepsilon}$, and this upper bound can be improved to $N_2^{\text{tor}}(X) \ll_{\varepsilon} X (\log X)^{1 + \varepsilon}$ under the assumption of the Cohen-Lenstra heuristics for $p=3$. |
| title | Counting algebraic tori over $\mathbb{Q}$ by Artin conductor |
| topic | Number Theory |
| url | https://arxiv.org/abs/2104.02855 |