Counting algebraic tori over $\mathbb{Q}$ by Artin conductor
Abstract
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 $p=3$.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2021
- DOI:
- 10.48550/arXiv.2104.02855
- arXiv:
- arXiv:2104.02855
- Bibcode:
- 2021arXiv210402855L
- Keywords:
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- Mathematics - Number Theory;
- 11R21;
- 11R29;
- 20G30
- E-Print:
- 20 pages, exposition improved