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 $p=3$.