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Counting algebraic tori over $\mathbb{Q}$ by Artin conductor
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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 $p=3$.
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An upper bound for counting algebraic tori over $\mathbb{Q}$ by Artin conductor
For every n ≥ 2, the number of n-dimensional rational tori with Artin conductor at most X is bounded by X^{exp(C(log n)^2)}, for an absolute constant C.
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