Counting algebraic tori over $\mathbb{Q}$ by Artin conductor
Jungin Lee
TL;DR
The paper develops a conductor-based framework for counting algebraic tori over $\mathbb{Q}$, unifying the study with classical counting of number fields by discriminant. It proposes Malle-type conjectures for tori, showing they follow from the existing conjectures for tori via EV05 and related work, and provides both unconditional and conditional (Cohen–Lenstra) bounds in dimension 2. By classifying 2D tori, computing their Artin conductors, and analyzing the asymptotics for the challenging $H_{12,A}$ case (a $D_6$-type torus) via a decomposition into a cubic and a quadratic field, the authors derive bounds for $N_2^{\mathrm{tor}}(X)$ and $N_2^{\mathrm{tor}}(X;H_{12,A})$. The results connect torus counting to discriminant and conductor theory of field extensions, offering conditional improvements under deep heuristics and illuminating the arithmetic-statistics landscape for tori.
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 $p=3$.