Malle-type conjecture for counting algebraic tori by finite monodromy group

Let $N_n^{\operatorname{tor}}(X;H)$ denote the number of isomorphism classes of $n$-dimensional algebraic tori over $\mathbb{Q}$ with Artin conductor at most $X$ and associated finite subgroup $1\neq H\leq \operatorname{GL}_n(\mathbb{Z})$. For $I_n\in\operatorname{GL}_n(\mathbb{Z})$, define

$$a(H):=\min_{h\in H\setminus\{I_n\}}\operatorname{rank}(h-I_n).$$

Let $b(H)$ be the number of orbits $\mathcal{C}$ of the action of $G_{\mathbb{Q}}$ on the conjugacy classes of $H$ via the cyclotomic character for which $\operatorname{rank}(h-I_n)=a(H)$ for some, equivalently all, $h\in\mathcal{C}$.

Malle-type conjecture for algebraic tori. For every $n\geq 1$ and every finite subgroup $1\neq H\leq\operatorname{GL}_n(\mathbb{Z})$, there are positive integers $a(H),b(H)$ and a constant $c_H>0$, depending only on $H$, such that

$$N_n^{\operatorname{tor}}(X;H)\sim c_H X^{\frac{1}{a(H)}}(\log X)^{b(H)-1}.$$

This is an analogue of Malle's conjecture for tori and follows from a more general conjecture of Ellenberg and Venkatesh according to the source. It also implies the preceding Linnik-type conjecture; its resolution is not established in the paper.