Counting $3$-dimensional algebraic tori over $\mathbb{Q}$

TL;DR

This paper estimates the number of 3-dimensional algebraic tori over  with bounded Artin conductor, providing upper bounds and verifying cases of Malle's conjecture under Cohen-Lenstra heuristics.

Contribution

It establishes new upper bounds for counting 3-dimensional algebraic tori and verifies Malle's conjecture for most conjugacy classes under certain heuristics.

Findings

Upper bound: $N_3^{ ext{tor}}(X) \,\ll\, X^{1 + \frac{\log 2 + \varepsilon}{\log \log X}}$

Improved bound: $N_3^{ ext{tor}}(X) \,\ll\, X (\log X)^4 \log \log X$ under Cohen-Lenstra heuristics

Verification of Malle's conjecture for 67 out of 72 conjugacy classes under heuristics

Abstract

In this paper we count the number $N_3^{\text{tor}}(X)$ of $3$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that $N_3^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \frac{\log 2 + \varepsilon}{\log \log X}}$, and this upper bound can be improved to $N_3^{\text{tor}}(X) \ll X (\log X)^4 \log \log X$ under the Cohen-Lenstra heuristics for $p=3$. We also prove that for $67$ out of $72$ conjugacy classes of finite nontrivial subgroups of $\operatorname{GL}_3(\mathbb{Z})$, Malle's conjecture for tori over $\mathbb{Q}$ holds up to a bounded power of $\log X$ under the Cohen-Lenstra heuristics for $p=3$ and Malle's conjecture for quartic $A_4$-fields.