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

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 $N_3^{\text{tor}}(X){\ll }_{\epsilon }{X}^{1+\frac{\log 2+\epsilon }{\log \log X}}$, 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 ${\mathrm{GL}}_3(\mathbb{Z})$, 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.

33 pages, to appear in J. Number Theory

References in corpus (1)

• Malle's Conjecture for $S_n\times A$ for $n=3,4,5$