Counting 3-dimensional algebraic tori over [formula omitted].

In this paper we count the number N 3 tor (X) of 3-dimensional algebraic tori over Q whose Artin conductor is bounded above by X. We prove that N 3 tor (X) ≪ ε X 1 + log ⁡ 2 + ε log ⁡ log ⁡ X , and this upper bound can be improved to N 3 tor (X) ≪ 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 GL 3 (Z) , Malle's conjecture for tori over 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. [ABSTRACT FROM AUTHOR]