Counting 3-dimensional algebraic tori over [formula omitted

Abstract

In this paper we count the number N3tor(X) of 3-dimensional algebraic tori over Q whose Artin conductor is bounded above by X. We prove that N3tor(X)≪εX1+log⁡2+εlog⁡log⁡X, and this upper bound can be improved to N3tor(X)≪X(log⁡X)4log⁡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 GL3(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 A4-fields.