An upper bound for counting algebraic tori over $\mathbb{Q}$ by Artin conductor

Abstract

Let $N_n^{\mathrm{tor}}(X)$ be the number of isomorphism classes of $n$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that there is an absolute constant $C>0$ such that, for every positive integer $n \ge 2$, $N_n^{\mathrm{tor}}(X)\ll_n X^{\exp(C(\log n)^2)}$. The proofs of the main results were developed through an iterative dialogue with ChatGPT 5.6 Pro.