Counting algebraic tori over ℚ by Artin conductor

2021/04/07 by Jungin Lee, Lee, Jungin
Mathematics · #math.NT

paper · pdf · doi:10.48550/arxiv.2104.02855

Abstract

In this paper we count the number N_n^tor(X) of n-dimensional algebraic tori over ℚ whose Artin conductor of the associated character is bounded by X. This can be understood as a generalization of counting number fields of given degree by discriminant. We suggest a conjecture on the asymptotics of N_n^tor(X) and prove that this conjecture follows from Malle's conjecture for tori over ℚ. We also prove that N₂^tor(X) ≪_ε X^{1 + ε}, and this upper bound can be improved to N₂^tor(X) ≪_ε X (log X)^{1 + ε} under the assumption of the Cohen-Lenstra heuristics for p=3.

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