On the asymptotics of cubic fields ordered by general invariants

TL;DR

This paper introduces generalized discriminants as invariants for cubic fields, derives their asymptotic distribution, and examines which families align with the Malle--Bhargava heuristic.

Contribution

It defines a new class of invariants for cubic fields and analyzes their asymptotic behavior and heuristic compatibility.

Findings

Asymptotic formulas for cubic fields ordered by generalized discriminants

Identification of families satisfying the Malle--Bhargava heuristic

Extension of understanding of cubic field invariants

Abstract

In this article, we introduce a class of invariants of cubic fields termed generalized discriminants. We then obtain asymptotics for the families of cubic fields ordered by these invariants. In addition, we determine which of these families satisfy the Malle--Bhargava heuristic.