Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the asymptotics of cubic fields ordered by general invariants

Arul Shankar, Frank Thorne

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.

On the asymptotics of cubic fields ordered by general 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.
Paper Structure (7 sections, 20 theorems, 71 equations)

This paper contains 7 sections, 20 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Families of cubic fields with a fixed invariant
  3. Counting cubic fields $K$ with fixed $\mathrm{F}(K)$
  4. Counting cubic fields $K$ with fixed $\mathrm{D}(K)$
  5. The asymptotics of cubic fields with bounded invariants
  6. Ordering cubic fields by generalized discriminants
  7. Numerical data

Key Result

Theorem 1

Let $N^{\pm}_{\operatorname{Disc}}(X)$ denote the number of cubic fields $K$, up to isomorphism, that satisfy $|\operatorname{Disc}(K)|<X$ and $\pm\operatorname{Disc}(K)>0$. Then

Theorems & Definitions (22)

  • Theorem 1: Davenport--Heilbronn
  • Theorem 2
  • Theorem 3: Belabas--Fouvry, Bhargava--Wood
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Theorem 8: BTT, Theorem 1.4
  • Proposition 9
  • Theorem 10: CT, Theorem 2.5
  • ...and 12 more