Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

There are consecutive cubic fields with large class numbers, when ordered by discriminant

Vitezslav Kala, Om Prakash

TL;DR

The article addresses the distribution of class numbers in cubic fields ordered by discriminant and proves that there exist arbitrarily long runs of consecutive discriminants $d<i\, (i=1,...,k)$ for which any cubic field with $\pm\Delta_K=d+i$ has class number exceeding a prescribed bound $H$. The main approach is to construct a fixed modulus $m$ and residue $a$ so that primes forced to ramify yield large genus numbers $g_K$, which in turn force $h(K)\ge g_K>H$; counting in arithmetic progressions is handled via the Taniguchi–Thorne density results, and the Maier matrix method translates discriminant-density into a dense set of suitable $t$. Key contributions include (i) a constructive framework producing $h(K)>H$ from congruence data, (ii) explicit density estimates for cubic discriminants in progressions, and (iii) a Maier-matrix-based argument giving at least $cX^{1-κ-ε}$ suitable integers $d\le X$ with blocks containing many admissible discriminants. The findings advance understanding of how large class numbers can appear in long sequences of cubic fields and illustrate the effectiveness of genus-theoretic methods paired with arithmetic-progression density results; the bounds hinge on the 3-torsion exponent $κ$ from Chan–Koymans, with $κ\approx 0.3193$.

Abstract

We consider cubic number fields ordered by their discriminants, and show that there exist arbitrarily long sequences that contain only fields with class numbers greater than a given bound.

There are consecutive cubic fields with large class numbers, when ordered by discriminant

TL;DR

The article addresses the distribution of class numbers in cubic fields ordered by discriminant and proves that there exist arbitrarily long runs of consecutive discriminants for which any cubic field with has class number exceeding a prescribed bound . The main approach is to construct a fixed modulus and residue so that primes forced to ramify yield large genus numbers , which in turn force ; counting in arithmetic progressions is handled via the Taniguchi–Thorne density results, and the Maier matrix method translates discriminant-density into a dense set of suitable . Key contributions include (i) a constructive framework producing from congruence data, (ii) explicit density estimates for cubic discriminants in progressions, and (iii) a Maier-matrix-based argument giving at least suitable integers with blocks containing many admissible discriminants. The findings advance understanding of how large class numbers can appear in long sequences of cubic fields and illustrate the effectiveness of genus-theoretic methods paired with arithmetic-progression density results; the bounds hinge on the 3-torsion exponent from Chan–Koymans, with .

Abstract

We consider cubic number fields ordered by their discriminants, and show that there exist arbitrarily long sequences that contain only fields with class numbers greater than a given bound.
Paper Structure (4 sections, 11 theorems, 29 equations)

This paper contains 4 sections, 11 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Cubic fields and their class numbers
  3. Counting cubic fields in an arithmetic progression
  4. Density estimates

Key Result

Theorem 1.1

Fix a sign choice $\pm$, $\varepsilon>0$, and positive integers $k, H$. There exists a positive constant $c$ (depending on $\pm, \varepsilon, k, H$) such that for all sufficiently large $X$, there are at least $cX^{1-\kappa-\varepsilon}$ positive integers $d\leq X$ such that the following hold:

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1: Hasse
  • proof
  • Theorem 2.2: Fröhlich
  • Theorem 2.3
  • proof
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1: TT
  • ...and 11 more