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Counting principal ideals of small norm in the simplest cubic fields

Mikuláš Zindulka

TL;DR

This work determines explicit, uniform asymptotics for the principal-ideal counting function $P(a,x)$ in the family of simplest cubic fields $K_a$, valid for all $x\ge 1$ and capturing small-norm regimes relative to the discriminant. The authors translate the problem into counting lattice points in a Shintani-type fundamental domain, decomposing the region into six three-dimensional cones via the unit group, and then bounding lattice counts on each cone with precise combinatorial parametrizations. The main result shows $P(a,x) \asymp \dfrac{(\log a)^2}{a^2}x + \left(\dfrac{x}{a}\right)^{2/3} + x^{1/3}$, while the primitive-principal count $P_p(a,x)$ satisfies $P_p(a,x) \asymp \dfrac{(\log a)^2}{a^2}x + \left(\dfrac{x}{a}\right)^{2/3} + 1$, revealing a surprisingly large number of small-norm principal ideals. A complementary lower-bound analysis for primitive ideals confirms the sharpness of the upper bounds, and the results hold in a regime where $x$ can be far below the discriminant. The methods combine explicit unit-theory geometry with meticulous lattice-point counting to yield quantitative, effective bounds with explicit dependence on $a$ and $x$.

Abstract

We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even when $ x $ is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm.

Counting principal ideals of small norm in the simplest cubic fields

TL;DR

This work determines explicit, uniform asymptotics for the principal-ideal counting function in the family of simplest cubic fields , valid for all and capturing small-norm regimes relative to the discriminant. The authors translate the problem into counting lattice points in a Shintani-type fundamental domain, decomposing the region into six three-dimensional cones via the unit group, and then bounding lattice counts on each cone with precise combinatorial parametrizations. The main result shows , while the primitive-principal count satisfies , revealing a surprisingly large number of small-norm principal ideals. A complementary lower-bound analysis for primitive ideals confirms the sharpness of the upper bounds, and the results hold in a regime where can be far below the discriminant. The methods combine explicit unit-theory geometry with meticulous lattice-point counting to yield quantitative, effective bounds with explicit dependence on and .

Abstract

We estimate the number of principal ideals of norm in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary , even when is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm.
Paper Structure (12 sections, 51 theorems, 396 equations)

This paper contains 12 sections, 51 theorems, 396 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Apriori estimate
  4. Setting up the counting problem
  5. Counting lattice points in a three-dimensional cone
  6. Upper bound
  7. Estimate for Q(a,S1(a))
  8. Estimate for Q(a,S2(a))
  9. Estimate for R(a,S3(a))
  10. Estimate for Q(a,S4(a))
  11. Proof of the upper bound
  12. Lower bound and proof of the main theorem

Key Result

Theorem 1.1

Let $K_a = \mathbb{Q}(\rho)$ be a simplest cubic field, where $\rho$ is the largest root of $f_a$. Assume that $\mathcal{O}_{K_a} = \mathbb{Z}[\rho]$. If $a \geq 8$ and $x \geq 1$, then and

Theorems & Definitions (100)

  • Theorem 1.1
  • proof
  • Lemma 2.1: LP
  • Lemma 3.1: De19
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 90 more