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Secondary terms in the distribution of genus numbers of cubic fields

Tatsuya Yamada

TL;DR

The paper proves the existence of a secondary term of order $X^{5/6}$ in the asymptotics for the sum of genus numbers and for the counts of cubic fields by genus, refining previous bounds. It develops a uniform counting framework for cubic fields with prescribed local specifications, including twisted counts by Dirichlet characters modulo $3$, and expresses the counts in terms of global constants and local data. The main contribution is the derivation of explicit secondary terms and uniform moment estimates for genus numbers across almost ordinary local specifications, extending the methods of McGown–Tucker and Bhargava–Taniguchi–Thorne. This advances arithmetic statistics for genus numbers and demonstrates a novel application of twisted counting techniques to genus-related problems in cubic fields.

Abstract

We prove the existence of secondary terms of order $X^{5/6}$ in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error estimates. These results refine the estimates obtained by McGown and Tucker. We also provide uniform estimates for the moments of the genus numbers of cubic fields.

Secondary terms in the distribution of genus numbers of cubic fields

TL;DR

The paper proves the existence of a secondary term of order in the asymptotics for the sum of genus numbers and for the counts of cubic fields by genus, refining previous bounds. It develops a uniform counting framework for cubic fields with prescribed local specifications, including twisted counts by Dirichlet characters modulo , and expresses the counts in terms of global constants and local data. The main contribution is the derivation of explicit secondary terms and uniform moment estimates for genus numbers across almost ordinary local specifications, extending the methods of McGown–Tucker and Bhargava–Taniguchi–Thorne. This advances arithmetic statistics for genus numbers and demonstrates a novel application of twisted counting techniques to genus-related problems in cubic fields.

Abstract

We prove the existence of secondary terms of order in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error estimates. These results refine the estimates obtained by McGown and Tucker. We also provide uniform estimates for the moments of the genus numbers of cubic fields.
Paper Structure (3 sections, 9 theorems, 61 equations, 1 table)

This paper contains 3 sections, 9 theorems, 61 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on Cubic Field Densities
  3. Proofs of Main Results

Key Result

Theorem 1.1

We have where the sum is taken over isomorphism classes of cubic fields $F$ with $0< \pm \mathop{\mathrm{Disc}}\nolimits \left( F \right) < X$, and Here, $C^{+} \coloneqq 1,C^{-} \coloneqq 3,K^{+} \coloneqq 1$, and $K^{-} \coloneqq \sqrt{3}$, the products are taken over all primes $p$, and $n_p$ is $3$ when $p \equiv 1 \pmod{3}$ and $1$ otherwise.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1: Fröhlich
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 3 more