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The existence of infinitely many cubic fields with class group of exact 2-rank 1

Manjul Bhargava, Arul Shankar, Artane Siad, Ashvin Swaminathan

TL;DR

The paper proves that there are infinitely many cubic fields whose class groups have exact $2$-rank $1$, supporting the $p=2$ predictions of the Cohen--Lenstra--Martinet--Malle heuristics. It presents two independent proof strategies: an anomaly-based approach via unit-monogenised cubic fields (MR4891959) yielding a $X^{1/2}$-type lower bound, and a moment-based approach via monogenised cubic fields (2506.05539) yielding a $X^{5/6}$-type lower bound; both approaches establish infinitude in the real cubic setting. The results demonstrate that one can obtain infinite families with fixed exact $2$-rank without genus-theoretic constraints, using distinct methodologies (anomaly and moments) and provide a framework for extending these ideas. Overall, the work advances understanding of fixed-rank class groups in thin families and connects to broader themes in Cohen–Lenstra heuristics and arithmetic statistics of cubic fields.

Abstract

We show that infinitely many cubic fields have class group of 2-rank 1.

The existence of infinitely many cubic fields with class group of exact 2-rank 1

TL;DR

The paper proves that there are infinitely many cubic fields whose class groups have exact -rank , supporting the predictions of the Cohen--Lenstra--Martinet--Malle heuristics. It presents two independent proof strategies: an anomaly-based approach via unit-monogenised cubic fields (MR4891959) yielding a -type lower bound, and a moment-based approach via monogenised cubic fields (2506.05539) yielding a -type lower bound; both approaches establish infinitude in the real cubic setting. The results demonstrate that one can obtain infinite families with fixed exact -rank without genus-theoretic constraints, using distinct methodologies (anomaly and moments) and provide a framework for extending these ideas. Overall, the work advances understanding of fixed-rank class groups in thin families and connects to broader themes in Cohen–Lenstra heuristics and arithmetic statistics of cubic fields.

Abstract

We show that infinitely many cubic fields have class group of 2-rank 1.
Paper Structure (4 sections, 5 theorems, 1 equation)

This paper contains 4 sections, 5 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Proofs
  3. Proof via MR4891959
  4. Proof via 2506.05539

Key Result

Theorem 1

There are infinitely many cubic fields with class groups of exact $2$-rank $1$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['thm: main theorem']}
  • proof : Proof of Theorem \ref{['thm: quantitative main theorem']} with $X^{1/2}$
  • proof : Proof of Theorems \ref{['thm: main theorem']} and \ref{['thm: quantitative main theorem']}
  • Theorem 3