The second moment of the size of the 2-class group of monogenized cubic fields

2025/06/05 by Manjul Bhargava, Arul Shankar, Bhargava, Manjul +3
Mathematics · #11R16 #11R29 #11R45 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.05539

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the 2-class group is at most 3 (resp., at most 6). In the totally real case, we further prove that the second moment of the size of the narrow 2-class group is at most 9. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.

Citations

• Conjectures for distributions of class groups of extensions of number fields containing roots of unity
• The second moment of the size of the 2-Selmer group of elliptic curves
• A new parametrization for ideal classes in rings defined by binary forms, and applications
• The mean number of 2-torsion elements in the class groups of n-monogenized cubic fields
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• The average number of elements in the 4-Selmer groups of elliptic curves is 7
• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• Minimisation and reduction of 5-coverings of elliptic curves
• Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0
• Binary quartic forms having bounded invariants, and the boundedness of\n the average rank of elliptic curves
• The density of discriminants of quintic rings and fields

Related

• The mean number of 2-torsion elements in the class groups of n-monogenized cubic fields
• Asymptotics of p-torsion subgroup sizes in class groups of monogenized cubic fields
• Effect of monogenicity on 2-torsion in the class group of number fields of odd degree
• The average sizes of two-torsion subgroups in quotients of class groups of cubic fields
• Hecke reciprocity and class groups