On the 2-torsion in class groups of number fields

2025/11/26 by Dante Bonolis, Bonolis, Dante
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2511.21899

openalex publication_date 2025/11/26 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

In 2020, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao proved that for a finite extension K/ℚ of degree n≥ 5, the size of the 2-torsion class group is bounded by # h₂(K)=O_n,ε(D_K^{(1)/(2)-(1)/(2n)+ε}), where D_K is the absolute discriminant of K. In the present paper, we improve their bound by proving that # h₂(K)=O_n,ε(D_K^(1)/(2)-(1)/(2n)-δ_K+ε), for a constant δ_K≥(1)/(28n)-(3)/(28n(n-1)).

Citations

• A new pointwise bound for 3-torsion of class groups
• Improving the trivial bound for ℓ-torsion in class groups
• The average size of 3-torsion in class groups of 2-extensions
• Uniform bounds for rational points on hyperelliptic fibrations
• Pointwise Bound for ℓ-torsion in Class Groups II: Nilpotent Extensions
• Pointwise Bound for ℓ-torsion in Class Groups: Elementary Abelian Extensions
• Averages and higher moments for the ℓ-torsion in class groups
• Average bounds for the ℓ-torsion in class groups of cyclic extensions
• An effective Chebotarev density theorem for families of number fields,\n with an application to \ℓ-torsion in class groups
• Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
• Averages and moments associated to class numbers of imaginary quadratic fields
• On the conductor of cohomological transforms
• Algebraic twists of modular forms and Hecke orbits
• Counting rational points on smooth cyclic covers
• Integral points on elliptic curves and 3-torsion in class groups

Related

• Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
• On the average size of 3-torsion in class groups of C₂ \wr H-extensions
• The average size of 3-torsion in class groups of 2-extensions
• Improving the trivial bound for ℓ-torsion in class groups
• Non-trivial bounds on 2, 3, 4, and 5-torsion in class groups of number fields, conditional on standard L-function conjectures