A parametrization of 3-class groups of quadratic rings over Dedekind domains

2025/09/01 by Eliot Hodges, Hodges, Eliot, Ashvin Swaminathan +1
Mathematics · #11E12 (Primary) 11R11 #11E76 #11R29 #13C20 (Secondary) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2509.01722

openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a Dedekind domain with field of fractions K and char(R)≠3. In this paper, we generalize Bhargava's parametrization of 3-torsion ideal classes by binary cubic forms to work over R. Specifically, we construct arithmetic subgroups of GL₂(K) whose actions on certain lattices of binary cubic forms over K parametrize 3-torsion ideal classes in class groups of quadratic rings over R.

Citations

• Geometry-of-numbers methods in the cusp
• The mean number of 2-torsion elements in the class groups of n-monogenized cubic fields
• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
• The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders
• Parametrization of ideal classes in rings associated to binary forms
• Parametrizing quartic algebras over an arbitrary base
• On the Davenport-Heilbronn theorems and second order terms
• The density of discriminants of quintic rings and fields

Related

• The mean number of 3-torsion elements in ray class groups of quadratic fields
• The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders
• A new parametrization for ideal classes in rings defined by binary forms, and applications
• A geometric approach to the Cohen-Lenstra heuristics
• A new pointwise bound for 3-torsion of class groups