Exploring the 2-Part of Class Groups in Quadratic Fields: Perspectives on the Cohen–Lenstra Conjectures

Yong Wang, Huili Zhang, Ying Zhou, Haopeng Deng and Lingyue Li ()
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Yong Wang: School of Arts and Sciences, Guangzhou Maritime University, Guangzhou 510725, China
Huili Zhang: School of Arts and Sciences, Guangzhou Maritime University, Guangzhou 510725, China
Ying Zhou: Institute of Visual Informatics (IVI), Universiti Kebangsaan Malaysia (UKM), Bangi 43600, Selangor, Malaysia
Haopeng Deng: School of Intelligent Transportation and Engineering, Guangzhou Jiaotong University, Guangzhou 510725, China
Lingyue Li: School of Arts and Sciences, Guangzhou Maritime University, Guangzhou 510725, China

Mathematics, 2024, vol. 13, issue 1, 1-26

Abstract: Cohen and Lenstra introduced conjectures concerning the distribution of class numbers in quadratic fields, though many of these conjectures remain unproven. This paper investigates the 2-part of class groups in imaginary quadratic fields and examines their alignment with the Cohen–Lenstra heuristics. We provide detailed proofs of key theorems related to ideal decompositions and modular homomorphisms, and we explore the distribution of class groups of imaginary quadratic fields. Our analysis includes constructing imaginary quadratic fields with prescribed 2-class groups and discussing the implications of these findings on the Cohen–Lenstra conjecture.

Keywords: quadratic fields; class numbers; class groups; Cohen–Lenstra conjecture (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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