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

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.