On the exponents of class groups of some families of imaginary quadratic fields

TL;DR

This paper investigates the structure of class groups of specific imaginary quadratic fields, proving the existence of subgroups of a given order under certain conditions and demonstrating the infinitude of such fields.

Contribution

It establishes the presence of particular subgroup structures in class groups of imaginary quadratic fields and proves the infinitude of such fields with these properties.

Findings

Existence of subgroups isomorphic to Z/nZ in class groups under certain conditions

Infinitely many imaginary quadratic fields have class groups with these subgroups

Unconditional results on divisibility of class numbers

Abstract

Let $a\geq 1$ and $n>1$ be odd integers. For a given prime $p$, we prove under certain conditions that the class groups of imaginary quadratic fields $\mathbb{Q}(\sqrt{a^2-4p^n})$ have a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. We also show that this family of fields has infinitely many members with the property that their class groups have a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. In addition, we deduce some unconditional results concerning the divisibility of the class numbers of certain imaginary quadratic fields. At the end, we provide some numerical examples to verify our results.