Exponents of class groups of certain imaginary quadratic fields

TL;DR

This paper demonstrates infinitely many imaginary quadratic fields with class groups containing elements of a specified order, providing counterexamples to a previous conjecture about their structure.

Contribution

It constructs a new infinite family of imaginary quadratic fields with prescribed class group elements, challenging existing conjectures.

Findings

Infinitely many such quadratic fields exist.

Counterexamples to Wada's conjecture are provided.

The class groups contain elements of order n in these fields.

Abstract

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a conjecture by H. Wada \cite{WA70} on the structure of ideal class groups.