A sharp upper bound for the $2$-torsion of class groups of multiquadratic fields

TL;DR

This paper establishes a sharp upper bound for the 2-torsion part of the narrow class group in multiquadratic fields, extending the result to ray class groups and demonstrating the bound's optimality in many cases.

Contribution

The authors prove the sharpness of a previously known bound for 2-torsion in narrow class groups of multiquadratic fields and extend the result to ray class groups.

Findings

The bound for 2-torsion is sharp in many multiquadratic fields.

Extension of the bound to ray class groups.

Validation of the bound's optimality in various cases.

Abstract

Let $K$ be a multiquadratic extension of $\mathbb{Q}$ and let $\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors \cite{KP} gave a bound for $|\text{Cl}^{+}(K)[2]|$ only in terms of the degree of $K$ and the number of ramifying primes. In the present work we show that this bound is sharp in a wide number of cases. Furthermore, we extend this to ray class groups.