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

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.

Some of the results in this paper were originally part of arXiv:1909.13871, which is now split in two parts