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

Abstract

Let K be a multiquadratic extension of Q $\mathbb {Q}$ and let Cl + ( K ) $\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors (Koymans and Pagano, Int. Math. Res. Not. 2022 (2022), no. 4, 2772–2823) gave a bound for | Cl + ( K ) [ 2 ] | $|\text{Cl}^{+}(K)[2]|$ 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.