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

Let Kappa be a multiquadratic extension of Q and let Cl+(Kappa) 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 vertical bar Cl+(Kappa)[2]vertical bar only in terms of the degree of Kappa 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.