The 4-ranks of the class groups of Q(sqrt(-d)) and Q(sqrt(-d0 d)) are asymptotically independent, each with the Cohen-Lenstra-Gerth distribution.
On Stevenhagen's conjecture
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abstract
We generalize a classical reciprocity law due to R\'edei using our recently developed description of the $2$-torsion of class groups of multiquadratic fields. This result is then used to prove a variety of new reflection principles for class groups, one of which involves a symbol similar to the spin symbol as defined in work of Friedlander, Iwaniec, Mazur and Rubin. We combine these reflection principles with Smith's techniques to prove Stevenhagen's conjecture on the solubility of the negative Pell equation.
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Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields
The 4-ranks of the class groups of Q(sqrt(-d)) and Q(sqrt(-d0 d)) are asymptotically independent, each with the Cohen-Lenstra-Gerth distribution.