On the strongly ambiguous classes of some biquadratic number fields

We study the capitulation of ideal classes in an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $k=Q(\sqrt{2pq},i)$, where $i=\sqrt{-1}$ and $p\equiv -q\equiv1 \pmod 4$aredifferentprimes\mathrm{.}Foreachofthethreequadraticextensions$K/k$insidetheabsolutegenusfield$k^{(*)}$of$k$, we compute the capitulation kernel of $K\mathrm{/}k$. Then we deduce that each strongly ambiguous class of $k\mathrm{/}Q(i)$ capitulates already in $k^{(\ast )}$, which is smaller than the relative genus field ${(k\mathrm{/}Q(i))}^{\ast }$.