We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\mathbf{ k} =\mathbb{ Q}(\sqrt{ p_1p_2q}, i)$, where $i=\sqrt{ -1}$ and $p_1\equiv p_2\equiv-q\equiv1 \pmod 4$ are different primes. For each of the three quadratic extensions $\mathbf{ K}/\mathbf{ k}$ inside the absolute genus field $\mathbf{ k}^{ (*)}$ of $\mathbf{ k}$, we compute the capitulation kernel of $\mathbf{ K}/\mathbf{ k}$. Then we deduce that each strongly ambiguous class of $\mathbf{ k}/\mathbb{ Q}(i)$ capitulates already in $\mathbf{ k}^{ (*)}$, which is smaller than the relative genus field $(\mathbf{ k}/\mathbb{ Q}(i))^*$.
Categories: math.NT