Initial layer of the anti-cyclotomic Zp-extension of Q(−m) and capitulation phenomenon

Let $k=Q\left(\sqrt{-m}\right)$ be an imaginary quadratic field. We consider the properties of capitulation of the p-class group of k in the anti-cyclotomic $Z_p$-extension $k^{\mathrm{ac}}$ of k; for this, using a new approach based on the ${\mathrm{Log}}_p$-function (Theorem 2.3, Theorem 3.4), we determine the first layer $k_1^{\mathrm{ac}}$ of $k^{\mathrm{ac}}$ over k, and we show that some partial capitulation may exist in $k_1^{\mathrm{ac}}$, even when $k^{\mathrm{ac}}/k$ is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the $Z_p$-extensions of k, distinct from the cyclotomic one. For $p=3$, we characterize a sub-family of fields k (Normal Split cases) for which $k^{\mathrm{ac}}$ is not linearly disjoint from the Hilbert class field (Theorem 5.1). No assumptions are made on the splitting of 3 in k and in $k^{⁎}=Q\left(\sqrt{3m}\right)$, nor on the structures of their 3-class groups. Four pari/gp programs (7.1, 7.2, 7.3, 7.4 depending on the classification of Definition 2.10) are given, computing a defining cubic polynomial of $k_1^{\mathrm{ac}}$, and the main invariants attached to the fields k, $k^{⁎}$, $k_1^{\mathrm{ac}}$; some relations with Iwasawa's invariants are discussed (Theorem 9.6).