Initial layer of the anti-cyclotomic $\mathbb{Z}_{3}$-extension of $\mathbb{Q}( \sqrt{-m})$ and capitulation phenomenon

Abstract: Let $k=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic field. We consider the properties of capitulation of the $p$-class group of $k$ in the anti-cyclotomic $\mathbb{Z}{p}$-extension $k^{ac}$ of $k$; for this, using a new method based on the Log$_p$-function (Theorems 1.3 and 2.3), we determine, for $p = 3$, the first layer $k{1}^{ac}$ of $k^{ac}$ over $k$, and we examine if, at least, some partial capitulation may exist in $k_{1}^{ac}$. The answer is obviously yes, even when $k^{ac}/k$ is totally ramified. We have conjectured that this phenomenon of capitulation is specific of all the $\mathbb{Z}_{p}$-extensions of $k$, distinct from the cyclotomic one. We characterize a sub-family of fields $k$ (Normal Split cases) for which $k^{ac}$ is not linearly disjoint from the Hilbert class field (Theorem 3.1). No assumptions are made on the structure of the 3-class group of $k$, nor on the splitting of 3 in $k$ and in its mirror field $k*=\mathbb{Q}(\sqrt{3m})$. Four pari/gp programs are given with numerical illustrations to cover all cases without any limitation.