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 $k^{\rm ac}$ of $k$; for this, using a new approach based on the Log$_p$-function (Theorems 2.3, 3.4), we determine the first layer $k_1^{\rm ac}$ of $k^{\rm ac}$ over $k$, and we show that some partial capitulation may exist in $k_1^{\rm ac}$, even when $k^{\rm ac}/k$ is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the $\mathbb{Z}_p$-extensions of $k$, distinct from the cyclotomic one. For $p=3$, we characterize a sub-family of fields $k$ (Normal Split cases) for which $k^{\rm ac}$ 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 $k^*=\mathbb{Q}(\sqrt{3m})$, 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 $k_1^{\rm ac}$, and the main invariants attached to the fields $k$, $k^*$, $k_1^{\rm ac}$; some relations with Iwasawa's invariants are discussed (Theorem 9.6).