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 $k^{\mathrm{ac}}$ of $k$; for this, using a new approach based on the Log${}_p$-function (Theorems 2.3, 3.4), we determine the first layer $k_1^{\mathrm{ac}}$ 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 ${\mathbb{Z}}_p$-extensions of $k$, distinct from the cyclotomic one. For $p=3$, we characterize a sub-family of fields $k$ (Normal Split cases) for which $k^{\mathrm{ac}}$ 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 $k^{\ast }=\mathbb{Q}(\sqrt{3m})$, 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 $k_1^{\mathrm{ac}}$, and the main invariants attached to the fields $k$, $k^{\ast }$, $k_1^{\mathrm{ac}}$; some relations with Iwasawa's invariants are discussed (Theorem 9.6).

Minor corrections from Referee's remarks . More complete proof of Theorem 8.1, page 33, and new writing of subsection 9.3

References in corpus (1)

• Invariant generalized ideal classes -- structure theorems for p-class groups in p-extensions