On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-Fran\c{c}ois Jaulent

Abstract

Let $k = \mathbb{Q}(\sqrt {-m})$ and $p \geq 3$ split in $k$. We prove new properties of the $\mathbb{Z}_p$-extensions $K/k$, distinct from the cyclotomic one; we do not assume $K/k$ totally ramified, nor the triviality of the $p$-class group of $k$. These properties are governed by the $p$-valuation $\delta_p(k)$ of a Fermat quotient of the fundamental $p$-unit $x$ of $k$, which also yields the order of the logarithmic class group $\# \mathcal{H}_k$ (Thm. 4.2 extended in App.A to the case of imaginary abelian fields of prime-to-$p$ degree), and allows to generalize the Gold-Sands criterion (Sec. 7). These results are related to the first two elements, $\mathcal{H}_{K_n}^1$ and $\mathcal{H}_{K_n}^2$, of the filtrations of the $p$-class groups in $K = \cup_n K_n$, without any argument of Iwasawa's theory, and provide new perspectives since $\# ( \mathcal{H}_{K_n}^2/ \mathcal{H}_{K_n}^1) = \# \widetilde {\mathcal{H}}_k$ for $n$ large enough (Thm 7.1). We give a short proof generalizing a result of Kundu-Washington (Thm. 7.8) on the $p$-class groups in the anti-cyclotomic $\mathbb{Z}_p$-extension $k^{\rm ac}$. We compute, Sec. 9, for $p = 3$, the first layer $k_1^{\rm ac}$ of $k^{\rm ac}$, using the Log$_p$-function, and show (Thms. 9.2,9.4) that capitulation of suitable ``classes'' is possible in $k^{\rm ac}$, suggesting Conjecture 7.10. Finally, we generalize (Thms.10.1,10.7) a result of Ozaki giving large $\lambda$'s invariants. Calculations and programs are gathered App. C.