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

Abstract

Let $k=\mathbb{Q}(\sqrt{-m})$ and $p\ge 3$ 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 $\#({\mathcal{H}}_{K_n}^2/{\mathcal{H}}_{K_n}^1)=\#{\tilde{\mathcal{H}}}_k$ 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 $k^{\mathrm{ac}}$. We compute, Sec. 9, for $p=3$, the first layer $k_1^{\mathrm{ac}}$ of $k^{\mathrm{ac}}$, using the Log${}_p$-function, and show (Thms. 9.2,9.4) that capitulation of suitable ``classes'' is possible in $k^{\mathrm{ac}}$, 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.

Version accepted by the Publ. Math. of Besançon. Final corrections