On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent (2403.16603v5)

Abstract: Let $k = \mathbb{Q}(\sqrt {-m})$ and $p > 2$ split in $k$. We prove new properties of the $\mathbb{Z}p$-extensions $K/k$, distinct from the cyclotomic; 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 suitable Fermat quotient of the fundamental $p$-unit $x$ of $k$, which also yields the order of the logarithmic class group (2.2, App. A), and generalize the Gold-Sands criterion (5.1, 5.3, 5.4, 5.5). These results use the second element $(\mathcal{H}{K_n}/\mathcal{H}_{K_n}^{G_n})^{G_n}$ of the filtration of the $p$-class groups in $K=\cup_n K_n$, without any argument of Iwasawa's theory, and provide new perspectives. We compute, Sec. 7, the first layer $K_1$, using the Log-function, then give a short proof of a result of Kundu-Washington (5.6), and show (7.4, 7.6) that capitulation is possible in $K$, suggesting Conjecture 5.8. Calculations are gathered App. B.