Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field

Abstract

Let $k$ be an imaginary quadratic field and let $p \ne 2$ be a prime number, split in $k$ into ${\mathfrak p}{\overline {\mathfrak p}}$. We assume that the $p$-class group of $k$ is trivial. Let $\delta \geq 0$ be the ${\mathfrak p}$-valuation of the ${\overline {\mathfrak p}}$-Fermat quotient of the fundamental ${\mathfrak p}$-unit $x$ of $k$. Let $K/k$ be any bi-ramified ${\mathbb{Z}}_p$-extension and let $p^e$ be the degree of the inertia field of ${\mathfrak p}$, $\overline {\mathfrak p}$ being totally ramified. We prove that if $e \geq \delta$, then $\lambda(K/k) = 1$, $\mu(K/k) = 0$; if $e<\delta$ a characterization is obtained from Iwasawa invariants of the $S^{\mathfrak p}$-class groups. This approach only uses generalizations of Chevalley-Herbrand formulas and the non-nullity of a $p$-adic regulator ${\mathcal R}^{\mathfrak p}_\delta$ in incomplete $p$-ramification. It provides effective and computable results that complement some aspects of Iwasawa theory. Conjecture states that only the cyclotomic ${\mathbb{Z}}_p$-extension is ''exceptional''; justifications are given. A pari/gp program computes $\delta$ and ${\mathcal R}^{\mathfrak p}_\delta$, for $p=3$, $e=1$.