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 $\delta \ge 0$ 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 $e\ge \delta$, 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 ${\mathcal{R}}_{\delta }^{\mathfrak{p}}$ 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 ${\mathcal{R}}_{\delta }^{\mathfrak{p}}$, for $p=3$, $e=1$.

Various minor corrections and rewriting of certain passages; addition of an appendix on the study of the p-class group of the compositum of the Zp-extensions of k. Addition of several references