On the Real Abelian Main Conjecture in the non semi-simple case

Abstract

Let $K/\mathbb{Q}$ be a real cyclic extension of degree divisible by $p$. We analyze the {\it statement} of the "Real Abelian Main Conjecture", for the $p$-class group $\mathcal{H}_K$ of $K$, in this non semi-simple case. The classical {\it algebraic} definition of the $p$-adic isotopic components $\mathcal{H}^{\rm alg}_{K,\varphi}$, for irreducible $p$-adic characters $\varphi$, is inappropriate with respect to analytical formulas, because of capitulation of $p$-classes in the $p$-sub-extension of $K/\mathbb{Q}$. In the 1970's we have given an {\it arithmetic} definition, $\mathcal{H}^{\rm ar}_{K,\varphi}$, and formulated the conjecture, still unproven, $\# \mathcal{H}^{\rm ar}_{K,\varphi} = \# (\mathcal{E}_K / \mathcal{E}^\circ_K \, \mathcal{F}_{\!K})_{\varphi_0}$, in terms of units $\mathcal{E}_K$ then $\mathcal{E}^\circ_K$ (generated by units of the strict subfields of $K$) and cyclotomic units $\mathcal{F}_K$, where $\varphi_0$ is the tame part of $\varphi$. We prove that the conjecture holds as soon as there exists a prime $\ell$, totally inert in $K$, such that $\mathcal{H}_K$ capitulates in $K(\mu_\ell^{})$, existence having been checked, in various circumstances, as a promising new tool.