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

Let $K\mathrm{/}\mathbb{Q}$ 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 ${\mathcal{H}}_{K,\phi }^{\mathrm{a}\mathrm{l}\mathrm{g}}$, for irreducible $p$-adic characters $\phi$, is inappropriate with respect to analytical formulas, because of capitulation of $p$-classes in the $p$-sub-extension of $K\mathrm{/}\mathbb{Q}$. In the 1970’s we have given an {\it arithmetic} definition, ${\mathcal{H}}_{K,\phi }^{\mathrm{a}\mathrm{r}}$, and formulated the conjecture, still unproven, $\mathrm{\#}{\mathcal{H}}_{K,\phi }^{\mathrm{a}\mathrm{r}}=\mathrm{\#}({\mathcal{E}}_K\mathrm{/}{\mathcal{E}}_K^{\circ }{\mathcal{F}}_{\text{\hspace{0.17em}⁣}K}{)}_{{\phi }_0}$, in terms of units ${\mathcal{E}}_K$ then ${\mathcal{E}}_K^{\circ }$ (generated by units of the strict subfields of $K$) and cyclotomic units ${\mathcal{F}}_K$, where ${\phi }_0$ is the tame part of $\phi$. We prove that the conjecture holds as soon as there exists a prime $\mathrm{\ell }$, totally inert in $K$, such that ${\mathcal{H}}_K$ capitulates in $K({\mu }_{\mathrm{\ell }}^{})$, existence having been checked, in various circumstances, as a promising new tool.