Application of the notion of $\varphi$-object to the study of $p$-class groups and $p$-ramified torsion groups of abelian extensions

We revisit, in an elementary way, the statement of the "Main Conjecture" for p-class groups and p-ramified torsion groups in abelian fields K in the non semi-simple case p divides [K : Q]; for this, we have used an "arithmetic" definition of p-adic isotopic components, different from the "algebraic" one used in the literature but inappropriate with respect to analytical formulas. The two notions coincide for relative class groups and real torsion groups of p-ramification theory, but not for real class groups. Numerical evidence of the gap between the two notions is given (Examples 3.12, 3.13). It would remain to make use of classical tools for this non semi-simple real context, still unproved as explained Section 1.4. In \S7.6, we shed new light on the classical proof of the Main Conjecture in the real semi-simple case. Two conjectures are proposed. PARI/GP programs of the tables are given.Comment: Addition of an important Section 7.6: "Mysterious link between cyclotomic units and ideal classes", with numerical examples. Some improvements and correction