Application of the notion of $\varphi$-object to the study of $p$-class groups and $p$-ramified torsion groups of abelian extensions
Résumé
We revisit, in an elementary way, the statement of the ``Main Conjecture for
p-class groups'', in abelian fields K, in the non semi-simple case p divides [K : Q];
for this, we use an ``arithmetic'' definition of the p-adic isotopic components,
different from the ``algebraic'' one used in the literature but not pertinent.
The two notions coincide for relative class groups and 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 some
classical tools (as Kolyvagin Euler systems) for this new non semi-simple
real context, still unproved as explained in Section 1.4 of the Introduction.
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