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

Georges Gras (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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 §7.6, 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.

Mots clés: cyclotomic polynomials; p-adic regulators; class field theory; class groups and units; p-adic characters; abelian fields

Domaines: Théorie des nombres [math.NT]

Georges Gras. Application of the notion of $\varphi$-object to the study of $p$-class groups and $p$-ramified torsion groups of abelian extensions. 2022. ⟨hal-03466431v4⟩