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

Application de la notion de $\varphi$-objet à l’étude du groupe des classes d’idéaux des extensions abéliennes

Georges Gras (1)

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

We revisit the statements of the ``Main Conjecture = Main Theorem'', for real abelian fields, 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, which is not pertinent in that case. The two notions coincide for relative class groups of imaginary cyclic fields and, of course, in the semi-simple case. Numerical evidence are given (Examples 3.13, 3.14). It would remain to write a proof using the classical tools (as Euler Systems) in this non semi-simple real context, still unproved as explained in §1.3 of the Introduction of the paper.

Mots clés: abelian fields; p-adic characters; class groups and units; cyclotomic polynomials; p-adic regulators; 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-03466431v2⟩