Notion of abelian arithmetic ϕ-objects for the study of p-class groups and p-ramified torsion groups

Abstract

We revisit, in an elementary way, the classical statement of various ``Main Conjectures'' for \(p\)-class groups \({\mathscr{H}}_{ K}\) and \(p\)-ramified torsion groups \({\mathscr{T}}_{ K}\) of abelian fields \(K\), in the non semi-simple case \(p \mid [K : \mathbb{Q}]\). The classical `algebraic'' definition of the \(p\)-adic isotypic components, \({\mathscr{H}}^{\scriptscriptstyle\text{alg}}_{K,\varphi}\), used in the literature, is inappropriate with respect to analytical formulas. For that reason we have introduced, in the 1970's, an ``arithmetic'' definition,  \({\mathscr{H}}^{\scriptscriptstyle \mathrm{ar}}_{K,\varphi}\) in perfect correspondence with all analytical formulas and giving a natural ``Main Conjecture'', still unproved for real fields in the non semi-simple case. The two notions coincide for relative class groups \({\mathscr{H}}_{ K}^-\) and groups \({\mathscr{T}}_{ K}\) since transfer maps are injective, in \(p\)-extensions for these groups, but not necessarily for real class groups. Numerical evidence of the gap between the two notions is given (Examples Appendix A.2 on p. 175, Appendix A.2 on p. 178) and PARI calculations corroborate that the true Real Abelian Main Conjecture writes \(\# {\mathscr{H}}^{\scriptscriptstyle\mathrm{ar}}_{ K,\varphi} = \# ({\mathscr{E}}_{ K} / \widehat{\mathscr{E}}_{ K} \, {\mathscr{F}}_{ K})^{e_{\varphi_0}}\) (\(\varphi = \varphi_0^{} \varphi_p\), \(\varphi_0^{}\) of prime-to-\(p\) order, \(\varphi_p\) of \(p\)-power order, \(e_{\varphi_0}\) being the corresponding idempotent), in terms of units \({\mathscr{E}}_{ K}\), \(\widehat{\mathscr{E}}_{ K}\) (units of the strict subfields) and \({\mathscr{F}}_{ K}\) (Leopoldt's cyclotomic units). A recent approach, conjecturing the capitulation of \({\mathscr{H}}_{ K}\) in some auxiliary cyclotomic extensions \(K(\mu_\ell^{})\), \(\ell \equiv 1 \pmod {2p^N}\) prime, proves the difficult non semi-simple real case.