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

TL;DR

This paper revisits classical conjectures on p-class groups and p-ramified torsion groups of abelian fields, proposing an arithmetic definition aligned with analytical formulas, and discusses the unproven real case of the Main Conjecture.

Contribution

It introduces an arithmetic approach to p-class groups, clarifies the distinction from algebraic definitions, and provides numerical evidence and formulations for the unproven real case.

Findings

Arithmetic definition aligns with analytical formulas

Numerical evidence shows gap between algebraic and arithmetic notions

Formulation of the true Real Main Conjecture involving units and cyclotomic units

Abstract

We revisit, in an elementary way, the classical statement of various ``Main Conjectures'' for $p$-class groups $\mathcal{H}_K$ and $p$-ramified torsion groups $\mathcal{T}_K$ of abelian fields $K$, in the non semi-simple case $p \mid [K : \mathbb{Q}]$. The classical ``algebraic'' definition of the $p$-adic isotopic components, $\mathcal{H}^{\rm alg}{K,\varphi}$, used in the literature, is inappropriate with respect to analytical formulas. For that reason we have introduced, in the 1970's, an ``arithmetic'' definition, $\mathcal{H}^{\rm ar}{K,\varphi}$, 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 $\mathcal{H}_K^-$ and groups $\mathcal{T}_K$ since, in $p$-extensions, transfer maps are injective for these groups but not necessarily for real class groups. Numerical evidence of the gap between the two notions is given (Examples A.2.2, A.2.3) and PARI calculations corroborate that the true Real Main Conjecture for $K$ writes on the form $\# \mathcal{H}^{\rm ar}_{K,\varphi} = \# (\mathcal{E}_K / \hat{\mathcal{E}}_K \, \mathcal{F}_{\!K})_{\varphi}$, in terms of units $\mathcal{E}_K$, $\hat{\mathcal{E}}_K$ (units of the strict subfields) and $\mathcal{F}_K$ (Leopoldt's cyclotomic units). A recent approach, conjecturing the capitulation of $\mathcal{H}_K$ in some auxiliary cyclotomic extensions $K(\mu_\ell^{})$, proves the difficult real case.