On the Real Abelian Main Conjecture in the non semi-simple case

Abstract

Let \(K/\mathbb{Q}\) be a real cyclic extension of degree divisible by \(p\). We analyze the {\it statement} of the "Real Abelian Main Conjecture", for the \(p\)-class group \(\mathcal{H}_K\) of \(K\), in this non semi-simple case. The classical {\it algebraic} definition of the \(p\)-adic isotopic components \(\mathcal{H}^{\rm alg}_{K,\varphi}\), for irreducible \(p\)-adic characters \(\varphi\), is inappropriate with respect to analytical formulas, because of capitulation of \(p\)-classes in the \(p\)-sub-extension of \(K/\mathbb{Q}\). In the 1970's we have given an {\it arithmetic} definition, \(\mathcal{H}^{\rm ar}_{K,\varphi}\), and formulated the conjecture, still unproven, \(\# \mathcal{H}^{\rm ar}_{K,\varphi} = \# (\mathcal{E}_K / \mathcal{E}^\circ_K \, \mathcal{F}_{\!K})_{\varphi_0}\), in terms of units \(\mathcal{E}_K\) then \(\mathcal{E}^\circ_K\) (generated by units of the strict subfields of \(K\)) and cyclotomic units \(\mathcal{F}_K\), where \(\varphi_0\) is the tame part of \(\varphi\). We prove that the conjecture holds as soon as there exists a prime \(\ell\), totally inert in \(K\), such that \(\mathcal{H}_K\) capitulates in \(K(\mu_\ell^{})\), existence having been checked, in various circumstances, as a promising new tool.