On the Real Abelian Main Conjecture in the non semi-simple case
Résumé
Let K/Q be a real cyclic extension of degree divisible by p. We analyze the statement of the ``Real Abelian Main Conjecture'', for the p-class group H_K of K, in this non semi-simple case p divides [K:Q]. The classical algebraic definition of the p-adic isotopic components H^alg_{K,ϕ}, for irreducible p-adic characters ϕ, is inappropriate with respect to analytical formulas, because of capitulation of p-classes in the p-sub-extension of K/Q. In the 1970's we have given an arithmetic definition H^ar_{K,ϕ} and formulated the conjecture, still unproven, #H^ar_{K,ϕ} = #(E_K/E^0_K·F_K)_{ϕ_0}, in terms of units E_K then E^0_K (generated by units of the strict subfields of K) and Leopoldt cyclotomic units F_K, where ϕ_0 is the tame part of ϕ. We prove the conjecture under the existence of an extension K(µ_ℓ), ℓ=1 (mod 2 p^N) totally inert in K, in which H_K capitulates, existence having been checked, in various circumstances, as a promising new tool.
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