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Pré-Publication, Document De Travail Année : 2022

The Chevalley--Herbrand formula and the real abelian Main Conjecture

Résumé

The Main Theorem for abelian fields (often called Main Conjecture in the literature) has found a solution by means of ``elementary arithmetic properties'', as detailed in Washington's book from Thaine's method having led to Kolyvagin's Euler systems. Analytic theory of real abelian fields K says (in the semi-simple case) that the order of the p-class group H_K is equal to the index (E_K : F_K) of cyclotomic units. We have conjectured (1977) the same relations for the isotypic p-adic components: #H_ϕ = (E_ϕ : F_ϕ) for the irreducible p-adic characters ϕ of K in the context of ϕ-objects allowing the non semi-simple case. We develop, in this article, new promising links between: (i) the Chevalley--Herbrand formula giving the order of the group of ``ambiguous classes'' in L/K, for p-extensions L
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Dates et versions

hal-03739451 , version 1 (27-07-2022)
hal-03739451 , version 2 (09-08-2022)
hal-03739451 , version 3 (02-09-2022)

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Georges Gras. The Chevalley--Herbrand formula and the real abelian Main Conjecture: New criterion using capitulation of the class group. 2022. ⟨hal-03739451v2⟩
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