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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 K (often called Main Conjecture in the literature despite proofs in most cases) has a long history which has found its solution by means of ``elementary arithmetic properties'', as detailed in Washington's book from Thaine's method having led to Kolyvagin's Euler systems and many developments. The analytic theory of real abelian fields says (in the semi-simple case [K : Q] not divisible by p) that the order of the p-class group H_K is equal to the index (E_K : F_K) of the subgroup of cyclotomic units in the group of units. Believing in the relevance of mathematical theories, 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 of degree p^N, for L=K(µ_ℓ), ℓ= 1 modulo p^N, and inert in K, (ii) the phenomenon of capitulation of H_K in L, (iii) the Main Conjecture #H_ϕ = (E_ϕ : F_ϕ) which is fulfilled as soon as H_K capitulates in L (Theorem 1.1). Many computations with PARI programs support this new philosophy of the Main Conjecture.
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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-03739451v1⟩
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