The Chevalley--Herbrand formula and the real abelian Main Conjecture
Résumé
The Main Theorem for abelian fields (often called Main Conjecture despite proofs in most cases) has a long history which has found a solution by means of ``elementary arithmetic'', 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 p-index of cyclotomic units (E_K : F_K). We have conjectured (1977) the relations #H_ϕ = (E_ϕ : F_ϕ) for the isotypic p-adic components using the irreducible p-adic characters ϕ of K. We develop, in this article, new promising links between: (i) the Chevalley--Herbrand formula giving the number of ``ambiguous classes'' in p-extensions L/K, L ≤ K(µ_ℓ) for the auxiliary prime numbers ℓ=1 mod 2p^N inert in K; (ii) the phenomenon of capitulation of H_K in L; (iii) the real Main Conjecture #H_ϕ = (E_ϕ : F_ϕ) for all ϕ. We prove that the real Main Conjecture is trivially fulfilled as soon as H_K capitulates in L (Theorem 1.1). Computations with PARI programs support this new philosophy of the Main Conjecture. The very frequent phenomenon of capitulation suggests Conjecture 1.2.
Origine | Fichiers produits par l'(les) auteur(s) |
---|