The Chevalley–Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group)

Abstract

Analytic theory of real abelian fields K says (in the cyclic semi-simple case) that the order of the p-class group HK is equal to the p-index of cyclotomic units (EK:FK). 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(μℓ) for auxiliary prime numbers ℓ≡1(mod2pN) inert in K; (ii) the phenomenon of capitulation of HK in L; (iii) the Main Theorem for the isotypic components using the irreducible p-adic characters φ of K, that we had conjectured (1977). We prove that this real Main Theorem is trivially fulfilled as soon as HK capitulates in L (Theorem 1.2). Computations with PARI programs support this new philosophy of the Main Theorem and the very frequent phenomenon of capitulation suggests Conjecture 1.1.