The Chevalley-Herbrand formula and the real abelian Main Conjecture

Abstract

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 $\mathcal{H}_K$ is equal to the $p$-index of cyclotomic units $(\mathcal{E}_K : \mathcal{F}_K)$. We have conjectured (1977) the relations $\# \mathcal{H}_\varphi = (\mathcal{E}_\varphi : \mathcal{F}_\varphi)$ for the isotypic $p$-adic components using the irreducible $p$-adic characters $\varphi$ 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 $L/K$, $L \subset K(\mu_\ell^{})$ for the auxiliary prime numbers $\ell \equiv 1 \pmod {2p^N}$ inert in $K$; (ii) the phenomenon of capitulation of $\mathcal{H}_K$ in $L$; (iii) the real Main Conjecture $\# \mathcal{H}_\varphi = (\mathcal{E}_\varphi : \mathcal{F}_\varphi)$ for all~$\varphi$. We prove that the real Main Conjecture is trivially fulfilled as soon as $\mathcal{H}_K$ capitulates in $L$ (Theorem \ref{thmppl}). Computations with PARI programs support this new philosophy of the Main Conjecture. The very frequent phenomenon of capitulation suggests Conjecture 1.2.