The Chevalley-Herbrand formula and the real abelian Main Conjecture

TL;DR

This paper explores the connections between the Chevalley-Herbrand formula, capitulation phenomena, and the real abelian Main Conjecture, providing new insights and computational evidence supporting the conjecture's validity in certain cases.

Contribution

It establishes new links between ambiguous class groups, capitulation, and the real Main Conjecture, proving the conjecture trivially when capitulation occurs, supported by computational evidence.

Findings

Capitulation of class groups implies the Main Conjecture holds trivially.

Computational evidence supports the conjecture in various cases.

New theoretical links between class number formulas and capitulation phenomena.

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.