On formulae for the class number of real Abelian fields

For a given real Abelian field $k$ and a given prime natural number $\ell$ we obtain an index formula for the order of the group $\operatorname{Cl}(k)_{\ell,\varphi}$, where $\operatorname{Cl}(k)_{\ell}$ is the $\ell$-component of the class group of $k$ $\operatorname{Cl}(k)_{\ell,\varphi}$ denotes the $\varphi$-component of $\operatorname{Cl}(k)_\ell$ corresponding to a ${\mathbf Q}_\ell$-irreducible character $\varphi$ of the Galois group $G(k/{\mathbf Q})$ that is trivial on the Sylow $\ell$-subgroup of $G(k/{\mathbf Q})$. This result generalizes a conjecture of Gras. The proofs rely on the “main conjecture” of Iwasawa theory.