Geodesic
On formulae for the class number of real Abelian fields
Izvestiya. Mathematics , Tome 60 (1996) no. 4, pp. 695-761.

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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. 
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L. V. Kuz'min. On formulae for the class number of real Abelian fields. Izvestiya. Mathematics , Tome 60 (1996) no. 4, pp. 695-761. http://geodesic.mathdoc.fr/item/IM2_1996_60_4_a1/

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