Let $p$ be an odd prime, and denote by $Cl\_p(H)$ the $p$-part of the ideal class group of a number field $H$. Let $\Delta:=Gal(\mathbb{Q}(\mu\_p)/\mathbb{Q})$ and $\omega : \Delta \longrightarrow \mathbb{Z}\_p^\times$ be the Teichmuller character. For a $\mathbb{Z}\_p[\Delta]$-module $C$ and an integer $i$, consider $C^{(\omega^i)}$ the $(\omega^i)$-isotypic component of $C$. 


The Iwasawa main conjecture states that for odd $i$, the characteristic ideal of the Iwasawa module $\varprojlim\_n Cl\_p(\mathbb{Q}(\mu\_{p^n})^{(\omega^i)}$ (where transition maps are norm maps) is generated by the Kubota-Leopoldt $p$-adic $L$-function (see [6.1](http://math.ucla.edu/~sharifi/iwasawa.pdf) for more details). It was proven by Mazur and Wiles, who also obtained the following formula : 


$$|\#Cl\_p(\mathbb{Q}(\mu\_p))^{(\omega^i)}|\_p = |B\_{1,\omega^{p-i}}|\_p$$


where $B\_{1,\omega^{p-i}}$ is a generalized Bernouilli number. 


Thus, we know the order of $Cl\_p(\mathbb{Q}(\mu\_p))^{(\omega^i)}$. My question is the following : can we (at least conjecturally) compute the full $\mathbb{Z}\_p$-structure of this finite module in terms of an analytic invariant such as the Kubota-Leopoldt $p$-adic $L$-function ?



Refinement of (classical) Iwasawa main conjecture
There is an answer which follows from Kolyvagin's theory of Euler Systems, and can be found in Theorem 4.4 of


K. Rubin, *Kolyvagin's System of Gauss Sums*, in *Arithmetic Algebraic Geometry*, van der Geer, Oort, Steenbrink (eds), Birkhäuser, Progress in Mathematics **89**, 1991.


The result is (in Rubin's words) "implicitly contained" in Kolyvagin's work and says that for each $i$ there is an isomorphism of $\mathbb{Z}\_p$-modules
$$
\bigg(Cl\bigl(\mathbb{Q}(\mu\_p)\bigr)\otimes\mathbb{Z}\_p\bigg)^{(\omega^i)}\cong \bigoplus\_{n=0}^{+\infty} S\_{n+1}^{(\omega^i)}/S\_n^{(\omega^i)}
$$
where $S\_k^{(\omega^i)}\subseteq \mathbb{Z}/M\mathbb{Z}[\Delta]^{(\omega^i)}$ is the "higher Stickelberger ideal" defined as
$$
S\_k^{(\omega^i)}:=\bigl\{\delta(n)\text{ s. t. }n\text{ is divisible by exactly }k\text{ primes all congruent to }1\!\!\!\mod{M}\bigr\}.
$$
In the above, $M$ is an arbitrary *large* power of $p$ (larger than $p^{\operatorname{ord}\_p(B\_{1,\omega^{-i}})}\cdot\lvert Cl\bigl(\mathbb{Q}(\mu\_p)\bigr)^{(\omega^i)}\rvert\_p$) and $\delta$ is obtained from the theory of Kolyvagin's derivative (see Rubin's paper, or Washington's book for details) applied to the Stickelberger element $\theta(n,\omega^i)$. One can look at the above result as an "analytic" statement by observing that the $p$-adic $L$-function is a limit of Stickelberger elements (see Washington's book, *infra* Proposition 7.9 and Theorem 7.10).


The above result has been generalized by Kurihara to arbitrary abelian CM fields in M. Kurihara, *On the structure of ideal class groups of CM fields*, Documenta Math. Extra Vol. Kato (2003), 539-563 (see the Erratum on page 3 of C. Greither, M. Kurihara, *Stickelberger elements, Fitting ideals of class groups of CM fields, and dualisation* in Math. Z., though).


It should be also observed that Iwasawa himself conjectured that the above eigenspace should be cyclic as a $\mathbb{Z}\_p$-module, so that only one summand should be non-trivial in Rubin's result. This is still unknown, but it follows from Vandiver's conjecture that $p\nmid \lvert Cl(\mathbb{Q}(\mu\_p)^+\rvert$ (for this implication, see Corollary 10.15 in Washington's book).


