# Refinement of (classical) Iwasawa main conjecture

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 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 ?

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).
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).