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 ?

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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).

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  • $\begingroup$ Your answer is very clear, thank you very much ! $\endgroup$ – Sharpsel Oct 19 '17 at 8:56

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