Let $K\_n = \Bbb Q(\mu\_{p^{n+1}})$ and let $A\_n$ be it's class group. Iwasawa theory tells us a lot about the $p$-part of $A\_n$. For instance, we know quite a lot about how it varies with $n$.


I am interested in the $\ell$-part of $A\_n$ where $\ell \neq p$ is a prime. What is known about these groups? For instance, do we know which primes can occur in the class numbers of these extensions? Do we know the order of growth of the prime powers that do occur?


Since none of the standard sources seem to cover these questions (despite being quite natural), I suspect this is quite a difficult question. What is the main difficulty in extending the methods of classical Iwasawa theory to this case? 


For instance, going through the proof of the p-part of the class number in a $\Bbb Z\_p$ extension (Chapter 13, Sections 1-3 of Washington's Cyclotomic fields), we have a problem while classifying modules over the corresponding Iwasawa algebra $\Bbb Z\_\ell[[\Bbb Z\_p]]$. If $\ell = p$, then this ring would be isomorphic to $\Bbb Z\_p[[t]]$ and one can classify modules over this field (upto finite kernel and cokernel). 


What other problems arise like this?



The $\ell$- part of the class groups of the $p$-cyclotomic fields
As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in *Invent. Math*. He proved that for each *abelian* number field $K/\mathbb{Q}$ and every pair $\ell\neq p$ of odd primes (I am not sure at what happens with the prime $2$), the $\ell$-part of the class group in the cyclotomic $\mathbb{Z}\_p$-extension of $K$ stays bounded. What is not known is if only finitely many $\ell$ can appear. 


Horie has made this explicit in his two papers


* K. Horie, Ideal Class Groups of the Iwasawa-theoretical extensions over the rationals. J. London Math. Soc. **66** 257–275 (2002)
* K. Horie, Triviality in ideal class groups of iwasawa-theoretical number fields. J. Math. Soc. Japan, **57** 827–857 (2005)


The main difficulty, as you say, is that the ring $\mathbb{Z}\_\ell[\![\mathbb{Z}\_p]\!]$ is not so nice. But one can study representations of $\mathbb{Z}\_p$ with values in larger and larger quotients $\mathrm{GL}\_n(\mathbb{Z}/\ell^k)$ for $k\to\infty$ and still get relevant information. 


