# The $\ell$- part of the class groups of the $p$-cyclotomic fields

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?

• Washington has a remark about this in his book. I don't have the book with me so I can't point to a definite page, but he proved a theorem about this: for $\ell \not= p$, the multiplicity of $\ell$ in the groups $A_n$ is bounded. See "The non-$p$-part of the class number in a cyclotomic ${\mathbf Z}_p$ -extension," Invent. Math., 49 (1978), 87–97. See this article at gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002094622. – KConrad May 20 '17 at 15:39
• Incidentally, just as a footnote to @Filippo's great answer: there is some very nice recent work of Jack Lamplugh which proves analogous statements to Washington's theorem with the cyclotomic $\mathbb{Z}_p$-extension replaced by $\mathbb{Z}_p^2$ extensions of imaginary quadratic fields. – David Loeffler May 23 '17 at 8:20

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.
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.
• Washington mentions in section 2 of this paper, that the case $p=2$ is solved in [Washington, L.: Class numbers and $\mathbb{Z}_p$-extensions, Math. Ann. 214, 177-193 (1975)] – debanjana Sep 10 '18 at 3:52