5
$\begingroup$

In Iwasawa theory, even if one is only interested in questions about a number field $K$ (e.g. class groups of $\mathbf{Q}(\mu_p)$, Selmer groups of abelian varieties over $\mathbf{Q}$), to prove deep theorems it seems that you often need to study arithmetic objects over $\mathbf{Z}_p$-extensions of $K$ to get results over $K$.

Are there conceptual explanations why this is the case?

For example, the $p$-adic $L$-functions live in the Iwasawa algebra, which depends on a $\mathbf{Z}_p$-extension, and the Euler systems I know are also defined over extensions, but is there a more high-level and better explanation?

The post of David Loeffler at Applications of Iwasawa Theory mentions http://staff.ustc.edu.cn/~yiouyang/colmez.pdf, but is there a short explanation?

Improve this question
$\endgroup$
8
  • 4
    $\begingroup$ Note that the Colmez notes you link to are distinct from Colmez’s excellent Bourbaki expose that David Loeffler recommended. $\endgroup$
    – Oli Gregory
    Commented Aug 7, 2023 at 11:17
  • 6
    $\begingroup$ One of the big differences is that in general $\mathbb{Z}_p[G]$ has zero-divisors for a finite $G$ and it is impossible to classify modules over it, but the Iwasawa algebra $\Lambda = \mathbb{Z}_p[\![\Gamma]\!]$ is a very nice ring. $\endgroup$
    – Chris Wuthrich
    Commented Aug 7, 2023 at 12:42
  • 2
    $\begingroup$ It is not only in Iwasawa theory that the importance of p-adic extensions is felt. This is the case in many fields including differential geometry. If you are interested in such broader ramifications, I can try to elaborate further. $\endgroup$
    – Mikhail Katz
    Commented Aug 7, 2023 at 12:54
  • 2
    $\begingroup$ @MikhailKatz: I would be interested to hear more about this, in particular in relation to DG. Could you elaborate further on that? $\endgroup$
    – M.G.
    Commented Aug 7, 2023 at 14:20
  • 3
    $\begingroup$ Since Mikhail has posted this reference to his work again, I am posting again the comment that this work does not actually involve $\mathbb{Z}_p$-extensions in the sense of the question (i.e. Galois extensions with Galois group isomorphic to $\mathbb{Z}_p$). $\endgroup$
    – David Loeffler
    Commented Aug 8, 2023 at 12:26

0

Reset to default

You must log in to answer this question.