I am a student learning Iwasawa theory. I am so sorry if this post is too trivial for this site. I posted it on math.stackexchange yesterday but obtained no responce.


A quite basic object is the Iwasawa algebra. A basic version is as follows:


Let $L/\mathbb{Q}\_p$ be a finite extension (i.e. a local number field) where $p$ is a prime (if necessary, we may assume $p \geq 3$ or $p \geq 5$.) Let $\mathcal{O}\_L$ be its ring of integer. Then we define $\Lambda := \mathcal{O}\_L[[T]]$ as the ring of formal power series of one indeterminate $T$ with $\mathcal{O}\_L$-coefficient, called an **Iwasawa algebra**.


I have see in many papers that people regard $\Lambda$ as a **disk**, or regard the "**rational Iwasawa algebra**" $\Lambda\_{\mathbb{Q}}:= \Lambda \otimes\_{\mathbb{Z}\_p} \mathbb{Q}\_p$ (I am still wondering if this is correct, as a tensor product over $\mathbb{Z}\_p$?) as a disk. Moreover, people often say that some kind of a $p$-adic L-function $\mathcal{L}\_p$ is in the disk, and **"a small neighborhood"** of the $\mathcal{L}\_p$ satisfies certain properties.


**MY QUESTION**: How to make such statements precise? In other words, how can people view the Iwasawa algebra as a disk, talk about its elements as points and the neighboorhood of the points.


Moreover, quite [recently](https://arxiv.org/abs/2304.09806), there is a notion called **"shrinked Iwasawa algebra"** $\Lambda\_{m}:= \mathcal{O}\_L[[p^{-m}T]] \subset \Lambda$. The author stated that this can be regarded as shrinking the disk into smaller radius, just as the name shows.


**My further question**: How do people regard this "shrinked Iwasawa algebra" as a disk of smaller radius?


Put a step further, in more applications, we may consider $\mathbb{Z}\_p$-extensions $\mathcal{K}\_{\infty}/\mathcal{K}$ of Galois group $\mathbb{Z}\_p^d$ for $d > 1$. Then the corresponding Iwasawa algebra should be $\Lambda\_{(d)} := \mathcal{O}\_L[[T\_1, \ldots, T\_d]]$. Then


**Can we also regard $\Lambda\_{(d)}$ as some kind of disk** and further generalize the "shrinked Iwasawa algebra"?


Thank you all for commenting and answering! Any references on explaining these are also welcome!



Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?
The correct viewpoint is not "$\Lambda$ is like a disc", but "$\Lambda$ is like the *functions on* a disc".


To see this, ask yourself: given an element $f \in \mathbb{Z}\_p[[T]]$, what values can we plug in for $T$ such that the series will converge? It's pretty easy to see that if we put $T = x$ for some $x \in \mathbb{Q}\_p$, then the series will converge if $|x| < 1$ but (usually) diverge otherwise.


With a little bit more effort, you can convince yourself that if we put $T = x$ for some $x$ lying in a *finite field extension* of $\mathbb{Q}\_p$, then again we will have convergence for $|x| < 1$ and divergence otherwise.


So we can interpret an element of $\mathbb{Z}\_p[[T]]$ as a function on the set of $x \in \overline{\mathbb{Q}}\_p$ (or its completion $\mathbb{C}\_p$) satisfying $|x| < 1$.


As for "shrinked" (shrunken?) Iwasawa algebras: $\mathbb{Z}\_p[[p^{-m}T]]$ is not a subring of $\mathbb{Z}\_p[[T]]$; it is a *bigger* ring, because it corresponds to functions satisfying a weaker condition -- that they converge on a smaller region. The exercise you should do is to work out for which $x \in \overline{\mathbb{Q}}\_p$ the "set $T = x$" map makes sense on $\mathbb{Z}\_p[[p^{-m}T]]$.


