Let $K$ be a local field and $D\_K$ the open unit disk, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D\_K$? Proposition 1.1 of [this article](http://www.math.ethz.ch/~pink/ftp/SigmaBd.pdf) describes functions on the punctured open disk as certain "Laurent series" (possibly with infinitely many terms of negative degree). Can we write functions on $D\_K$ as power series satisfying some convergence condition?


A related question: suppose $K = k((t))$ for some field $k$. If I am not mistaken, $D\_K = \text{Spa } k[[t]] \times \text{Spa } k((t))$ with the product taken in the category of adic spaces over $k$. So it seems to me that $D\_K$ will have a "diagonal" $K$-point. Does this make sense?



The rigid-analytic open disk
About the first part of your question:


What do you mean by "analytic function"?


If I translate "closed unit disc" by "maximal spectrum of the Tate algebra" (but they are not the same), then the open unit disc is the union of the closed discs with radius smaller than one. Let $A\_\varepsilon$ denote the algebra of analytic functions on $B\_\varepsilon (0)$. In our case, $A\_\varepsilon$ equals the rescaled Tate algebra $K \langle \varepsilon ^{-1} T \rangle$. For $\alpha < \beta$ we get an inclusion of function algebras $A\_{\beta} \subset A\_{\alpha}$ and the algebra of analytic functions on $D\_K$ is the (projective) limit of those.


As far as I see, rescaling does not affect whether a point only exists after a field extension. So, depending on what you mean exactly, I guess that the answer is: You can write it as a projective limit of power series algebras with convergence conditions.


