My question originates from the book of Silverman "The Aritmetic of Elliptic Curves", 2nd edition (call it [S]). On p. 273 of [S] the author is considering an elliptic curve $E/K$ defined over a number field $K$ and he introduces the notion of a $v$-adic distance from $P$ to $Q$. This is done as follows:

Firstly, let's fix an absolute value (archimedean or not) $v$ of $K$ and a point $Q\in E(K_v)$ (here $K_v$ is the completion of $K$ at $v$). Next let's pick a function $t_Q \in K_v(E)$ defined over $K_v$ which vanishes at $Q$ to the order $e$ but has no other zeroes. Now the $v$-adic distance from $P \in E(K_v)$ to $Q$ is defined to be $d_v(P, Q) := \min (|t_Q(P)|_v^{1/e}, 1)$. We will say that $P$ goes to $Q$, written $P~\xrightarrow{v}~ Q$, if $d_v(P, Q) \rightarrow 0$. Later in the text (among other places in the proof of IX.2.2) the author considers a function $\phi\in K_v(E)$ which is regular at $Q$ and claims that this means that $|\phi(P)|_v$ is bounded away from $0$ and $\infty$ if $P~\xrightarrow{v}~ Q$.

I have a couple of questions about this:

- How does one choose a $t_Q$ that works? In the footnote in [S] it is demonstrated how one could use Riemann-Roch to pick a $t_Q$ that has a zero only at $Q$. It seems to me however that such a procedure will not make sure that $t_Q$ is defined over $K_v$ since $K_v$ is not algebraically closed.
- For $\phi$ as above which does not vanish nor has a pole at $Q$, how does one see that $|\phi(P)|_v$ is bounded away from $0$ and $\infty$ as $P~\xrightarrow{v}~ Q$?
- Do these $d_v$ have anything to do with defining a topology on $E(K_v)$? I assume not, since I don't see how to make sense of it; but then on the other hand they are called "distance functions"...