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:


1. 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.
2. 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 $P~\xrightarrow{v}~ Q$?
3. 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"...



Distance functions on elliptic curves over number fields
* You can choose $t\_Q$ to be defined over $K\_v$, since the divisor $n(Q\_v)$ is defined over $K\_v$, and for large enough $n$ there will be a global section. Note that Riemann-Roch works over non-algebraically closed fields this way. Or you can choose a basis defined over some finite Galois extension of $K\_v$, and then taking appropriate linear combinations of the Galois conjugates, get a function defined over $K\_v$. See Proposition II.5.8 in [S].
* The function defined in the text is only a reasonable "distance function" in the sense that it measures the distance from $P$ to the fixed point $Q$. For the purposes of this proof, that's fine. If you want to define the $v$-adic topology, you need to be a little more careful. Locally around $Q$ you could use $$d\_v(P\_1,P\_2)=min(|t\_Q(P\_1)-t\_Q(P\_2)|^{1/e},1)$$, but that still only works in a neighborhood of $Q$, i.e., in a set $$\{P : d\_v(P,Q)<\epsilon\}$$ for a sufficiently small $\epsilon$. Using local height functions, more precisely the local height relative to the diagonal in $E(K\_v)\times E(K\_v)$, one gets a "good" distance function that is defined everywhere. See for example Lang's *Fundamentals of Diophantine Geometry* or the book *Diophantine Geometry* that Hindry and I wrote for the general construction of local height functions.


