Distance functions on elliptic curves over number fields

Question

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"...

Answer 1

• 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.

Answer 2

Some complement to Joe Silverman's answer. Any algebraic variety over $K_v$ (or any topological field) has a canonical topology induced by that of the base field. This topology can be defined by a distance (far from to be unique). Over $\mathbb{P}^n_{K_v}$, a distance can be given (once a system de coordinates is fixed) by $$ d((x_0,\dots, x_n), \ (y_0, \dots, x_n))= \dfrac{\max_{i, j} \lbrace |x_iy_j-x_jy_i|_v \rbrace}{(\max_i\lbrace |x_i|_v\rbrace\max_j\lbrace |y_j|_v \rbrace)} $$ This is a non-archimedean distance. Concretely, one can see that if there exists an index $r$ such that $|x_i|_v\le |x_r|_v$ and $|y_j|_v\le |y_r|_v$ for all $i, j$, then $$d(x,y)=\max_i \lbrace |x_i/x_r - y_i/y_r|_v \rbrace. $$ Otherwise $d(x,y)=1$.

One can describe this distance as following: there is a canonical reduction map $\pi: \mathbb P^n(K_v) \to \mathbb P^n(\mathbb k_v)$ where $k_v$ is the residue field of $K_v$. This map consists, after dividing by a coordinate of maximal absolute value, in reducing the coordinates mod $m_v$. The fiber $\pi^{-1}(P)$ of a rational point $P\in \mathbb P^n(\mathbb k_v)$ is just an open unit polydisk. Now if $\pi(x)=\pi(y)$, then $d(x,y)$ is the usual distance in the unit polydisk (maximum of the $|x_i-y_i|_v$), and $d(x,y)=1$ otherwise.

For any quasi-projective variety $X$ over $K_v$, the embedding in some projective space induces a distance on $X$ with the above distance on projective spaces.

If $X$ has a smooth quasi-projective model $\mathcal X$ such that canonical map $\mathcal X(O_v)\to X(K_v)$ is surjective (hence bijective), one can define a distance using the reduction map $\pi: X(K_v)\simeq \mathcal X(O_v)\to \mathcal X(k_v)$ similarly to the projective space (the fibers of $\pi$ are analytically isomorphic to a polydisk). If we embedd $\mathcal X$ into a projective space $\mathbb P^n_{O_v}$, then this distance is induced by the above distance on $\mathbb P^n$.

This applies to abelian varieties with their Néron models and the distance is canonical (compatible with homomorphisms of abelian varieties). I don't have access to the books of Lang and of Hindry-Silverman at home, I guess the distance described here has something to do with the good distance function that Joe alludes to.