How can the number of rational points depend on the choice of height function?

Question

Let $V/\mathbb{Q}$ be a subvariety of $\mathbb{P}^n$. There are many plausible choices of height function, some differing only by constant factors: $\max |x_i|$ (for $(x_0,x_1,\dotsc,x_n)$, $\gcd(x_1,\dotsc,x_n)=1$, representing a given point $P$), or also $\sqrt{\sum_i |x_i|^2}$, etc.

Given a height function $H$, we can define $N_H(V;x)$ to be the number of points $P$ in $V(\mathbf{Q})$ with $H(P)\leq x$. For $H_1\ll H_2\ll H_1$, we clearly have $N_{H_1}(V;x)\ll O(1)^n N_{H_2}(V;x)$ and $N_{H_2}(V;x)\ll O(1)^n N_{H_1}(V;x)$. Could the behavior of $N_{H_1}(V;x)$ and $N_{H_2}(V;x)$ nevertheless be qualitatiely different? For example -- could it be that $N_{H_1}(V;x)$ has an asymptotic, whereas $N_{H_2}(V;x)$ does not (i.e. it oscillates)?

Answer 1

Surely this behaviour can never happen, but it will be near impossible to prove this. Conjectures of Manin and others predict that there is an asymptotic formula for these functions in many cases, and the shape of the asymptotic only depends on the choice of embedding and not the choice of norm. (Changing the norm just changes the leading constant in the asymptotic formula).

A nice trick however is that if you can prove an asymptotic for a "dense" set of norms, then you get an asymptotic for all norms via standard approximation arguments. In particular, e.g. it suffices to prove an asymptotic for all norms which are given by infinitely differentiable functions.