**Background on heights**


Consider $P = [a: b] \in \mathbb{P}^1(\mathbb{Q})$, where $a,b$ are coprime integers. We define the naive (multiplicative) height as
$$H(P) = \max \{|a|, |b|\}$$


We can change the coordinates of $\mathbb{P}^1$, which would induce a different height function. For example, consider the automorphism $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ that sends $[x: y] \to [x: x+y]$. Then pullback height $H' = H \circ \varphi$ is then
$$H'(P) = \max\{|a|, |a+b|\}$$


These two heights are related; we can show that
$$\log H'(P) = \log H(P) + O\_{\varphi}(1)$$


**Questions**
In point counting questions, it is pretty reasonable to count points in the set


$$N(H'; B) := \# \{P \in \mathbb{P}^1(\mathbb{Q}): H'(P) \leq B \}$$


for some parameter $B$. The $O(1)$ relation above tells us that $N(H'; B)$ and $N(H, B)$ should be of same order of magnitude as $B \to \infty$, but does not say anything about how the leading coefficient changes. This seems to mean the naive height is not intrinsic in some sense.


* Is there a sense where the naive height $H$ is "canonical"/"intrinsic", where it is invariant under automorphisms of $\mathbb{P}^1$? The main thing I am shooting for here is the preservation of leading constant, not just the order of magnitude.
* If there's no such sense, does it mean we should always work with log height instead of multiplicative height?
	+ As an example, Schanuel's theorem says that as $B \to \infty$,
	$$N(H; B) := \#\{P \in \mathbb{P}^1(\mathbb{Q}): H(P) \leq B\} \sim \frac{2}{\zeta(2)} B^2$$
	In that case, is there any arithmetic interpretation for the leading coefficient $\frac{2}{\zeta(2)}$, or is that the wrong framing and I should just look at $\log N(H; e^B)$ instead?


Thanks.



How should multiplicative height on projective space interact with automorphisms?
The naive height is not at all intrinsic. It is just a convenient choice to work with for notational simplicity. If one is doing things properly one should be counting rational points of bounded height with respect to every choice of adelic metric on the line bundle $\mathcal{O}(1)$. For different choices of adelic metric one obtains a different leading constant in general; this is expected and closely related to Peyre's notation of equidistribution of rational points.


Logarithmic heights are only used on varieties with very sparse sets of rational points, for example elliptic curves or abelian varieties. For Fano varieties the natural choice of height are the usual exponential heights. But again one should really do this with respect to every choice of adelic metric.


I would recommend reading some of the survey papers of Peyre on Manin's conjecture for Fano varieties, they are very informative.


