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 of the same order of magnitude; 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.

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.