How should multiplicative height on projective space interact with automorphisms?

Question

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.

Answer 1

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.

Answer 2

Daniel Loughran is correct that the naive height may be regarded as merely one among many possible heights, and for naive counting problems, which one you choose will not affect the order of growth, but it will affect your constants (often in interesting ways). However, there is a sense in which the "naive" Weil height on $\mathbb P^1(\mathbb Q)$, or more generally on $\mathbb P^N\bigl(\overline{\mathbb Q}\bigr)$, is special. Namely, for all $d\ge0$, it is canonical with respect to the $d$-power maps $$f_d([x_0,\dots,x_N])=[x_0^d,\dots,x_N^d]$$ in the sense that $$ h\bigl(f_d(P)\bigr) = d\cdot h(P)\quad\text{exactly, with no $O(1)$ term.} $$ More generally, for any morphism $f:\mathbb P^N\to\mathbb P^N$ of degree $d\ge2$, there is a unique canonical height $\hat h_f$ satisfying

1. $\hat h_f = h + O(1)$.
2. $\hat h_f\bigl(f(P)\bigr) = d \cdot\hat h_f(P)$.

Notice that in Property 2, there is no $O(1)$. The canonical height associated to $f$ may be constructed via the formula $$ \hat h_f(P) = \lim_{n\to\infty} \frac{1}{d^n} h\bigl(f^n(P)),$$ where $f^n=f\circ f\circ\cdots\circ f$ is the $n$th iterate of $f$.