Average height of rational points on a curve

Question

I am seeking a formalism to define the average height of the rational points on a curve. This is straightforward if the number of points is finite, but (to me) not straightforward when the rational points are dense along the curve. I will stick to $\mathbb{R}^2$ but all generalizes to $\mathbb{R}^d$.

The height of rational number $x=a/b$ in lowest terms is $h(x)= \max \{ |a|,|b| \}$. A point $p \in \mathbb{R}^2$ is rational if both coordinates are rational, and the height $h(p)$ is the maximum height of its coordinates.

Example. $$ \left( x-\tfrac{1}{2}\right)^2+\left( 2 y-\sqrt{2}+1\right)^2=3 $$ has (I believe) exactly two rational points, $$ (-\tfrac{1}{2},-\tfrac{1}{2}) \;,\; (\tfrac{3}{2},-\tfrac{1}{2}) $$ of heights $2$ and $3$, and so average rational height $2\frac{1}{2}$.

The challenge is to define average height for curves that are dense in rational points, for example, $x^2 + y^2 = 1$. One crude & clumsy attempt follows.

For $p \in \mathbb{R}^2$, define $r(p)$ as $$ r(p)= \begin{cases} 1 &\text{if $p$ is a rational point},\\ 0 &\text{if $p$ is not a rational point}. \end{cases} $$ and define $H(p)$ to be the same as $h(p)$ but extended to all points of $\mathbb{R}^2$: $$ H(p)= \begin{cases} h(p) &\text{if $p$ is a rational point},\\ 0 &\text{if $p$ is not a rational point}. \end{cases} $$

The average height of the rationals on a curve $C$ should be something like $$ \frac { \int_C H(p) \, r(p) \, ds } { \int_C r(p) \, ds } $$ but the denominator is zero. (Credit but no blame to GlenO in an MSE posting.)

Q. Is there a natural and well-defined notion of the average height of an infinite set of rational points on a curve? Or is it impossible to make sense of this concept?

Answer 1

First, may I change your notation a bit? Usually one uses $H(p/q)=\max\{|p|,|q|\}$ for the (multiplicative) height of a rational number, and $h(p/q)=\log H(p/q)$ is the logarithmic height. So I'll use that notation.

One natural way to study the distribution of the infinitely many rational points on a curve is to use a Dirichlet series. So if your curve is $C$ and if we write $C(\mathbb Q)$ for the set of rational points on $C$, then define $$ D(C,s) = \sum_{P\in C(\mathbb Q)} \frac{1}{H(P)^s}. $$ For example, if $C$ is the affine line (what you're writing as $\mathbb R$), then $C(\mathbb Q)=\mathbb Q$ and $$ D(\mathbb Q,s) = \sum_{a/b\in\mathbb Q} \frac{1}{\max\{|a|,|b|\}^s}. $$ This is equal to something like $2\zeta(s-1)+1$ (I may well have made a minor mistake here), but the point is that the residue at $s=1$ gives you information about the distribution of the heights. These height zeta functions play a prominent role in the conjectures of Batyrev and Manin that have attracted much attention over the past couple of decades. (The conjecture as described at https://en.wikipedia.org/wiki/Manin_conjecture is in terms of the height counting function, which is an alternative way to study the distribution of points.)

Or you could look at something like $$ N(C,x) := \sum_{\substack{P\in C(\mathbb Q)\\ H(P)\le x\\}} H(P). $$ Then the goal would be to determine the growth rate as a function of $x$. For $C(\mathbb Q)=\mathbb Q$, this is a nice exercise.

ADDENDUM: On further thought, I guess the "average height of a point on a curve" would be $$ \lim_{x\to\infty} \frac{\displaystyle\sum_{\substack{P\in C(\mathbb Q)\\ H(P)\le x\\}} H(P)}{\displaystyle\sum_{\substack{P\in C(\mathbb Q)\\ H(P)\le x\\}} 1}. $$ But most likely this limit is going to diverge. So more interesting, I think, is to determine the growth rate of either $N(C,x)$, or of the ratio in the limit, as a function of $x$.