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Let $C$ be a smooth projective geometrically connected curve over $\mathbb{Q}$. Assume that $g(C)=3$ and that $C$ is not hyperelliptic. Then the canonical sheaf defines a closed immersion $C\to\mathbb{P}^2_{\mathbb{Q}}$. Tensoring with $\mathbb{R}$ and analytifying we get an isotopy class of a nullhomologous one-dimensional submanifold of $\mathbb{R}P^2$ that is either

  • empty
  • one oval
  • two ovals not encircling each other
  • one oval surrounded by another
  • three ovals not encircling each other
  • four ovals not encircling each other.

By Faltings's theorem $C$ has finitely many rational points. In fact assuming the weak Lang conjecture there is an absolute bound on how many rational points a curve of fixed genus may have. Has the distribution of the rational points with respect to the real locus been studied?

A similar question has been asked for elliptic curves.

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    $\begingroup$ I don't know the answer to this question, but the Bombieri-Pila/Heath-Brown method suggests that low-height points like to repel each other p-adically (including oo-adically) so one might imagine that, statistically speaking, the lowest-height points would be biased towards lying on different components $\endgroup$
    – JSE
    Aug 6, 2020 at 1:52
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    $\begingroup$ Also, taking P_0 to be a basepoint, I believe you can tell which component you're on by looking at the class of P - P_0 in J(Q) / 2J(Q). $\endgroup$
    – JSE
    Aug 6, 2020 at 1:53
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    $\begingroup$ It is likely that the average number of rational points on curves of genus 3 is zero, so it is very hard to talk about a "distribution" in such a setting. On average, each component will have no rational points. $\endgroup$
    – Stanley Yao Xiao
    Aug 6, 2020 at 12:08
  • $\begingroup$ @StanleyYaoXiao yes, you are right. I didn't mean that in a statistical sense necessarily, just that there is finitely many unordered tuples of non-negative integers so one can ask which are realizable. $\endgroup$
    – user158636
    Aug 6, 2020 at 12:18

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