# Statistics for rational points on curves of genus $g$ over $\mathbb{F}_q$, $g\gg q$

Consider the distribution of the number of $$\mathbb{F}_q$$ points as I range over smooth projective curves of genus $$g$$ (defined over $$\mathbb{F}_q$$). If $$q\gg g,$$ the Hasse-Weil bounds give me a lot of information. The other extreme case, $$g\gg q$$, came up in a conversation recently, and here as far as I can tell etale cohomology tells me nothing useful about the expected distribution of rational points.

Instead, I'm tempted to conjecture that this distribution will be a Poisson distribution with mean $$q+1+\frac{1}{q}+\frac{1}{q^2}\cdots.$$ Here is a heuristic. For a random plane curve of degree $$d$$, each point in $$\mathbb{P}^2_{\mathbb{F}_q}$$ has probability $$\frac{1}{q}$$ of lying on your curve. As $$d$$ goes to infinity, these events (for different points) become independent, and so your distribution will have a generating function of $$(1+\frac{1}{q}x)^{q^2+q+1}$$ - which is a close-to-Poisson distribution of mean $$q+1+\frac{1}{q}.$$

Plane curves are quite special, so lets play the same game for an arbitrary curve. Take the canonical embedding into $$\mathbb{P}^{g-1}$$. If I assume again 1) that each point has a probability of $$\frac{1}{q^{g-2}}$$ of lying on the curve and 2) as for any fixed number $$k$$, $$k$$-tuples of these events become close to independent for large $$g$$, then the same heuristic gives my prediction above.

Of course, this is a pretty fragile thought experiment, and I'm not sure that you should believe my guess. Has this question been considered anywhere in the literature, and if so, has anybody proposed (or even better, proved) an answer?

• This is quite related to another MathOverflow question from some years back: mathoverflow.net/questions/187116/… – Jason Starr Nov 8 '18 at 21:00
• See in this paper that confirm your intuition and more arxiv.org/abs/1410.7373 – Vlad Matei Nov 8 '18 at 23:12
• @Vlad: An interesting paper, but somewhat puzzling. While their Conjecture 1, involving Poisson distribution, is stated for fixed $q$ and $g\to\infty$, their purported evidence for it is based on proved results for $q>g^k, g\to\infty$. But it is well known that the asymptotic behavior of the number of points for genus large compared with $q$ (e.g., the Drinfeld-Vladut bound, exploiting a strong correlation between the Frobenius eigenvalues) drastically differs from the case of small genus (say, below $\sqrt{q}$, where such restrictions are not known to arise). – Victor Protsak Nov 9 '18 at 5:20
• (cont) I guess that the last section, comparing with the matrix models, concedes this point, but lacks lucidity as to wheher, let alone why, these two regimes can exhibit similar statistical properties. – Victor Protsak Nov 9 '18 at 5:27
• @dhy: Where does the heuristic probability $\frac{1}{q^{g-2}}$ for a random point to lie on a canonical curve of genus $g$ over $\Bbb {F}_q$ come from? For a complete intersection of codimension $d$, we can pretend that the values of some $d$ defining polynomials are i.i.d. variables uniformly distributed over $\Bbb {F}_q$. But the general canonical curve is not a complete intersection except for a few small genus values. – Victor Protsak Nov 9 '18 at 6:00