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?