I was reading about the Hasse-Weil bound for the number of points in on a curve over the finite field $\mathbb{F}_q$.
$$ \big| |C(\mathbb{F}_q)| - (q+1) \big| \leq 2g \sqrt{q} $$
However, this reminded me quite a bit of the Chebyshev inequality.
$$ \mathbb{P}\big[ |X - \mathbb{E}[X]| \geq k \sigma \big] \leq \frac{1}{k^2} $$
Is there a way to prove - or at least interpret - the Hasse-Weil bound as an estimate of the "Expected" number of points on a curve?
Maybe if we do the correspondence $X \mapsto |C(\mathbb{F}_q))| $ , so the random variable is the number of points of the curve.
Then $k \mapsto g$ so the genus can be read as the number of "standard deviations" away from the norm.
We can even justify $\mathbb{E}[X] \mapsto q+1$ since a generic projective curve $f(x,y,z) = 0$ should have one solution for every $x \in \mathbb{F}_qP^1$ which has $q+1$ elements.
Even more far-fetched is to declare the probability measure $\mu \mapsto \chi$ the Euler characteristic of the curve. This is suggested since apparently the Lefschetz trace formula plays a role.
$$ |C(\mathbb{F}_{p^n})| = \big( p^n + 1 \big)- \sum_{i=1}^{2g} a_i^n $$
So the "variants" amounts to controlling the numbers, $|a_i| \leq \sqrt{p}$. At this point the details are beyond me anyway.
