I wonder if there is some way to prove the following result:
given a (possibly big) genus $g$ and a prime number $p$, let $k$ be the field with $p$ elements. Let $S_p$ be the set of all the possibile Frobenius characteristic polynomials of curves $C/k$ of genus $g$. Then the elements of $S_p$, as $p \rightarrow \infty$ become equidistributed in some reasonable space.
I think there might be some result by Katz and Sarnak on this, but, by now, I've just found results on a slightly different problem: if the curve $C/\mathbb{Q}$ is fixed, by Deligne's equidistribution theorem we know that the reduction of $C$ modulo $p$, as $p \rightarrow \infty$ has equidistributed roots of the Frobenius and this would suffice to prove the statement above by symmetric sums means.
Is there a way (maybe using Katz-Sarnak work) to prove this with $g$ fixed and varying the curve for $p \rightarrow \infty$?
Trying to explain better: is there some function $f$ (depending on $\epsilon$) such that if $p>f(g)$ than $S_p$ approximates well (in the sense of $\epsilon$) an equidistribution?
I think the only Deligne's theorem is not enough, since it fixes the curve. I've tried to use Deligne's theorem applied to a family of curves but that doesn't tell me much.
Edited