If $X$ is a smooth, geometrically connected, projective curve of genus $g$
over $\mathbb{F}\_q$, then the [zeta function](https://en.wikipedia.org/wiki/Weil_conjectures) of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of degree $2g$. Denote the $P(s)$ of $X$ by $P\_X$.


By the Riemann hypothesis for curves, the zeroes of $P\_X$ all have complex absolute value $1/\sqrt q$.


**Question:** Is it possible to find a family of curves $\{X\_n\}\_{n=1}^{\infty}$ with $g(X\_n) \rightarrow \infty$ ($g(X\_n)$ is the genus of $X\_n$) such that the zeros of $P\_{X\_n}$ tends to being evenly distributed, i.e. if $Y\_n$ denotes the set of phase angle differences (normalized into $[0,2\pi)$) of consecutive zeros of $P\_{X\_n}$, then $\max{Y\_n}-\min{Y\_n} = o(1/g(X\_n))$?



Could the Weil zeroes of curves be evenly distributed?
If $q$ is a prime for which $2$ is a primitive root, then I claim that the Frobenius eigenvalues of the curve $y^2-y = x^q$ over $\mathbb{F}\_2$ have spacing exactly $\tfrac{2 \pi}{q-1}$. This, if [Artin's conjecture](https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots) holds, such examples exist over $\mathbb{F}\_2$.


Let $C$ be the affine curve $y^2-y = x^q$ over $\mathbb{F}\_2$. We note that $C$ is smooth, by taking the partial derivative with respect to $y$; that $C$ has one point at $\infty$ and that $C$ has genus $\tfrac{q-1}{2}$. So the number of points on $C$ over $\mathbb{F}\_{2^k}$ is
$$2^k - \sum\_{i=1}^{q-1} \lambda\_i^k$$
where $\lambda\_i$ are the eignvalues of Frobenius. To show that the $\lambda\_i$ are spaced as claimed, we must show that $\#C(\mathbb{F}\_{2^k}) = 2^k$ for $1 \leq k < q-1$.


The hypothesis that $2$ is a primitive root modulo $q$ means that $x \mapsto x^q$ is bijective on $\mathbb{F}\_{2^k}$ for $k$ not divisible by $q-1$. So, for each $y \in \mathbb{F}\_{2^k}$, there is exactly one solution to $y^2-y = x^q$ in $\mathbb{F}\_{2^k}$. $\square$


The same argument works if $p$ is an odd prime which is a primitive root for infinitely many $q$ and we consider the hyperelliptic curves $y^2 = x^q-1$ over $\mathbb{F}\_p$.


