Let $p$ be a prime number and let $q = p^2$. Let $C$ be a separated scheme of finite type over $\mathbb F\_q$ of dimension $1$.


If we know that for every $\alpha \in \mathbb Z\_{>0}$, "the number of $Spec(\mathbb F\_{q^{\alpha}})$-points on $C$" $= C\_1 q^{\alpha} + C\_2p^{\alpha}$ for some $C\_1, C\_2 \in \mathbb Z$.


Then can we read off some geometric information of $C$ from the above point counting formula (for example, $C\_1$ is related to the number of geometrically irreducible components of $C$)?



If we have a nice formula for number of points on a curve over finite fields, can we get some geometric information of the curve from the formula?
This formula implies that the zeta function of $C$ is given by the formula


$$\zeta\_C( u) = e^{ \sum\_{\alpha=1}^{\infty} | C(\mathbb F\_{q^\alpha})| u^\alpha / \alpha } = e^{ \sum\_{\alpha=1}^{\infty} (C\_1 q^{\alpha} + C\_2 p^\alpha) u^\alpha / \alpha } = \frac{1}{ (1 - q u)^{C\_1} (1 - p u)^{C\_2}} $$


Now, the zeta function of the resolution of singularities of $C$ differs from the zeta function of $C$ itself. However, this difference is only by factors of the form $1/ (1 - u^n)$. For example, if we resolve a point of degree $n$ to obtain two points of degree $n$, we get a factor of $1/(1-u^n)$. Compactifying will similarly only multiply or divide by factors of this type.


Thus the zeroes and poles of the zeta function at points satisfying $|u|<1$ will be unchanged by the resolution process.


We know the zeta function of a smooth projective geometrically irreducible curve of genus $g$ has a pole of order $1$ at $u= q^{-1}$, a pole of order $1$ at $u = 1$, and zeroes of total order $2g$ with absolute value $\sqrt{q}^{-1}$. So taking a union of curves, we see the order of the pole at $q^{-1}$ is the total number of irreducible components, and the total degree of the zeroes of absolute value $\sqrt{q}^{-1}$ is twice the sum of the genuses of the irreducible (geometric) components.


In your case, that implies $C\_1$ is the number of irreducible components and $-C\_2/2$ is the total genus of the components, since we have a zero of order $-C\_2$ at $p^{-1} = \sqrt{q}^{-1}$ and no other zeroes on the circle of radius $\sqrt{q}^{-1}$.


