I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I am mostly interested in irreducible projective plane curves $C$ over finite fields $\mathbb{F}_q$. We define the zeta function by
$$Z(C, t) = \exp\left(\sum^{\infty}_{n = 1}{\frac{\# C(\mathbb{F}_{q^n})}{n} t^n}\right)$$
My supervisor has told me some facts about the zeta function for plane curves, and I am seeking a reference for these facts. I would also like to know under what conditions these facts hold true, in particular whether they are still true even when the curve is singular, or if they are still true if the curve is an affine plane curve.
The facts I want information about are:
For a plane curve $C \subseteq \mathbb{A}^2$ given by $f(x,y) = 0$ with $f \in \mathbb{F}_q[x,y]$, the zeta function $Z(C,t)$ is rational. Also, for a plane curve $C \subseteq \mathbb{P}^2$, $Z(C,t)$ is rational.
$Z(C,t) = \frac{f(t)}{(1-t)(1-qt)}$, i.e. the denominator has the form $g(t) = (1-t)(1-qt)$. My understanding is that this the case for irreducible projective plane curves, whether or not the curve is singular or non-singular. Is this also true for affine plane curves?
When the (irreducible projective plane) curve is non-singular, we have $f(t) = 1 + a_1 t + \dots + a_g t^g + q a_{g-1} t^{g+1} + \dots + q^g t^{2g}$. My understanding is that this is not the case when the curve is singular.