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?

EDIT: I have phrased (2) incorrectly. What I want to know is if $Z(C,t)$ can be *written* in the form $\frac{f(t)}{(1-t)(1-qt)}$ where $f(t) \in \mathbb{Z}[t]$, or more precisely, if it is true that $$Z(C,t) = \frac{f(t)}{(1-t)(1-qt)} (1-t)^i$$ for some $i \geq 0$. Is there anywhere I can find a reference for this?

- 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*.