Definition of congruent Zeta function

DEFINITION 4.1   Let $q$ be a power of a prime. Let $\{f_1,f_2,\dots,f_m\}$ be a set of polynomial equations in $n$-variables over $\mathbb{F}_q$. Recall that we have defined in section 2 the affine variety $V=\operatorname{Spec}(\mathbb{F}_q[X_1,\dots,X_n]/(f_1,\dots,f_m)$. We may identify $V(\mathbb{F}_{q^s})$ with the set of solutions of $\{f_1,\dots,f_m\}$ in $(\mathbb{F}_{q^s})^n$. That means,

$$V(\mathbb{F}_{q^s})=\{x\in ( \mathbb{F}_{q^s})^n; f_1(x)=0,f_2(x)=0,\dots, f_m(x)=0\}.$$

Then we define

$$Z(V/\mathbb{F}_q,T)=\exp(\sum_{s=1}^\infty (\frac{1}{s} \char93 V(\mathbb{F}_{q^s}) T^s)).$$

EXERCISE 4.1   Compute congruent zeta function for $V=\operatorname{Spec}(\mathbb{F}_q[X,Y](X Y))$.

EXERCISE 4.2   Compute congruent zeta function for $V=\operatorname{Spec}(\mathbb{F}_q[X,Y]/(X^2+ Y^2-1))$.