Definition of congruent Zeta function

DEFINITION 3.1   Let $q$ be a power of a prime. Let $V=\{f_1,f_2,\dots,f_m\}$ be a set of polynomial equations in $n$ -variables over $\mathbb{F}_q$ . We denote by $V(\mathbb{F}_{q^s})$ the set of solutions of $V$ 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 3.1   Compute congruent zeta function for $V=\{X Y\}$ (an equation on two variables).

EXERCISE 3.2   Compute congruent zeta function for $V=\{X^2+ Y^2-1\}$ (an equation on two variables).