Definition of congruent Zeta function

In this section we define the congruent Zeta function $Z(V/\mathbb{F}_q,T)$ . To avoid assuming too much knowledge on algebraic geometry, we only define it for ''affine schemes of finite type'' (although we do not use that terminology) for now. For a considerably good account of the theory of the congruent Zeta functions, see [3]. We also recommend [1] which also has a brief explanation on the topic.

DEFINITION 5.11   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

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