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Definition of congruent Zeta function

DEFINITION 3.1   Let % latex2html id marker 734
$ 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 % latex2html id marker 740
$ \mathbb{F}_q$ . We denote by % latex2html id marker 742
$ V(\mathbb{F}_{q^s})$ the set of solutions of $ V$ in % latex2html id marker 746
$ (\mathbb{F}_{q^s})^n$ . That means,

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$\displaystyle 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

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$\displaystyle 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).



2015-04-23