CONGRUENT ZETA FUNCTIONS. NO.3

YOSHIFUMI TSUCHIMOTO

3.1. Definition of congruent Zeta function.

Definition 3.1. Let q be a power of a prime. Let V = {f_{1}, f_{2}, …, f_{m}} be a set of polynomial equations in n-variables over 𝔽_{q}. We denote by V (𝔽_{q^{s}}) the set of solutions of V in (𝔽_{q^{s}})^{n}. That means,

V (𝔽_{q^{s}}) = {x ∈ (𝔽_{q^{s}})^{n}; f_{1}(x) = 0, f_{2}(x) = 0, …, f_{m}(x) = 0}.

Then we define

Z(V/𝔽_{q}, T) = exp(∑_{s=1}^{∞}(\frac{1}{s}#V (𝔽_{q^{s}})T^{s})).

Exercise 3.1. Compute congruent zeta function for V = {XY } (an equation on two variables).

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