ZETA FUNCTIONS. NO.3

YOSHIFUMI TSUCHIMOTO

In this lecture we define and observe some properties of conguent zeta functions.

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).

3.2. First properties of congruent Zeta function. Let us first recall an elementary formula

Lemma 3.2.

∑_{k=1}^{∞} \frac{1}{k}T^{k} = − log(1 − T)

Definition 3.3. We denote by 𝔸_{n} the “void set of equation” in n-variables. That means, for any field (or ring) k, we put

𝔸_{n}(k) = {x ∈ k^{n}}.

YOSHIFUMI TSUCHIMOTO

Proposition 3.4.

Z(𝔸_{n}/𝔽_{q}, T) = \frac{1}{1 − q^{n}T}

Proposition 3.5. Let V, W, W_{1}, W_{2} be sets of equations.

(1) If #V (𝔽_{q^{s}}) = #W (𝔽_{q^{s}}) for any s ,then Z(V/𝔽_{q}, T) = Z(W/𝔽_{q}, T).

(2) If #V (𝔽_{q^{s}}) = #W_{1}(𝔽_{q^{s}}) + #W_{2}(𝔽_{q^{s}}) for any s ,then:

Z(V/𝔽_{q}, T) = Z(W_{1}/𝔽_{q}, T)Z(W_{2}/𝔽_{q}, T).

Proposition 3.6. Let f ∈ 𝔽_{q}[X] be an irreducible polynomial in one variable of degree d. Let us consider V = {f}, an equation in one variable. Then:

(1)

V (𝔽_{q^{s}}) = \begin{cases}d & if d|s \\ 0 & otherwise \end{cases}

(2)

Z(V/𝔽_{q}, T) = \frac{1}{1 − T^{d}}

Exercise 3.3. Describe what happens when we omit the assumption of f being irreducible in Proposition 3.6.