CONGRUENT ZETA FUNCTIONS. NO.4

YOSHIFUMI TSUCHIMOTO

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

Lemma 4.1.

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

Definition 4.2. 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}}.

Proposition 4.3.

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

Proposition 4.4. 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 4.5. 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 4.1. Describe what happens when we omit the assumption of f being irreducible in Proposition 4.5.