ZETA FUNCTIONS. NO.7

YOSHIFUMI TSUCHIMOTO

Congruent zeta as a zeta of a dynamical system. The definition of Artin Mazur zeta function is valid without assuming

the number of the base space M to be a finite set.

Definition 7.1. Let M be a set. Let f : M → M be a map such that # Fix(f^{n}) is finite for any n > 0. We define the Artin-Mazur zeta function of a dynamical system (M, f) as

Z((M, f), T) = exp(∑_{j=1}^{∞} \frac{# Fix(f^{j})T^{j}}{j})

Let q be a power of a prime p. We may consider an automorphism Frob_{q} of 𝔽̄_{q} over 𝔽_{q} by

Frob_{q}(x) = x^{q}

Proposition 7.2. Frob_{q} : 𝔽_{q^{r}} → 𝔽_{q^{r}} is an automorphism of order r. It is a generator of the Galois group Gal(𝔽_{q^{r}}/𝔽_{q}).

For any projective variety X defined over 𝔽_{q}, we may define a Frobenius action Frob_{q} on X(𝔽̄_{q}):

Frob_{q}([x_{0} : x_{1} : … x_{N}]) = ([x_{0}^{q} : x_{1}^{q} : … x_{N}^{q}]).

For any 𝔽̄_{q}-valued point x ∈ X(𝔽̄_{q}), We have

Frob_{q}^{r}(x) = x ⟺ x ∈ X(𝔽_{q^{r}}).

Proposition 7.3. The Artin Mazur zeta function of the dynamical system (X(𝔽̄_{q}), Frob_{q}) conincides with the congruent zeta function Z(X/𝔽_{q}, t).