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 $f:M\to M$ be a map such that $\char93 {\mathrm{Fix}}(f^n)$ is finite for any $n>0$ . We define the Artin-Mazur zeta function of a dynamical system $(M,f)$ as

$$Z((M,f),T)= \exp(\sum_{j=1}^\infty \frac{\char93 {\mathrm{Fix}}(f^j) T^j}{j})$$

Let $q$ be a power of a prime $p$ . We may consider an automorphism ${\mathrm{Frob}}_q$ of $\bar{\mathbb{F}_{q}}$ over $\mathbb{F}_q$ by

$${\mathrm{Frob}}_q(x)=x^q$$

PROPOSITION 7.2   ${\mathrm{Frob}}_q: \mathbb{F}_{q^r}\to \mathbb{F}_{q^r}$ is an automorphism of order $r$ . It is a generator of the Galois group ${\mathrm{Gal}}(\mathbb{F}_{q^r}/\mathbb{F}_q)$ .

For any projective variety $X$ defined over $\mathbb{F}_q$ , we may define a Frobenius action ${\mathrm{Frob}}_q$ on $X(\bar{\mathbb{F}_q})$ :

$${\mathrm{Frob}}_q([x_0:x_1:\dots x_N]) = ([x_0^q:x_1^q:\dots x_N^q]).$$

For any $\bar{\mathbb{F}_q}$ -valued point $x \in X(\bar{\mathbb{F}_q})$ , We have

$${\mathrm{Frob}}_q^r(x)=x \iff x \in X(\mathbb{F}_{q^r}).$$

PROPOSITION 7.3   The Artin Mazur zeta function of the dynamical system $(X(\bar{\mathbb{F}_q}),{\mathrm{Frob}}_q)$ conincides with the congruent zeta function $Z(X/\mathbb{F}_q,t)$ .