Zeta functions. No.4

Yoshifumi Tsuchimoto

PROPOSITION 4.1   Let $f\in \mathbb{F}_q[X]$ be an irreducible polynomial in one variable of degree $d$ . Let us consider $V=\{f\}$ , an equation in one variable. Then:

1. $$% latex2html id marker 635V(\mathbb{F}_{q^s})= \begin{cases} d & \text {if } d \vert s \\ 0 & \text {otherwise} \end{cases}$$

2. $$Z(V/\mathbb{F}_q,T) = \frac{1}{1-T^d}$$

DEFINITION 4.2   Let $R$ be a ring. A polynomial $f(X_0,X_1,\dots,X_n)\in R[X_0,X_1,\dots, X_n]$ is said to be homogenius of degree $d$ if an equality

$$f(\lambda X_0,\lambda X_1,\dots, \lambda X_n) = \lambda^d f(X_0,X_1,\dots,X_n)$$

holds as a polynomial in $n+2$ variables $X_0,X_1,X_2,\dots, X_n, \lambda$ .

DEFINITION 4.3   Let $k$ be a field.

1. We put
   $$\P ^n(k)=(k^{n+1}\setminus \{0\}) /k^\times$$
   and call it (the set of $k$ -valued points of) the projective space. The class of an element $(x_0,x_1,\dots,x_n)$ in $\P ^n(k)$ is denoted by $[x_0:x_1:\dots:x_n]$ .
2. Let $f_1,f_2,\dots, f_l \in k[X_0,\dots, X_n]$ be homogenious polynomials. Then we set
   $$V_h(f_1,\dots,f_l)= \{ [x_0:x_1:x_2:\dots x_n] ; f_j (x_0,x_1,x_2,\dots,x_n)=0 \qquad(j=1,2,3,\dots,l) \}.$$
   and call it (the set of $k$ -valued point of) the projective variety defined by $\{f_1,f_2,\dots,f_l\}$ .

(Note that the condition $f_j(x)=0$ does not depend on the choice of the representative $x\in k^{n+1}$ of $[x]\in \P ^n(k)$ .)

LEMMA 4.4   We have the following picture of $\P ^2$ .

1. $$\P ^2=\mathbb{A}^2\coprod \P ^1.$$
   That means, $\P ^2$ is divided into two pieces $\{Z\neq 0\}=\complement V_h(Z)$ a nd $V_h(Z)$ .

2. $$\P ^2=\mathbb{A}^2\cup \mathbb{A}^2 \cup \mathbb{A}^2.$$
   That means, $\P ^2$ is covered by three ``open sets'' $\{Z\neq 0\}, \{Y\neq 0\}, \{X \neq 0\}$ . Each of them is isomorphic to the plane (that is, the affine space of dimension 2).