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. $$% latex2html id marker 819V(\mathbb{F}_{q^s})= \begin{cases} d & \text {if } d \vert s \\ 0 & \text {otherwise} \end{cases}$$

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).

We quote the famous

CONJECTURE 4.5 (Now a theorem ¹)   Let $X$ be a projective smooth variety of dimension $d$ . Then:

W1..

(Rationality)

$$Z(X,T)= \frac{P_1(X,T)P_3(X,T)\dots P_{2 d-1} (X,T)} {P_0(X,T)P_2(X,T) \dots P_{2 d}(X,T)}$$

W2..

(Integrality) $P_0(X,T)=1-T$ , $P_{2 d}( X,T)=1-q^d T$ , and for each $r$ , $P_r$ is a polynomial in $\mathbb{Z}[T]$ which is factorized as

$$P_r(X,T)=\prod (1-a_{r,i} T)$$

where $a_{r,i}$ are algebraic integers.

W3..

(Functional Equation)

$$Z(X,1/q^d T)=\pm q^{d \chi/2 }T^\chi Z(t)$$

where $\chi=(\Delta.\Delta)$ is an integer.

W4..

(Rieman Hypothesis) each $a_{r,i}$ and its conjugates have absolute value $q^{r/2}$ .

W5..

If $X$ is the specialization of a smooth projective variety $Y$ over a number field, then the degeee of $P_r(X,T)$ is equal to the $r$ -th Betti number of the complex manifold $Y(\mathbb{C})$ . (When this is the case, the number $\chi$ above is equal to the ``Euler characteristic'' $\chi=\sum_i (-1)^i b_i$ of $Y(\mathbb{C})$ .)

It is a profound theorem, relating rational points $X(\mathbb{F}_q)$ of $X$ over finite fields and topology of $Y(\mathbb{C})$ .

The following proposition (which is a precursor of the above conjecture) is a special case

PROPOSITION 4.6 (Weil)   Let $E$ be an elliptic curve over $\mathbb{F}_q$ . Then we have

$$Z(E/\mathbb{F}_q,T) = \frac{1-a T+ q T^2} {(1-T)(1-q T)}.$$

where $a$ is an integer which satisfies $\vert a\vert\leq 2 \sqrt{q}$ .

Note that for each $E$ we have only one unknown integer $a$ to determine the Zeta function. So it is enough to compute $\char93 E(\mathbb{F}_q)$ . to compute the Zeta function of $E$ . (When $q=p$ then one may use the result in the preceding section.)

For a further study we recommend [1, Appendix C],[2].