Zeta functions. No.5

Yoshifumi Tsuchimoto

We quote the famous

CONJECTURE 5.1 (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 5.2 (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].