Congruent zeta functions. No.5

Yoshifumi Tsuchimoto

For any projecvite variety $V$ over a field $\mathbb{F}_q$, we may define its congruent zeta function $Z(V/\mathbb{F}_q,T)$ likewise for the affine varieties.

We quote the famous

CONJECTURE 5.1 (Now a theorem ¹)   Let $X$ be a projective smooth variety of dimension $d$. Then:

1. (Rationality) There exists polynomials $\{P_i\}$ such that
   $$Z(X,T)= \frac{P_1(X,T)P_3(X,T)\dots P_{2 d-1} (X,T)} {P_0(X,T)P_2(X,T) \dots P_{2 d}(X,T)} .$$

2. (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.
3. (Functional Equation)
   $$Z(X,\frac{1}{q^d T})=\pm q^{\frac{d \chi}{2} }T^\chi Z(t)$$
   where $\chi=(\Delta.\Delta)$ is an integer.
4. (Rieman Hypothesis) each $a_{r,i}$ and its conjugates have absolute value $q^{r/2}$.
5. If $X$ is the specialization of a smooth projective variety $X$ over a number field, then the degeee of $P_r(X,T)$ is equal to the $r$-th Betti number of the complex manifold $X(\mathbb{C})$. (When this is the case, the number $\chi$ above is equal to the “Euler characteristic” $\chi=\sum_i (-1)^i b_i$ of $X(\mathbb{C})$.)

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

For a further study we recommend [1, Appendix C],[2], [3].