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 [#!Ha!#, Appendix C],[#!Milne!#], [#!milneLEC!#].

DEFINITION 5.2   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 5.3   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 5.4   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).