Congruent zeta functions. No.8

Yoshifumi Tsuchimoto

LEMMA 8.1   Let $p$ be an odd prime. Let $\zeta$ be a primitive $8$ -th root of unity in $\overline{\mathbb{F}_p}$ . That means, $\zeta$ is a root of $X^4+1\in {\mathbb{F}}_p[X]$ . Let us put $x=\zeta+\zeta^{-1}$ . Then:

1. $x^2=2$ .
2. $x^p-x=0$ if $p=\pm 1 \mod 8$ .
3. $x^p+x=0$ if $p=\pm 3 \mod 8$ .
4. ${\left(\frac{2}{p}\right)}=(-1)^{(p^2-1)/8}$ .

DEFINITION 8.2   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 8.3   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 $[x]\in \P ^n(k)$ .)

THEOREM 8.4   Let $p$ be a prime, $q$ be a power of $p$ . Let $f(X_1,\dots,X_n)\in \mathbb{F}_q[X_1,\dots,X_n]$ be a polynomial of degree $d$ . Let us put

$$N=\char93 \{x=(x_1,x_2,\dots,x_n)\in \mathbb{F}_q^n; f(x)=0 \}.$$

Then we have

$$n>d \implies p\vert N.$$

For the proof we use the following lemma

LEMMA 8.5   Let $k$ be a positive integer. Then we have

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