Congruent zeta functions. No.8

Yoshifumi Tsuchimoto

DEFINITION 8.1   Let $k$ be a field. A projective transformation of $\P ^n=\P ^n(k)$ is a map

$$f:\P ^n \to \P ^n$$

which is given by a non-degenerate matrix $A \in \operatorname{GL}_{n+1}(k)$ as follows:

$$f([v])= [A.v] \qquad (v\in k^{n+1})$$

where [v] is the class of $v\in k^{n+1}$ in $\P ^n$.

We would like to prove the following proposition.

PROPOSITION 8.2   Let $F=F(X,Y,Z)\in \mathbb{F}_q[X,Y,Z]$ be a homogenious polynomial of degree $2$. We assume $F$ is irreducible over $\overline{\mathbb{F}_q}$. Let us put $C=V_h(F)$. Then:

1. There exists at least one $\mathbb{F}_q$-valued point $P$ in $V_h(F)$.
2. For any line $L$ passing through $P$ defined over $\mathbb{F}_q$, the intersection $L\cap C$ consists of two $\mathbb{F}_q$-valued points $P$ and $Q_L$ except for a case where $L$ contacts $C$.
3. There exists a projective change of coordinate $f:\P ^2 \to \P ^2$ such that $f (V_h(F))=V_h( XY-Z^2)$.
4. The congruent zeta function of $C$ is always equal to the congruent zeta function of $\P ^1$.

LEMMA 8.3   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 $\{Z\neq 0\}=\complement V_h(Z)$ and $V_h(Z)$.

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).

Using the Lemma and the Proposition, we may easily compute the zeta function of a non-degenerate cubic equation

$$a_1 X^2+a_2 XY+a_3 Y^2+b_1 X+b_2 Y+ c$$

in $\mathbb{A}^2$. (See the exercise below.)

EXERCISE 8.1   Let $p$ be a prime. Compute the congruent zeta functions of the following two equations (varieties) over $\mathbb{F}_p$.

1. $V(X^2+Y^2-1)\subset \mathbb{A}^2$.
2. $V(1+Y^2)\subset \mathbb{A}^1$.
3. $V_h(X^2+Y^2-Z^2)\subset \P ^2$.

Is there any relation between them? (Why?)

For the sake of completeness, we should have shown the following lemma.

LEMMA 8.4   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. ${\left(\frac{2}{p}\right)}=(-1)^{(p^2-1)/8}$.

See the book of Serre (cited in No.01) for a proof.