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Congruent zeta functions. No.9

Yoshifumi Tsuchimoto

\fbox{plane conic}

DEFINITION 9.1   Let $ k$ be a field. A projective transformation of $ \P ^n=\P ^n(k)$ is a map

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

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

% latex2html id marker 631
$\displaystyle 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 9.2   Let % latex2html id marker 642
$ 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 % latex2html id marker 648
$ \overline{\mathbb{F}_q}$ . Let us put $ C=V_h(F)$ . Then:
  1. There exists at least one % latex2html id marker 652
$ \mathbb{F}_q$ -valued point $ P$ in $ V_h(F)$ .
  2. For any line $ L$ passing through $ P$ defined over % latex2html id marker 662
$ \mathbb{F}_q$ , the intersection $ L\cap C$ consists of two % latex2html id marker 666
$ \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 9.3   We have the following picture of $ \P ^2$ .
  1. $\displaystyle \P ^2=\mathbb{A}^2\coprod \P ^1.
$

    That means, $ \P ^2$ is divided into two pieces % latex2html id marker 695
$ \{Z\neq 0\}=\complement V_h(Z)$ and $ V_h(Z)$ .
  2. $\displaystyle \P ^2=\mathbb{A}^2\cup \mathbb{A}^2 \cup \mathbb{A}^2.
$

    That means, $ \P ^2$ is covered by three ``open sets'' % latex2html id marker 703
$ \{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

$\displaystyle 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 9.1   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?)


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2007-06-22