CONGRUENT ZETA FUNCTIONS. NO.9

YOSHIFUMI TSUCHIMOTO

plane conic

Definition 9.1. Let k be a field. A projective transformation of ℙ^{n} = ℙ^{n}(k) is a map

f : ℙ^{n} → ℙ^{n}

which is given by a non-degenerate matrix A ∈ GL_{n+1}(k) as follows:

f([v]) = [A.v] (v ∈ k^{n+1})

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

We would like to prove the following proposition.

Proposition 9.2. Let F = F (X, Y, Z) ∈ 𝔽_{q}[X, Y, Z] be a homogenious polynomial of degree 2. We assume F is irreducible over \overline{𝔽_{q}}. Let us put C = V_{h}(F). Then:

(1) There exists at least one 𝔽_{q}-valued point P in V_{h}(F).

(2) For any line L passing through P defined over 𝔽_{q}, the intersection L∩C consists of two 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} → 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)

ℙ^{2} = 𝔸^{2} ∐ ℙ^{1}.

That means, ℙ^{2} is divided into two pieces {Z ≠ 0} = ∁V_{h}(Z) and V_{h}(Z).

(2)

ℙ^{2} = 𝔸^{2} ∪ 𝔸^{2} ∪ 𝔸^{2}.

That means, ℙ^{2} is covered by three “open sets” {Z ≠ 0}, {Y ≠ 0}, {X ≠ 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 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 𝔽_{p}.

(1) V (X^{2} + Y^{2} − 1) ⊂ 𝔸^{2}.

(2) V (1 + Y^{2}) ⊂ 𝔸^{1}.

(3) V_{h}(X^{2} + Y^{2} − Z^{2}) ⊂ ℙ^{2}. Is there any relation between them? (Why?)