ZETA FUNCTIONS. NO.4

YOSHIFUMI TSUCHIMOTO

Proposition 4.1. Let f ∈ 𝔽_{q}[X] be an irreducible polynomial in one variable of degree d. Let us consider V = {f}, an equation in one variable. Then:

(1)

V (𝔽_{q^{s}}) = \begin{cases}d & if d|s \\ 0 & otherwise \end{cases}

(2)

Z(V/𝔽_{q}, T) = \frac{1}{1 − T^{d}}

projective space and projective varieties.

Definition 4.2. Let R be a ring. A polynomial f(X_{0}, X_{1}, …, X_{n}) ∈ R[X_{0}, X_{1}, …, X_{n}] is said to be homogenius of degree d if an equality

f(λX_{0}, λX_{1}, …, λX_{n}) = λ^{d}f(X_{0}, X_{1}, …, X_{n})

holds as a polynomial in n + 2 variables X_{0}, X_{1}, X_{2}, …, X_{n}, λ.

Definition 4.3. Let k be a field.

(1) We put

ℙ^{n}(k) = (k^{n+1} \ {0})/k^{×}

and call it (the set of k-valued points of) the projective space. The class of an element (x_{0}, x_{1}, …, x_{n}) in ℙ^{n}(k) is denoted by [x_{0} : x_{1} : ⋯: x_{n}].

(2) Let f_{1}, f_{2}, …, f_{l} ∈ k[X_{0}, …, X_{n}] be homogenious polynomials. Then we set

V_{h}(f_{1}, …, f_{l}) = {[x_{0} : x_{1} : x_{2} : … x_{n}]; f_{j}(x_{0}, x_{1}, x_{2}, …, x_{n}) = 0 (j = 1, 2, 3, …, l)}.

and call it (the set of k-valued point of) the projective variety defined by {f_{1}, f_{2}, …, f_{l}}.

(Note that the condition f_{j}(x) = 0 does not depend on the choice of the representative x ∈ k^{n+1} of [x] ∈ ℙ^{n}(k).)

Lemma 4.4. We have the following picture of ℙ^{2}.

YOSHIFUMI TSUCHIMOTO

(1)

ℙ^{2} = 𝔸^{2} ∐ ℙ^{1}.

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