CONGRUENT ZETA FUNCTIONS. NO.8

YOSHIFUMI TSUCHIMOTO

Lemma 8.1. Let p be an odd prime. Let ζ be a primitive 8-th root of unity in \overline{𝔽_{p}}. That means, ζ is a root of X^{4} + 1 ∈ 𝔽_{p}[X]. Let us put x = ζ + ζ^{−1}. Then:

(1) x^{2} = 2.

(2) x^{p} − x = 0 if p = ±1 mod 8.

(3) x^{p} + x = 0 if p = ±3 mod 8.

(4) (\frac{2}{p}) = (−1)^{(p^{2}−1)/8}.

projective space and projective varieties.

Definition 8.2. 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 8.3. 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).)

Theorem 8.4. Let p be a prime, q be a power of p. Let f(X_{1}, …, X_{n}) ∈ 𝔽_{q}[X_{1}, …, X_{n}] be a polynomial of degree d. Let us put

N = #{x = (x_{1}, x_{2}, …, x_{n}) ∈ 𝔽_{q}^{n}; f(x) = 0}.

Then we have

n > d ⟹ p|N.

For the proof we use the following lemma

Lemma 8.5. Let k be a positive integer. Then we have

∑_{c∈𝔽_{q}} c^{k} = \begin{cases}−1 & if (q − 1)|k \\ 0 & otherwise. \end{cases}