CONGRUENT ZETA FUNCTIONS. NO.6

YOSHIFUMI TSUCHIMOTO

6.1. Legendre symbol.

Definition 6.1. Let p be an odd prime. Let a be an integer which is not divisible by p. Then we define the Legendre symbol (\frac{a}{p}) by the following formula.

(\frac{a}{p}) = \begin{cases}1 & if (X^{2} − a) is irreducible over 𝔽_{p} \\ −1 & otherwise \end{cases}

We further define

(\frac{a}{p}) = 0 if a ∈ pℤ.

Lemma 6.2. Let p be an odd prime. Then:

(1) (\frac{a}{p}) = a^{(p−1)/2} mod p

(2) (\frac{ab}{p}) = (\frac{a}{p})(\frac{b}{p})

We note in particular that (\frac{−1}{p}) = (−1)^{(p−1)/2}.

Definition 6.3. Let p, ℓ be distinct odd primes. Let λ be a primitive ℓ-th root of unity in an extension field of 𝔽_{p}. Then for any integer a, we define a Gauss sum τ_{a} as follows.

τ_{a} = ∑_{t=1}^{ℓ−1} (\frac{t}{ℓ})λ^{at}

τ_{1} is simply denoted as τ.

Lemma 6.4. (1) τ_{a} = (\frac{a}{ℓ})τ.

(2) ∑_{a=0}^{l−1} τ_{a}τ_{−a} = ℓ(ℓ − 1).

(3) τ^{2} = (−1)^{(ℓ−1)/2}ℓ ( = ℓ^{∗} (say)).

(4) τ^{p−1} = (ℓ^{∗})^{(p−1)/2}.

(5) τ^{p} = τ_{p}.

Theorem 6.5. (\frac{p}{ℓ}) = (\frac{ℓ^{∗}}{p} ) ( where ℓ^{∗} = (−1)^{(ℓ−1)/2}ℓ ) (\frac{−1}{ℓ}) = (−1)^{(ℓ−1)/2} (\frac{2}{ℓ}) = (−1)^{(ℓ^{2}−1)/8}