CONGRUENT ZETA FUNCTIONS. NO.10

YOSHIFUMI TSUCHIMOTO

elliptic curves There is diverse deep theories on elliptic curves. Let k be a field of characteristic p ≠ 0, 2, 3. We consider a curve E

in ℙ(k) of the following type:

y^{2} = x^{3} + ax + b (a, b ∈ k, 4a^{3} + 27b^{2} ≠ 0).

(The equation, of course, is written in terms of inhomogeneous coordinates. In homogeneous coordinates, the equation is rewritten as:

Y^{2} = X^{3} + aXZ^{2} + bZ^{3}.)

Such a curve is called an elliptic curve. It is well known (but we do not prove in this lecture) that

Theorem 10.1. The set E(k) of k-valued points of the elliptic curve E carries a structure of an abelian group. The addition is so defined that

P + Q + R = 0 ⟺ P,Q,R are colinear.

We would like to calculate congruent zeta function of E.

For the moment, we shall be content to prove:

Proposition 10.2. Let p be and odd prime. Let us fix λ ∈ 𝔽_{p} and consider an elliptic curve E : y^{2} = x(x − 1)(x − λ). Then

#E(𝔽_{p}) = the coefficient of x^{\frac{p−1}{2}} in the polynomial expansion of [(x − 1)(x − λ)]^{\frac{p−1}{2}} .