Congruent zeta functions. No.10

Yoshifumi Tsuchimoto

There is diverse deep theories on elliptic curves.

Let $k$ be a field of characteristic $p\neq 0,2,3$ . We consider a curve $E$ in $\P (k)$ of the following type:

$$y^2=x^3+a x+b\quad (a,b\in k, 4 a^3+27b^2\neq 0).$$

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

$$Y^2=X^3+a XZ^2+bZ^3.)$$

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

$$P+Q+R=0 \iff$$   $P$,$Q$,$R$ 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 $\lambda \in \mathbb{F}_p$ and consider an elliptic curve $E: y^2=x(x-1)(x-\lambda)$ . Then

$$\char93 E(\mathbb{F}_p) = \text{the coefficient of $x^{\fr... ...polynomial expansion of $[(x-1)(x-\lambda)]^{\frac{p-1}{2}}$.}$$