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

$$P+Q+R=0 \iff$$   the points $$P,\ Q,\ R$$ 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

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

See [1] for more detail and a further story.

The following proposition is a special case of the Weil conjecture. (It is actually a precursor of the conjecture)

PROPOSITION 10.3 (Weil)   Let $E$ be an elliptic curve over $\mathbb{F}_q$. Then we have

$$Z(E/\mathbb{F}_q,T) = \frac{1-d_E T+ q T^2} {(1-T)(1-q T)}.$$

where $d_E$ is an integer which satisfies $\vert d_E\vert\leq 2 \sqrt{q}$.

Note that for each $E$ we have only one unknown integer $d_E$ to determine the Zeta function. So it is enough to compute $\char93 E(\mathbb{F}_q)$ to compute the Zeta function of $E$. (When $q=p$ then one may use Proposition 10.2 to do that.)

$$\char93 E(\mathbb{F}_q)=1+q-d_E.$$

EXERCISE 10.1   compute the congruent zeta function $Z(E,T)$ for an elliptic curve $E: y^2=x(x-1)(x+1)$.