Apparently the Riemann Hypothesis part of the Weil conjectures (which are now theorems) imply, in the case of elliptic curves, that if we have an elliptic curve defined over $\mathbb{F}_p$, and $q = p^n$, then the number of $\mathbb{F}_q$-points of this curve equals
$$ q + 1 - \alpha^n - \beta^n $$
where $\alpha, \beta \in \mathbb{C}$ have $|\alpha| = |\beta| = q^{1/2}$. Is this correct? And if so, can someone show me some simple examples of elliptic curves and primes $p$ for which $\alpha$ and $\beta$ are known?
Yes, for $E$ an elliptic curve defined over the finite field $\Bbb F_p$, $p$ prime, the shape of the number of $\Bbb F_{p^n}$-rational points is $$ |E(\Bbb F_{p^n})| = (p^n+1)- \alpha^n-\beta^n\ , $$ where $\alpha,\beta$ depend (only) on $E$, are complex numbers, are conjugated, and $\alpha\beta=p$.
This answer uses computer aid, here sage, to produce some examples in some small characteristic. The "do it yourself" code was written to be easily changed, and experiment with other similar situations. Since i need some $\pm 2\sqrt p$ marge, i will use a decent prime, $p=7$.
If this is not wanted, please ignore the answer.
Sage comes with a lot of algorithms related to elliptic curves over finite fields. A brute force loop collects the elliptic curves in a dictionary, the keys being their orders. The following orders can be realized:
So as expected, all integers in the range between $(7+1)\pm 2\sqrt 7$, i.e. in the range delimited by $8\pm 5$, can be realized. For each such possible order $N$ we list the equation of one elliptic curve realizing $N$, and compute the corresponding $\alpha$ and $\beta$, and check the formula. To have a first detailed situation, let us consider $N=10$ with more lines of code. We initialize in
Ethe first found elliptic curve with order ten. We ask for the minimal polynomial of the Frobenius morphism. Its roots are the needed values for $\alpha,\beta$. It turns out that the polynomial is $x^2 + 2x + 7$, its roots live in $K=\Bbb Q(\sqrt{-6})$, it is handy to use a notation for $\sqrt{-6}$, we will use $a$ for it. Sage offers than the values $$ \begin{aligned} \alpha &= -1+a=-1+\sqrt{-6}\ ,\\ \beta &= -1-a=-1-\sqrt{-6}\ . \end{aligned} $$We asked explicitly for the numbers $p^n -\alpha^n -\beta^n + 1^n$ for some first small values of $n$, and obtained some natural numbers after computations in the imaginary quadratic field $K$, then we asked for the orders of $E$ after a base change to $\Bbb F_{p^n}$, i.e. for the number of $\Bbb F_{p^n}$-rational points on $E$. Well, the orders came immediately, so no computation of some $282450300$ was performed in the last case. So to have a small check, the ten points over $\Bbb F_p$ were printed.
We can also ask for the action of the Frobenius morphism, defined first on $\Bbb F_{p^n}$, then on (the components of) a point in $E(\Bbb F_{p^n})$ in a particular example. To have a non-trivial action, let us work over the field with $7^4=2401$ elements.
So after the random choice of a point in $E(\Bbb F_{2401})$, and the point was complicated enough, we have checked the relation $f^2(P) + 2f(P) + 7P=0$. Here, $f:E(\Bbb F_{2401})\to E(\Bbb F_{2401})$ is the morphism $(x,y)\to(x^7,y^7)$, and the operation of addition and multiplication in the above relation are the algebraic ones, coming from the group structure of the elliptic curve. It is clear that the trace of the Frobenius morphism on $E$,
determines the orders. The norm is of course
Let us give some messy information for some other elliptic curves realizing the orders in $[3,4,\dots,14,15]$. This time i've got: