I need a little help understanding Hasse's theorem for elliptic curves over finite fields, as well as the proof of this theorem. (Sorry about my editing)
Hasse’s Theorem: Let $E$ be an elliptic curve defined over $F_q$.
Then |#$E(K)-q-1|≤2\sqrt{q}$
So Hasse's theorem is estimating the number of points of the E over $F_q$. It says that there are approximately q points with an error of no more than $2\sqrt{q}$ right?
I also read somewhere that "Hasse’s theorem on elliptic curves, provides a bound for the number of points on an elliptic curve when it is reduced modulo a prime p. It’s also referred to as the Hasse bound, because as a result the value is bounded both above and below." but I don't completely understand this result.
Proof: Consider the Frobenius endomorphism on $E$ in $F_q$ where $p$
is prime. This is the map $ϕ:(x,y)↦(x^p,y^p)$.
I guess I also am struggling with the concept of the Froebnius endomorphism. i can see that (x,y) gets mapped to a power p of itself $x^p,y^p)$, but I don't really see the important of it, although I see Frobenius endomorphism constantly being mentioned in elliptic curve literature.
From Fermat’s little theorem we have $x^p≡x modp$.
I understand this.
So the map fixes $E$ pointwise, that is, $ϕ(P)=P$.
This just means every x maps to an $x^p$ which is congruent to x (modp) right? so actually every x maps to itself…?
Then $ϕ(P)-P=0$, so $(ϕ-1)(P)=0$.
Just alegbra I think.
$P∈ker(ϕ-1)$. Thus $E$ is isomorphic to the kernel of the map $(ϕ-1)$.
I don't get where this result comes from, the concept of kernel has always confused me.
The isomorphism yields $N_p (E)=ker(ϕ-1)=(ϕ-1)$.
is $N_p$ just a different notation for the # of points?
So $|(ϕ-1)-deg(ϕ)-deg91)|≤√(deg(ϕ) deg(1) )$.
Totally confused.
But $deg(ϕ-1)=N_p
> (E)$,$deg(ϕ)=p$ and $deg1=1$ so $|N_p (E)-p-1|≤2√p$.
Where did these values for the degrees come from??
Any help would be greatly appreciated. Thanks
Here is the proof I used:
Alternatively, here is Silverman's proof, which I understand even less:
Best Answer
The idea is, $E$ has lots of points in $\overline{\mathbb F_p}$, not all of which are in $\mathbb F_p$. However, if I have an $\overline{\mathbb F_p}$-point, then it is actually an $\mathbb F_p$-point iff it is fixed by the $p^{th}$ power map. This follows because the solutions to $x^p=x$ in $\overline{\mathbb{F_p}}$ (of which there are $p$) are canonically identified with $\mathbb F_p$.
So computing the number of points of $E$ mod $p$ is the same as counting solutions to $\phi(P)=P$ which is the same as counting solutions to $(\phi-1)P=0$, where $(\phi-1)P = \phi(P)-P$.
Saying that $(\phi-1)P=0$ is exactly saying that $P$ is in the kernel of $(\phi-1)$.
The value of the degrees of these maps is worked out elsewhere in Silverman. Does this help?