For the circle $(C): x^2 + y^2=1$ over finite field, we can use simple method to count the number of points. The case $p\equiv 1\mod 4$ is not difficult to find, because $-1$ is a square on $F_p$. When $p\equiv 3\mod 4$. Let $i^2=-1$, then $F_p(i)/F_p$ is quadratic extension. And one can realize that $\#(C)=\#\ker(Norm)=\#\{u\in F_q(i)^\times| u^{q+1} = 1\}=(q-1)$, because $F_q(i)^\times$ is a cyclic group of order $q^2-1$.
In the method above, actually, we can see the Frobenius automorphism plays an important role, because it is the generator of the Galois group. Also, the theorem of Hasse-Weil is obtained by representation of Frobenius endomorphism.
However, I don't see the role of Frobenius in the following famous example, which is the number of points on the curve $x^3 + y^3 = 1$ over $F_p$, where $p\equiv 1 \mod 3$. The only solution I know comes from Jacobi's sum in the book of Rosen and Ireland.
And I wonder if there is any other method to obtain the solution of the example above by Frobenius automorphism?
I already found an answer via the comment of mercio here. And I also found similarity of his comment and Theorem 14.16 in the book of D. A. Cox "Primes of Form $x^2 + ny^2$".
The main idea here is the isomorphism that preserve the degree from $\text{End}_{\mathbb{C}}(E)$ to $\text{End}_{\overline{\mathbb{F}_p}}(E)$, where $\text{End}_{\mathbb{C}}(E)$ is an order in an quadratic field $K$. And hence, there exists a prime $\pi\in\text{End}_{\mathbb{C}}(E)$ such that the image of $\pi$ is the $Frob_p$. This shows the degree of $\pi$ is $p$, which is also its norm. That means, $p=\pi\overline{\pi}$. And
$\#E(\mathbb{F}_p)=\#\ker(Frob_p-1)=\#\deg(Frob_p-1)=(1-\pi)(1-\overline{\pi})=(p+1)-(\pi+\overline{\pi})$
Now, the theorem of Gauss follows.