TITLE: The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements
QUESTION [5 upvotes]: Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$.
We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$.
I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. 
I was wondering if we could use the rank of the group.. Do you have an idea?
EDIT:
There are the following possibilities: 
If the point $(x,y)$ is in $E(\mathbb{F}_p)$, then the point $(x,-y)$ is also in $E(\mathbb{F}_p)$. 
If the point $(-x,y)$ is in $E(\mathbb{F}_p)$, then the point $(-x,-y)$ is also in $E(\mathbb{F}_p)$.  
$y^2=f(x)$
$f(x)=x^3+Ax \Rightarrow f(-x)=-f(x)$
Let $(x,  y)$ be a point in $E(\mathbb{F}_p)$. Then $f(x)$ is a square. If $(-x,  y)$ would also be a point in $E(\mathbb{F}_p)$, then $f(-x)$ would also be a square, and then $-f(x)$ would be a square, so $-1$ would be a square. That cannot be true, since $p \equiv 3 \pmod 4$. Is this correct? 
But I still don't understand how we can count the solutions. Could you explain it to me?

REPLY [2 votes]: Trying to clarify the step that seems to be troubling the OP.
We partition the elements $x\in\Bbb{F}_p$ into three subsets $S_1,S_2$ and $S_3$:

Subset $S_1$ consists of those elements $x\in\Bbb{F}_p$ such that $x^3+Ax=0$. You should notice that if $x\in S_1$, then $y=0$ is the only solution of $y^2=x^3+Ax$. Therefore to each $x\in S_1$ there is a single point $(x,0)$ on the curve.
Subset $S_2$ consists of those elements $x\in\Bbb{F}_p$ such that $x^3+Ax$ is a non-zero square in the field $\Bbb{F}_p$. If $x\in S_2$, and $y^2=x^3+Ax$ then $y\neq0$, and $(-y)^2=x^3+Ax$ is also a solution. And the only other solution. Therefore to each $x\in S_2$
there are two points, $(x,y)$ and $(x,-y)$ on the curve.
Subset $S_3$ consists of those elements $x\in\Bbb{F}_p$ such that $x^3+Ax$ is not a square in the field $\Bbb{F}_p$. So if $x\in S_3$, then there are no solutions $y$ to the equation $y^2=x^3+Ax$.

If we denote by $|S_1|$ (resp. $|S_2|, |S_3|$) the number of elements in each of the three sets, then the number of pairs $(x,y)\in\Bbb{F}_p^2$ such that $y^2=x^3+Ax$ is
$$
N=|S_1|+2\cdot |S_2|+0\cdot |S_3|.
$$
The key observation is that if $x^3+Ax$ is a non-zero square, then $(-x)^3+A(-x)=-(x^3+Ax)$ is a non-square, for otherwise their ratio $=-1$ would also be a square which is not the case. Therefore if $x\in S_2$, then $-x\in S_3$ and vice versa, if $x\in S_3$, then $-x\in S_2$. This means that $|S_2|=|S_3|$. Thus $2\cdot|S_2|=|S_2|+|S_3|$, so
$$
N=|S_1|+|S_2|+|S_3|=|\Bbb{F}_p|=p.
$$
Taking into account the point at infinity we see that there are $p+1$ points altogether on your elliptic curve.