I am self reading theory of elliptic curves over finite fields. The following question has come to my mind.
Let $E$ be Weierstrass form of an elliptic curve over $\mathbb{F}_p$, the point at infinity of $E$ in the projective space of $E$ is $(0,1,0)$. Can we represent this point at infinity of $E$ as a point in $\mathbb{F}_q \times \mathbb{F}_q$?
If anyone can provide any reference regarding this will be of great help.
Any kind of help is appreciated. Thanks in advance.