My question is about an exercise from "arithmetic of elliptic curves":
Let $$ V : X^{2} + Y^{2} = pZ^{2}$$ be a projective vareity in $\mathbb{P}^2$ and $p$ be a prime number.
prove that $V$ is isomorphic to $\mathbb{P}^{1}$ iff $ p \equiv 1$(mod $4$).
and for $p \equiv 3$(mod $4$) no two of such vareities are isomorphic.
I was trying to solve this exercise but I have stuck in defining the morphism. And as it is the case that $p \equiv 1$(mod $4$) I think we should use an integer solution of $a^{2} + b^{2} =p$. but most of my efforts didn't turn out to be a morphism. So if anyone could help with this, it would be great.