Definition 1.5 (Elliptic curve (temporary definition)).

• (i) An elliptive curve $E∕K$ is the projective closure of the plane affine curve

  $$y^2=f(x)$$

  where $f\in K[x]$ is a monic cubic polynomial with distinct roots in $\overline{K}$. We call this equation “a Weierstrass equation”.

• (ii) For $L∕K$ any field extension

  $$E(L)=\{(x,y)\in L^2|y^2=f(x)\}\cup \{0\}.$$

  $0$ is the “point at infinity” that we get because we take the projective closure.