Consider an additive subgroup $\Lambda$ of $\mathbb{C}$ so that $\mathbb{C}/\Lambda$ is compact (indeed, a torus). The corresponding Weierstrass $\wp$-function satisfies the ODE $\wp'(z)^2=4\wp(z)^3-g_2z-g_3$, and so if we write $x=\wp(z)$ and $y=\wp'(z)$ then $(x,y)$ lies on the elliptic curve $y^2=4z^3-g_2z-g_3$ (indeed, this parametrizes the entire curve). Addition in $\mathbb{C}/\Lambda$ is well defined, of course, and the addition theorem states that if $z_1+z_2+z_3=0$ then the corresponding $(x_j,y_j)$ satisfy $$\begin{vmatrix}x_1&y_1&1\\\\x_2&y_2&1\\\\x_3&y_1&3\end{vmatrix}=0,$$ i.e., the three points $(x_j,y_j)$ lie on a line. Thus the map $z\mapsto(x,y)$ maps the usual addition on $\mathbb{C}/\Lambda$ to the elliptic curve addition on the curve.

Consider an additive subgroup $\Lambda$ of $\mathbb{C}$ so that $\mathbb{C}/\Lambda$ is compact (indeed, a torus). The corresponding Weierstrass $\wp$-function satisfies the ODE $\wp'(z)^2=4\wp(z)^3-g_2z-g_3$, and so if we write $x=\wp(z)$ and $y=\wp'(z)$ then $(x,y)$ lies on the elliptic curve $y^2=4z^3-g_2z-g_3$ (indeed, this parametrizes the entire curve). Addition in $\mathbb{C}/\Lambda$ is well defined, of course, and the addition theorem states that if $z_1+z_2+z_3=0$ then the corresponding $(x_j,y_j)$ satisfy $$\begin{vmatrix}x_1&y_1&1\\\\x_2&y_2&1\\\\x_3&y_3&1\end{vmatrix}=0,$$ i.e., the three points $(x_j,y_j)$ lie on a line. Thus the map $z\mapsto(x,y)$ maps the usual addition on $\mathbb{C}/\Lambda$ to the elliptic curve addition on the curve.