 Complex uniformization of an elliptic curve

Every complex elliptic curve, as a complex Lie group, is the quotient of $\mathbb{C}$ by a rank-two lattice. The coordinate functions are built from the Weierstrass elliptic function and its derivative. Holomorphic group homomorphisms lift to scalar multiplication on $\mathbb{C}$, and their endomorphism condition is $aL\subseteq L$. This turns complex multiplication into a statement about lattice stabilizers.