Definition of Lattice (Complex Torus)

Definition 1. (lattice in $\mathbb{C}$) A lattice in $\mathbb{C}$ is a discrete subgroup \(\Lambda = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2\) where $\omega_1,\omega_2 \in \mathbb{C}$ are linearly independent over $\mathbb{R}$.

Definition 2. (Homothety) Two lattices $\Lambda$ and $\Lambda^{\prime}$ in $\mathbb{C}$ are homothetic if there exists $\alpha \in \mathbb{C}^{\times}$ such that \(\Lambda^{\prime} = \alpha \Lambda.\)

Definition 3. (Elliptic curve) An elliptic curve is $\mathbb{C}/\Lambda$ for some lattice $\Lambda \subset \mathbb{C}$.