Complex lattice ($\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2\subset \mathbb{C}$)

A complex lattice is a discrete rank-two subgroup of the complex plane, generated by two real-linearly independent periods. An oriented basis can be chosen with $\mathrm{Im}({\omega }_1/{\omega }_2)>0$. Scaling by ${\omega }_2^{-1}$ then gives $\mathbb{Z}\tau +\mathbb{Z}$ for a point in the complex upper half-plane.

Two complex lattices are homothetic when multiplication by one nonzero complex number takes one onto the other. Changes between oriented integral bases have determinant one, so normalized period ratios classify these classes by $S{L}_2(\mathbb{Z})\backslash \mathbb{H}$. The lattice Eisenstein sums scale as $G_k(a\Lambda )=a^{-k}{G}_k(\Lambda )$.