Period Lattice

Although the concept of a two-dimensional lattice is quite simple, there is a considerable amount of specialized notation and language concerning the lattice that occurs in mathematical literature. This article attempts to review this notation, as well as to present some theorems that are specific to the two-dimensional case. DEFINITION The fundamental pair of periods is a pair of complex numbers $\omega_1,\omega_2 \in \Complex$ such that their ratio $\omega_2/\omega_1$ is not real. In other words, considered as vectors in $\mathbb{R}^2$ , the two are not Collinear . The lattice generated by $\omega_1$ and $\omega_2$ is
	:<math>U	\left\{ z \in H: \left z ight > 1,\, \left \,\mbox{Re}(z) \, ight < rac{1}{2} ight\}</math>
	:<math>D	U\cup\left\{ z \in H: \left z ight \geq 1,\, \mbox{Re}(z)=-rac{1}{2} ight\} \cup \left\{ z \in H: \left z ight = 1,\, \mbox{Re}(z)<0 ight\}</math>

	CATEGORIES ABOUT FUNDAMENTAL PAIR OF PERIODS

	riemann surfaces

	modular forms

	elliptic functions

	lattice points

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