Fundamental pair of periods

In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined.



Definition
A 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. If considered as vectors in $$\R^2$$, the two are linearly independent. The lattice generated by $$\omega_1$$ and $$\omega_2$$ is


 * $$\Lambda = \left\{ m\omega_1 + n\omega_2 \mid m,n\in\Z \right\}.$$

This lattice is also sometimes denoted as $$\Lambda(\omega_1, \omega_2)$$ to make clear that it depends on $$\omega_1$$ and $$\omega_2.$$ It is also sometimes denoted by $$\Omega\vphantom{(}$$ or $$\Omega(\omega_1, \omega_2),$$ or simply by $$(\omega_1, \omega_2).$$ The two generators $$\omega_1$$ and $$\omega_2$$ are called the lattice basis. The parallelogram with vertices $$(0, \omega_1, \omega_1+\omega_2, \omega_2)$$ is called the fundamental parallelogram.

While a fundamental pair generates a lattice, a lattice does not have any unique fundamental pair; in fact, an infinite number of fundamental pairs correspond to the same lattice.

Algebraic properties
A number of properties, listed below, can be seen.

Equivalence
Two pairs of complex numbers $$(\omega_1, \omega_2)$$ and $$(\alpha_1, \alpha_2)$$ are called equivalent if they generate the same lattice: that is, if $$\Lambda(\omega_1, \omega_2) = \Lambda(\alpha_1, \alpha_2).$$

No interior points
The fundamental parallelogram contains no further lattice points in its interior or boundary. Conversely, any pair of lattice points with this property constitute a fundamental pair, and furthermore, they generate the same lattice.

Modular symmetry
Two pairs $$(\omega_1,\omega_2)$$ and $$(\alpha_1,\alpha_2)$$ are equivalent if and only if there exists a $ω_{1}$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with integer entries $$a,$$ $$b,$$ $$c,$$ and $$d$$ and determinant $$ad - bc = \pm 1$$ such that


 * $$\begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} =

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix},$$

that is, so that


 * $$\begin{align}

\alpha_1 = a\omega_1+b\omega_2, \\[5mu] \alpha_2 = c\omega_1+d\omega_2. \end{align}$$

This matrix belongs to the modular group $$\mathrm{SL}(2,\Z).$$ This equivalence of lattices can be thought of as underlying many of the properties of elliptic functions (especially the Weierstrass elliptic function) and modular forms.

Topological properties
The abelian group $$\Z^2$$ maps the complex plane into the fundamental parallelogram. That is, every point $$z \in \Complex$$ can be written as $$z = p+m\omega_1+n\omega_2$$ for integers $$m,n$$ with a point $$p$$ in the fundamental parallelogram.

Since this mapping identifies opposite sides of the parallelogram as being the same, the fundamental parallelogram has the topology of a torus. Equivalently, one says that the quotient manifold $$\C/\Lambda$$ is a torus.

Fundamental region
Define $$\tau = \omega_2/\omega_1$$ to be the half-period ratio. Then the lattice basis can always be chosen so that $$\tau$$ lies in a special region, called the fundamental domain. Alternately, there always exists an element of the projective special linear group $$\operatorname{PSL}(2,\Z)$$ that maps a lattice basis to another basis so that $$\tau$$ lies in the fundamental domain.

The fundamental domain is given by the set $$D,$$ which is composed of a set $$U$$ plus a part of the boundary of $U$:


 * $$U = \left\{ z \in H: \left| z \right| > 1, \, \left| \operatorname{Re}(z) \right| < \tfrac{1}{2} \right\}.$$

where $$H$$ is the upper half-plane.

The fundamental domain $$D$$ is then built by adding the boundary on the left plus half the arc on the bottom:


 * $$D = U \cup \left\{ z \in H: \left| z \right| \geq 1,\, \operatorname{Re}(z) = -\tfrac{1}{2} \right\} \cup \left\{ z \in H: \left| z \right| = 1,\, \operatorname{Re}(z) \le 0 \right\}.$$

Three cases pertain:
 * If $$\tau \ne i$$ and $\tau \ne e^{i\pi/3}$, then there are exactly two lattice bases with the same $$\tau$$ in the fundamental region: $$(\omega_1,\omega_2)$$ and $$(-\omega_1,-\omega_2).$$
 * If $$\tau=i$$, then four lattice bases have the same $\tau$: the above two $$(\omega_1,\omega_2)$$, $$(-\omega_1,-\omega_2)$$ and $$(i\omega_1,i\omega_2)$$, $$(-i\omega_1,-i\omega_2).$$
 * If $\tau=e^{i\pi/3}$, then there are six lattice bases with the same $\tau$: $$(\omega_1,\omega_2)$$, $$(\tau \omega_1, \tau \omega_2)$$, $$(\tau^2 \omega_1, \tau^2 \omega_2)$$ and their negatives.

In the closure of the fundamental domain: $$\tau=i$$ and $\tau=e^{i\pi/3}.$