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## Summary

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. Fundamental parallelogram defined by a pair of vectors in the complex plane.

## Definition

A fundamental pair of periods is a pair of complex numbers $\omega _{1},\omega _{2}\in \mathbb {C}$  such that their ratio $\omega _{2}/\omega _{1}$  is not real. If considered as vectors in $\mathbb {R} ^{2}$ , the two are not collinear. The lattice generated by $\omega _{1}$  and $\omega _{2}$  is

$\Lambda =\left\{m\omega _{1}+n\omega _{2}\mid m,n\in \mathbb {Z} \right\}.$

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

A lattice spanned by periods ω1 and ω2, showing an equivalent pair of periods α1 and α2.

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 2 × 2 matrix ${\textstyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$  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{aligned}\alpha _{1}=a\omega _{1}+b\omega _{2},\\[5mu]\alpha _{2}=c\omega _{1}+d\omega _{2}.\end{aligned}}

This matrix belongs to the modular group $\mathrm {SL} (2,\mathbb {Z} ).$  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 $\mathbb {Z} ^{2}$  maps the complex plane into the fundamental parallelogram. That is, every point $z\in \mathbb {C}$  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 $\mathbb {C} /\Lambda$  is a torus.

## Fundamental region

The grey depicts the canonical fundamental domain.

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 $\operatorname {PSL} (2,\mathbb {Z} )$  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:

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

Three cases pertain:

• If $\tau \neq i$  and ${\textstyle \tau \neq e^{i\pi /3}}$ , 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 ${\textstyle \tau =e^{i\pi /3}}$ , 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 ${\textstyle \tau =e^{i\pi /3}.}$