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Fundamental pair of periods

## 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 ${\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }$  such that their ratio ${\displaystyle \omega _{2}/\omega _{1}}$  is not real. If considered as vectors in ${\displaystyle \mathbb {R} ^{2}}$ , the two are not collinear. The lattice generated by ${\displaystyle \omega _{1}}$  and ${\displaystyle \omega _{2}}$  is

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

This lattice is also sometimes denoted as ${\displaystyle \Lambda (\omega _{1},\omega _{2})}$  to make clear that it depends on ${\displaystyle \omega _{1}}$  and ${\displaystyle \omega _{2}.}$  It is also sometimes denoted by ${\displaystyle \Omega {\vphantom {(}}}$  or ${\displaystyle \Omega (\omega _{1},\omega _{2}),}$  or simply by ${\displaystyle (\omega _{1},\omega _{2}).}$  The two generators ${\displaystyle \omega _{1}}$  and ${\displaystyle \omega _{2}}$  are called the lattice basis. The parallelogram with vertices ${\displaystyle (0,\omega _{1},\omega _{1}+\omega _{2},\omega _{2})}$  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 ${\displaystyle (\omega _{1},\omega _{2})}$  and ${\displaystyle (\alpha _{1},\alpha _{2})}$  are called equivalent if they generate the same lattice: that is, if ${\displaystyle \Lambda (\omega _{1},\omega _{2})=\Lambda (\alpha _{1},\alpha _{2}).}$

### 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 ${\displaystyle (\omega _{1},\omega _{2})}$  and ${\displaystyle (\alpha _{1},\alpha _{2})}$  are equivalent if and only if there exists a 2 × 2 matrix ${\textstyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$  with integer entries ${\displaystyle a,}$  ${\displaystyle b,}$  ${\displaystyle c,}$  and ${\displaystyle d}$  and determinant ${\displaystyle ad-bc=\pm 1}$  such that

${\displaystyle {\begin{pmatrix}\alpha _{1}\\\alpha _{2}\end{pmatrix}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}\omega _{1}\\\omega _{2}\end{pmatrix}},}$

that is, so that

{\displaystyle {\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 ${\displaystyle \mathrm {SL} (2,\mathbb {Z} ).}$  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 ${\displaystyle \mathbb {Z} ^{2}}$  maps the complex plane into the fundamental parallelogram. That is, every point ${\displaystyle z\in \mathbb {C} }$  can be written as ${\displaystyle z=p+m\omega _{1}+n\omega _{2}}$  for integers ${\displaystyle m,n}$  with a point ${\displaystyle p}$  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 ${\displaystyle \mathbb {C} /\Lambda }$  is a torus.

## Fundamental region

The grey depicts the canonical fundamental domain.

Define ${\displaystyle \tau =\omega _{2}/\omega _{1}}$  to be the half-period ratio. Then the lattice basis can always be chosen so that ${\displaystyle \tau }$  lies in a special region, called the fundamental domain. Alternately, there always exists an element of the projective special linear group ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}$  that maps a lattice basis to another basis so that ${\displaystyle \tau }$  lies in the fundamental domain.

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

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

where ${\displaystyle H}$  is the upper half-plane.

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

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

Three cases pertain:

• If ${\displaystyle \tau \neq i}$  and ${\textstyle \tau \neq e^{i\pi /3}}$ , then there are exactly two lattice bases with the same ${\displaystyle \tau }$  in the fundamental region: ${\displaystyle (\omega _{1},\omega _{2})}$  and ${\displaystyle (-\omega _{1},-\omega _{2}).}$
• If ${\displaystyle \tau =i}$ , then four lattice bases have the same ${\displaystyle \tau }$ : the above two ${\displaystyle (\omega _{1},\omega _{2})}$ , ${\displaystyle (-\omega _{1},-\omega _{2})}$  and ${\displaystyle (i\omega _{1},i\omega _{2})}$ , ${\displaystyle (-i\omega _{1},-i\omega _{2}).}$
• If ${\textstyle \tau =e^{i\pi /3}}$ , then there are six lattice bases with the same ${\displaystyle \tau }$ : ${\displaystyle (\omega _{1},\omega _{2})}$ , ${\displaystyle (\tau \omega _{1},\tau \omega _{2})}$ , ${\displaystyle (\tau ^{2}\omega _{1},\tau ^{2}\omega _{2})}$  and their negatives.

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