Fundamental pair of periods
# 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 $\omega _{1},\omega _{2}\in \mathbb {C}$ 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

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 (abcd){\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

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 $τ≠e^{iπ/3}{\textstyle$ \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 $τ=e^{iπ/3}{\textstyle$ \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 $τ=e^{iπ/3}.{\textstyle$ \tau =e^{i\pi /3}.}

