Fundamental pair of periods

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.

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 ${\displaystyle \Lambda (\omega _{1},\omega _{2})=\Lambda (\alpha _{1},\alpha _{2}).}$

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.

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

• A number of alternative notations for the lattice and for the fundamental pair exist, and are often used in its place. See, for example, the articles on the nome, elliptic modulus, quarter period and half-period ratio.
• Elliptic curve
• Modular form
• Eisenstein series

• Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 (See chapters 1 and 2.)
• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See chapter 2.)