Fundamental pair of periods

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

Although the concept of a two-dimensional lattice is quite simple, there is a considerable amount of specialized notation and language concerning the lattice that occurs in mathematical literature. This article attempts to review this notation, as well as to present some theorems that are specific to the two-dimensional case.

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

Definition

The fundamental pair of periods is a pair of complex numbers $\omega_1,\omega_2 \in \Complex$ such that their ratio ω₂/ω₁ is not real. In other words, considered as vectors in $\mathbb{R}^2$ , the two are not collinear. The lattice generated by ω₁ and ω₂ is

\Lambda=\{m\omega_1+n\omega_2 \,\,|\,\, m,n\in\mathbb{Z} \}

This lattice is also sometimes denoted as Λ(ω₁, ω₂) to make clear that it depends on ω₁ and ω₂. It is also sometimes denoted by Ω or Ω(ω₁, ω₂), or simply by 〈ω₁, ω₂〉. The two generators ω₁ and ω₂ are called the lattice basis.

The parallelogram defined by the vertices 0, $\omega_1$ and $\omega_2$ is called the fundamental parallelogram.

It is important to note that, while a fundamental pair generates a lattice, a lattice does not have any unique fundamental pair, that is, many (in fact, an infinite number) fundamental pairs correspond to the same lattice.

Algebraic properties

A number of properties, listed below, obtain.

Equivalence

A lattice spanned by periods ω₁ and ω₂, showing equivalent points and edges.

Two pairs of complex numbers (ω₁,ω₂) and (α₁,α₂) are called equivalent if they generate the same lattice: that is, if ⟨ω₁,ω₂⟩ = ⟨α₁,α₂⟩.

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 $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with integer entries a, b, c and d and determinant ad − bc = ±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

\alpha_1 = a\omega_1+b\omega_2\,

and

\alpha_2 = c\omega_1+d\omega_2.\,

Note that this matrix belongs to the matrix group $SL(2,\mathbb{Z})$ , which, with slight abuse of terminology, is known as the modular group. 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 $\Complex/\Lambda$ is a torus.

Fundamental region

The grey depicts the canonical fundamental domain.

Define τ = ω₂/ω₁ to be the half-period ratio. Then the lattice basis can always be chosen so that τ lies in a special region, called the fundamental domain. Alternately, there always exists an element of PSL(2,Z) that maps a lattice basis to another basis so that τ 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| \,\mbox{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,\, \mbox{Re}(z)=-\tfrac{1}{2} \right\} \cup \left\{ z \in H: \left| z \right| = 1,\, \mbox{Re}(z) \le 0 \right\}.

If τ is not i and is not t=exp(1/3*pi*i), then there are exactly two lattice bases with the same τ in the fundamental region: namely, $(\omega_1,\omega_2)$ and $(-\omega_1,-\omega_2)$ . If $\tau=i$ then four lattice bases have the same τ: the above two and $(i\omega_1,i\omega_2)$ . If t=exp(1/3*pi*i) then there are six lattice bases with the same τ: $(\omega_1,\omega_2)$ , $(\tau \omega_1,\tau \omega_2)$ , $(\tau^2 \omega_1, \tau^2 \omega_2)$ and their negatives. Note that $\tau=i$ and t=exp(1/3*pi*i) in the closure of the fundamental domain.

References

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.)