Fundamental pair of periods

From Wikipedia, the free encyclopedia

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

Contents

[edit] 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 \langle\omega_1,\omega_2 \rangle. The two generators ω1 and ω2 are called the lattice basis.

The parallelogram defined by the vertices 0, ω1 and ω2 is called the fundamental parallelogram.

[edit] Algebraic properties

A number of properties, listed below, should be noted.

[edit] Equivalence

Two pairs of complex numbers (ω1,ω2) and (α1,α2) are called equivalent if they generate the same lattice: that is, if \langle\omega_1,\omega_2\rangle = \langle\alpha_1,\alpha_2\rangle.

[edit] 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.

[edit] Modular symmetry

Two pairs (ω1,ω2) and (α1,α2) 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=\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

α1 = aω1 + bω2

and

α2 = cω1 + dω2.

Note that this matrix belongs to the matrix group S^{*}L(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.

[edit] 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 + mω1 + nω2 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.

[edit] Fundamental region

The grey depicts the canonical fundamental domain.
The grey depicts the canonical fundamental domain.

Define \tau=\frac{\omega_2}{\omega_1} 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| < \frac{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)=-\frac{1}{2} \right\} \cup \left\{ z \in H: \left| z \right| = 1,\, \mbox{Re}(z)<0 \right\}.

If τ is not i and is not \exp\left(\frac{i\pi}{3}\right), then there are exactly two lattice basis with the same τ in the fundamental region: namely, (ω1,ω2) and ( − ω1, − ω2). If τ = i then four lattice basis have the same τ: the above two and (iω1,iω2). If \tau=\exp\left(\frac{i\pi}{3}\right) then there are six lattice basis with the same τ: (ω1,ω2), (τω1,τω2), (τ2ω1,τ2ω2) and their negatives. Note that τ = i and \tau=\exp\left(\frac{i\pi}{3}\right) are exactly the fixed points of PSL(2,Z) in the closure of the fundamental domain.

[edit] See also

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.

[edit] 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.)