Fundamental pair of periods

Summary

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.

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

DefinitionEdit

A fundamental pair of periods is a pair of complex numbers   such that their ratio ω21 is not real. If considered as vectors in  , the two are not collinear. The lattice generated by ω1 and ω2 is

 

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,   and   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 propertiesEdit

A number of properties, listed below, can be seen.

EquivalenceEdit

 
A lattice spanned by periods ω1 and ω2, showing an equivalent pair of periods α1 and α2.

Two pairs of complex numbers (ω1,ω2) and (α12) are called equivalent if they generate the same lattice: that is, if ⟨ω12⟩ = ⟨α12⟩.

No interior pointsEdit

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 symmetryEdit

Two pairs   and   are equivalent if and only if there exists a 2 × 2 matrix   with integer entries abc and d and determinant adbc = ±1 such that

 

that is, so that

 

and

 

This matrix belongs to the matrix group  , which 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 propertiesEdit

The abelian group   maps the complex plane into the fundamental parallelogram. That is, every point   can be written as   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   is a torus.

Fundamental regionEdit

 
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:

 

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:

 

Three cases pertain:

  • If   and  , then there are exactly two lattice bases with the same τ in the fundamental region:   and  
  • If  , then four lattice bases have the same τ: the above two  ,   and  ,  
  • If  , then there are six lattice bases with the same τ:  ,  ,   and their negatives.

In the closure of the fundamental domain:   and  

See alsoEdit

ReferencesEdit

  • 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.)