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  such that their ratio  is not real. If considered as vectors in , the two are not collinear. The lattice generated by  and  is

This lattice is also sometimes denoted as  to make clear that it depends on  and  It is also sometimes denoted by  or  or simply by  The two generators  and  are called the lattice basis. The parallelogram with vertices  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  and   are called equivalent if they generate the same lattice: that is, if

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  and  are equivalent if and only if there exists a  matrix  with integer entries    and  and determinant  such that

that is, so that

This matrix belongs to 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  maps the complex plane into the fundamental parallelogram. That is, every point  can be written as  for integers  with  a point  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 region

Define  to be the half-period ratio.  Then the lattice basis can always be chosen so that  lies in a special region, called the fundamental domain.  Alternately, there always exists an element of the projective special linear group  that maps a lattice basis to another basis so that  lies in the fundamental domain.

The fundamental domain is given by the set  which is composed of a set  plus a part of the boundary of 

where  is the upper half-plane.

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

In the closure of the fundamental domain:  and

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.
 Elliptic curve
 Modular form
 Eisenstein series

References
 Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York.  (See chapters 1 and 2.)
 Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York.  (See chapter 2.)

Riemann surfaces
Modular forms
Elliptic functions
Lattice points