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.

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


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 propertiesEdit

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


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

Two pairs of complex numbers   and   are called equivalent if they generate the same lattice: that is, if  

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       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 propertiesEdit

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 regionEdit

The grey depicts the canonical fundamental domain.

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 alsoEdit


  • 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.)