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 \R^2, the two are linearly independent. The lattice generated by \omega_1 and \omega_2 is This lattice is also sometimes denoted as to make clear that it depends on \omega_1 and \omega_2. It is also sometimes denoted by or or simply by The two generators \omega_1 and \omega_2 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

[[Image:A lattice spanned by periods.svg|right|thumb|250px|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 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 2 × 2 matrix with integer entries a, b, c, and d 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 \Z^2 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 \C/\Lambda is a torus.

Fundamental region

Define to be the half-period ratio. Then the lattice basis can always be chosen so that \tau 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 \tau 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: In the closure of the fundamental domain: \tau=i and