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.
A fundamental pair of periods is a pair of complex numbers
, the two are linearly independent.
The lattice generated by
to make clear that it depends on
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.
A number of properties, listed below, can be seen.
Two pairs of complex numbers
are called equivalent if they generate the same lattice: that is, if
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.
are equivalent if and only if there exists a 2 × 2 matrix
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.
maps the complex plane into 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
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
lies in the fundamental domain.
The fundamental domain is given by the set
which is composed of a set
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: In the closure of the fundamental domain:
