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 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. A number of properties, listed below, can be seen. Two pairs of complex numbers and 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. 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. 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. Define to be the half-period ratio.