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In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Mathematics is the study of quantity, structure, space and change. ...
An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
See lattice for other meanings of this term, both within and without mathematics. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ...
Modular form - Wikipedia /**/ @import /skins-1. ...
Although the concept of a two-dimensional lattice is quite simple, there is a considerable amount of specialized notation and language concerning the lattice that occurs in mathematical literature. This article attempts to review this notation, as well as to present some theorems that are specific to the two-dimensional case.
 Fundamental parallelogram defined by a pair of vectors in the complex plane. Image File history File links Fundamental Parallelogram Image shows a pair of vectors in the complex plane, the fundamental pair of periods, defining a lattice. ...
Image File history File links Fundamental Parallelogram Image shows a pair of vectors in the complex plane, the fundamental pair of periods, defining a lattice. ...
Definition
The fundamental pair of periods is a pair of complex numbers such that their ratio ω2 / ω1 is not real. In other words, considered as vectors in , the two are not collinear. The lattice generated by ω1 and ω2 is In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...
This lattice is also sometimes denoted as Λ(ω1,ω2) to make clear that it depends on ω1 and ω2. It is also sometimes denoted by Ω or Ω(ω1,ω2), or simply by . The two generators ω1 and ω2 are called the lattice basis.
The parallelogram defined by the vertices 0, ω1 and ω2 is called the fundamental parallelogram. A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. ...
Algebraic properties A number of properties, listed below, should be noted.
Equivalence Two pairs of complex numbers (ω1,ω2) and (α1,α2) are called equivalent if they generate the same lattice: that is, if . In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
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 (ω1,ω2) and (α1,α2) are equivalent if and only if there exists a 2 × 2 matrix with integer entries a,b,c and d and determinant such that In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
- ,
that is, so that - α1 = aω1 + bω2
and - α2 = cω1 + dω2.
Note that this matrix belongs to the matrix group , which, with slight abuse of terminology, is known as 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. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ...
In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ...
Topological properties The abelian group maps the complex plane into the fundamental parallelogram. That is, every point can be written as z = p + mω1 + nω2 for integers m,n, with a point p in the fundamental parallelogram. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...
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. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...
// Geometry In geometry, a torus (pl. ...
Fundamental region
 The grey depicts the canonical fundamental domain. Define to be the half-period ratio. Then the lattice basis can always be chosen so that τ lies in a special region, called the fundamental domain. Alternately, there always exists an element of PSL(2,Z) that maps a lattice basis to another basis so that τ lies in the fundamental domain. Image File history File links Not the best picture, but its the only one I have right now. ...
Image File history File links Not the best picture, but its the only one I have right now. ...
In mathematics, the half-period ratio τ of an elliptic function j is the ratio of the two half-periods ω1 and ω2 of j, where j is defined in such a way that See also Modular form Categories: Math stubs ...
In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...
The fundamental domain is given by the set D, which is composed of a set U plus a part of the boundary of U:
The fundamental domain D is then built by adding the boundary on the left plus half the arc on the bottom: If τ is not i and is not , then there are exactly two lattice basis with the same τ in the fundamental region: namely, (ω1,ω2) and ( − ω1, − ω2). If τ = i then four lattice basis have the same τ: the above two and (iω1,iω2). If then there are six lattice basis with the same τ: (ω1,ω2), (τω1,τω2), (τ2ω1,τ2ω2) and their negatives. Note that τ = i and are exactly the fixed points of PSL(2,Z) in the closure of the fundamental domain.
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. In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ...
In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ...
In mathematics, the half-period ratio τ of an elliptic function j is the ratio of the two half-periods ω1 and ω2 of j, where j is defined in such a way that See also Modular form Categories: Math stubs ...
References - 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.)
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