In Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that...
mathematics, a **fundamental pair of periods** is an 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. An ordered pair with first element a and second element b is usually written as (a, b). The notation (a, b) is also used...
ordered pair of The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit, with , i.e., is a square root of . Every complex number can be represented in the form , where and are real numbers called...
complex numbers that define a See lattice for other meanings of this term, both within and without mathematics. In mathematics, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by considering all linear...
lattice in the The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit, with , i.e., is a square root of . Every complex number can be represented in the form , where and are real numbers called...
complex plane. This type of lattice is the underlying object with which In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only). Formally, an elliptic function is a meromorphic function f defined...
elliptic functions and A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory lies in its connections with number theory. As a function on...
modular forms are defined. 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. ## Definition
The **fundamental pair of periods** is a pair of complex numbers such that their ratio ω_{2} / ω_{1} is not real. In other words, considered as vectors in , the two are not 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. For instance, in three-dimensional Euclidean space R3, the three vectors (1, 0...
colinear. The lattice generated by ω_{1} and ω_{2} is This lattice is also sometimes denoted as Λ(ω_{1},ω_{2}) to make clear that it depends on ω_{1} and ω_{2}. It is also sometimes denoted by Ω or Ω(ω_{1},ω_{2}), or simply by < ω_{1},ω_{2} > . The two generators ω_{1} and ω_{2} are called the **lattice basis**. The A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. Every parallelogram is a polygon, and more specifically a quadrilateral. Special cases of a parallelogram are the rhombus, in which all four sides are of equal length, the rectangle, in which the two sets...
parallelogram defined by the vertices 0, ω_{1} and ω_{2} is called the **fundamental parallelogram**.
## Algebraic properties A number of properties, listed below, should be noted.
### Equivalence Two pairs of complex numbers (ω_{1},ω_{2}) and (α_{1},α_{2}) are called In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i.e., if the relation is written as ~ it holds for all a, b and c in X that (Reflexivity) a ~ a (Symmetry) if a ~ b then b ~ a...
equivalent if they generate the same lattice: that is, if < ω_{1},ω_{2} > = < α_{1},α_{2} > .
### 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 (ω_{1},ω_{2}) and (α_{1},α_{2}) are equivalent if and only if there exists a 2 × 2 matrix with integer entries *a*,*b*,*c* and *d* and 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. Determinants are important both in calculus, where they enter...
determinant such that - ,
that is, so that - α
_{1} = *a*ω_{1} + *b*ω_{2} and - α
_{2} = *c*ω_{1} + *d*ω_{2}. Note that this matrix belongs to the matrix In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste Galois (1830), concerning the...
group , which, with slight abuse of terminology, is known as the 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. The modular group can be represented as a group of geometric transformations or as a group of matrices. Definition The modular...
modular group. This equivalence of lattices can be thought of as underlying many of the properties of In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only). Formally, an elliptic function is a meromorphic function f defined...
elliptic functions (especially the In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. Definitions Consider two complex numbers and , with not purely real; the definition depends on these. There are some significant choices of convention, and the literature is not consistent in its usage...
Weierstrass elliptic function) and modular forms.
## Topological properties The In mathematics, an abelian group is a commutative group, i.e. a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions for abelian groups -- additive and multiplicative. Examples Every...
abelian group maps the complex plane into the fundamental parallelogram. That is, every point can be written as *z* = *p* + *m*ω_{1} + *n*ω_{2} 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 (Greek topos = place and logos = word) is a branch of mathematics concerned with the study of topological spaces. Topology is concerned with the study of the so-called topological properties of figures, that is to say properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms...
topology of a See also torus (nuclear physics). In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle. The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the...
torus. Equivlently, one says that the quotient manifold is a torus.
## Fundamental region The grey depicts the canonical fundamental domain. Define τ = ω_{2} / ω_{1} to be the 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 ...
half-period ratio. Then the lattice basis can always be chosen so that τ lies in a special region, called the 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. One typical condition is...
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. 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 exp(*i*π / 3), then there are exactly two lattice basis with the same τ in the fundamental region: namely, (ω_{1},ω_{2}) and ( − ω_{1}, − ω_{2}). If τ = *i* then four lattice basis have the same τ: the above two and (*i*ω_{1},*i*ω_{2}). If τ = exp(*i*π / 3) then there are six lattice basis with the same τ: (ω_{1},ω_{2}), (τω_{1},τω_{2}), (τ^{2}ω_{1},τ^{2}ω_{ 2}) and their negatives. Note that τ = *i* and τ = exp(*i*π / 3) 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 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 ...
half-period ratio.
## 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.)* |