In
mathematics, a fundamental pair of periods is an
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
of
complex numbers that define a
lattice in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
. This type of lattice is the underlying object with which
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
s and
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
s are defined.
Definition
A fundamental pair of periods is a pair of complex numbers
such that their ratio ω
2/ω
1 is not real. If considered as vectors in
, the two are not
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
. 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
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equ ...
defined by the vertices 0,
and
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
Two pairs of complex numbers (''ω''
1,''ω''
2) and (α
1,α
2) are called
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
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
and
are equivalent if and only if there exists a 2 × 2 matrix
with integer entries ''a'', ''b'', ''c'' and ''d'' and
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
such that
:
that is, so that
:
and
:
This matrix belongs to the matrix
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, which is known as the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
. This equivalence of lattices can be thought of as underlying many of the properties of
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
s (especially the
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
) and modular forms.
Topological properties
The
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
of a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
. Equivalently, one says that the quotient manifold
is a torus.
Fundamental region
Define ''τ'' = ''ω''
2/''ω''
1 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
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of ...
. 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'':
:
where ''H'' is the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds t ...
.
The fundamental domain ''D'' is then built by adding the boundary on the left plus half the arc on the bottom:
:
Three cases pertain:
* If
and
, then there are exactly two lattice bases with the same τ in the fundamental region:
and
* If
, then four lattice bases have the same τ: the above two
,
and
,
* If
, then there are six lattice bases with the same τ:
,
,
and their negatives.
In the closure of the fundamental domain:
and
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
In mathematics, the modular lambda function λ(τ)\lambda(\tau) is not a modular function (per the Wikipedia definition), but every modular function is a rational function in \lambda(\tau). Some authors use a non-equivalent definition of "modular f ...
,
quarter period In mathematics, the quarter periods ''K''(''m'') and i''K'' ′(''m'') are special functions that appear in the theory of elliptic functions.
The quarter periods ''K'' and i''K'' ′ are given by
:K(m)=\int_0^ \frac
and
:K'(m) ...
and
half-period ratio.
*
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
*
Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
*
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
References
*
Tom M. Apostol
Tom Mike Apostol (August 20, 1923 – May 8, 2016) was an American analytic number theorist and professor at the California Institute of Technology, best known as the author of widely used mathematical textbooks.
Life and career
Apostol was bor ...
, ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ''(See chapters 1 and 2.)''
* Jurgen Jost, ''Compact Riemann Surfaces'' (2002), Springer-Verlag, New York. ''(See chapter 2.)''
{{DEFAULTSORT:Fundamental Pair Of Periods
Riemann surfaces
Modular forms
Elliptic functions
Lattice points