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In [[mathematics]], a '''fundamental pair of periods''' is an [[ordered pair]] of [[complex number]]s that define a [[lattice (group)|lattice]] in the [[complex plane]]. This type of lattice is the underlying object with which [[elliptic function]]s and [[modular form]]s 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.
[[Image:Fundamental parallelogram.png|thumb|right|Fundamental parallelogram defined by a pair of vectors in the complex plane.]]

==Definition==
The '''fundamental pair of periods''' is a pair of complex numbers <math>\omega_1,\omega_2 \in \Complex</math> such that their ratio ω2/ω1 is not real. In other words, considered as vectors in <math>\mathbb{R}^2</math>, the two are not [[linearly independent|collinear]]. The lattice generated by ω1 and ω2 is

:<math>\Lambda=\{m\omega_1+n\omega_2 \,\,|\,\, m,n\in\mathbb{Z} \}</math>

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 〈ω1, ω2〉. The two generators ω1 and ω2 are called the '''lattice basis'''.

The [[parallelogram]] defined by the vertices 0, <math>\omega_1</math> and <math>\omega_2</math> is called the '''fundamental parallelogram'''.

It is important to note that, while a fundamental pair generates a lattice, a lattice does not have any unique fundamental pair, that is, many (in fact, an infinite number) fundamental pairs correspond to the same lattice.

==Algebraic properties==
A number of properties, listed below, obtain.

===Equivalence===
[[Image:Lattice torsion points.svg|right|thumb|250px|A lattice spanned by periods ω1 and ω2, showing equivalent points and edges. ]]
Two pairs of complex numbers (ω1,ω2) and (α1,α2) are called [[equivalence relation|equivalent]] if they generate the same lattice: that is, if ⟨ω1,ω2⟩ = ⟨α1,α2⟩.

===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 <math>(\omega_1,\omega_2)</math> and <math>(\alpha_1,\alpha_2)</math> are equivalent if and only if there exists a 2 × 2 matrix <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math> with integer entries ''a'', ''b'', ''c'' and ''d'' and [[determinant]] ''ad'' − ''bc'' = ±1 such that

:<math>\begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} =
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix},</math>

that is, so that

:<math>\alpha_1 = a\omega_1+b\omega_2\,</math>

and

:<math>\alpha_2 = c\omega_1+d\omega_2.\,</math>

Note that this matrix belongs to the matrix [[group (mathematics)|group]] <math>SL(2,\mathbb{Z})</math>, 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 function]]s (especially the [[Weierstrass elliptic function]]) and modular forms.

== Topological properties ==
The [[abelian group]] <math>\mathbb{Z}^2</math> maps the complex plane into the fundamental parallelogram. That is, every point <math>z \in \mathbb{C}</math> can be written as <math>z=p+m\omega_1+n\omega_2</math> 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]] of a [[torus]]. Equivalently, one says that the quotient manifold <math>\Complex/\Lambda</math> is a torus.

==Fundamental region==
[[Image:ModularGroup-FundamentalDomain-01.png|thumb|400px|The grey depicts the canonical fundamental domain.]]
Define τ = ω2/ω1 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.

The fundamental domain is given by the set ''D'', which is composed of a set ''U'' plus a part of the boundary of ''U'':

:<math>U = \left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \tfrac{1}{2} \right\}.</math>

where ''H'' is the [[upper half-plane]].

The fundamental domain ''D'' is then built by adding the boundary on the left plus half the arc on the bottom:

:<math>D=U\cup\left\{ z \in H: \left| z \right| \geq 1,\, \mbox{Re}(z)=-\tfrac{1}{2} \right\} \cup \left\{ z \in H: \left| z \right| = 1,\, \mbox{Re}(z) \le 0 \right\}.</math>

If τ is not ''i'' and is not t=exp(1/3*pi*i), then there are exactly two lattice bases with the same τ in the fundamental region:
namely, <math>(\omega_1,\omega_2)</math> and <math> (-\omega_1,-\omega_2)</math>. If <math>\tau=i</math> then four lattice bases have the same τ: the above two and <math>(i\omega_1,i\omega_2)</math>. If t=exp(1/3*pi*i) then there are six lattice bases with the same τ: <math>(\omega_1,\omega_2)</math>, <math>(\tau \omega_1,\tau \omega_2)</math>, <math>(\tau^2 \omega_1, \tau^2 \omega_2)</math> and their negatives. Note that <math>\tau=i</math> and t=exp(1/3*pi*i) 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 (mathematics)|nome]], [[elliptic modulus]], [[quarter period]] and [[half-period ratio]].
* [[Elliptic curve]]
* [[Modular form]]
* [[Eisenstein series]]

==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.)''

{{DEFAULTSORT:Fundamental Pair Of Periods}}
[[Category:Riemann surfaces]]
[[Category:Modular forms]]
[[Category:Elliptic functions]]
[[Category:Lattice points]]

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en>Quibik: Moved the "Encoding" paragraph to the bottom.

90.195.173.220: Grammar Correction

en>Sapphorain: Undid revision 584622913 by 173.2.21.207 (talk)