Fundamental pair of periods

 ( Riemann Surfaces ) 

In mathematics, a fundamental pair of periods is an ordered pair of that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and are defined.

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Definition A fundamental pair of periods is a pair of complex numbers $\backslash omega\_1,\backslash omega\_2\; \backslash in\; \backslash Complex$ such that their ratio $\backslash omega\_2\; /\; \backslash omega\_1$ is not real. If considered as vectors in $\backslash R^2$, the two are linearly independent. The lattice generated by $\backslash omega\_1$ and $\backslash omega\_2$ is

$\backslash Lambda\; =\; \backslash left\backslash \{\; m\backslash omega\_1\; +\; n\backslash omega\_2\; \backslash mid\; m,n\backslash in\backslash Z\; \backslash right\backslash \}.$

This lattice is also sometimes denoted as $\backslash Lambda(\backslash omega\_1,\; \backslash omega\_2)$ to make clear that it depends on $\backslash omega\_1$ and $\backslash omega\_2.$ It is also sometimes denoted by $\backslash Omega\backslash vphantom\{(\}$ or $\backslash Omega(\backslash omega\_1,\; \backslash omega\_2),$ or simply by $(\backslash omega\_1,\; \backslash omega\_2).$ The two generators $\backslash omega\_1$ and $\backslash omega\_2$ are called the lattice basis. The parallelogram with vertices $(0,\; \backslash omega\_1,\; \backslash omega\_1+\backslash omega\_2,\; \backslash omega\_2)$ 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.

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Algebraic properties A number of properties, listed below, can be seen.

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Equivalence Two pairs of complex numbers $(\backslash omega\_1,\; \backslash omega\_2)$ and $(\backslash alpha\_1,\; \backslash alpha\_2)$ are called equivalent if they generate the same lattice: that is, if $\backslash Lambda(\backslash omega\_1,\; \backslash omega\_2)\; =\; \backslash Lambda(\backslash alpha\_1,\; \backslash alpha\_2).$

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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.

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Modular symmetry Two pairs $(\backslash omega\_1,\backslash omega\_2)$ and $(\backslash alpha\_1,\backslash alpha\_2)$ are equivalent if and only if there exists a matrix with integer entries $a,$ $b,$ $c,$ and $d$ and determinant $ad\; -\; bc\; =\; \backslash pm\; 1$ such that

$\backslash begin\{pmatrix\}\; \backslash alpha\_1\; \backslash \backslash \; \backslash alpha\_2\; \backslash end\{pmatrix\}\; =$

\begin{pmatrix} a & b \\ c & d \end{pmatrix}
     

\begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix},

that is, so that

$\backslash begin\{align\}$

\alpha_1 = a\omega_1+b\omega_2, \\5mu \alpha_2 = c\omega_1+d\omega_2. \end{align}

This matrix belongs to the modular group $\backslash mathrm\{SL\}(2,\backslash Z).$ 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.

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Topological properties The abelian group $\backslash Z^2$ maps the complex plane into the fundamental parallelogram. That is, every point $z\; \backslash in\; \backslash Complex$ can be written as $z\; =\; p+m\backslash omega\_1+n\backslash omega\_2$ 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 of a torus. Equivalently, one says that the quotient manifold $\backslash C/\backslash Lambda$ is a torus.

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Fundamental region Define $\backslash tau\; =\; \backslash omega\_2/\backslash omega\_1$ to be the half-period ratio. Then the lattice basis can always be chosen so that $\backslash tau$ lies in a special region, called the fundamental domain. Alternately, there always exists an element of the projective special linear group $\backslash operatorname\{PSL\}(2,\backslash Z)$ that maps a lattice basis to another basis so that $\backslash tau$ 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

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:

$D\; =\; U\; \backslash cup\; \backslash left\backslash \{\; z\; \backslash in\; H:\; \backslash left|\; z\; \backslash right|\; \backslash geq\; 1,\backslash ,\; \backslash operatorname\{Re\}(z)\; =\; -\backslash tfrac\{1\}\{2\}\; \backslash right\backslash \}\; \backslash cup\; \backslash left\backslash \{\; z\; \backslash in\; H:\; \backslash left|\; z\; \backslash right|\; =\; 1,\backslash ,\; \backslash operatorname\{Re\}(z)\; \backslash le\; 0\; \backslash right\backslash \}.$

Three cases pertain:

• If $\backslash tau\; \backslash ne\; i$ and $\backslash tau\; \backslash ne\; e^\{i\backslash pi/3\}$, then there are exactly two lattice bases with the same $\backslash tau$ in the fundamental region: $(\backslash omega\_1,\backslash omega\_2)$ and $(-\backslash omega\_1,-\backslash omega\_2).$
• If $\backslash tau=i$, then four lattice bases have the same the above two $(\backslash omega\_1,\backslash omega\_2)$, $(-\backslash omega\_1,-\backslash omega\_2)$ and $(i\backslash omega\_1,i\backslash omega\_2)$, $(-i\backslash omega\_1,-i\backslash omega\_2).$
• If $\backslash tau=e^\{i\backslash pi/3\}$, then there are six lattice bases with the same $(\backslash omega\_1,\backslash omega\_2)$, $(\backslash tau\; \backslash omega\_1,\; \backslash tau\; \backslash omega\_2)$, $(\backslash tau^2\; \backslash omega\_1,\; \backslash tau^2\; \backslash omega\_2)$ and their negatives.

In the closure of the fundamental domain: $\backslash tau=i$ and $\backslash tau=e^\{i\backslash pi/3\}.$

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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.
• Elliptic curve
• Modular form
• Eisenstein series

• Tom M. Apostol, 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.)

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