- Fundamental pair of periods
In
mathematics , a fundamental pair of periods is anordered pair ofcomplex number s that define a lattice in thecomplex plane . This type of lattice is the underlying object with whichelliptic function s andmodular 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.
Definition
The fundamental pair of periods is a pair of complex numbers omega_1,omega_2 in Complex such that their ratio omega_2/omega_1 is not real. In other words, considered as vectors in mathbb{R}^2, the two are not collinear. The lattice generated by omega_1 and omega_2 is
:Lambda={momega_1+nomega_2 ,,|,, m,ninmathbb{Z} }
This lattice is also sometimes denoted as Lambda(omega_1,omega_2) to make clear that it depends on omega_1 and omega_2. It is also sometimes denoted by Omega or Omega(omega_1,omega_2), or simply by langleomega_1,omega_2 angle. The two generators omega_1 and omega_2 are called the lattice basis.
The
parallelogram defined by the vertices 0, omega_1 and omega_2 is called the fundamental parallelogram.Algebraic properties
A number of properties, listed below, should be noted.
Equivalence
Two pairs of complex numbers omega_1,omega_2) and alpha_1,alpha_2) are called equivalent if they generate the same lattice: that is, if langleomega_1,omega_2 angle = langlealpha_1,alpha_2 angle.
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 omega_1,omega_2) and alpha_1,alpha_2) are equivalent if and only if there exists a 2 × 2 matrix egin{pmatrix} a & b \ c & d end{pmatrix} with integer entries "a","b","c" and "d" and
determinant ad-bc=pm 1 such that :egin{pmatrix} alpha_1 \ alpha_2 end{pmatrix} = egin{pmatrix} a & b \ c & d end{pmatrix}egin{pmatrix} omega_1 \ omega_2 end{pmatrix},that is, so that:alpha_1 = aomega_1+bomega_2and:alpha_2 = comega_1+domega_2.Note that this matrix belongs to the matrix group S^{*}L(2,mathbb{Z}), which, with slight abuse of terminology, is known as themodular group . This equivalence of lattices can be thought of as underlying many of the properties ofelliptic function s (especially theWeierstrass elliptic function ) and modular forms.Topological properties
The
abelian group mathbb{Z}^2 maps the complex plane into the fundamental parallelogram. That is, every point z in mathbb{C} can be written as z=p+momega_1+nomega_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 atorus . Equivalently, one says that the quotient manifold Complex/Lambda is a torus.Fundamental region
Define au=frac{omega_2}{omega_1} to be the
half-period ratio . Then the lattice basis can always be chosen so that τ lies in a special region, called thefundamental 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":
:U = left{ z in H: left| z ight| > 1,, left| ,mbox{Re}(z) , ight| < frac{1}{2} ight}.
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=Ucupleft{ z in H: left| z ight| geq 1,, mbox{Re}(z)=-frac{1}{2} ight} cup left{ z in H: left| z ight| = 1,, mbox{Re}(z)<0 ight}.
If τ is not "i" and is not expleft(frac{ipi}{3} ight), then there are exactly two lattice basis with the same τ in the fundamental region: namely, omega_1,omega_2) and omega_1,-omega_2). If au=i then four lattice basis have the same τ: the above two and iomega_1,iomega_2). If au=expleft(frac{ipi}{3} ight) then there are six lattice basis with the same τ: omega_1,omega_2), au omega_1, au omega_2), au^2 omega_1, au^2 omega_2) and their negatives. Note that au=i and au=expleft(frac{ipi}{3} ight) are exactly the fixed points of PSL(2,Z) in the closure of the fundamental domain.
ee 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 andhalf-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.)"
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