In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined.
Definition A fundamental pair of periods is a pair of complex numbers ω 1 , ω 2 ∈ C {\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} } such that their ratio ω 2 / ω 1 {\displaystyle \omega _{2}/\omega _{1}} is not real. If considered as vectors in R 2 {\displaystyle \mathbb {R} ^{2}} , the two are linearly independent. The lattice generated by ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} is
Λ = { m ω 1 + n ω 2 ∣ m , n ∈ Z } . {\displaystyle \Lambda =\left\{m\omega _{1}+n\omega _{2}\mid m,n\in \mathbb {Z} \right\}.}
This lattice is also sometimes denoted as Λ ( ω 1 , ω 2 ) {\displaystyle \Lambda (\omega _{1},\omega _{2})} to make clear that it depends on ω 1 {\displaystyle \omega _{1}} and ω 2 . {\displaystyle \omega _{2}.} It is also sometimes denoted by Ω ( {\displaystyle \Omega {\vphantom {(}}} or Ω ( ω 1 , ω 2 ) , {\displaystyle \Omega (\omega _{1},\omega _{2}),} or simply by ( ω 1 , ω 2 ) . {\displaystyle (\omega _{1},\omega _{2}).} The two generators ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} are called the lattice basis. The parallelogram with vertices ( 0 , ω 1 , ω 1 + ω 2 , ω 2 ) {\displaystyle (0,\omega _{1},\omega _{1}+\omega _{2},\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.
Algebraic properties A number of properties, listed below, can be seen.
Equivalence
Two pairs of complex numbers ( ω 1 , ω 2 ) {\displaystyle (\omega _{1},\omega _{2})} and ( α 1 , α 2 ) {\displaystyle (\alpha _{1},\alpha _{2})} are called equivalent if they generate the same lattice: that is, if Λ ( ω 1 , ω 2 ) = Λ ( α 1 , α 2 ) . {\displaystyle \Lambda (\omega _{1},\omega _{2})=\Lambda (\alpha _{1},\alpha _{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.
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