Fundamental pair of periods

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.