Weierstrass elliptic function

In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.

Image:Weierstrass p.svg|100px|Symbol for Weierstrass P function

Symbol for Weierstrass -function

Motivation

A cubic of the form, where are complex numbers with, cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.
For the quadric ; the unit circle, there exists a parameterization using the sine function and its derivative the cosine function:
Because of the periodicity of the sine and cosine is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of by means of the doubly periodic -function and its derivative, namely via. This parameterization has the domain, which is topologically equivalent to a torus.
There is another analogy to the trigonometric functions. Consider the integral function
It can be simplified by substituting and :
That means. So the sine function is an inverse function of an integral function.
Elliptic functions are the inverse functions of elliptic integrals. In particular, let:
Then the extension of to the complex plane equals the -function. This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.

Definition

Let be two complex numbers that are linearly independent over and let be the period lattice generated by those numbers. Then the -function is defined as follows:
This series converges locally uniformly absolutely in the complex torus.
It is common to use and in the upper half-plane as generators of the lattice. Dividing by maps the lattice isomorphically onto the lattice with. Because can be substituted for, without loss of generality we can assume, and then define. With that definition, we have.

Properties

is a meromorphic function with a pole of order 2 at each period in.
is a homogeneous function in that:
is an even function. That means for all, which can be seen in the following way:
The derivative of is given by:
and are doubly periodic with the periods and. This means: It follows that and for all.
Laurent expansion

Let. Then for the -function has the following Laurent expansion
where
for are so called Eisenstein series.

Differential equation

Set and. Then the -function satisfies the differential equation
This relation can be verified by forming a linear combination of powers of and to eliminate the pole at. This yields an entire elliptic function that has to be constant by Liouville's theorem.

Invariants

The coefficients of the above differential equation and are known as the invariants. Because they depend on the lattice they can be viewed as functions in and.
The series expansion suggests that and are homogeneous functions of degree and. That is
for.
If and are chosen in such a way that, and can be interpreted as functions on the upper half-plane.
Let. One has:
That means g₂ and g₃ are only scaled by doing this. Set
and
As functions of, and are so called modular forms.
The Fourier series for and are given as follows:
where
is the divisor function and is the nome.

Modular discriminant

The modular discriminant is defined as the discriminant of the characteristic polynomial of the differential equation as follows:
The discriminant is a modular form of weight. That is, under the action of the modular group, it transforms as
where with.
Note that where is the Dedekind eta function.
For the Fourier coefficients of, see Ramanujan tau function.

The constants ''e''₁, ''e''₂ and ''e''₃

, and are usually used to denote the values of the -function at the half-periods.
They are pairwise distinct and only depend on the lattice and not on its generators.
, and are the roots of the cubic polynomial and are related by the equation:
Because those roots are distinct the discriminant does not vanish on the upper half plane. Now we can rewrite the differential equation:
That means the half-periods are zeros of.
The invariants and can be expressed in terms of these constants in the following way:
, and are related to the modular lambda function:

Relation to Jacobi's elliptic functions

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.
The basic relations are:
where and are the three roots described above and where the modulus k of the Jacobi functions equals
and their argument w equals

Relation to Jacobi's theta functions

The function can be represented by Jacobi's theta functions:
where is the nome and is the period ratio. This also provides a very rapid algorithm for computing.

Relation to elliptic curves

Consider the embedding of the cubic curve in the complex projective plane
where is a point lying on the line at infinity. For this cubic there exists no rational parameterization, if. In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the -function and its derivative :
Now the map is bijective and parameterizes the elliptic curve.
is an abelian group and a topological space, equipped with the quotient topology.
It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair with there exists a lattice, such that
and.
The statement that elliptic curves over can be parameterized over, is known as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof of Fermat's Last Theorem.

Addition theorem

The addition theorem states that if and do not belong to, then
This states that the points and are collinear, the geometric form of the group law of an elliptic curve.
This can be proven by considering constants such that
Then the elliptic function
has a pole of order three at zero, and therefore three zeros whose sum belongs to. Two of the zeros are and, and thus the third is congruent to.

Alternative form

The addition theorem can be put into the alternative form, for :
As well as the duplication formula:

Proofs

This can be proven from the addition theorem shown above. The points and are collinear and lie on the curve. The slope of that line is
So,, and all satisfy a cubic
where is a constant. This becomes
Thus
which provides the wanted formula
A direct proof is as follows. Any elliptic function can be expressed as:
where is the Weierstrass sigma function and are the respective zeros and poles in the period parallelogram. Considering the function as a function of, we have
Multiplying both sides by and letting, we have, so
By definition the Weierstrass zeta function: therefore we logarithmically differentiate both sides with respect to obtaining:
Once again by definition thus by differentiating once more on both sides and rearranging the terms we obtain

Knowing that has the following differential equation and rearranging the terms one gets the wanted formula

Typography

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863. It should not be confused with the normal mathematical script letters P: ? and ?.
In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is, with the more correct alias. In HTML, it can be escaped as &weierp; or &wp;.