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 are 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. 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. 
Symbol for Weierstrass formula_1-function
Definition.
Let formula_2 be two complex numbers that are linearly independent over formula_3 and let formula_4 be the lattice generated by those numbers. Then the formula_1-function is defined as follows:
formula_6
This series converges locally uniformly absolutely in formula_7. Oftentimes instead of formula_8 only formula_9 is written.
The Weierstrass formula_1-function is constructed exactly in such a way that it has a pole of the order two at each lattice point.
Because the sum formula_11 alone would not converge it is necessary to add the term formula_12.
It is common to use formula_13 and formula_14 in the upper half-plane formula_15 as generators of the lattice. Dividing by formula_16 maps the lattice formula_17 isomorphically onto the lattice formula_18 with formula_19. Because formula_20 can be substituted for formula_14, without loss of generality we can assume formula_22, and then define formula_23.
Motivation.
A cubic of the form formula_24, where formula_25 are complex numbers with formula_26, can not be rationally parameterized. Yet one still wants to find a way to parameterize it.
For the quadric formula_27, the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
formula_28
Because of the periodicity of the sine and cosine formula_29 is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of formula_30 by means of the doubly periodic formula_31-function (see in the section "Relation to elliptic curves"). This parameterization has the domain formula_32, which is topologically equivalent to a torus.
There is another analogy to the trigonometric functions. Consider the integral function
formula_33
It can be simplified by substituting formula_34 and formula_35:
formula_36
That means formula_37. So the sine function is an inverse function of an integral function.
Elliptic functions are also inverse functions of integral functions, namely of elliptic integrals. In particular the formula_31-function is obtained in the following way:
Let
formula_39
Then formula_40 can be extended to the complex plane and this extension equals the formula_31-function.
Properties.
It follows that formula_52 and formula_53 for all formula_54. Functions which are meromorphic and doubly periodic are also called elliptic functions.
Laurent expansion.
Let formula_55
is the divisor function and formula_56 is the nome.
Modular discriminant.
The "modular discriminant" Δ is defined as the discriminant of the polynomial on the right-hand side of the above differential equation:
formula_57
The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as
formula_58
where formula_59 with "ad" − "bc" = 1.
Note that formula_60 where formula_61 is the Dedekind eta function.
For the Fourier coefficients of formula_62, see Ramanujan tau function.
The constants "e"1, "e"2 and "e"3.
formula_63, formula_64 and formula_65 are usually used to denote the values of the formula_1-function at the half-periods.
formula_67
formula_68
formula_69
They are pairwise distinct and only depend on the lattice formula_70 and not on its generators.
formula_63, formula_64 and formula_65 are the roots of the cubic polynomial formula_74 and are related by the equation:
formula_75
Because those roots are distinct the discriminant formula_62 does not vanish on the upper half plane. Now we can rewrite the differential equation:
formula_77
That means the half-periods are zeros of formula_48.
The invariants formula_79 and formula_80 can be expressed in terms of these constants in the following way:
formula_81
formula_82
formula_63, formula_64 and formula_65 are related to the modular lambda function:
formula_86
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:
formula_87
where formula_88and formula_65 are the three roots described above and where the modulus "k" of the Jacobi functions equals
formula_90
and their argument "w" equals
formula_91
Relation to Jacobi's theta functions.
The function formula_92 can be represented by Jacobi's theta functions:
formula_93
where formula_56 is the nome and formula_14 is the period ratio formula_96. This also provides a very rapid algorithm for computing formula_97.
Relation to elliptic curves.
Consider the projective cubic curve
formula_98
For this cubic, also called Weierstrass cubic, there exists no rational parameterization, if formula_99. In this case it is also called an elliptic curve. Nevertheless there is a parameterization that uses the formula_1-function and its derivative formula_48:
formula_102
Now the map formula_103 is bijective and parameterizes the elliptic curve formula_104.
formula_32 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 formula_25 with formula_107 there exists a lattice formula_17, such that
formula_109 and formula_110.
The statement that elliptic curves over formula_111 can be parameterized over formula_111, is known as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem.
Addition theorems.
Let formula_113, so that formula_114. Then one has:
formula_115
As well as the duplication formula:
formula_116
These formulas also have a geometric interpretation, if one looks at the elliptic curve formula_117 together with the mapping formula_118 as in the previous section.
The group structure of formula_119 translates to the curve formula_117and can be geometrically interpreted there:
The sum of three pairwise different points formula_121is zero if and only if they lie on the same line in formula_122.
This is equivalent to:
formula_123
where formula_124, formula_125 and formula_126.
Typography.
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘.
In computing, the letter ℘ is available as codice_1 in TeX. In Unicode the code point is , with the more correct alias . In HTML, it can be escaped as codice_2.