Weierstrass's elliptic functions

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In mathematics, Weierstrass's 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 p-functions and they are usually denoted by the symbol ℘. They play an important role in 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 P function

Symbol for Weierstrass -function

Model of Weierstrass -function

Definition[edit]

Visualization of the -function with invariants and in which white corresponds to a pole, black to a zero.

Let be two complex numbers that are linear independent over and let be the lattice generated by those numbers. Then the -function is defined as follows:

This series converges locally uniformly absolutely in . Oftentimes instead of only is written.

The Weierstrass -function is constructed exactly in such a way that it has a pole of the order two at each lattice point.

Because the sum alone would not converge it is necessary to add the term .[1]

It is common to use and as generators of the lattice. Multiplying by maps the lattice isomorphically onto the lattice with . By possibly substituting by it can be assumed that . One sets .

Motivation[edit]

A cubic of the form , where are complex numbers with , can not be rationally parameterized.[2] Yet one still wants to find a way to parameterize it.

For the quadric , the unit circle, there exists a (non-rational) 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 (see in the section "Relation to ellitpic curves"). This parameterization has the domain , which is topologically equivalent to a torus.[3]

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.[4]

Elliptic functions are also inverse functions of integral functions, namely of elliptic integrals. In particular the -function is obtained in the following way:

Let

.

Then can be extended to the complex plane and this extension equals the -function.[5]

Properties[edit]

  • ℘ is an even function. That means for all , which can be seen in the following way:

The second last equality holds because . Since the sum converges absolutely this rearrangement does not change the limit.

  • ℘ is meromorphic and its derivative is[6]
.
  • and are doubly periodic with the periods und .[6] This means:
and .

It follows that and for all . Functions which are meromorphic and doubly periodic are also called elliptic functions.

Laurent expansion[edit]

Let . Then for the -function has the following Laurent expansion

where

for are so called Eisenstein series.[6]

Differential equation[edit]

Set and . Then the -function satisfies the differential equation[6]

.

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 .[6]

Invariants[edit]

The real part of the invariant g3 as a function of the nome q on the unit disk.
The imaginary part of the invariant g3 as a function of the nome q on the unit disk.

The coefficients of the above differential equation g2 and g3 are known as the invariants. Because they depend on the lattice they can be viewed as functions in and .

The series expansion suggests that g2 and g3 are homogeneous functions of degree −4 and −6. That is[7]

for .

If and are chosen in such a way that g2 and g3 can be interpreted as functions on the upper half-plane .

Let . One has:[8]

,
.

That means g2 and g3 are only scaled by doing this. Set

, .

As functions of are so called modular forms.

The Fourier series for and are given as follows:[9]

where is the divisor function and .

Modular discriminant[edit]

The real part of the discriminant as a function of the nome q on the unit disk.

The modular discriminant Δ is defined as the discriminant of the polynomial at right-hand side of the above differential equation:

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as

where with ad − bc = 1.[10]

Note that where is the Dedekind eta function.[11]

For the Fourier coefficients of , see Ramanujan tau function.

The constants e1, e2 and e3[edit]

, 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.[12]

, 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.[13] 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:[14]

Relation to elliptic curves[edit]

Consider the projective cubic curve

.

For this cubic, also called Weierstrass cubic, there exists no rational parameterization, if .[2] In this case it is also called an elliptic curve. Nevertheless there is a parameterization that uses the -function and its derivative :[15]

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 .[16]

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 (1995) of Fermat's Last Theorem.

Addition theorems[edit]

Let , so that . Then one has:[17]

.

As well as the duplication formula:[17]

.

These formulas also have a geometric interpretation, if one looks at the elliptic curve together with the mapping as in the previous section.

The group structure of translates to the curve and can be geometrically interpreted there:

The sum of three pairwise different points is zero if and only if they lie on the same line in .[18]

This is equivalent to:

,

where , and .[19]

Relation to Jacobi's elliptic functions[edit]

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.

The basic relations are:[20]

where and are the three roots described above and where the modulus k of the Jacobi functions equals

and their argument w equals

Typography[edit]

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘.[footnote 1]

In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118 SCRIPT CAPITAL P (HTML ℘ · ℘, ℘), with the more correct alias weierstrass elliptic function.[footnote 2] In HTML, it can be escaped as ℘.

Character information
Preview
Unicode name SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION
Encodings decimal hex
Unicode 8472 U+2118
UTF-8 226 132 152 E2 84 98
Numeric character reference ℘ ℘
Named character reference ℘, ℘

See also[edit]

Footnotes[edit]

  1. ^ This symbol was used already at least in 1890. The first edition of A Course of Modern Analysis by E. T. Whittaker in 1902 also used it.[21]
  2. ^ The Unicode Consortium has acknowledged two problems with the letter's name: the letter is in fact lowercase, and it is not a "script" class letter, like U+1D4C5 𝓅 MATHEMATICAL SCRIPT SMALL P, but the letter for Weierstrass's elliptic function. Unicode added the alias as a correction.[22][23]

References[edit]

  1. ^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 9. ISBN 0-387-90185-X. OCLC 2121639.
  2. ^ a b Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 8, ISBN 978-3-8348-2348-9
  3. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
  4. ^ Jeremy Gray (2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, p. 71, ISBN 978-3-319-23715-2
  5. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294, ISBN 978-3-540-32058-6
  6. ^ a b c d e Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 11, ISBN 0-387-90185-X
  7. ^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 14. ISBN 0-387-90185-X. OCLC 2121639.
  8. ^ Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 14, ISBN 0-387-90185-X
  9. ^ Apostol, Tom M. (1990). Modular functions and Dirichlet series in number theory (2nd ed.). New York: Springer-Verlag. p. 20. ISBN 0-387-97127-0. OCLC 20262861.
  10. ^ Apostol, Tom M. (1976). Modular functions and Dirichlet series in number theory. New York: Springer-Verlag. p. 50. ISBN 0-387-90185-X. OCLC 2121639.
  11. ^ Chandrasekharan, K. (Komaravolu), 1920- (1985). Elliptic functions. Berlin: Springer-Verlag. p. 122. ISBN 0-387-15295-4. OCLC 12053023.CS1 maint: multiple names: authors list (link)
  12. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 270, ISBN 978-3-540-32058-6
  13. ^ Tom M. Apostol (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 13, ISBN 0-387-90185-X
  14. ^ K. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 33, ISBN 0-387-15295-4
  15. ^ Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 12, ISBN 978-3-8348-2348-9
  16. ^ Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 111, ISBN 978-3-8348-2348-9
  17. ^ a b Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286, ISBN 978-3-540-32058-6
  18. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 287, ISBN 978-3-540-32058-6
  19. ^ Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 288, ISBN 978-3-540-32058-6
  20. ^ Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw–Hill. p. 721. LCCN 59014456.
  21. ^ teika kazura (2017-08-17), The letter ℘ Name & origin?, MathOverflow, retrieved 2018-08-30
  22. ^ "Known Anomalies in Unicode Character Names". Unicode Technical Note #27. version 4. Unicode, Inc. 2017-04-10. Retrieved 2017-07-20.
  23. ^ "NameAliases-10.0.0.txt". Unicode, Inc. 2017-05-06. Retrieved 2017-07-20.

External links[edit]