Modular Discriminant

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 $\wp$-function

Definition

Let $\omega_1,\omega_2\in\mathbb$ be two complex numbers that are linearly independent over $\mathbb$ and let $\Lambda:=\mathbb\omega_1+\mathbb\omega_2:=\$ be the lattice generated by those numbers. Then the $\wp$-function is defined as follows:

$\weierp(z,\omega_1,\omega_2):=\weierp(z,\Lambda) := \frac + \sum_\left(\frac 1 - \frac 1 \right).$

This series converges locally uniformly absolutely in $\mathbb\setminus\Lambda$. Oftentimes instead of $\wp(z,\omega_1,\omega_2)$ only $\wp(z)$ is written.

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

Because the sum $\sum_ \frac 1$ alone would not converge it is necessary to add the term $-\frac 1$.

It is common to use $1$ and $\tau$ in the upper half-plane $:=\$ as generators of the lattice. Dividing by $\omega_1$ maps the lattice $\mathbb\omega_1+\mathbb\omega_2$ isomorphically onto the lattice $\mathbb+\mathbb\tau$ with $\tau=\tfrac$. Because $-\tau$ can be substituted for $\tau$, without loss of generality we can assume $\tau\in\mathbb$, and then define $\wp(z,\tau) := \wp(z, 1,\tau)$.

Motivation

A cubic of the form $C_^\mathbb=\$, where $g_2,g_3\in\mathbb$ are complex numbers with $g_2^3-27g_3^2\neq0$, can not be rationally parameterized. Yet one still wants to find a way to parameterize it.

For the quadric $K=\left\$, the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
$\psi:\mathbb/2\pi\mathbb\to K, \quad t\mapsto(\sin t,\cos t).$

Because of the periodicity of the sine and cosine $\mathbb/2\pi\mathbb$ is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of $C_^\mathbb$ by means of the doubly periodic $\wp$-function (see in the section "Relation to elliptic curves"). This parameterization has the domain $\mathbb/\Lambda$, which is topologically equivalent to a torus.

There is another analogy to the trigonometric functions. Consider the integral function
$a(x)=\int_0^x\frac .$

It can be simplified by substituting $y=\sin t$ and $s=\arcsin x$:
$a(x)=\int_0^s dt = s = \arcsin x .$

That means $a^(x) = \sin x$. 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 $\wp$-function is obtained in the following way:

Let
$u(z)=-\int_z^\infin\frac .$

Then $u^$ can be extended to the complex plane and this extension equals the $\wp$-function.

Properties

* ℘ is an even function. That means $\wp(z)=\wp(-z)$ for all $z \in \mathbb \setminus \Lambda$, which can be seen in the following way: \begin
\weierp(-z) & =\frac+\sum_\left(\frac1-\frac1\right) \\ pt& =\frac+\sum_\left(\frac1-\frac1\right) \\ pt& =\frac+\sum_\left(\frac1-\frac1\right)=\wp(z).
\end The second last equality holds because $\=\Lambda$. Since the sum converges absolutely this rearrangement does not change the limit.
* ℘ is meromorphic and its derivative is $\wp'(z)=-2\sum_\frac1.$
* $\wp$ and $\wp'$ are doubly periodic with the periods $\omega_1$and $\omega_2$. This means: \begin
\wp(z+\omega_1) &= \wp(z) = \wp(z+\omega_2),\ \textrm \\ mu\wp'(z+\omega_1) &= \wp'(z) = \wp'(z+\omega_2).
\end

It follows that $\wp(z+\lambda)=\wp(z)$ and $\wp'(z+\lambda)=\wp'(z)$ for all $\lambda \in \Lambda$. Functions which are meromorphic and doubly periodic are also called elliptic functions.

Laurent expansion

Let $r:=\min\$. Then for 0<, z, the $\wp$-function has the following Laurent expansion
$\wp(z)=\frac1+\sum_^\infin (2n+1)G_z^$
where
$G_n=\sum_\lambda^$ for $n \geq 3$ are so called Eisenstein series.

Differential equation

Set $g_2=60G_4$ and $g_3=140G_6$. Then the $\wp$-function satisfies the differential equation
$\wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3.$

This relation can be verified by forming a linear combination of powers of $\wp$ and $\wp'$ to eliminate the pole at $z=0$. This yields an entire elliptic function that has to be constant by Liouville's theorem.

Invariants

The coefficients of the above differential equation ''g''₂ and ''g''₃ are known as the ''invariants''. Because they depend on the lattice $\Lambda$ they can be viewed as functions in $\omega_1$and $\omega_2$.

The series expansion suggests that ''g''₂ and ''g''₃ are homogeneous functions of degree −4 and −6. That is
$g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^ g_2(\omega_1, \omega_2)$
$g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^ g_3(\omega_1, \omega_2)$ for $\lambda \neq 0$.

If $\omega_1$and $\omega_2$ are chosen in such a way that $\operatorname\left( \tfrac \right)>0$, ''g''₂ and ''g''₃ can be interpreted as functions on the upper half-plane $\mathbb:=\$.

Let $\tau=\tfrac$. One has:
$g_2(1,\tau)=\omega_1^4g_2(\omega_1,\omega_2),$
$g_3(1,\tau)=\omega_1^6 g_3(\omega_1,\omega_2).$
That means ''g''₂ and ''g''₃ are only scaled by doing this. Set
$g_2(\tau):=g_2(1,\tau)$ and $g_3(\tau):=g_3(1,\tau).$

As functions of $\tau\in\mathbb$ $g_2,g_3$ are so called modular forms.

The Fourier series for $g_2$ and $g_3$ are given as follows:
g_2(\tau)=\frac43\pi^4 \left 1+ 240\sum_^\infty \sigma_3(k) q^ \right
g_3(\tau)=\frac\pi^6 \left 1- 504\sum_^\infty \sigma_5(k) q^ \right
where
$\sigma_a(k):=\sum_d^\alpha$
is the divisor function and $q=e^$ is the nome.

Modular discriminant

The ''modular discriminant'' Δ is defined as the discriminant of the polynomial at right-hand side of the above differential equation:
$\Delta=g_2^3-27g_3^2.$

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as
$\Delta \left( \frac \right) = \left(c\tau+d\right)^ \Delta(\tau)$
where $a,b,d,c\in\mathbb$ with ''ad'' − ''bc'' = 1.

Note that $\Delta=(2\pi)^\eta^$ where $\eta$ is the Dedekind eta function.

For the Fourier coefficients of $\Delta$, see Ramanujan tau function.

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

$e_1$, $e_2$ and $e_3$ are usually used to denote the values of the $\wp$-function at the half-periods.

$e_1\equiv\wp\left(\frac\right)$
$e_2\equiv\wp\left(\frac\right)$
$e_3\equiv\wp\left(\frac\right)$

They are pairwise distinct and only depend on the lattice $\Lambda$ and not on its generators.

$e_1$, $e_2$ and $e_3$ are the roots of the cubic polynomial $4\wp(z)^3-g_2\wp(z)-g_3$ and are related by the equation:
$e_1+e_2+e_3=0.$

Because those roots are distinct the discriminant $\Delta$ does not vanish on the upper half plane. Now we can rewrite the differential equation:
$\wp'^2(z)=4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3).$

That means the half-periods are zeros of $\wp'$.

The invariants $g_2$ and $g_3$ can be expressed in terms of these constants in the following way:
$g_2 = -4 (e_1 e_2 + e_1 e_3 + e_2 e_3)$
$g_3 = 4 e_1 e_2 e_3$

$e_1$, $e_2$ and $e_3$ are related to the modular lambda function:
$\lambda (\tau)=\frac,\quad \tau=\frac.$

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:
$\wp(z) = e_3 + \frac = e_2 + ( e_1 - e_3 ) \frac = e_1 + ( e_1 - e_3 ) \frac$
where $e_1,e_2$and $e_3$ are the three roots described above and where the modulus ''k'' of the Jacobi functions equals
$k = \sqrt\frac$
and their argument ''w'' equals
$w = z \sqrt.$

Relation to Jacobi's theta functions

The function $\wp (z,\tau)=\wp (z,1,\omega_2/\omega_1)$ can be represented by Jacobi's theta functions:
$\wp (z,\tau)=\left(\pi \theta_2(0,q)\theta_3(0,q)\frac\right)^2-\frac\left(\theta_2^4(0,q)+\theta_3^4(0,q)\right)$
where $q=e^$ is the nome and $\tau$ is the period ratio $(\tau\in\mathbb)$. This also provides a very rapid algorithm for computing $\wp (z,\tau)$.

Relation to elliptic curves

Consider the projective cubic curve
$\bar C_^\mathbb = \\cup\\subset\mathbb_\mathbb^2 .$

For this cubic, also called Weierstrass cubic, there exists no rational parameterization, if $\Delta \neq 0$. In this case it is also called an elliptic curve. Nevertheless there is a parameterization that uses the $\wp$-function and its derivative $\wp'$:
$\varphi: \mathbb/\Lambda\to\bar C_^\mathbb, \quad \bar\mapsto \begin (\wp(z),\wp'(z),1) & \bar\neq0\\ \infin \quad &\bar=0 \end$

Now the map $\varphi$ is bijective and parameterizes the elliptic curve $\bar C_^\mathbb$.

$\mathbb/\Lambda$ 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 $g_2,g_3\in\mathbb$ with $\Delta = g_2^3 - 27g_3^2 \neq 0$ there exists a lattice $\mathbb\omega_1+\mathbb\omega_2$, such that

$g_2=g_2(\omega_1,\omega_2)$ and $g_3=g_3(\omega_1,\omega_2)$.

The statement that elliptic curves over $\mathbb$ can be parameterized over $\mathbb$, 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 $z,w\in\mathbb$, so that $z,w,z+w,z-w\notin\Lambda$. Then one has:
\wp(z+w)=\frac14 \left frac\right2-\wp(z)-\wp(w).

As well as the duplication formula:
\wp(2z)=\frac14\left frac\right2-2\wp(z).

These formulas also have a geometric interpretation, if one looks at the elliptic curve $\bar C_^\mathbb$ together with the mapping $:\mathbb/\Lambda\to\bar C_^\mathbb$ as in the previous section.

The group structure of $(\mathbb/\Lambda,+)$ translates to the curve $\bar C_^\mathbb$and can be geometrically interpreted there:

The sum of three pairwise different points $a,b,c\in\bar C_^\mathbb$is zero if and only if they lie on the same line in $\mathbb_\mathbb^2$.

This is equivalent to:
$\det\left(\begin 1&\wp(u+v)&-\wp'(u+v)\\ 1&\wp(v)&\wp'(v)\\ 1&\wp(u)&\wp'(u)\\ \end\right) =0 ,$
where $\wp(u) = a$, $\wp(v)=b$ and $u,v\notin\Lambda$.

Typography

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

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 ℘.

See also

* Weierstrass functions
* Jacobi elliptic functions
* Lemniscate elliptic functions

Footnotes

References

*
* N. I. Akhiezer, ''Elements of the Theory of Elliptic Functions'', (1970) Moscow, translated into English as ''AMS Translations of Mathematical Monographs Volume 79'' (1990) AMS, Rhode Island
* Tom M. Apostol, ''Modular Functions and Dirichlet Series in Number Theory, Second Edition'' (1990), Springer, New York (See chapter 1.)
* K. Chandrasekharan, ''Elliptic functions'' (1980), Springer-Verlag
* Konrad Knopp, ''Funktionentheorie II'' (1947), Dover Publications; Republished in English translation as ''Theory of Functions'' (1996), Dover Publications
* Serge Lang, ''Elliptic Functions'' (1973), Addison-Wesley,
* E. T. Whittaker and G. N. Watson, '' A Course of Modern Analysis'', Cambridge University Press, 1952, chapters 20 and 21

External links

* {{springer, title=Weierstrass elliptic functions, id=p/w097450

Weierstrass's elliptic functions on Mathworld

* Chapter 23
Weierstrass Elliptic and Modular Functions
in DLMF ( Digital Library of Mathematical Functions) by W. P. Reinhardt and P. L. Walker.

Weierstrass P function and its derivative implemented in C by David Dumas
Modular forms
Algebraic curves
Elliptic functions