Editing Weierstrass elliptic function (section)

== Relation to elliptic curves ==
{{see also|Elliptic curve#Elliptic curves over the complex numbers}}
Consider the embedding of the cubic curve in the [[complex projective plane]]
:<math>\bar C_{g_2,g_3}^\mathbb{C} = \{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x-g_3\}\cup\{O\}\subset \mathbb{C}^{2} \cup \mathbb{P}_1(\mathbb{C}) =  \mathbb{P}_2(\mathbb{C}).</math>

where <math>O</math> is a point lying on the [[line at infinity]] <math>\mathbb{P}_1(\mathbb{C})</math>. For this cubic there exists no rational parameterization, if <math>\Delta \neq 0</math>.<ref name=":5">{{citation|surname1=Hulek, Klaus.|title=Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen|edition=2., überarb. u. erw. Aufl. 2012|publisher=Vieweg+Teubner Verlag|publication-place=Wiesbaden|at=p.&nbsp;8|isbn=978-3-8348-2348-9|date=2012|language=German}}</ref> In this case it is also called an elliptic curve. Nevertheless there is a parameterization in [[homogeneous coordinates]] that uses the <math>\wp</math>-function and its derivative <math>\wp'</math>:<ref>{{citation|title=Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen|date=2012|at=p.&nbsp;12|edition=2., überarb. u. erw. Aufl. 2012|publication-place=Wiesbaden|publisher=Vieweg+Teubner Verlag|language=German|isbn=978-3-8348-2348-9|surname1=Hulek, Klaus.}}</ref>
:<math> \varphi(\wp,\wp'): \mathbb{C}/\Lambda\to\bar C_{g_2,g_3}^\mathbb{C}, \quad
z \mapsto \begin{cases}
\left[\wp(z):\wp'(z):1\right] & z \notin \Lambda \\
\left[0:1:0\right] \quad & z \in \Lambda 
\end{cases} </math>

Now the map <math>\varphi</math> is [[Bijection|bijective]] and parameterizes the elliptic curve <math>\bar C_{g_2,g_3}^\mathbb{C}</math>.

<math>\mathbb{C}/\Lambda </math> 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 <math>g_2,g_3\in\mathbb{C}</math> with <math>\Delta = g_2^3 - 27g_3^2 \neq 0 </math> there exists a lattice <math>\mathbb{Z}\omega_1+\mathbb{Z}\omega_2</math>, such that

<math>g_2=g_2(\omega_1,\omega_2) </math> and <math>g_3=g_3(\omega_1,\omega_2) </math>.<ref>{{citation|surname1=Hulek, Klaus.| title=Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen| edition=2., überarb. u. erw. Aufl. 2012|publisher=Vieweg+Teubner Verlag|publication-place=Wiesbaden|at=p.&nbsp;111| isbn=978-3-8348-2348-9| date=2012|language=German}}</ref>

The statement that elliptic curves over <math>\mathbb{Q}</math> can be parameterized over <math>\mathbb{Q}</math>, is known as the [[modularity theorem]]. This is an important theorem in [[number theory]]. It was part of [[Andrew Wiles|Andrew Wiles']] proof (1995) of [[Fermat's Last Theorem]].