Editing Elliptic curve (section)

==Elliptic curves over the complex numbers==
{{Further|Complex multiplication}}
[[Image:Lattice torsion points.svg|right|thumb|upright=1.2|An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice {{math|Λ}}, here spanned by two fundamental periods {{math|''ω''<sub>1</sub>}} and {{math|''ω''<sub>2</sub>}}. The four-torsion is also shown, corresponding to the lattice {{math|{{sfrac|1|4}}Λ}} containing {{math|Λ}}.]]
The formulation of elliptic curves as the embedding of a [[torus]] in the [[complex projective plane]] follows naturally from a curious property of [[Weierstrass's elliptic functions]].  These functions and their first derivative are related by the formula

:<math>\wp'(z)^2 = 4\wp(z)^3 -g_2\wp(z) - g_3</math>

Here, {{math|''g''<sub>2</sub>}} and {{math|''g''<sub>3</sub>}} are constants; {{math|℘(''z'')}} is the [[Weierstrass elliptic function]] and {{math|℘{{prime}}(''z'')}} its derivative. It should be clear that this relation is in the form of an elliptic curve (over the [[complex number]]s). The Weierstrass functions are doubly periodic; that is, they are [[fundamental pair of periods|periodic]] with respect to a [[Lattice_(group)|lattice]] {{math|Λ}}; in essence, the Weierstrass functions are naturally defined on a torus  {{math|1=''T'' = '''C'''/Λ}}. This torus may be embedded in the complex projective plane by means of the map

:<math>z \mapsto \left[1 : \wp(z) : \tfrac12\wp'(z)\right]</math>

This map is a [[group isomorphism]] of the torus (considered with its natural group structure) with the chord-and-tangent group law on the cubic curve which is the image of this map.  It is also an isomorphism of [[Riemann surface]]s from the torus to the cubic curve, so topologically, an elliptic curve is a torus.  If the lattice {{math|Λ}} is related by multiplication by a non-zero complex number {{mvar|c}} to a lattice {{math|''c''Λ}}, then the corresponding curves are isomorphic. Isomorphism classes of elliptic curves are specified by the [[j-invariant|{{mvar|j}}-invariant]].

The isomorphism classes can be understood in a simpler way as well. The constants {{math|''g''<sub>2</sub>}} and {{math|''g''<sub>3</sub>}}, called the [[j-invariant|modular invariant]]s, are uniquely determined by the lattice, that is, by the structure of the torus. However, all real polynomials factorize completely into linear factors over the complex numbers, since the field of complex numbers is the [[algebraic closure]] of the reals. So, the elliptic curve may be written as

:<math>y^2 = x(x - 1)(x - \lambda)</math>

One finds that
:<math>\begin{align}
g_2' &= \frac{\sqrt[3]4}{3} \left(\lambda^2 - \lambda + 1\right) \\[4pt]
g_3' &= \frac{1}{27} (\lambda + 1)\left(2\lambda^2 - 5\lambda + 2\right)
\end{align}</math>

and

:<math>j(\tau) = 1728\frac{{g_2'}^3}{{g_2'}^3 - 27{g_3'}^2} = 256\frac{ \left(\lambda^2 - \lambda + 1\right)^3}{\lambda^2\left(\lambda - 1\right)^2}</math>

with [[J-invariant|{{mvar|j}}-invariant]] {{math|''j''(''τ'')}} and {{math|''λ''(''τ'')}} is sometimes called the [[modular lambda function]]. For example, let {{math|1=''τ'' = 2''i''}}, then {{math|1=''λ''(2''i'') = (−1 + {{sqrt|2}})<sup>4</sup>}} which implies {{math|''g''{{prime}}<sub>2</sub>}}, {{math|''g''{{prime}}<sub>3</sub>}}, and therefore {{math|''g''{{prime}}<sub>2</sub>{{su|p=3}} − 27''g''{{prime}}<sub>3</sub>{{su|p=2}}}} of the formula above are all [[algebraic numbers]] if {{mvar|τ}} involves an [[imaginary quadratic field]]. In fact, it yields the integer {{math|1=''j''(2''i'') = 66<sup>3</sup> = {{val|287496}}}}. 

In contrast, the [[modular discriminant]]
:<math>\Delta(\tau) = g_2(\tau)^3 - 27g_3(\tau)^2 = (2\pi)^{12}\,\eta^{24}(\tau)</math>
is generally a [[transcendental number]]. In particular, the value of the [[Dedekind_eta_function#Special_values|Dedekind eta function]] {{math|''η''(2''i'')}} is

:<math>\eta(2i)=\frac{\Gamma \left(\frac14\right)}{2^\frac{11}{8} \pi^\frac34}</math>

Note that the [[uniformization theorem]] implies that every [[Compact space|compact]] Riemann surface of genus one can be represented as a torus. This also allows an easy understanding of the [[torsion subgroup|torsion points]] on an elliptic curve: if the lattice {{math|Λ}} is spanned by the fundamental periods {{math|''ω''<sub>1</sub>}} and {{math|''ω''<sub>2</sub>}}, then the {{mvar|n}}-torsion points are the (equivalence classes of) points of the form
:<math> \frac{a}{n} \omega_1 + \frac{b}{n} \omega_2</math>

for integers {{mvar|a}} and {{mvar|b}} in the range {{math|0 ≤ (''a'', ''b'') < ''n''}}.

If

:<math>E : y^2=4(x-e_1)(x-e_2)(x-e_3)</math>

is an elliptic curve over the complex numbers and

:<math>a_0=\sqrt{e_1-e_3}, \qquad b_0=\sqrt{e_1-e_2}, \qquad c_0=\sqrt{e_2-e_3},</math>

then a pair of fundamental periods of {{mvar|E}} can be calculated very rapidly by

:<math>\omega_1=\frac{\pi}{\operatorname{M}(a_0,b_0)}, \qquad \omega_2=\frac{\pi}{\operatorname{M}(c_0,ib_0)}</math>

{{math|M(''w'', ''z'')}} is the [[arithmetic–geometric mean]] of {{mvar|w}} and {{mvar|z}}. At each step of the arithmetic–geometric mean iteration, the signs of {{mvar|z<sub>n</sub>}} arising from the ambiguity of geometric mean iterations are chosen such that {{math|{{abs|''w<sub>n</sub>'' − ''z<sub>n</sub>''}} ≤ {{abs|''w<sub>n</sub>'' + ''z<sub>n</sub>''}}}} where {{mvar|w<sub>n</sub>}} and {{mvar|z<sub>n</sub>}} denote the individual arithmetic mean and geometric mean iterations of {{mvar|w}} and {{mvar|z}}, respectively. When {{math|1={{abs|''w<sub>n</sub>'' − ''z<sub>n</sub>''}} = {{abs|''w<sub>n</sub>'' + ''z<sub>n</sub>''}}}}, there is an additional condition that {{math|Im<big>(</big>{{sfrac|''z<sub>n</sub>''|''w<sub>n</sub>''}}<big>)</big> > 0}}.<ref>{{Cite web|url=http://etheses.whiterose.ac.uk/20887/1/Final.pdf|title=The Arithmetic-Geometric Mean and Periods of Curves of Genus 1 and 2|last=Wing Tat Chow|first=Rudolf|date=2018|website=White Rose eTheses Online|page=12}}</ref>

Over the complex numbers, every elliptic curve has nine [[inflection point]]s. Every line through two of these points also passes through a third inflection point; the nine points and 12 lines formed in this way form a realization of the [[Hesse configuration]].