Editing Elliptic curve (section)

{{Short description|Algebraic curve in mathematics}}
{{Distinguish|Ellipse}}
{{Redirect|Elliptic Equation|the type of partial differential equation|Elliptic partial differential equation}}
[[Image:EllipticCurveCatalog.svg|right|thumb|392px|A catalog of elliptic curves. The region shown is {{math|''x'', ''y'' ∈ [−3,3]}}.<br>(For {{math|1=(''a'', ''b'') = (0, 0)}} the function is not smooth and therefore not an elliptic curve.)]]
{{Group theory sidebar |Algebraic}}

In [[mathematics]], an '''elliptic curve''' is a [[Smoothness|smooth]], [[Projective variety|projective]], [[algebraic curve]] of [[Genus of an algebraic curve|genus]] one, on which there is a specified point {{mvar|O}}. An elliptic curve is defined over a [[field (mathematics)|field]] {{mvar|K}} and describes points in {{math|''K''{{isup|2}}}}, the [[Cartesian product]] of {{mvar|K}} with itself. If the field's [[characteristic (algebra)|characteristic]] is different from 2 and 3, then the curve can be described as a [[plane algebraic curve]] which consists of solutions {{math|(''x'', ''y'')}} for:

:<math>y^2 = x^3 + ax + b</math>

for some coefficients {{mvar|a}} and {{mvar|b}} in {{mvar|K}}. The curve is required to be [[Singular point of a curve|non-singular]], which means that the curve has no [[cusp (singularity)|cusps]] or [[Self-intersection|self-intersections]]. (This is equivalent to the condition {{math|4''a''<sup>3</sup> + 27''b''<sup>2</sup> ≠ 0}}, that is, being [[square-free polynomial|square-free]] in {{mvar|x}}.)  It is always understood that the curve is really sitting in the [[projective plane]], with the point {{mvar|O}} being the unique [[point at infinity]]. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the [[coefficient field]] has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular [[cubic plane curve|cubic curves]]; see {{section link|#Elliptic curves over a general field}} below.)

An elliptic curve is an [[abelian variety]] – that is, it has a group law defined algebraically, with respect to which it is an [[abelian group]] – and {{mvar|O}} serves as the identity element.

If {{math|1=''y''<sup>2</sup> = ''P''(''x'')}}, where {{mvar|P}} is any polynomial of degree three in {{mvar|x}} with no repeated roots, the solution set is a nonsingular plane curve of [[genus (mathematics)|genus]] one, an elliptic curve. If {{mvar|P}} has degree four and is [[Square-free polynomial|square-free]] this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example the intersection of two [[quadric (algebraic geometry)|quadric surfaces]] embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity.

Using the theory of [[elliptic function]]s, it can be shown that elliptic curves defined over the [[complex number]]s correspond to embeddings of the [[torus]] into the [[complex projective plane]]. The torus is also an [[abelian group]], and this correspondence is also a [[group isomorphism]].

Elliptic curves are especially important in [[number theory]], and constitute a major area of current research; for example, they were used in [[Wiles's proof of Fermat's Last Theorem|Andrew Wiles's proof of Fermat's Last Theorem]]. They also find applications in [[elliptic curve cryptography]] (ECC) and [[integer factorization]].

An elliptic curve is ''not'' an [[ellipse]] in the sense of a projective conic, which has genus zero: see [[elliptic integral]] for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant {{math|''j'' ≥ 1}} as ellipses in the hyperbolic plane <math>\mathbb{H}^2</math>. Specifically, the intersections of the Minkowski hyperboloid with quadric surfaces characterized by a certain constant-angle property produce the Steiner ellipses in <math>\mathbb{H}^2</math> (generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise the elliptic curves with {{math|''j'' ≤ 1}}, and any ellipse in <math>\mathbb{H}^2</math> described as a locus relative to two foci is uniquely the elliptic curve sum of two Steiner ellipses, obtained by adding the pairs of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as the identity on each trajectory curve.<ref>{{Cite journal |last=Sarli |first=J. |date=2012 |title=Conics in the hyperbolic plane intrinsic to the collineation group |journal=J. Geom. |volume=103 |pages=131–148 |doi=10.1007/s00022-012-0115-5|s2cid=119588289 }}</ref>

[[Topologically]], a complex elliptic curve is a [[torus]], while a complex ellipse is a [[sphere]].