Elliptic curve: Difference between revisions
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An '''elliptic curve''' over a [[field (mathematics)|field]] <math>K</math> is a one dimensional [[Abelian variety]] over <math>K</math>. Alternatively it is a smooth [[algebraic curve]] of [[genus (mathematics)|genus]] one together with marked point. | |||
If <math>f(x,y,z)</math> is a [[homogenous]] cubic polynomial in three variables, such that at no point <math>(x:y:z)\in \mathbb{P}^2</math> | ==Curves of genus 1 as smooth plane cubics== | ||
If <math>f(x,y,z)</math> is a [[homogenous]] degree 3 (also called "cubic") polynomial in three variables, such that at no point <math>(x:y:z)\in \mathbb{P}^2</math> | |||
all the three derivatives of f are simultaneously zero, then the [[Null set]] <math>E:=\{(x:y:z)\in\mathbb{P}^2|f(x,y,z)=0\}\subset\mathbb{P}^2</math> | all the three derivatives of f are simultaneously zero, then the [[Null set]] <math>E:=\{(x:y:z)\in\mathbb{P}^2|f(x,y,z)=0\}\subset\mathbb{P}^2</math> | ||
is a smooth curve of genus 1. Smoothness follows from the condition on derivatives, and the genus can be computed in various ways; e.g.: | is a smooth curve of genus 1. Smoothness follows from the condition on derivatives, and the genus can be computed in various ways; e.g.: | ||
* Let <math>H</math> be the class of line in the [[Picard group]] <math>Pic(P^2)</math>, then <math>E</math> is [[rationally equivalent]] to <math>3H</math>. Then by the [[adjunction formula]] we have <math>\#K_E=(K_{\mathbb{P}^2}+[E])[E]=(-3H+3H)3H=0</math>. | * Let <math>H</math> be the class of line in the [[Picard group]] <math>Pic(P^2)</math>, then <math>E</math> is [[rationally equivalent]] to <math>3H</math>. Then by the [[adjunction formula]] we have <math>\#K_E=(K_{\mathbb{P}^2}+[E])[E]=(-3H+3H)3H=0</math>. | ||
* By the [[genus degree formula]] for plane curves we see that <math>genus(E)=(3-1)(3-2)/2=1</math> | * By the [[genus-degree formula]] for plane curves we see that <math>genus(E)=(3-1)(3-2)/2=1</math> | ||
* If we choose a point <math>p\in E</math> and a line <math>L\subset\mathbb{P}^2</math> such that <math>p\not\in L</math>, we may project <math>E</math> to <math>L</math> by sending a point <math>q\in E</math> to the intersection point <math>\overline{pq}\cap L</math> (if <math>p=q</math> take the line <math>T_p(E)</math> instead of the line <math>\overline{pq}</math>). This is a double cover of a line | * If we choose a point <math>p\in E</math> and a line <math>L\subset\mathbb{P}^2</math> such that <math>p\not\in L</math>, we may project <math>E</math> to <math>L</math> by sending a point <math>q\in E</math> to the intersection point <math>\overline{pq}\cap L</math> (if <math>p=q</math> take the line <math>T_p(E)</math> instead of the line <math>\overline{pq}</math>). This is a double cover of a line with four [[ramification points]]. Hence by the [[Riemann-Hurwitz formula]] <math>genus(E)-1=-2+4/2=0</math> | ||
with four [[ramification points]]. Hence by the [[Riemann-Hurwitz formula]] <math>genus(E)-1=-2+4/2=0</math> | |||
On the other hand, if <math>C</math> is a smooth algebraic curve of genus 1, and <math>p,q,r</math> are points on <math>C</math>, then by the [[Riemann-Roch theorem|Riemann-Roch formula]] we have <math>h^0(O_C(p+q+r))=3-(1-1)-h^0(-(p+q+r))=3.</math>. Choosing a basis <math>g_0,g_1,g_2</math> to the three dimensional vector space <math>H^0(O_C(p+q+r))=\{g:C\to\mathbb{P}^1</math> such that <math>g</math> is algebraic and <math>g^{-1}(\infty)=\{p,q,r\}\}</math>, the map given by <math>s\in C\mapsto (g_0(s):g_1(s):g_2(s))\in\mathbb{P}^2</math> is an embedding. | |||
=== The group operation on a pointed smooth plane cubic === | |||
[[Image:Cubic_addition.png|thumb|400px|Addition on cubic with a marked point <math>O</math>]]Let <math>E</math> be as above, and <math>O</math> point on <math>E</math>. If <math>p</math> and <math>q</math> are two points on <math>E</math> we set <math>p*q:=\overline{pq}\cap E\setminus\{p,q\},</math> where if <math>p=q</math> we take the line <math>T_p(E)</math> instead, and the intersection is to be understood with multiplicities. The addition on the elliptic curve <math>E</math> is defined as <math>p+q:=O*(p*q)</math>. Both the commutativity and the existence of inverse follow from the definition. The proof of the associativity of this operation is more delicate. | |||
=== Weierstrass forms === | |||
Suppose that the cubic curve <math>E</math> admits a [[flex]] defined over ''K'', that is, a line <math>l</math> which is tri-tangent to <math>E</math> at a point <math>p</math>: this will happen, for example, if the field <math>K</math> is [[algebraically closed field|algebraically closed]]). In this case there is a change of coordinates on the projective plane which takes the line <math>l</math> to the line <math>\{z=0\}</math> and the point <math>p</math> to the point <math>(0:1:0)</math>: we may thus assume that the only terms in the cubic polynomial <math>f</math> which include <math>y</math>, are <math>y^2z,xyz,yz^2</math>. The equation can then be put in generalised Weierstrass form | |||
:<math>Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6 .</math> | |||
If the characteristic of <math>K</math> is not 2 or 3 then by another change of coordinates, the cubic polynomial can be changed to the form | |||
:<math>Y^2 = X^3 - 27c_4 X - 54c_6 . </math> | |||
In this case the [[Discriminant of a polynomial|discriminant]] of the cubic polynomial on the right hand side of the equation is given by <math>\Delta=(c_4^3 - c_6^2)/1728</math>, and is non-zero because the curve is non-singular. The <math>j</math> invariant of the curve <math>E</math> is defined to be <math>c_4^3/\Delta</math>. Two elliptic curves are isomorphic over an algebraically closed field if and only if they have the same <math>j</math> invariant. | |||
== Elliptic curves over the complex numbers == | |||
=== One dimensional complex tori and lattices in the complex numbers=== | |||
An elliptic curve over the [[complex numbers]] is a [[Riemann surface]] of genus 1, or a two dimensional [[torus]] over the [[real numbers]]. The [[universal cover]] of this torus, as a [[complex manifold]], is the [[complex line]] <math>\mathbb{C}</math>. Hence the elliptic curve is isomorphic to a quotient of the complex numbers by some [[lattice (geometry)|lattice]]; moreover two elliptic curves are isomorphic if and only the two corresponding lattices are isomorphic. Hence the [[moduli]] of elliptic curves | |||
over the complex numbers is identified with the moduli of lattices in <math>\mathbb{C}</math> up to [[homothety]]. For each homothety class there is a lattice such that one of the points of the lattice is 1, and the other is some point <math>\tau</math> in the [[upper half plane]] <math>\mathcal{H}</math>. | |||
[[Image:SL2_fundamental_domain.png|thumb|400px|Each triangular region is a free regular set of <math>\mathcal{H}/SL_2(\mathbb{Z})</math>;; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.]]Hence the moduli of lattices in <math>\mathbb{C}</math> is the quotient <math>\mathcal{H}/PSL_2(\mathbb{Z})</math>, where a group element | |||
<math>\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in SL_2(\mathbb{Z})</math> acts on the upper half plane via the [[mobius transformation]] <math>z\mapsto\frac{az+b}{cz+d}</math>. The standard [[fundamental domain]] for this action is the set: <math>\{\tau|-\frac{1}{2}\leq Im(\tau)\leq\frac{1}{2},|\tau|\geq 1\}</math>. | |||
=== Modular forms === | |||
For the main article see [[Modular forms]] | |||
Modular forms are functions on the upper half plane, such that for any <math>\gamma=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in SL_2(\mathbb{Z})</math> we have | |||
<math>f(\gamma(\tau))=(c\tau+d)^k f(\tau)</math> for some <math>k</math> which is called the "weight" of the form. | |||
=== Theta functions === | |||
For the main article see [[Theta function]] | |||
=== Weierstrass's <math>\wp</math> function === | |||
Let <math>\Lambda=\omega_1\mathbb{Z}+\omega_2\mathbb{Z}</math> be a lattice. The Weirstrass <math>\wp</math>-function is the absolutely convergent series | |||
<math>\wp(z,\Lambda)=\frac{1}{z^2}+\sum_{\omega\in\Lambda}' \frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}</math> | |||
where the sum is taken over all nonzero lattice points. It is an elliptic function having poles of order two at each lattice point. | |||
=== Application: elliptic integrals=== | |||
== Elliptic curves over number fields == | |||
Let ''K'' be an [[algebraic number field]], a finite [[field extension|extension]] of '''Q''', and ''E'' an elliptic curve defined over ''K''. Then ''E''(''K''), the points of ''E'' with coordinates in ''K'', is an abelian group. The structure of this group is determined by the Mordell-Weil theorem, which states that ''E''(''K'') is finitely generated. By the [[fundamental theorem of finitely generated abelian groups]] we have | |||
:<math>E(K) \cong \mathbf{Z}^r \oplus T ,\,</math> | |||
where the torsion-free part has finite rank ''r'', and the torsion group ''T'' is finite. | |||
It is not known whether the rank of an elliptic curve over '''Q''' is bounded. The elliptic curve | |||
:<math>y^2 + xy + y = x^3 - x^2 - 20067762415575526585033208209338542750930230312178956502x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429 </math> | |||
has rank at least 28, due to Noam Elkies <ref>N. Elkies, Posting to NMBRTHRY list, May 2006</ref>. | |||
The torsion group of a curve over '''Q''' is determined by Mazur's theorem; over a general number field ''K'' a result of Merel<ref>{{ cite journal | author=Loïc Merel | title=Bornes pour la torsion des courbes elliptiques sur les corps de nombres | journal=Invent. Math. | volume=124 | year=1996 | pages=437-449 }}</ref> shows that the torsion group is bounded in terms of the degree of ''K''. | |||
The rank of an elliptic curve over a number field is related to the [[L-function]] of the curve by the [[Birch-Swinnerton-Dyer conjecture]]s. | |||
===Mordell-Weil theorem=== | |||
The proof of the Mordell-Weil theorem combines two main parts. The "weak Mordell-Weil theorem" states that the quotient <math>E(K) / 2E(K)</math> is finite: this is combined with an argument involving the [[height function]]. | |||
The theorem also applies to an [[abelian variety]] ''A'' of higher dimension over a number field. The [[Lang-Néron theorem]] implies that ''A''(''K'') is finitely generated when ''K'' is finitely generated over its [[prime field]]. | |||
===Mazur's theorem=== | |||
Mazur's theorem<ref>{{cite journal | author=Barry C. Mazur | title=Rational isogenies of prime degree | journal=Invent. Math. | volume=44 | year=1978 | pages=129-162 }}</ref> shows that the torsion subgroup of an elliptic curve over '''Q''' must be one of the following | |||
:<math>C_1; C_2; C_3; C_4; C_5; C_6; C_7; C_8; C_9; C_{10}; C_{12}; \,</math> | |||
:<math>C_2 \times C_2; C_2 \times C_4; C_2 \times C_6; C_2 \times C_8 .\,</math> | |||
= | Each of these torsion structures is parametrizable.<ref>{{cite journal | author=D.S. Kubert | title=Universal bounds on the torsion of elliptic curves | journal=J. London Maths Soc. | volume=33 | year=1976 | pages=193-227 }}</ref> | ||
== Elliptic curves over finite fields == | |||
=== Application:cryptography=== | |||
==Elliptic curves over local fields== | |||
== References == | |||
{{reflist}}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 11:00, 11 August 2024
An elliptic curve over a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is a one dimensional Abelian variety over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} . Alternatively it is a smooth algebraic curve of genus one together with marked point.
Curves of genus 1 as smooth plane cubics
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y,z)} is a homogenous degree 3 (also called "cubic") polynomial in three variables, such that at no point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x:y:z)\in \mathbb{P}^2} all the three derivatives of f are simultaneously zero, then the Null set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E:=\{(x:y:z)\in\mathbb{P}^2|f(x,y,z)=0\}\subset\mathbb{P}^2} is a smooth curve of genus 1. Smoothness follows from the condition on derivatives, and the genus can be computed in various ways; e.g.:
- Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} be the class of line in the Picard group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Pic(P^2)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is rationally equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3H} . Then by the adjunction formula we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \#K_E=(K_{\mathbb{P}^2}+[E])[E]=(-3H+3H)3H=0} .
- By the genus-degree formula for plane curves we see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle genus(E)=(3-1)(3-2)/2=1}
- If we choose a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in E} and a line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\subset\mathbb{P}^2} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\not\in L} , we may project Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} by sending a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\in E} to the intersection point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{pq}\cap L} (if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=q} take the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p(E)} instead of the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{pq}} ). This is a double cover of a line with four ramification points. Hence by the Riemann-Hurwitz formula Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle genus(E)-1=-2+4/2=0}
On the other hand, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is a smooth algebraic curve of genus 1, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p,q,r} are points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , then by the Riemann-Roch formula we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(O_C(p+q+r))=3-(1-1)-h^0(-(p+q+r))=3.} . Choosing a basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_0,g_1,g_2} to the three dimensional vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0(O_C(p+q+r))=\{g:C\to\mathbb{P}^1} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is algebraic and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^{-1}(\infty)=\{p,q,r\}\}} , the map given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s\in C\mapsto (g_0(s):g_1(s):g_2(s))\in\mathbb{P}^2} is an embedding.
The group operation on a pointed smooth plane cubic
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} be as above, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} point on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} are two points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p*q:=\overline{pq}\cap E\setminus\{p,q\},} where if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=q} we take the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p(E)} instead, and the intersection is to be understood with multiplicities. The addition on the elliptic curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p+q:=O*(p*q)} . Both the commutativity and the existence of inverse follow from the definition. The proof of the associativity of this operation is more delicate.
Weierstrass forms
Suppose that the cubic curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} admits a flex defined over K, that is, a line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} which is tri-tangent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} : this will happen, for example, if the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is algebraically closed). In this case there is a change of coordinates on the projective plane which takes the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} to the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{z=0\}} and the point to the point : we may thus assume that the only terms in the cubic polynomial which include Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2z,xyz,yz^2} . The equation can then be put in generalised Weierstrass form
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6 .}
If the characteristic of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is not 2 or 3 then by another change of coordinates, the cubic polynomial can be changed to the form
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^2 = X^3 - 27c_4 X - 54c_6 . }
In this case the discriminant of the cubic polynomial on the right hand side of the equation is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta=(c_4^3 - c_6^2)/1728} , and is non-zero because the curve is non-singular. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} invariant of the curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is defined to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_4^3/\Delta} . Two elliptic curves are isomorphic over an algebraically closed field if and only if they have the same Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} invariant.
Elliptic curves over the complex numbers
One dimensional complex tori and lattices in the complex numbers
An elliptic curve over the complex numbers is a Riemann surface of genus 1, or a two dimensional torus over the real numbers. The universal cover of this torus, as a complex manifold, is the complex line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} . Hence the elliptic curve is isomorphic to a quotient of the complex numbers by some lattice; moreover two elliptic curves are isomorphic if and only the two corresponding lattices are isomorphic. Hence the moduli of elliptic curves over the complex numbers is identified with the moduli of lattices in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} up to homothety. For each homothety class there is a lattice such that one of the points of the lattice is 1, and the other is some point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} in the upper half plane Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} .
Hence the moduli of lattices in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} is the quotient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}/PSL_2(\mathbb{Z})} , where a group element
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in SL_2(\mathbb{Z})} acts on the upper half plane via the mobius transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\mapsto\frac{az+b}{cz+d}} . The standard fundamental domain for this action is the set: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\tau|-\frac{1}{2}\leq Im(\tau)\leq\frac{1}{2},|\tau|\geq 1\}} .
Modular forms
For the main article see Modular forms Modular forms are functions on the upper half plane, such that for any we have for some which is called the "weight" of the form.
Theta functions
For the main article see Theta function
Weierstrass's function
Let be a lattice. The Weirstrass -function is the absolutely convergent series where the sum is taken over all nonzero lattice points. It is an elliptic function having poles of order two at each lattice point.
Application: elliptic integrals
Elliptic curves over number fields
Let K be an algebraic number field, a finite extension of Q, and E an elliptic curve defined over K. Then E(K), the points of E with coordinates in K, is an abelian group. The structure of this group is determined by the Mordell-Weil theorem, which states that E(K) is finitely generated. By the fundamental theorem of finitely generated abelian groups we have
where the torsion-free part has finite rank r, and the torsion group T is finite.
It is not known whether the rank of an elliptic curve over Q is bounded. The elliptic curve
has rank at least 28, due to Noam Elkies [1].
The torsion group of a curve over Q is determined by Mazur's theorem; over a general number field K a result of Merel[2] shows that the torsion group is bounded in terms of the degree of K.
The rank of an elliptic curve over a number field is related to the L-function of the curve by the Birch-Swinnerton-Dyer conjectures.
Mordell-Weil theorem
The proof of the Mordell-Weil theorem combines two main parts. The "weak Mordell-Weil theorem" states that the quotient is finite: this is combined with an argument involving the height function.
The theorem also applies to an abelian variety A of higher dimension over a number field. The Lang-Néron theorem implies that A(K) is finitely generated when K is finitely generated over its prime field.
Mazur's theorem
Mazur's theorem[3] shows that the torsion subgroup of an elliptic curve over Q must be one of the following
Each of these torsion structures is parametrizable.[4]
Elliptic curves over finite fields
Application:cryptography
Elliptic curves over local fields
References
- ↑ N. Elkies, Posting to NMBRTHRY list, May 2006
- ↑ Loïc Merel (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres". Invent. Math. 124: 437-449.
- ↑ Barry C. Mazur (1978). "Rational isogenies of prime degree". Invent. Math. 44: 129-162.
- ↑ D.S. Kubert (1976). "Universal bounds on the torsion of elliptic curves". J. London Maths Soc. 33: 193-227.