Elliptic curve: Difference between revisions
imported>David Lehavi (first draft) |
imported>David Lehavi m (add categories) |
||
| Line 8: | Line 8: | ||
* 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 | 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 | ||
| Line 19: | Line 18: | ||
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 he 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. | 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 he 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. | ||
[[Category:Mathematics Workgroup]] | |||
[[Category:CZ Live]] | |||
Revision as of 00:14, 16 February 2007
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 - the identity element.
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 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 we 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.}
Hence the complete linear system 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_C(p+q+r)|} is two dimensional, and the map from 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} to the dual linear system 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 he 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.